The Dynamics of High-Frequency Nanoelectromechanical

The Dynamics of High-Frequency

Nanoelectromechanical

Resonators in Fluid

Douglas Richard Brumley

Thesis submitted in partial requirement of the degree

of

Bachelor of Science (Honours)

November 2008

The University of Melbourne

Department of Mathematics and Statistics

Supervisor: Prof. John E. Sader

Abstract

When a silicon carbide beam clamped at both ends is excited thermoelastically, it ex-

hibits a spectrum of in-plane and out-of-plane resonances. We present several original

formulations to determine the quality factors associated with the in-plane modes when

the beam is immersed in a gas. We analyse several different fluid-beam boundary con-

ditions, and account for the effects of slip. The model predictions agree quantitatively

with the experimental values obtained by the Roukes group from California Institute

of Technology.

Contents

1 Introduction 1

2 Nanoelectromechanical Systems (NEMS) 5

2.1 Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.2 Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.3 Hydrodynamic Function . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.4 Quality Factors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.5 Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

3 Resonant Frequencies in Vacuum 15

3.1 Free Vibrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

3.2 Free Vibrations - Intrinsic Tension . . . . . . . . . . . . . . . . . . . . . 18

3.2.1 Euler Buckling Formula . . . . . . . . . . . . . . . . . . . . . . . 23

4 High-Frequency Limit 25

4.1 No-Slip Boundary Condition . . . . . . . . . . . . . . . . . . . . . . . . 26

4.2 Second-Order Slip Boundary Condition . . . . . . . . . . . . . . . . . . 27

4.3 Matched Asymptotic Expansion . . . . . . . . . . . . . . . . . . . . . . . 29

4.4 Progressive Comparison With Experiment . . . . . . . . . . . . . . . . . 33

5 Beam of Zero Thickness 35


5.1.1 Numerical Solution Method . . . . . . . . . . . . . . . . . . . . . 40

5.1.2 Calculation of Hydrodynamic Function . . . . . . . . . . . . . . . 43

5.2 Second-Order Slip Boundary Condition . . . . . . . . . . . . . . . . . . 46


i


5.3 Alternative Solution Method for First-Order Slip . . . . . . . . . . . . . 53

5.4 High-Frequency Limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

5.5 Convergence With Mesh Discretization . . . . . . . . . . . . . . . . . . . 58


6 Beam of Non-Zero Thickness 63




6.1.3 Alternative Scaling For Out-of-Plane Modes . . . . . . . . . . . . 79

6.2 First-Order Slip Boundary Condition . . . . . . . . . . . . . . . . . . . . 82



6.2.3 Comparison with Free Molecular Solution . . . . . . . . . . . . . 96


6.4 Determination of Surface Properties . . . . . . . . . . . . . . . . . . . . 99

7 Concluding Remarks 103

A Richardson Extrapolation 105

B Matched Asymptotic Solutions 107

B.1 Beam of Zero Thickness . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

B.2 Beam of Non-Zero Thickness . . . . . . . . . . . . . . . . . . . . . . . . 112

ii

List of Figures

1.1 Scanning electron micrograph of device . . . . . . . . . . . . . . . . . . . 2

1.2 Schematic diagram of clamped-clamped beam . . . . . . . . . . . . . . . 3

3.1 Predicted resonant frequencies . . . . . . . . . . . . . . . . . . . . . . . 21

4.1 Asymptotic solutions for infinite plate . . . . . . . . . . . . . . . . . . . 32

4.2 Predicted quality factors: high-frequency limit . . . . . . . . . . . . . . 34

5.1 Cross-section of beam of zero thickness . . . . . . . . . . . . . . . . . . . 35

5.2 Vorticity distribution for beam of zero thickness: no-slip condition . . . 43

5.3 Vorticity distribution for beam of zero thickness: no-slip condition . . . 49

5.4 Vorticity distribution for beam of zero thickness: first-order slip condition 50

5.5 Vorticity distribution for beam of zero thickness: second-order slip con-

dition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

5.6 Hydrodynamic function for beam of zero thickness . . . . . . . . . . . . 52

5.7 Convergence of hydrodynamic function with β . . . . . . . . . . . . . . 56

5.8 Convergence of hydrodynamic function with N . . . . . . . . . . . . . . 58

5.9 Beam of zero thickness: perturbation solutions . . . . . . . . . . . . . . 60

5.10 Predicted quality factors: beam of zero thickness . . . . . . . . . . . . . 62

6.1 Cross-section of beam of non-zero thickness . . . . . . . . . . . . . . . . 63

6.2 Integration contour enclosing entire fluid . . . . . . . . . . . . . . . . . . 66

6.3 Integration contour around beam of non-zero thickness . . . . . . . . . . 68

6.4 Vorticity and pressure distribution on front face of beam: no-slip . . . . 74

6.5 Vorticity and pressure distribution on top face of beam: no-slip . . . . . 75

6.6 Hydrodynamic function for beam of non-zero thickness . . . . . . . . . . 78

iii

6.7 Convergence of hydrodynamic function with aspect ratio A . . . . . . . 80

6.8 Vorticity and pressure distribution on front face of beam: first-order slip 89

6.9 Vorticity and pressure distribution on top face of beam: first-order slip . 90

6.10 Behaviour of hydrodynamic function with Knudsen number . . . . . . . 91

6.11 Linear components of hydrodynamic function . . . . . . . . . . . . . . . 93

6.12 Hydrodynamic function as a Pade approximant . . . . . . . . . . . . . . 94

6.13 Convergence of hydrodynamic function with N . . . . . . . . . . . . . . 95

6.14 Comparison between computed hydrodynamic function and free molec-

ular solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

6.15 Predicted quality factors: beam of non-zero thickness . . . . . . . . . . . 100

B.1 Asymptotic solutions for infinitely thin blade of finite width . . . . . . . 110

B.2 Three distinct flow regimes around blade . . . . . . . . . . . . . . . . . . 111

iv

List of Tables

2.1 Experimental quality factors . . . . . . . . . . . . . . . . . . . . . . . . . 13

3.1 Resonant frequencies for beam of length 8µm . . . . . . . . . . . . . . . 18

3.2 Resonant frequencies for beam of length 16µm . . . . . . . . . . . . . . 18

4.1 Quality factors: high-frequency limit . . . . . . . . . . . . . . . . . . . . 33

5.1 Quality factors: beam of zero thickness . . . . . . . . . . . . . . . . . . . 61

6.1 Quality factors: beam of non-zero thickness (unfitted) . . . . . . . . . . 98

6.2 Quality factors: beam of non-zero thickness (fitted) . . . . . . . . . . . . 100

v

“The choice is always the same. You can make your model more complex and more

faithful to reality, or you can make it simpler and easier to handle. Only the most

naive scientist believes that the perfect model is the one that perfectly represents reality.

Such a model would have the same drawbacks as a map as large and detailed as the city

it represents, a map depicting every park, every street, every building, every tree, every

pothole, every inhabitant, and every map. Were such a map possible, its specificity

would defeat its purpose: to generalize and abstract. Mapmakers highlight such features

as their clients choose. Whatever their purpose, maps and models must simplify as

much as they mimic the world.” – James Gleick

vi

Acknowledgements

I would sincerely like to thank John Sader for the many exciting discussions. The

support and encouragement I have received has been phenomenal.

Thanks must also go to Igor Bargatin and the Roukes Group from California Institute

of Technology for their amazing experimental work.

I am indebted to my wonderful family and friends – in particular Naomi Clarke,

Harrison Wraight and Jason Nassios – for their patience and understanding over the

past year.

viii

ix

Chapter 1

Introduction

Nanoelectromechanical systems (NEMS) enable exquisitely sensitive measurements

to be taken at the nano-scale. The dynamics of fluid systems at this level can be vastly

different to those of the macroscopic world with which we are most familiar. Much re-

search has been undertaken in an attempt to gain an appreciation of how nano-devices

function, as well as the principles governing fluid motion, at this scale. The practical

applications are very exciting indeed. For example, an understanding of the dynamics

of thin cantilever beams has led to the widespread use of the atomic force microscope,

capable of identifying individuals atoms. Furthermore, the extreme sensitivity of other

NEMS devices has facilitated zeptogram-scale (10−21g) mass sensing [1] .

A comprehensive knowledge of the underlying physical principles is essential in

harnessing the enormous potential that NEMS devices offer. In this thesis, we explore

the dynamics of one such device. When a clamped-clamped silicon carbide beam is

excited thermoelastically, it exhibits a spectrum of in-plane and out-of-plane resonances.

Recent experiments conducted by Bargatin et al. [2] involved immersing such beams

in a gas and measuring the quality factors associated with the exhibited resonances.

The quality factor Q is a dimensionless parameter that relates the stored energy in the

beam to the rate of energy dissipation. The beams used by Bargatin et al. [2] have

a width of 400nm, a thickness of 80nm and vary in length between 8µm and 24µm.

Consider Fig. 1.1, which shows a scanning electron micrograph of one such device.

Motion of the beam is initiated electrically through thermal expansion. The electrical

resistance of components of the beam depend on the mechanical stresses present and

1

2

thus the deflection. This allows for the motion to be detected and processed using a

metal piezoresistor.

Figure 1.1: Scanning electron micrograph of one of the devices used by Bargatin et al. [2](oblique view). The top two insets are close-up images showing the drive (right) and detection(left) loops.

Consider the illustrations in Fig. 1.2 which clearly show both the nature of the

apparatus and the motions it can execute. We emphasise that the in-plane and out-of-

plane movements depicted are the fundamental modes only. The beams exhibit many

mode shapes in the experiments [2].

As the beam vibrates, it loses energy due to both internal dissipation and damp-

ing by the surrounding fluid. The rate at which energy is lost from the beam can be

measured, and is found to exhibit a strong dependence on the frequency of oscillation.

The problem with which we are faced is to develop an understanding of the energy

dissipation in these systems.

We will begin by analysing the beam as a linearly elastic body and predict the

resonant frequencies in vacuum. In Chapter 4 we assess the dynamics of the beam in a

viscous fluid, operating in the high-frequency limit. Chapters 5 and 6 account for the

finite dimensions of the beam, and successfully incorporate the effects of slip. These

final two chapters constitute original research and provide a significant contribution

towards the understanding of NEMS devices. At the end of each chapter we will assess

the accuracy of the model by comparing the predictions with experimental results.

Chapter 1. Introduction 3

h

b

L

x

y

z

(a)

(b) (c)

Figure 1.2: (a) Diagram of clamped-clamped beam. The origin of the coordinate systemis located at the geometric centre of the cross-section of the beam at the left clamped end(depicted by a small black dot). (b) Schematic diagram showing out-of-plane motion. (c)Schematic diagram showing in-plane motion.

Chapter 2

Nanoelectromechanical Systems

(NEMS)

Micro-cantilever beams have been used in atomic force microscopy (AFM) for a

number of years. Significant work — both experimental and theoretical — has been un-

dertaken in an attempt to understand the behavior of such elastic devices. Knowledge

of the frequency spectra is paramount in application to the atomic force microscope

[3]. The frequency response of an elastic beam is highly sensitive to the nature of the

fluid in which it is immersed. In order to accurately predict the frequency spectrum,

we must take into account the physical properties of the beam, as well as the nature of

the surrounding fluid.

In 1851 Stokes [4] provided an analytical solution to the problem involving a cylin-

der of circular cross-section oscillating in a viscous fluid. He found that during such

oscillations, the cylinder experienced viscous losses due to the surrounding fluid, as well

as an added (or “virtual”) mass component. In 1969, Tuck [5] presented a boundary in-

tegral formulation whereby the hydrodynamic loading could, in principle, be calculated

for a cylinder of arbitrary cross-section. Whilst Tuck’s solution was quite general, the

absence of powerful computers limited the circumstances to which it could be applied.

We will use this boundary integral technique extensively throughout the course of our

problem. It must be noted that the assumptions made by Tuck in deriving his result

are also adopted in our problem formulation.

5

6

Since the publication of Tuck’s paper in 1969, much work has been done in de-

termining the frequency response of cantilevers subject to a driving force. The driving

force is often due to thermal oscillations and can be modified by the presence of a nearby

surface e.g. in the case of the atomic force microscope. Typically, the surrounding fluid

is considered to be infinite in extent. However, Green [6] looked at the effects associated

with the presence of a nearby boundary in the fluid. Sader [3] analysed the frequency

response and hydrodynamic loading of both rectangular and circular cantilever beams

immersed in viscous fluids. The modes exhibited by cantilevers can be either transverse

or torsional in nature. Green and Sader [7] discussed the torsional frequency response

of rectangular cantilever beams in viscous fluids. This is of particular importance in

understanding how the atomic force microscope works.

Until recently, only out-of-plane and torsional modes have received attention. From

a practical point of view, it is relatively easy to detect and measure out-of-plane motion.

In experiments conducted by Sader et al. [8], a laser beam was used to monitor the

deflection of the cantilever. This method cannot be employed for in-plane modes since

the plane of the cantilever does not move during oscillations. In 2007, Bargatin et al. [2]

were able to detect both the in-plane and out-of-plane modes for silicon-carbide-based

beams. The devices were actuated thermoelastically at room temperature and their

motion detected piezoresistively. The piezoresistor could be used since the electrical

resistance of device-integrated metal loops depend on the mechanical stresses present.

The amplitude and frequency of oscillation could thus be calculated by analysing the

electrical signal passed through the loop.

Cantilevers of many different shapes and sizes have been studied. These include

rectangular beams, triangular and v-shaped cantilevers. The nominal width of such

devices is usually considered to greatly exceed the thickness. Consequently, the devices

studied by Green and Sader were modelled as infinitely thin.

Although cantilevers used in atomic force microscopy are typically of order 100µm

in length [8], the no-slip boundary condition traditionally used in fluid dynamics is still

applied. The mean free path of air molecules is approximately 68nm at room temper-

ature and pressure. Consequently, at small length scales, the treatment of the fluid

as a continuous medium warrants questioning. For practical applications, cantilevers

Chapter 2. Nanoelectromechanical Systems (NEMS) 7

exhibiting out-of-plane motion have been traditionally used. In such circumstances, the

effects associated with the rarefied nature of the fluid are minimal, and so treatment of

the fluid as continuous is justified.

Various slip models have, in the past been adopted in an attempt to quantify

the effects of slip. We define the Knudsen number as Kn = η/b, where η and b are

the mean free path of the fluid molecules and the dominant length scale respectively.

Hadjiconstantinou [9] outlined a second-order slip model which is claimed to extend

the applicability of Navier-Stokes solutions beyond Kn ≈ 0.1 where the accuracy of

the first-order slip model deteriorates. Hadjiconstantinou [9] presented the following

second-order boundary condition:

U∣∣wall

− uw = γη∂U

∂z

∣∣∣∣wall

− δη2 ∂2U

∂z2

∣∣∣∣wall

. (2.1)

In the above equation, γ and δ are non-adjustable parameters and η is the mean free

path of the fluid molecules. The fluid velocity field and beam velocity are denoted by U

and uw respectively. The z-direction is normal to the surface and pointing into the fluid.

For one-dimensional flows with hard spheres, we have γ = 1.11 and δ = 0.61 [9]. The

beauty of this boundary condition is that it enables the use of Navier-Stokes solutions

in a flow regime where they are usually invalid. The relative ease of determining such a

solution, compared to implementing a rigorous molecular simulation, makes this method

particularly appealing. The above values of γ and δ assume diffuse reflection of fluid

molecules from the surface. This involves the fluid molecules temporarily adhering

to the surface and subsequently being re-emitted at various velocities according to a

Maxwellian distribution [10]. If we allow for specular reflection, whereby fluid particles

simply bounce off the surface like light from a mirror, we get the following correction

for the first-order slip condition [11]:

γ →(

1.79σ

− 0.65− 0.14σ)γ.

In the above equation, 0 ≤ σ ≤ 1 is the thermal accommodation coefficient and

represents the relative proportion of diffuse encounters. The value of σ = 1 corresponds

to purely diffuse reflection while σ = 0 corresponds to purely specular reflection.

8

2.1 Assumptions

Before we continue, it is necessary to make some preliminary assumptions regarding

the system. Henceforth, we will assume that the clamped-clamped beam satisfies the

following properties:

1. The length of the beam greatly exceeds its nominal width;

2. The amplitude of vibration of the beam is extremely small compared to any otherphysical length scale of the beam;

3. The beam does not adopt any torsional modes. Only in-plane and out-of-planemotion is permitted;

4. The beam is composed of a linearly elastic, isotropic material.

Due to the presence of the actuation and detection loops visible in Fig. 1.1, it is

clear that the beam is not entirely isotropic. Nevertheless, the effects of this will be

minimal. Assumption 4 can thus be made, and is indeed necessary, though only for

Chapter 3. We will adopt the material properties in accordance with the values used

by Bargatin et al. [2]. The density and Young’s modulus of the silicon carbide beams

will be taken to be ρ = 3.2 g/cm3 and E = 430 GPa respectively.

We also state some assumptions regarding the nature of the fluid. We assert that

the fluid surrounding the nanomechanical resonator is incompressible. This assumption

is justified provided the following two conditions are satisfied:

• u c i.e. the characteristic velocity of the fluid must be very small compared

to the speed of sound.

• λ = c/ω b i.e. the “wavelength” in the fluid must be significantly larger than

the dominant length scale, which for our problem is equal to b.

A beam oscillating with displacement amplitude A and angular frequency ω will

have an associated velocity amplitude of order Aω. We thus see that u c by as-

sumption 2. Furthermore, we note that for the resonances observed by Bargatin et al.,

λ ∼ 10−6m b. Consequently, we find that the treatment of the surrounding fluid as

incompressible is indeed justified [3]. This is particularly useful since we can now write


∇ · u = 0.

The stress tensor associated with the fluid is given by

T = µ

(∇u + (∇u)T

).

The density, viscosity and molecular mean free path of air will be assumed to be

ρair = 1.20 kg/m3, µ = 1.78× 10−5 Pa·s and η = 68nm respectively. By assumption 2,

the nonlinear convective effects in the fluid surrounding the beam can be ignored. Fur-

thermore, since the length of the beam greatly exceeds its nominal width, the fluid

velocity varies quite slowly along the x-direction. A direct consequence of this is that

locally, the beam can be treated as rigid and infinitely long, undergoing tangential and

normal oscillations (y and z directions respectively) [3].

2.2 Boundary Conditions

Historically, the no-slip boundary condition has been adopted when solving prob-

lems involving viscous fluids. Quite simply, this means that at the fluid-object interface,

the fluid must not be moving relative to the boundary. Recall that the Knudsen num-

ber Kn = η/b is defined as the ratio between the molecular mean free path η and the

dominant length scale b. For a blade of width b = 400nm in air, we have Kn ≈ 0.17. At

this scale, it becomes inappropriate to use continuum mechanics everywhere. Working

in this regime typically requires the use of molecular simulations. Such simulations

are extremely demanding from a computational point of view. In order to analytically

account for the effects of slip, we perform a matched asymptotic expansion between the

two distinct flow regimes: close to, and far away from the beam, operating formally

in the limits of Kn 1 and Kn 1. We write the fluid velocity as a perturbation

expansion for small η = η/a, where η is the mean free path of the fluid molecules and

a is half of the blade width:

U = U (0) + ηU (1) +O(η2).

The term U (0) is the solution corresponding to the no-slip boundary condition,

while the subsequent terms are corrections based on the effects of slip. We note that

10

solving the problem rigorously involves sequentially determining U (0), U (1), etc. The

numerous corrections can be evaluated recursively. This procedure provides excellent

results for an oscillating infinite plate as we will see in Chapter 4, but not for the

infinitely thin blade. The problems encountered in the latter case are outlined in Ap-

pendix B. Instead of performing the matched asymptotic expansion, we thus choose to

implement the second-order slip model produced by Hadjiconstantinou [9]. This model

allows us to capture both the flow and stress fields at the surface of the blade without

using any fitted or adjustable parameters. The beauty of this approach is that we can

apply Navier-Stokes solutions simply by adopting a different boundary condition. The

first-order slip condition is obtained simply by setting δ = 0 in Eq. (2.1), i.e.

U∣∣wall

− uw = γη∂U

∂z

∣∣∣∣wall

. (2.2)

From above, the slip velocity is given by

Uslip = γη∂U

∂z

∣∣∣∣wall

.

In the continuum limit as η → 0, the slip velocity approaches zero as it should and

the no-slip condition is recovered. Furthermore, in the free molecular regime represented

by η →∞, we see that ∂U/ ∂z → 0 and so the stress vector at the surface is zero. The

Navier slip condition will be very useful since it is valid in both the η → 0 and η →∞limiting cases, as we shall discuss.


2.3 Hydrodynamic Function

Consider the linearized, Fourier transformed equations of motion of the fluid:

∇ · u = 0, −∇P + µ∇2u = −iρωu, (2.3)

where

X =∫ ∞

−∞Xe−iωtdt.

The nonlinear convective term vanishes because we are considering oscillations of

a sufficiently small amplitude. For a cantilever beam, the width b of which greatly

exceeds its thickness h, the dominant length scale is b. Similarly, for the clamped-

clamped blade undergoing in-plane oscillations, the dominant length scale is its width

b. Sader [3] showed that if we solve Eq. (2.3), we obtain

Fhydro(x|ω) =π

4ρω2b2Γ(ω)W (x|ω), (2.4)

where Γ(ω) is the “hydrodynamic function” which is a complex-valued dimensionless

function. The real and imaginary components of the hydrodynamic function represent

the added mass and viscous losses respectively of the beam as it vibrates. At no point

in the derivation of Eq. (2.4) from equation (2.3) does one need to distinguish between

in-plane and out-of-plane oscillations. We can thus use Eq. (2.4) for the case where the

blade undergoes in-plane oscillations.

12

2.4 Quality Factors

The quality factor Q is a dimensionless parameter that compares the stored energy

in the system to the rate of energy dissipation. We define

Q = 2π × Energy StoredEnergy dissipated per cycle

∣∣∣∣ωR

, (2.5)

where ωR denotes evaluation at a resonant frequency. Equation (2.5) is an expression

relevant to any oscillating physical system. A larger quality factor represents a system

with a lower rate of energy dissipation. We consider a vibrating beam immersed in a

viscous fluid. The stored energy is thus the sum of the elastic potential and kinetic

energy associated with the beam. The two mechanisms responsible for dissipation of

energy from the system are viscous losses due to the surrounding fluid and internal

friction in the beam. Sader [3] derived an expression for the quality factors associated

with the deflection of a cantilever :

Qcantilever =4λπρb2

+ Γr(ωR)

Γi(ωR), (2.6)

where λ is the linear mass density of the cantilever and ρ is the density of the fluid.

Γr(ωR) and Γi(ωR) can be found by taking the real and imaginary parts of the hy-

drodynamic function. The subscript R denotes resonance. Equation (2.6) has been

derived for a general cantilever. We will be predominantly concerned with a beam that

is oscillating in its own plane, rather than perpendicular to it as in the case of the can-

tilever. Furthermore, unlike a cantilever, our apparatus is clamped at each end. It can

be shown that adopting the same scaling as in Eq. (2.4), the expression for the quality

factors associated with the in-plane modes is exactly the same as Eq. (2.6). That is,

Qbeam = Qcantilever. The general procedure for calculating the fluid-only quality factors

is as follows:

1. Calculate the hydrodynamic force acting on the beam;

2. Find the hydrodynamic function using Eq. (2.4);

3. Determine the associated quality factor using Eq. (2.6).


2.5 Experimental Results

The total quality factor Qtotal associated with a blade oscillating in a viscous fluid

can be written as a combination of the quality factors associated with the two different

avenues of energy loss,1

Qtotal=

1Qbeam

+1

Qfluid. (2.7)

If a vibrating beam were not subject to any driving force then viscous losses would

diminish the amplitude of oscillation. Bargatin et al. [2] excite the blades thermoelasti-

cally and an equilibrium state is achieved where the driving power is equal to the total

power loss. In such a situation, all transient effects have decayed and the quality factors

can be measured. Bargatin et al. [2] successfully measured the quality factors associ-

ated with beams in vacuum as well as in air. Using Eq. (2.7) we can thus determine

the experimental quality factor related to the viscous losses only,

Qfluid =QbeamQtotal

Qbeam −Qtotal. (2.8)

We do not attempt to model in any way the intrinsic quality factor associated with

the beam. We will use Eq. (2.8) to isolate the fluid contribution. These values will serve

as our benchmark with which we hope to obtain good theoretical agreement. Using the

results published by Bargatin et al. [2], we present the quality factors associated with

the first five in-plane resonances of the 16µm-long beam as well as the second and third

in-plane modes of the 24µm-long beam.

Mode Qtotal Qbeam Qfluid

1 in 130 3111 1362 in 300 2627 3393 in 431 2214 5354 in 550 1743 8045 in 700 1765 11602 in 150 3300 1573 in 280 2800 311

Table 2.1: Experimental quality factors measured for the first five in-plane resonances ofthe 16µm-long beam and the second and third in-plane modes of the 24µm-long beam. Qfluid

represents the contribution to the quality factor from the fluid only, as calculated using Eq. (2.8).

Chapter 3

Resonant Frequencies in Vacuum

In order to identify the various mechanisms involved in both the in-plane and

out-of-plane oscillations of the beam, it is important to gain an understanding of the

frequency response of the beam itself. Bargatin et al. [2] performed experiments in

which they recorded the resonant frequencies of various blades. The in-plane and out-

of-plane vibrational modes were recorded for cases where the blade was in vacuum as

well as in air. For the time being we will concern ourselves with the modes exhibited

in vacuum, since this corresponds to the absence of any hydrodynamic loading.

3.1 Free Vibrations

We initially consider the case of a clamped-clamped beam exhibiting free vibrations.

The governing equation for the deflection of an isotropic beam is

∂2

∂x2

(EI

∂2W (x, t)∂x2

)= w. (3.1)

We assume that the forces acting on the beam cause it to bend, but not to stretch

or twist. The curve W (x, t) describes the deflection of the beam at a particular position

x and time t. The value of w is the distributed load, which for a freely vibrating beam

is constituted entirely of inertial forces. The parameters E and I are Young’s modulus

and the area moment of inertia respectively. The inertial force per unit length on the

beam is given by Newton’s second law:

w = −λ ∂2W (x, t)∂t2

,

15

16

where λ is the linear mass density of the beam. The governing equation can thus be

written as a partial differential equation involving both spatial and time derivatives

∂2

∂x2

(EI

∂2W

∂x2

)+ λ

∂2W

∂t2= 0.

We are primarily concerned with determining the resonant frequencies of the vibrat-

ing beam. We care very little about the actual mode shapes and their time evolution.

It will thus be much more appropriate to work in the frequency domain as opposed to

the time domain. We proceed to take the Fourier transform of the above equation to

obtaind2

dx2

(EI

d2

dx2

(W (x|ω)

))− λω2W (x|ω) = 0. (3.2)

We note that E and I are constant throughout the beam. Upon introduction of

the scaled variable x = x/L, Eq. (3.2) simplifies substantially to become

d4W

dx4− n4W = 0, (3.3)

where we have identified

n4 =λω2L4

EI. (3.4)

The general solution of Eq. (3.3) is given by

W (x|ω) = A cosh(nx) +B sinh(nx) + C cos(nx) +D sin(nx).

Since the beam is clamped at each end, the displacement and its derivative at these

points must be equal to zero. The boundary conditions for W are thus as follows:

W (0|ω) = 0, W (1|ω) = 0,dWdx (0|ω) = 0, dW

dx (1|ω) = 0.(3.5)

Application of the boundary conditions to the general solution yields the following

dispersion relation:

coshn cosn = 1. (3.6)

The positive roots of Eq. (3.6) correspond to the allowable modes of vibration. Un-

fortunately, a closed analytical solution does not exist and so we resort to a numerical

Chapter 3. Resonant Frequencies in Vacuum 17

method. Below is a table containing the first five positive roots of Eq. (3.6), correct to

four decimal places.

n1 4.7300

n2 7.8532

n3 10.9956

n4 14.1372

n5 17.2788

The area moment of inertia for the in-plane and out-of-plane oscillations are

Iin = 112hb

3, Iout = 112bh

3.

The area moment of inertia essentially measures the resistance of the beam to bend-

ing. Since b > h, we note that Iin > Iout, i.e., the in-plane modes are “stiffer” than the

out-of-plane modes. This is in perfect accordance with our intuition and indeed with

personal experience: it is easy to bend a ruler by hand in the out-of-plane direction yet

exceedingly difficult to bend the ruler in its own plane. Using Eq. (3.4) we can calculate

the first five resonant frequencies of the beams used in the experiments conducted by

Bargatin et al. [2]. We present these values alongside the experimental results for both

the in-plane (in) and out-of-plane (out) modes of the 8-µm-long and 16-µm-long beams.

18

Mode Number f pred (MHz) f exp (MHz)1 in 74.5 56.42 in 205 1453 in 402 272

Mode Number f pred (MHz) f exp (MHz)1 out 14.9 23.52 out 41.1 50.43 out 80.5 84.24 out 133 1245 out 199 172

Table 3.1: Predicted and experimental values of the lowest three in-plane and five out-of-planemodes. Dimensions of the beam: L = 8µm, b = 400nm, h = 80nm. Material properties ofsilicon carbide beam: Young’s modulus E = 430 GPa, mass density ρ = 3.2 g/cm3.

Mode Number f pred (MHz) f exp (MHz)1 in 18.6 20.82 in 51.3 54.03 in 101 1034 in 166 1675 in 248 243

Mode Number f pred (MHz) f exp (MHz)1 out 3.72 9.522 out 10.3 20.03 out 20.1 32.34 out 33.3 46.85 out 49.7 63.9

Table 3.2: Predicted and experimental values of the lowest five in-plane and out-of-planemodes. Dimensions of the beam: L = 16µm, b = 400nm, h = 80nm. Material properties ofsilicon carbide beam: Young’s modulus E = 430 GPa, mass density ρ = 3.2 g/cm3.

Notice that the fundamental in-plane mode is considerably higher in frequency than

the fundamental out-of-plane mode for each blade. This can be directly attributed to

the difference in the area moment of inertia for the two cases. In fact,

fin

fout=

IinIout

=b

h= 5.

The predictions presented in Tables 3.1 and 3.2 are reasonably accurate.

3.2 Free Vibrations - Intrinsic Tension

There are certain aspects of the process of beam fabrication that are difficult to

control. Whilst the dimensions and other physical characteristics of each beam are

often known quite accurately, Bargatin et al. [2] allude to the possibility that each

silicon carbide beam may possess an intrinsic strain that arises during its construction.

Indeed, one can appreciate the challenges faced in attempting to manufacture a beam

with a thickness in the vicinity of 1/250 times the diameter of a human hair. Intuitively,


the inclusion of some intrinsic strain in the beam should affect the subsequent resonant

frequencies. Indeed, this is precisely the principle that allows for a guitar string to be

tuned. Recall that in Eq. (3.1), w represents the distributed load on the beam. In the

case of free vibrations, this consisted solely of the inertial forces. We now allow it to

incorporate the intrinsic tension T , which is unknown at this stage. Upon inclusion of

this new term, the governing equation for the deflection of the beam becomes

∂2

∂x2

(EI

∂2W (x, t)∂x2

)+ λ

∂2W (x, t)∂t2

= T∂2W (x, t)

∂x2.

Taking the Fourier transform of the above equation and again treating E and I as

constant, we obtain the following fourth-order ordinary differential equation:

d4W

dx4−

(TL2

EI

)d2W

dx2−

(λω2L4

EI

)W = 0, (3.7)

where the scaled variable x = x/L has again been adopted. Setting T = 0 recovers

Eq. (3.3) as expected. The coefficients in the above equation are dimensionless quanti-

ties and are defined to be

H =TL2

EI, σ =

λω2L4

EI. (3.8)

Notice that H and σ encompass the tension and frequency of the beam respectively.

It will be convenient to work with these dimensionless parameters and evaluate the

dimensional quantities where necessary. If we attempt a solution of the form W (x|ω) =

eax we find that

a2 =12(H ±

√H2 + 4σ

),

and thus

a = ±χ, a = ±iγ,

where

χ = 1√2

√√H2 + 4σ +H, (3.9)

γ = 1√2

√√H2 + 4σ −H. (3.10)

20

The general solution is then

W (x|ω) = A1eχx +B1e

−χx + C1eγix +D1e

−γix,

or equivalently

W (x|ω) = A cosh(χx) +B sinh(χx) + C cos(γx) +D sin(γx),

where A, B, C and D are arbitrary constants. The boundary conditions are precisely

the same as those from Eq. (3.5). Application of such conditions to the above general

solution yields the following dispersion relation:

2 coshχ cos γ +(γ

χ− χ

γ

)sinhχ sin γ − 2 = 0. (3.11)

Equation (3.11) is too complicated to solve analytically, however a suitable software

package such as mathematica R© can easily find numerical solutions. Recall that χ and

γ are functions of both H and σ which are in turn functions of T and ω respectively.

Thus, Eq. (3.11) relates the tension T in the beam with the permissible vibrational

modes. It is important to note that the only difference between solving for the in-plane

modes and the out-of-plane modes lies in the value of I which is different for the two

cases. The intrinsic tension is not something that is generally known given the man-

ufacturing procedure. It is left as a tuneable parameter which must be determined

numerically. The value of T will depend only on the beam itself and not on the nature

of any deflection which it undergoes since it is an inherent property of the material.

The actual resonant frequencies of the blade are well-known. Using Eq. (3.11) it is

possible to solve for the value of H and thus T given the experimental values of ω. If all

aspects of the model are correct and we have not missed anything, then the subsequent

values of T should turn out to be the same in all cases. However, the calculated values

of the parameter T turn out to vary by a factor of approximately 5 over the range of

exhibited vibrational modes. This is hardly surprising given how crude our model is

at this stage. To perform a regression in order to determine an appropriate value of

T is unwarranted, since it is far more likely that there are other aspects of the system

responsible for this behaviour which we have failed to include.


Based on finite-element simulations performed using femlab, Bargatin et al. [2]

conclude that the 16µm-beam possesses an intrinsic tensile strain of ε = 2.8 × 10−4

which would correspond to T = Ebhε = 3.85 × 10−6 N. Using the dispersion relation

that is Eq. (3.11), the values of ω and thus f corresponding to this choice of T can be

found. In Fig. 3.1 we present the predicted resonant frequencies for such a situation

alongside the experimental values.

1 2 3 4 5 6 7 8 9 10 11 1210

0

101

102

103

Mode Number

Fre

quen

cy (

MH

z)

IN−PLANE

OUT−OF−PLANE

ExperimentT = 3.85µNT = 0N

Figure 3.1: Plot of predicted and experimental resonant frequencies for in-plane and out-of-plane modes of 16µm-long beam in vacuum. Free vibrations (dotted); Intrinsic tension T =3.85× 10−6 N (dashed); Experimental values (solid).

We note that the inclusion of intrinsic tension into the model has a much more

pronounced effect on the out-of-plane modes than on the in-plane modes. Indeed, this

can be accounted for by the fact that in-plane movement of the beam is much stiffer

than out-of-plane motion. Thus, the in-plane modes are less susceptible to changes in

tension of the beam.

The difficulty faced in choosing an appropriate value of T across all modes now

becomes apparent. Increasing the tension in the beam causes a subsequent increase in

the resonant frequency of all modes. If we increase T beyond 3.85 × 10−6 N in order

to match up the low frequency predictions with experiment, we do so at the expense

of the higher modes. Indeed, it is impossible to find a value of T which yields truly

accurate predictions across the range of exhibited modes. This problem is even more

significant for the out-of-plane modes of the 8µm-long beam, where the predictions for

22

the high and low frequencies (see Table 3.1) are seen to significantly overestimate and

underestimate respectively. Let us not be disheartened by this fact. We have managed

to obtain reasonably accurate estimates of the resonant frequencies given the crudeness

of the model. There are several aspects of the model which must be revisited and

questioned:

• We have assumed that the boundary conditions given in Eq. (3.5) are appropriate

for all modes. It is likely that in the high-frequency regime, the validity of these

boundary conditions becomes questionable. Indeed we must bear in mind that we

are attempting to model a real situation. It is likely that, particularly for higher

frequencies, the beam is subject to imperfect clamping. This would obviously

affect the calculated resonances in a fashion that is non-uniform across the fre-

quency range. Inspection of the data in Tables 3.1 and 3.2 reveals that the error

in our predictions is much more significant for the 8µm-long beam. For the higher

mode numbers, the free vibrating beam model yields substantial overestimates.

It appears that the consequences of imperfect clamping are more pronounced for

the shorter beam.

• The beams used by Bargatin et al. [2] contained gold actuation and gold-palladium

detection loops. These were used to thermally initiate the motion of the blade,

and detect the frequency of vibration respectively. These features mean that the

apparatus is anisotropic. The loops are relatively small, and so the corresponding

effects will be minimal. However, we must acknowledge that the model we have

adopted to predict the resonances does assume isotropy of the beam.

We emphasise that the resonant frequencies predicted in this chapter will not be

used in any of the subsequent chapters. In attempting to predict the quality factors

associated with the in-plane resonances, we will use the actual resonant frequencies

measured by Bargatin et al. [2].


3.2.1 Euler Buckling Formula

We now seek to verify the Euler formula for column buckling. We know that

applying tension to the beam causes a subsequent increase in the allowable modes of

vibration. Intuition would suggest that if a compressive force was applied, the converse

would happen. For a sufficiently large compressive force, we anticipate that the beam

will buckle. Physically this corresponds to the value of T where the frequency of vi-

bration of the beam approaches zero. Recall the definitions of H and σ from Eq. (3.8).

Close to the onset of buckling we obviously have σ → 0. We can thus consider |H| σ

from now on. We note thatH < 0 since we are considering the beam under compression.

It follows that √H2 + 4σ ≈ −H

(1 +

12

4σH2

). (3.12)

Equations (3.9) and (3.10) thus become

χ ≈√− σ

H, γ ≈

√− σ

H−H.

After substitution of the above equations into Eq. (3.11) and performing a Taylor

series expansion about σ = 0, we see that to leading order,

2 cos√−H +

√−H sin

√−H − 2 = 0. (3.13)

By inspection, solutions to Eq. (3.13) are given by −H = 4π2n2 for n = 1, 2, 3, . . ..

There are many buckling modes possible, but we are primarily interested in the lowest

one, corresponding to n = 1. In this situation,

T = − π2EI

(KL)2,

where we have defined K = 1/2. This corresponds precisely to the buckling formula of

an ideal column.

Chapter 4

High-Frequency Limit

In the limit of high-frequency, oscillations of an infinitely thin blade in the fluid will

generate an extremely thin viscous boundary layer near its surface. In this limit, the

fluid does not see the width or length of the blade. This facilitates the use of an infinite

plate solution. To begin with, let us consider the linearized Navier-Stokes equation.

We justify ignoring the non-linear convective term since for an infinite plate the fluid

exhibits unidirectional flow. In addition, the amplitude of oscillation is considered to

be small (see Section 2.1) and so the non-linear effects are negligible. The governing

equation for the fluid is then∂u

∂t= ν∇2u, (4.1)

where u = uy is the velocity and ν is the kinematic viscosity of the fluid above the

plate. The plate oscillates with angular frequency ω so it is natural to Fourier transform

Eq. (4.1) and work in the frequency domain. We have

−iωU = ν∇2U .

This is an ordinary differential equation for U , and is given by

d2U(z|ω)dz2

= − iωνU(z|ω). (4.2)

25

26

Solving this and requiring that U → 0 as z →∞, we obtain the following general

solution for the fluid velocity above the plate:

U(z|ω) = A exp(i

√iω

νz

). (4.3)

4.1 No-Slip Boundary Condition

If we wish to apply the no-slip boundary condition, we simply identify A in Eq. (4.3)

as the velocity amplitude of the plate. If W (ω) is the displacement amplitude of the

oscillating plate, then A can be written as −iωW (ω). We note that although our

vibrating beam appears locally infinite, the displacement varies along its length. ie.

W = W (x|ω). The fluid velocity above the plate thus becomes

U(x, z|ω) = −iωW (x|ω) exp(i

√iω

νz

). (4.4)

The stress tensor for incompressible flow can be easily calculated and is given by

T = µωW (x|ω)

√iω

ν

(zy + yz

)− µiωdW (x|ω)

dx

(xy + yx

).

The stress vector,

t = n ·T = µωW (x|ω)

√iω

νy,

is the force per unit area exerted by the fluid on the surface, with normal vector n

pointing into the fluid. The net force per unit length on the vibrating blade exerted by

the fluid is then given by

Fhydro = 2bµωW (x|ω)

√iω

ν, (4.5)

where the factor of 2 accounts for the top and bottom faces and b is the width of the

blade. Comparing Eqs. (4.5) and (2.4), we see that

Γ(ω) =8πb

√iν

ω. (4.6)

If λ is the linear mass density of the beam, ρ is the density of the fluid and Γr and

Γi are the real and imaginary components of the hydrodynamic function respectively,

Chapter 4. High-Frequency Limit 27

then the associated quality factor for a given frequency ω is given by Eq. (2.6):

Q =4λπρb2

+ Γr(ω)

Γi(ω)

= 1 +λ

b

√ω

2µρ. (4.7)

There are a couple of interesting points to note from Eq. (4.7). Recall that the

quality factor relates to the rate of energy dissipation from the blade due to viscous

losses. A large quality factor corresponds to a system where such energy losses are

relatively small. For the current system, we note in particular the frequency dependence

of Q. We see that high-frequency modes have relatively large associated quality factors.

Note also that Q is a decreasing function of blade width b and surrounding fluid viscosity

µ and density ρ. This result is in perfect accordance with intuition. It is important

to note that although we have assumed an infinitely thin viscous boundary layer and

used the corresponding infinite plate solution, we have certainly not treated the blade

as an infinite plate itself. Indeed, quantities such as b and λ do not make sense for an

infinitely large surface.

We emphasise that the expression given for the quality factor in Eq. (4.7) is the

contribution from the fluid only. This does not include any intrinsic quality factor

associated with the blade itself.

4.2 Second-Order Slip Boundary Condition

To this point we have treated the fluid surrounding the resonator as continuous

and have thus adopted the no-slip boundary condition at the fluid-blade interface. The

Knudsen number for the 400nm-wide blade in air is Kn ≈ 0.17 and so the use of contin-

uum mechanics near the blade surface is actually inappropriate. Recall the second-order

slip boundary condition presented in Eq. (2.1). We can rewrite this equation in terms

of the Fourier transformed velocity:

U∣∣z=0

− uw = γη∂U

∂z

∣∣∣∣z=0

− δη2 ∂2U

∂z2

∣∣∣∣z=0

, (4.8)

where uw is the velocity of the wall. Observe that the no-slip boundary condition used

previously corresponds precisely to the case where η = 0 m. For one-dimensional flows,

28

the hard-sphere second-order slip model can be used with γ = 1.11 and δ = 0.61. These

parameter values are fixed for the purposes of this section, though we can set δ = 0 to

adopt the first-order slip condition. We begin with the fluid velocity given in Eq. (4.3)

and identify C = i√

iων . For a blade oscillating at angular frequency ω, the wall velocity

is uw = −iωW (x|ω) as before. Equation (4.8) thus becomes

A+ iωW (x|ω) = γηAC − δη2AC2,

and so

A =−iωW (x|ω)

1− γηC + δη2C2. (4.9)

Since the fluid velocity field is known, we can proceed to find the net force per unit

length exerted on the blade by the fluid:

Fhydro = 2bµAC,

or equivalently

Fhydro =π

4ρω2b2Γ(ω)W (x|ω),

where the hydrodynamic function Γ(ω) is given by

Γ(ω) =8νπb

√i

νω× 1

1− γηC + δη2C2(4.10)

=8ν√iν

πb√ω(ν − iγη

√iων − iδη2ω

) . (4.11)

The real and imaginary components of the hydrodynamic function are then

Γr(ω) =4ν

√2νω

(ν − δη2ω)

πb

[ν2 + γην

(γηω +

√2νω

)+ δη3ω

(δηω + γ

√2νω

)] , (4.12)

Γi(ω) =4ν

√νω

[√2ν + η

(√2δηω + 2γ

√νω

)]πb

[ν2 + γην

(γηω +

√2νω

)+ δη3ω

(δηω + γ

√2νω

)] . (4.13)


4.3 Matched Asymptotic Expansion

We now attempt to calculate explicitly the correction to the no-slip results arising

through inclusion of first-order slip. The two flow regimes are the kinetic layer close to

the beam and the region far away from the beam. We begin by writing the solution as

an asymptotic expansion for small values of the scaled mean free path η = η/a = 2Kn;

U = U (0) + ηU (1) +O(η2) (4.14)

Recall that the Fourier transformed linearized equation of motion of the fluid is

given byd2U

dz2= − iω

νU .

Upon substitution of Eq. (4.14) into the governing equation and equating orders of η,

we see thatd2U (n)

dz2= − iω

νU (n) for n = 0, 1, 2, . . . .

It follows that the general solution for each order of η is the same and is given by

U (n) = An exp(i

√iω

νz

)for n = 0, 1, 2, . . . (4.15)

where An must be determined for each order. The first-order slip condition can be

rewritten as

U (0) + ηU (1) +O(η2) = uw + aγη

(dU (0)

dz+ η

dU (1)

dz+O(η2)

),

where each side of the above equation is evaluated at z = 0. We can equate the

coefficients of different orders of η to obtain the boundary conditions for the different

orders. We find these to be

O(1) : U (0)∣∣z=0

= uw, (4.16)

O(η) : U (1)∣∣z=0

= aγdU (0)

dz. (4.17)

Notice that Eq. (4.16) is quite simply the no-slip boundary condition. To leading

order, the perturbation expansion in η yields the no-slip boundary condition as it should.

30

Furthermore, we note that the right hand side of Eq. (4.17) can be written in terms of

the zeroth-order vorticity. Since dU (0)/dz = −ω(0), Eq. (4.17) becomes

U (1)∣∣z=0

= −aγω(0)∣∣z=0

.

The first-order correction to the fluid velocity can be found using the zeroth-order

solution. The function ω(0) is simply the fluid vorticity at the surface in the no-slip

case, for which which we have already solved. It can be found directly from Eq. (4.4)

and the corresponding boundary condition for the slip velocity becomes

U (1)∣∣z=0

= γaωW (x|ω)

√iω

ν.

Upon application of the above boundary condition, the general solution presented

in Eq. (4.15) yields the slip velocity,

U (1) = γaωW (x|ω)

√iω

νexp

(i

√iω

νz

).

Thus, to first-order in η, the fluid velocity is given by

U = U (0) + ηU (1)

= ωW (x|ω)(− i+ γη

√iω

ν

)exp

(i

√iω

νz

).

Since the fluid velocity field is known, we can proceed to find the net force per unit

length exerted on the blade by the fluid,

Fhydro = 2bµωW (x|ω)(√

iω

ν− γηω

ν

). (4.18)

Notice that if we set η = 0, the force corresponding to the no-slip condition is

indeed recovered as in Eq. (4.5). Furthermore, we see that the inclusion of slip serves

to reduce the force exerted on the blade by the fluid; a mathematical result consistent

with our intuition. We can evaluate the hydrodynamic function associated with the slip

case by equating Eq. (2.4) and Eq. (4.18). We see that

Γ(ω) =8πb

(√iν

ω− γη

). (4.19)


This solution is only appropriate for η 1 (or equivalently Kn 1). We note that

the first-order correction to the hydrodynamic function is actually quite large. Indeed,

unphysical solutions can be obtained for fairly modest values of Kn. We know that as

Kn →∞, the force should approach zero and so Γ → 0 in this limit. In order to satisfy

this condition, we construct a Pade approximant [12] from Eq. (4.19) by noting that

1− x ≈ 11+x for small x. This can be written as

Γ(ω) =8πb

√iν

ω

(1− γbKn

√ω

iν

)

≈8πb

√iνω

1 + γbKn√

ωiν

. (4.20)

For Kn 1, Eqs. (4.19) and (4.20) have the same asymptotic form, whilst the

latter has the advantage of also possessing the correct asymptotic form as Kn →∞. In

fact, for large Kn, Eq. (4.20) becomes

Γ(ω) ∼ 8νiπγωb2Kn

for Kn →∞.

This is precisely the same form as Eq. (4.10) for large Kn (with δ = 0). We now

plot the solutions in Eqs. (4.19) and (4.20) which have been obtained through matched

asymptotic expansion, alongside the hydrodynamic function in Eq. (4.10) which was

derived using the Navier slip condition. We note that although the second-order slip

condition was used in Section 4.2, we use this solution with δ = 0 to recover the first-

order solution.

Very interestingly, the matched asymptotic expansion initially derived for small Kn

and subsequently modified to give the appropriate behaviour at Kn → ∞, agrees per-

fectly with the results obtained through simple application of the Navier slip condition.

Another interesting feature of Fig. 4.1 is that the first-order correction for small Kn

diverges rapidly from the exact solution as Kn increases. It follows that this expansion,

depicted by the dashed line, is only valid for quite small Kn. Indeed, for the fundamen-

tal mode of the 16µm-long beam used by Bargatin et al. [2], we have Kn ≈ 0.17. Even

for this value the accuracy of the first-order correction has deteriorated.

32

Pade approximantAsymptotic

expansion

for Kn<<1

0.0 0.5 1.0 1.5 2.0 2.5 3.00.0

0.5

1.0

1.5

2.0

Kn

Re@GD

(a)

Asymptotic

expansion

for Kn<<1

Pade approximant

0.0 0.2 0.4 0.6 0.8 1.00.0

0.5

1.0

1.5

2.0

Kn

Im@GD

(b)

Figure 4.1: Real and imaginary components of the hydrodynamic function for the fundamentalmode of the 16µm-long beam used by Bargatin et al. [2]. The plots show the matched asymptoticexpansion (dashed) and modified asymptotic expansion (solid). The latter coincides perfectlywith the results from Section 4.2 where the Navier slip condition was used.


4.4 Progressive Comparison With Experiment

We emphasise that for the thin boundary layer approximation, Q does not depend

on the length of the blade being considered, nor on the shape of the mode. The width of

the blade and the material properties of both the blade and surrounding fluid influence

Q, but these are fixed across the range of beams used by Bargatin et al. [2]. The

quality factor is thus a function only of frequency. We now present the calculated

quality factors alongside the experimental values. Note that the experimental quality

factors presented here correspond to the fluid contribution only.

Experiment Theoryβ Qfluid only Qno-slip Qfirst-order slip Qsecond-order slip

0.3527 136 448 465 4590.9223 339 721 783 7681.7477 535 996 1141 11192.8268 804 1268 1537 15234.1315 1160 1529 1960 19770.4346 157 497 519 5120.8115 311 678 731 717

Table 4.1: Quality factors predicted by infinitely thin boundary layer model for first five in-plane resonances of 16µm-long beam and second and third in-plane resonances of 24µm-longbeam.

The frequencies used to predict the quality factors are the actual resonant fre-

quencies and not the values presented in Chapter 3. From Table 4.1, we see that the

predicted values are significantly higher than the experimental quantities. In accordance

with intuition, incorporating the effects of slip increases the calculated quality factors.

Notice also that the effect of slip is more pronounced in the high-frequency regime as

Qno-slip and Qslip begin to diverge. In addition, Qfirst-order slip > Qsecond-order slip for low

frequencies and Qfirst-order slip < Qsecond-order slip for high frequencies. Indeed, consider

the following plot, which shows the predicted and experimentally determined quality

factors.

34

æ

æ

æ

æ

æ

+

+

Second- order slip

No - slip

First- order slip

Experiment

10 50 100 500 1000 5000f HMHzL10

100

1000

104

Q

Figure 4.2: Infinitely thin boundary layer model : Predicted and experimental quality factors.No-slip (solid), first-order slip (dashed) and second-order slip (dotted) results are depictedalongside experiment values for the 16µm-long (dots) and 24µm-long (+) beams.

Chapter 5

Beam of Zero Thickness

So far we have considered the beam to be infinitely thin, and have associated with

it an infinitely thin boundary layer, thereby neglecting any edge effects. We now explore

the case where we have a blade of finite width. In order to do so we will adopt and

extend the boundary integral technique of Tuck [5]. Furthermore, we will again employ

the second-order slip model and examine the consequences in terms of the calculated

quality factors.

ω(y,0+)

−a a

ω(y,0-)

y

z

Figure 5.1: Cross-section of beam of zero thickness

35

36

We consider an infinitely thin beam, the length of which greatly exceeds its width.

To justify ignoring the non-linear convective term in the equation of motion of the fluid,

we insist that the amplitude of oscillation of the blade is extremely small; a condition

which is typically satisfied during experiment [3]. It then follows that the fluid velocity

has components only in the y and z directions (refer to Figs. 1.2 and 5.1 for orientation

of coordinate axes). It follows that u = vy +wz where v and w are the components of

the fluid velocity in the y and z directions respectively. Since the fluid velocity in the

x direction vanishes, i.e. u = 0, we can express the fluid vorticity as

ω =(∂w

∂y− ∂v

∂z

)x− ∂w

∂xy +

∂v

∂xz.

The vorticity above and below the oscillating blade can be written as

ωtop =(∂wtop

∂y− ∂vtop

∂z

)x− ∂wtop

∂xy +

∂vtop∂x

z, (5.1)

ωbottom =(∂wbottom

∂y− ∂vbottom

∂z

)x− ∂wbottom

∂xy +

∂vbottom

∂xz. (5.2)

However, by symmetry, Eq. (5.2) becomes

ωbottom =(∂wtop

∂y+∂vtop∂z

)x− ∂wtop

∂xy +

∂vtop∂x

z. (5.3)

It then follows that the vorticity difference across the face of the blade is given by

∆ω = ωtop

∣∣z=0+ − ωbottom

∣∣z=0−

= −2∂vtop∂z

∣∣∣∣z=0

x. (5.4)

Since the vorticity jump has a non-zero component only in the x-direction, we will

abandon the vector notation and proceed to write it as ∆ω(x, y), i.e. ∆ω = ∆ω(x, y)x.

For the long, infinitely thin blade with cross-section from (y, z) = (−a, 0) to (a, 0), we

can also write the pressure difference across the face of the blade as

∆p(x, y) = p(x, y, z)∣∣z=0+ − p(x, y, z)

∣∣z=0−

. (5.5)

Chapter 5. Beam of Zero Thickness 37

Tuck [5] showed that for a cylinder of arbitrary cross-section undergoing sufficiently

small oscillations, the two-dimensional streamfunction can be written as

ψ =∫C

[ψGn − ψnΩ− ωΨn +

1µpΨl

]dl, (5.6)

provided the pressure p is continuous on the cross-section contour C. In the above

equation, ∂/ ∂l and ∂/ ∂n represent the derivatives tangential and normal to C, re-

spectively, and ω is the fluid vorticity. The streamfunction ψ satisfies the equation

∇4ψ = α2∇2ψ. The function Ψ is the corresponding Green’s function for this equation

and thus satisfies ∇4Ψ− α2∇2Ψ = δ. Tuck [5] gives the solution to this equation as

Ψ = − 12πα2

(logR+K0(αR)),

where K0 is the modified Bessel function of the third kind, zeroth-order. In the above

equation, α2 = iων and R =

√(y − y′)2 + (z − z′)2. We note that α has dimensions of

inverse length. In order to satisfy the requirement that the arguments of the transcen-

dental functions are dimensionless, we rewrite Ψ as

Ψ = − 12πα2

(log (αR) +K0(αR)

).

This is a minor modification to Tuck’s expression, and has no bearing on the

calculated results since the factor disappears upon differentiation of Ψ. Nevertheless,

for consistency we proceed with the latter expression. Recall that ν is the kinematic

viscosity of the surrounding fluid and ω is the angular frequency at which the cylinder

oscillates. It is important not to confuse this angular frequency with the vorticity in

Eq. (5.6). We also have

G =12π

log(αR),

Ω = − 12πK0(αR),

where ∇2Ω − α2Ω = δ(y − y′)δ(z − z′) and Ψ = 1α2 (Ω − G). The Green’s function

G satisfies Laplace’s equation in two dimensions, i.e. ∇2G = δ(y − y′)δ(z − z′). In

the above equation we have inserted a factor of α in the expression for G in order to

maintain dimensional integrity. For the infinitely thin blade, Tuck [5] showed that the

38

two-dimensional streamfunction presented in Eq. (5.6) reduces substantially to become

ψ(y′, z′) =∫ a

−a

[∆ω(y)Ψz(y, 0; y′, z′)− 1

µ∆p(y)Ψy(y, 0; y′, z′)

]dy. (5.7)

One subtlety that is not mentioned by Tuck is that the above expression is only

valid when the streamfunction is evaluated away from the surface of the cylinder. In

evaluating ψ(y′, z′) at the surface, there exists a factor of one half. This is a direct

consequence of a Dirac delta function being “split in half” when a volume integral is

evaluated during the derivation. Thus, Eq. (5.6) should actually read

On contour C :12ψ(y′, z′) =

∫C


1µpΨl

]dl, (5.8)

Off contour C : ψ(y′, z′) =∫C


1µpΨl

]dl. (5.9)

In the subsequent sections and chapters we will be required to take derivatives of

ψ with respect to both the normal and tangential coordinates to the surface of C. In

evaluating ψl and ψn, we will use Eqs. (5.8) and (5.9) respectively. If the surface C is

a streamline, then the factor of 1/2 will not contribute at all since it will only manifest

itself in ψl = 0, in which case it can be removed.


For the infinitely thin blade immersed in a viscous fluid and subject to the no-slip

boundary condition, the fluid velocity calculated using the streamfunction in Eq. (5.7)

must match the blade velocity at the surface. From Eq. (5.7), it is easy to show

that the normal and tangential oscillations of the blade are uncoupled, and that the

two components of the motion of the blade can be expressed as the following integral

equations:

ψ(y′, 0) = − 1µ

∫ a

−a∆p(y)Ψy(y, 0; y′, 0)dy, (5.10)

U(y′) = ψz′(y′, 0) =∫ a

−a∆ω(y)Ψzz′(y, 0; y′, 0)dy. (5.11)

Note that we are considering the cross-section of the blade at a fixed value of x.

Thus, Eqs. (5.10) and (5.11) are independent of x. We will consider the x-direction

later, however for the time being, we restrict ourselves to motion in the y − z plane.


We are particularly interested in the tangential component of the blade’s velocity since

we are considering the in-plane modes. The Green’s function Ψ is known, so we can

directly evaluate Ψzz′(y, 0; y′, 0).

Ψz =∂Ψ∂z

=dΨdR

× ∂R

∂z,

Ψzz′ =∂

∂z′

(dΨdR

∂R

∂z

),

=(dΨdR

)(∂2R

∂z ∂z′

)−

(d2ΨdR2

)(∂R

∂z

)2

.

Upon setting z = z′ = 0, the second term vanishes and

∂2R

∂z ∂z′

∣∣∣∣z=z′=0

=−1√

(y − y′)2=

−1|y − y′|

.

We can then write

Ψzz′(y, 0; y′, 0) =−1

|y − y′|dΨdR

∣∣∣∣z=z′=0

.

Let Z = αR∣∣z=z′=0

= α|y − y′|. The above equation becomes

Ψzz′(y, 0; y′, 0) =1

2πZd

dZ

(log (Z) +K0(Z)

).

If we define the kernel function as

L(Z) =1

2πZd

dZ

(logZ +K0(Z)

),

then our integral equation becomes

U(y′) = ψz′(y′, 0) =∫ a

−a∆ω(y)L(α|y − y′|)dy. (5.12)

We would now like to consider the behaviour of L(Z) for small and large values of Z:

L(Z) = − 14π

(logZ + γ − 1

2− log 2

)+O(Z2) for Z 1, (5.13)

L(Z) =1

2πZ2− 1√

8πZe−Z

(1Z

+O(

1Z2

))for Z 1, (5.14)

where γ is Euler’s constant and is approximately equal to 0.5772. You can see from

40

Eq. (5.13) that the kernel function has a logarithmic singularity for small values of the

argument, i.e. as Z → 0, L(Z) ∝ logZ. Remember that Z = α|y − y′| and so the

logarithmic singularity will need to be acknowledged and treated with care for y ≈ y′.

The equation to be solved is (5.12). Before we proceed, it is a good idea to scale the

problem through the introduction of dimensionless parameters. We know that the cross-

section of the blade is simply the line joining (y, z) = (−a, 0) to (a, 0). The natural

length scale is thus a, and so we scale position across the blade as

ξ =y

a. (5.15)

We also introduce the dimensionless frequency and vorticity as

β =ωa2

ν, (5.16)

Λ(ξ) =(a

U0

)∆ω(y). (5.17)

We will consider the long-term behaviour of the system only. After the transients

have died away, the vorticity jump due to U(y)e−iωt is ∆ω(y)e−iωt where U and ∆ω

are related through Eq. (5.12). For this case, we take U(y) = U0 = constant. Thus,

Eq. (5.12) becomes

U0e−iωt =

∫ a

−a∆ω(y)e−iωtL(α|y − y′|)dx,

which simplifies substantially once the scaled variables are introduced to become∫ 1

−1Λ(ξ)L(−i

√iβ|ξ − ξ′|)dξ = 1. (5.18)

5.1.1 Numerical Solution Method

We know the kernel function L and wish to solve for the complex-valued vorticity

distribution Λ(ξ). We cannot solve the integral equation (5.18) analytically and so will

seek a numerical solution. This will involve discretizing the domain ξ ∈ [−1, 1]. The

integral equation (5.18) can then be transformed into a summation using an appropriate

quadrature method. The corresponding matrix-vector equation that will eventuate from

this process can then be easily solved. Before we attempt to solve Eq. (5.18), there are


a few concerns which must first be addressed:

1. The presence of the logarithmic singularity in the kernel function at ξ = ξ′.

2. The blade under consideration is infinitely thin. At the leading edges, we expect

the presence of a square-root singularity [5]. That is, we anticipate square-root

singularities in Λ(ξ) at ξ = ±1.

3. Consider the exponential term in Eq. (5.14). If β is large, then this term oscillates

rapidly as it approaches zero.

These are the same issues that Tuck [5] faced in solving the case of normal oscil-

lations. However, we will see that these sources of potential problems can be resolved

quite easily. The presence of the logarithmic singularity in the kernel function will ac-

tually turn out to be beneficial. As a result of the singularity, the kernel matrix that

will be formed once an appropriate quadrature rule is decided upon will be diagonally

dominant, and hence easy to invert [5]. The square-root singularity in Λ(ξ) can be

easily accounted for by taking unequal intervals in the quadrature formula, with a bias

towards the ends. We know that the vorticity will exhibit a square-root singularity at

the edges of the blade. Though an accurate numerical solution can be found by breaking

up the interval ξ ∈ [−1, 1] evenly, into many pieces, the convergence would be slow and

would require N to be very large indeed. It is much more efficient to break up the inter-

val using the points ξ = ξj = − cos(πjN

), j = 0, . . . , N . The third issue can be resolved

by observing that we do not expect Λ(ξ) to vary significantly. We will approximate

Λ(ξ) as a constant on each interval, but will not approximate the kernel function L in

the quadrature method. Using Eq. (5.18) and approximating Λ(ξ) = Λj = constant on

each interval ξj < ξ < ξj+1, we obtain the following equation:

N−1∑j=0

(Λj

∫ ξj+1

ξj

L(−i√iβ|ξ − ξ′|)dξ

)= 1. (5.19)

We then demand that Eq. (5.19) hold at the mid-point of each segment. In other

words, we set ξ = ξ′k = 12(ξk + ξk+1) for k = 0, 1, . . . , N − 1 and substitute this into

Eq. (5.19). This yields

N−1∑j=0

(Λj

∫ ξj+1

ξj

L(−i√iβ|ξ − ξ′k|)dξ

)= 1 k = 0, 1, . . . , N − 1. (5.20)

42

Equation (5.20) can actually be expressed as the following matrix equation:

N−1∑j=0

MkjΛj = 1, k = 0, 1, . . . , N − 1, (5.21)

where

Mkj =∫ ξj+1

ξj

L(−i√iβ|ξ − ξ′k|)dξ. (5.22)

Since the kernel function L is known, we can evaluate the N ×N matrix M . Aside

from N , the matrix M will depend solely on β, the dimensionless frequency defined

in Eq. (5.16). Constructing this matrix is relatively straightforward using a software

package such as mathematica R© . Once this is done, the matrix-vector system can

be solved. We note that the solutions can be found by built-in linear solvers or by

direct inversion of the matrix M . Both methods yield the same results, though from

a computational point of view, the former method is more robust. We readily find the

vector Λ containing the numerical solutions Λj at each of the points ξ′j . Since we have

discretized the problem by turning the integral equation into a matrix-vector equation,

we cannot find a continuous solution for Λ(ξ). However, increasing N will increase the

number of computed points. Obviously, N must be sufficiently large so as to ensure

that any numerical solution which is obtained is accurate. We will consider this in more

depth shortly.

Recall that Λ(ξ) = Λj has been treated as constant on each of the intervals ξj <

ξ < ξj+1. Λ(ξ) is a complex-valued function and so in plotting it over the appropriate

domain, we must consider both its real and imaginary components. Below is a plot of

the vorticity distribution for the first in-plane resonance of the blade of length 16µm.

The only independent variable that explicitly determines the shape of this graph is the

frequency of oscillation of the blade.


−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1−6

−4

−2

0

2

4

6

8

10

Position ξ

Vor

ticity

Λ(ξ

)

Re(Λ(ξ))

Im(Λ(ξ))

Figure 5.2: Vorticity difference across the infinitely thin blade subject to no-slip boundarycondition. β = 0.3527, N = 60.

5.1.2 Calculation of Hydrodynamic Function

We have found that the complex-valued vorticity distribution is a function of both

frequency and position, i.e. Λ = Λ(ξ, β). However, we do not have an analytic expres-

sion for Λ directly. It is very important to note that at this stage, the vorticity profile is

known across the face of the blade for a given velocity amplitude U0. As of yet, we do

not know how it varies along the length of the beam. Ultimately, we seek to know the

frequency dependence of the quality factors associated with this beam. Recall Eq. (5.4),

where we obtained an expression for the vorticity difference across the surface of the

blade. The stress tensor for incompressible flow is given by

T = 2µe = µ

(∇u + (∇u)T

). (5.23)

44

The stress vector,

t = n ·T, (5.24)

is the force per unit area exerted by the fluid on the surface, with normal vector n

pointing into the fluid. Recall that the fluid velocity can be written as u = vy + wz.

Combining this with Eqs. (5.23) and (5.24), the component of the stress vector in the

y-direction for the two faces of the blade can be written as

ttop · y =(z ·Ttop

)· y,

tbottom · y =(− z ·Tbottom

)· y.

The total pressure exerted on the blade in the y-direction is then given by

t · y =(ttop + tbottom

)· y,

which simplifies substantially to become

t · y = 2µ∂vtop∂z

∣∣∣∣z=0

= −µ∆ω(x, y).

We are considering motion where the velocity amplitude is U0. Earlier we solved for

the complex vorticity in 2-dimensions, namely the y-z plane. We introduced a velocity

amplitude U0 that was taken to be constant in y. Now that we extend this problem to

3-dimensions, we need to account for the fact that the velocity amplitude of the beam

is actually a function of x. We know that U(x, t) = U0(x)e−iωt, and so

U0(x) = −iωW (x|ω), (5.25)

where W (x|ω) is the displacement amplitude as a function of distance x along the

beam. We can write the net force per unit length that the fluid exerts on the beam in

the y-direction as

Fhydro(x|ω) = −∫ 1

−1µU0(x)Λ(ξ, β)dξ. (5.26)


Substituting Eq. (5.25) into Eq. (5.26) yields

Fhydro(x|ω) = µiωW (x|ω)J(ω), (5.27)

where J(ω) =∫ 1−1 Λ(ξ, β)dξ has been written as a function of the dimensional frequency

ω rather than the scaled frequency β. In calculating the integral J(ω), we use the

midpoint rule given by

J(ω) =∫ 1

−1Λ(ξ, β)dξ ∼

N−1∑j=0

Λj(ξj+1 − ξj

). (5.28)

We see from Eq. (5.27) that the hydrodynamic force on the blade at position x is

proportional to the displacement amplitude of the blade at that point. In addition, the

frequency dependence manifests itself in two different ways. Firstly, in the mode shape

W (x|ω) as just mentioned. For larger amplitude of vibration, the force will be larger.

Secondly, the term J(ω) is simply the integral of the complex vorticity difference over the

width of the blade. Something which is important to note is that J(ω) knows nothing

about the in-plane resonances of the blade. All information regarding the resonances is

encompassed in the term W (x|ω). Now that an expression for the hydrodynamic force

on the blade has been found, the hydrodynamic function can be calculated. Equating

Eqs. (2.4) and (5.27), we see that

µiωW (x|ω)J(ω) =π

4ρω2b2Γ(ω)W (x|ω). (5.29)

The displacement function cancels from both sides and so the hydrodynamic func-

tion will depend only on the frequency of oscillation of the blade and not on the mode

shape explicitly. We will find that the quality factors associated with the oscillations

are also independent of W (x|ω). From Eq. (5.29) the hydrodynamic function can be

written as

Γ(ω) =4µiπρωb2

J(ω) =i

πβJ(ω). (5.30)

The real and imaginary components of the hydrodynamic function represent the

added mass of the beam and the viscous losses respectively. In modelling the interaction

of the blade with the fluid, we have assumed zero thickness. However, it is important

to realise that the beam still has a non-zero linear mass density.

46

5.2 Second-Order Slip Boundary Condition

Recall the expression for the two-dimensional streamfunction in Eq. (5.7). For

in-plane modes, ∆p(y) = 0 by symmetry and so

ψ(y′, z′) =∫ a

−a

[∆ω(y)Ψz(y, 0; y′, z′)

]dy.

The Green’s function Ψ is analytic everywhere in the y-z plane except at y = y′,

z = z′. For this reason, we avoid setting z′ = 0, but work very close to this limit. The

fluid velocity and its first two derivatives with respect to the normal coordinate are

given by

U(y′, z′) = ψz′(y′, z′) =∫ a

−a∆ω(y)Ψzz′(y, 0; y′, z′)dy,

∂U

∂z′(y′, z′) = ψz′z′(y′, z′) =

∫ a

−a∆ω(y)Ψzz′z′(y, 0; y′, z′)dy,

∂2U

∂z′2(y′, z′) = ψz′z′z′(y′, z′) =

∫ a

−a∆ω(y)Ψzz′z′z′(y, 0; y′, z′)dy.

Recall from Eq. (2.1) that the second-order slip boundary condition at the surface

of the blade is of the form

U∣∣wall

− uw = γη∂U

∂z′

∣∣∣∣wall

− δη2 ∂2U

∂z′2

∣∣∣∣wall

. (5.31)

Recall also that γ = 1.11 and δ = 0.61 are dimensionless parameters for one-

dimensional flows with hard-spheres. The molecular mean free path is given by η and

uw is the velocity of the blade surface. The physical length scale for the infinitely thin

blade is simply the width of the blade b = 2a. Since we are working in the regime where

the Knudsen number, Kn = η2a ≈ 0.17 < 1, we apply the second-order slip condition in

Eq. (5.31) at the surface of the blade. This is given by

uw = U∣∣wall

− γη ∂U∂z′

∣∣∣∣wall

+ δη2 ∂2U

∂z′2

∣∣∣∣wall

=∫ a

−a∆ω(y)

[Ψzz′(y, 0; y′, z′)− γηΨzz′z′(y, 0; y′, z′) + δη2Ψzz′z′z′(y, 0; y′, z′)

]dy.

(5.32)


We wish to evaluate the above equation at z′ = 0. However, since the Green’s

function Ψ is not analytic at this point, we instead choose to evaluate Eq. (5.32) at

z′ = ε, 0 < ε 1. That is, we proceed to calculate the fluid velocity and its subsequent

derivatives just above the surface of the blade. We scale η according to

η =η

a= 2×Kn,

where Kn is the Knudsen number for the system. The dimensionless frequency β is

defined as earlier,

β =ωa2

ν.

The tangential and normal coordinates can be scaled as

ξ =y

a, ξ′ =

y′

a, χ =

z

a, χ′ =

z′

a.

Since α has dimensions of inverse length, it seems natural to want to scale it to become

dimensionless. Since α2 = iω/ν = iβ/a2, we define

α2 = a2 × α2 = iβ,

and so

α = −i√iβ.

The Green’s Function Ψ thus can be written as

Ψ = − a2

2πα2(log (αR/a) +K0(αR/a)). (5.33)

Then, the dimensionless Green’s Function is

Ψ(κ) =Ψa2

= − 12πα2

(log (ακ) +K0(ακ)

). (5.34)

Scaling Eq. (5.32) in the same fashion as in Section 5.1 and identifying the kernel

function as

L(ξ, 0; ξ′, χ′) = Ψχχ′(ξ, 0; ξ′, χ′)− 2γKnΨχχ′χ′(ξ, 0; ξ′, χ′) + 4δKn2Ψχχ′χ′χ′(ξ, 0; ξ′, χ′),

(5.35)

48

we see that ∫ 1

−1Λ(ξ)L(ξ, 0; ξ′, χ′)dξ = 1. (5.36)


The complex vorticity distribution is given by Λ(ξ) =(aU0

)∆ω(y). In solving

Eq. (5.36), we will need to take a sufficiently small χ′ > 0 to accurately capture the

behaviour at the surface of the blade. Before we solve this integral equation, there are

several important points which must be raised. The kernel function is comprised of

various derivatives of the Green’s function Ψ, which has a singularity at R = 0. This

seems concerning upon first glance, however this property will actually work to our

advantage. The presence of this singularity will ensure that the kernel matrix obtained

after discretization will be diagonally dominant and thus relatively easy to invert. In

the discretization of Eq. (5.36) we will again use the Chebyshev points defined by

ξ = ξj = − cos(πjN ), j = 0, . . . , N.

Armed with the same tools used for the no-slip case, we attack Eq. (5.36) and

obtain the following matrix-vector equation:

N−1∑j=0

MkjΛj = 1, k = 0, 1, . . . , N − 1,

where

Mkj =∫ ξj+1

ξj

L(ξ, 0, ξ′k, χ′)dξ.

The values Λj correspond to the complex vorticity which is taken to be constant

in each region ξ ∈ [ξj , ξj+1]. The points ξ′k = 12(ξk + ξk+1) are the midpoints of each

domain element. The resultant matrix equation can be constructed and solved quite

easily using mathematica R© . Recall that when the no-slip condition was used, the

corresponding matrix equation had a dependence only on dimensionless frequency β.

However, since the first and second-order slip conditions manifest themselves through

incorporation of the mean free path of the surrounding fluid, the system is no longer

independent of scale. That is, the absolute width of the blade is now significant, as

evident in the dependence of Eq. (5.35) on Knudsen number.


We now present the numerical solutions for the vorticity distribution across the

face of the blade. We can adopt any of the no-slip, first-order slip or second-order slip

conditions simply by changing the parameter values γ and δ. Setting γ = δ = 0 should

yield precisely the same results as obtained earlier since this would correspond to appli-

cation of the no-slip condition at the blade surface. Consider the following three plots,

which illustrate the vorticity distribution on the interval ξ ∈ [−1, 1] for several differ-

ent frequencies. The dots represent the actual solution points from the matrix-vector

equation. Linear interpolation has been performed between these values.

ReHLL

ImHLL

Increasing

frequency

Increasing

frequency

-1.0 -0.5 0.5 1.0Ξ

-10

-5

5

10

15

L

Figure 5.3: Real and imaginary components of vorticity jump across the blade face subject tono-slip boundary condition (β = 0.01, 5, 20 and Kn = 0.17).

As expected, we see that the vorticity exhibits a singularity at the leading edges

of the blade. This is consistent with our results from Section 5.1. As the frequency is

increased, both Re(Λ) and Im(Λ) increase in magnitude. Consider now the numerical

results for the case where the first-order slip condition is applied. This is obtained by

using the kernel function evaluated with γ = 1.11 and δ = 0.

50

Increasing

frequency

ImHLLIncreasing

frequency

ReHLL

-1.0 -0.5 0.5 1.0Ξ

-2

2

4

L

Figure 5.4: Real and imaginary components of vorticity jump across the blade face subject tofirst-order slip condition (β = 0.01, 5, 20 and Kn = 0.17).

Qualitatively, Fig. 5.4 is very different to Fig. 5.3. We note that the vorticity is

not as strongly peaked at the leading edges of the blade. Indeed, the singularities at

ξ = ±1 have been removed. Vorticity is a measure of the average local angular velocity

of a fluid element. This rotational motion is initiated through the tangential motion of

the blade. By incorporating the effects of slip into our model, we diminish the ability

of the blade to introduce vorticity into the surrounding fluid. In addition, note that

in Fig. 5.4, |Im(Λ)| does not increase monotonically with frequency like in Fig. 5.3.

Consider now the effects of adopting the second-order slip boundary condition, which

is done by taking γ = 1.11 and δ = 0.61.


Increasing

frequency

ImHLL

ReHLL

-1.0 -0.5 0.5 1.0Ξ

-1

1

2

3L

Figure 5.5: Real and imaginary components of vorticity jump across the blade face subject tosecond-order slip condition (β = 0.01, 0.2, 5 and Kn = 0.17).

The second-order slip condition has quite a profound effect on the subsequent

vorticity distribution. As a result of the increased slip at the surface of the blade, we

see that the vorticity at the leading edges is very small indeed.


We now turn our attention to the calculation of the hydrodynamic function associ-

ated with the various in-plane resonances. Recall from Eq. (5.30) that the hydrodynamic

function can be written as

Γ(ω) =i

πβJ(ω),

where we have defined

J(ω) =∫ 1

−1Λ(ξ, β)dξ.

We can use this same equation, since the incorporation of slip affects only the value

of J(ω). The hydrodynamic function can be readily computed for different frequencies.

As in the no-slip case, we evaluate J(ω) using the midpoint rule given in Eq. (5.28).

The real and imaginary components of the hydrodynamic function can be compared

52

for the three cases that we are considering. We will examine the behaviour of such

functions in both the high-frequency and low-frequency regimes.

10−1

10010

−2

10−1

100

101

102

Dimensionless Frequency β

Re(

Γ)

2nd order slip

No−slip

1st order slip

(a)

10−4

10−3

10−2

10−1

100

10110

−2

10−1

100

101

102

103

104

105

Dimensionless Frequency β

Im(Γ

)

2nd order slip

1st order slipNo−slip

(b)

Figure 5.6: (a) Real and (b) imaginary components of hydrodynamic function


From Fig. 5.6, it can be seen that the hydrodynamic functions for each of the three

cases agree in the low frequency limit. From Eq. (5.31), we see that the first and second-

order effects of slip depend on the first and second derivatives of the fluid velocity with

respect to the normal coordinate. For low frequencies, we expect that these derivatives

will be smaller in magnitude and thus the corresponding terms in Eq. (5.31) will be less

significant. In the high-frequency regime, the thickness of the viscous boundary layer

decreases. This becomes the characteristic length scale and so the appropriate Knudsen

number is quite large. Since the second-order slip model works only for Kn < 1 [9],

we see that it can be no longer applied in the high-frequency limit. This is precisely

the reason why the curves in Fig. 5.6 begin to diverge at β ≈ 1. In fact, Im(Γ) < 0

for β > 15 in the second-order slip case. This is clearly unphysical since it would

suggest that the viscous “losses” are negative. We attribute this inconsistency to the

breakdown of the model for large Kn. For high frequencies, the viscous penetration

depth diminishes and so the effective Knudsen number increases.

5.3 Alternative Solution Method for First-Order Slip

As a small aside, we present an alternative method for finding the vorticity distri-

bution across the blade face where the fluid velocity is subject to the first-order Navier

slip condition. Recall this boundary condition is of the form

U∣∣wall

− uw = γη∂U

∂z′

∣∣∣∣wall

,

and can be written in terms of the streamfunction ψ(y′, z′),

∂ψ

∂z′− γη ∂

2ψ

∂z′2= uw. (5.37)

The general form of the streamfunction is known and is presented as a bound-

ary integral in Eq. (5.7). For in-plane modes, the pressure contribution vanishes by

symmetry and we are left with

ψ(y′, z′) =∫ a

−a∆ω(y)Ψz(y, 0; y′, z′)dy. (5.38)

54

The approach in Section 5.2 was to substitute Eq. (5.38) into Eq. (5.37) and rear-

range to obtain an integral equation with the unknown ∆ω(y) located entirely inside

the integrand. While this is an effective method and indeed is necessary to solve for

the second-order slip boundary condition, it is possible to be much more efficient in the

case of first-order slip. Construction of the kernel functions in the previous section is

computationally demanding since the introduction of slip introduces additional terms

into the integral equation. Notice that Eq. (5.37) involves ∂2ψ∂z′2 . Instead of differentiat-

ing Eq. (5.38) twice to obtain this term, we note that it can be written in terms of the

primary unknown. The vorticity on the top face of the blade is ω(y′) = − ∂2ψ∂z′2 and so

combining Eqs. (5.37) and (5.38), we see that[ ∫ a

−a∆ω(y)Ψzz′(y, 0; y′, z′)dy

]+

12γη∆ω(y′) = uw.

The factor of 1/2 accounts for the fact that we are solving for the vorticity difference

across the face of the blade. Upon scaling and discretization of the integral in the same

way as Section 5.2, we obtain the following matrix-vector equation

(M +

12γηI

)Λ =

1

1...

1

, (5.39)

where M is precisely the same matrix as in Eq. (5.21) and I is the N × N identity

matrix. The beauty of using Eq. (5.39) to solve for the first-order slip case is that it is

computationally inexpensive. The only modification necessary from the no-slip case is

the inclusion of some constants to the diagonal elements of the matrix M . Furthermore,

these terms serve to make the matrix even more diagonally dominant and thus more

docile in its inversion properties. The results obtained using this method agree perfectly

with the results from Section 5.2 and so we need not present them.

It must be noted that this method of solving for the vorticity distribution only

works for the first-order slip scenario since ∂2ψ∂z′2 = −ω(y′). In the case of second-order

slip this process cannot be adopted since we cannot write ∂3ψ∂z′3 in terms of the vorticity.

Nevertheless, the approach used in this section is extremely important and will turn

out to be valuable when dealing with a beam of non-zero thickness.


5.4 High-Frequency Limit

While oscillating, the blade will generate a thin viscous boundary layer. In the

high-frequency limit, the viscous penetration depth will be extremely thin. In such

circumstances, the infinite plate solution derived in Chapter 4 should agree with the

results for the beam of zero thickness. From Eqs. (5.35) and (5.36), it is evident that

the matrix-vector equation corresponding to first-order slip can be written as

(M0 + KnM1

)Λ =

1

1...

1

, (5.40)

where M0 and M1 are independent of Knudsen number. Once the matrices M0 and M1

have been constructed, we can solve the above equation repeatedly for various Knudsen

numbers. In order to compare results from the two different models, we will compute

the hydrodynamic function, and identify the linear component in Knudsen number for

Kn 1. That is, we look for the region where

Γ = Γ0 + Γ1Kn.

The hydrodynamic function for the infinite plate subject to first-order slip is ob-

tained by setting δ = 0 in Eq. (4.11). Performing a Taylor series expansion for small

Kn = η/b, we obtain

Γplate(β) =4√i

π√β− 8γ

πKn for Kn 1, (5.41)

where we have used the dimensionless frequency β = ωa2/ν. This is complex-valued

and linear in Knudsen number. As β →∞, we can identify

Γ0 → 0 and Γ1 → −8γπ. (5.42)

In order to compare this result with the blade of finite width, we require a solution

of the aforementioned form. Since we can repeatedly solve Eq. (5.40) with ease, it is

possible to identify the linear region of the hydrodynamic function for the beam of

56

zero thickness, ΓZT(β) by choosing sufficiently small values of Kn. Solving the matrix-

vector equation several times and performing linear interpolation, we can readily find

the linear form:

ΓZT(β) = Γ0 + Γ1Kn for Kn 1.

The above equation requires not only that Kn 1, but that the mean free path

is significantly smaller than all other physical length scales. We now quantitatively

compare the results obtained for the beam of zero thickness with those of the infinite

plate solution given by Eq. (5.41). The results have been computed with a mesh cor-

responding to N = 100 and are plotted as a function of β. The process of Richardson

extrapolation has been used to improve the accuracy of the β →∞ results; the explicit

method for this process is outlined in Appendix A.

æ

æ

æ

æ

æ

0 500 1000 15000.00

0.02

0.04

0.06

0.08

0.10

Β

Re@G0D

(a)

æ

æ

æ

æ

0 500 1000 15000.02

0.03

0.04

0.05

0.06

0.07

0.08

Β

Im@G0D

(b)

æ

æ

æ

æ

æ

0 500 1000 1500-5.0

-4.5

-4.0

-3.5

-3.0

-2.5

Β

Re@G1D

(c)

æ

æ

æ

æ

æ

0 500 1000 1500-2.0

-1.5

-1.0

-0.5

0.0

0.5

Β

Im@G1D

(d)

Figure 5.7: Real and imaginary components of the no-slip hydrodynamic function Γ0 as well asthe first-order correction Γ1 . The solid and dotted curves represent the beam of zero thicknessand infinite plate solutions respectively.


The solid curves depicted in Figs. 5.7(a)-5.7(d) are given respectively by

Re(Γ0) = 0.000669 + 0.99β0.52 , Im(Γ0) = 0.003119 + 1.65

β0.59 ,

Re(Γ1) = −2.8287 + −15.00β0.50 , Im(Γ1) = −0.0157 + −17.11

β0.52 .

It can be seen both graphically and numerically that the results are very similar

indeed to that of the infinite plate solution. In particular, we have successfully captured

the behaviour with respect to β exhibited by the components of ΓZT(β). Notice that

the zeroth-order components are very similar in form to those of Γplate(β), which behave

as β−1/2. The predicted value of ΓZT(∞) obtained using Richardson extrapolation is

given by

ΓZT(∞) = (3.092× 10−7 − 7.41× 10−5i) + (−2.8264 + 0.00035i)Kn,

which is extremely close to the asymptotic value of −(8γ/π)Kn ≈ −2.8266Kn given

by Eq. (5.42). In performing this extrapolation, we have assumed that the functions

converge as β−1/2. It is particularly interesting to see that the beam of zero thickness

and infinite plate possess similar values of Γ0 across the range of frequencies, whilst the

first-order corrections do not agree until β →∞.

So far we have looked only at the case where N = 100. We now consider the

behaviour of the hydrodynamic function for N = 20 and N = 50. The solution points

for the various values of β are slightly different, however, the real and imaginary com-

ponents of Γ0 and Γ1 each behave as ∼ β−1/2. Richardson extrapolation can thus be

again used, and the following results obtained:

N = 20 : ΓZT(∞) = (3.489× 10−6 − 7.29× 10−5i) + (−2.8225 + 0.0021i)Kn,

N = 50 : ΓZT(∞) = (5.046× 10−7 − 7.40× 10−5i) + (−2.8254 + 0.00057i)Kn.

Although very different mesh discretizations have been used, they all give approx-

imately the same answer in the high-frequency limit. We attribute this to the fact that

for high frequencies, the viscous boundary layer is extremely thin and smooth. In such

circumstances, even a modest value of N can quantitatively capture the solution.

58

5.5 Convergence With Mesh Discretization

It is important to understand precisely how the computed solutions converge as the

number of subdivisions on the blade is increased. We will again analyse the behaviour

of Γ0 and Γ1 for the beam of zero thickness, but this time with β fixed and N being

the independent variable. As previously, the calculation of Γ0 and Γ1 requires the

matrix-vector equation to be repeatedly solved for various small Knudsen numbers.

Interpolation can then be performed and the hydrodynamic function correct to O(Kn)

can be determined. In order to determine this linear expression, we perform a cubic

interpolation and subsequently extract the linear terms. Consider the following plots

which show how the computed values of Γ0 and Γ1 vary as a function of N . Note that

for this example, we have considered β = 1.

æ

æ

æ

0 20 40 60 801.0965

1.0970

1.0975

1.0980

1.0985

1.0990

1.0995

1.1000

N

Re@G0D

(a)

æ

æ

æ

0 20 40 60 80

2.359

2.360

2.361

2.362

N

Im@G0D

(b)

æ

æ

ææ

0 20 40 60 80-10.9

-10.8

-10.7

-10.6

-10.5

-10.4

N

Re@G1D

(c)

æ

æ

æ

0 20 40 60 80

-10.65

-10.60

-10.55

N

Im@G1D

(d)

Figure 5.8: Real and imaginary components of the no-slip hydrodynamic function Γ0 as wellas the first-order correction Γ1


Upon fitting each of the above curves to the form Γ = a+ b/N c, we find that the

calculated values of c range between 1.92 and 1.94. Since the method of discretization is

based on the midpoint rule which has an associated error of O(1/N2), we conclude that

the actual values of c are equal to 2. The slight deviation can likely be attributed to

the fact that we have not yet reached the asymptotic behaviour with N = 10, 20, 40, 80.

Since the convergence of the above components are known, we can perform Richardson

extrapolation as before. This enables us to accurately determine the solution corre-

sponding to N → ∞ which is of course the ultimate objective. The values obtained

using Richardson extrapolation are depicted as dotted lines in Fig. 5.8.

Let us now consider the problem in a slightly different way. Since we are looking for

a first-order correction to the no-slip solution, it is natural to rewrite the matrix-vector

equation given in Eq. (5.40) as follows:(M0 + KnM1

)(Λ0 + KnΛ1

)= b. (5.43)

In the above equation, b is the vector containing only ones. We perform a per-

turbation expansion of the solution for small Kn. Equating different orders of Kn in

Eq. (5.43), we find that the desired solutions can be written as

Λ0 = M−10 b,

Λ1 = −M−10 M1M

−10 b.

It is relatively easy to find these two solutions since we have already evaluated

the matrices M0 and M1. We emphasise that the two solutions corresponding to the

no-slip case and first-order correction, are independent of Knudsen number. Consider

the following plots which show Λ0 and Λ1 for N = 60 and β = 1.

60

-1.0 -0.5 0.5 1.0Ξ

-20

-10

10

20

Re@L0D

(a)

-1.0 -0.5 0.5 1.0Ξ

-20

-10

10

20

Im@L0D

(b)

-1.0 -0.5 0.5 1.0Ξ

-500

500

Re@L1D

(c)

-1.0 -0.5 0.5 1.0Ξ

-400

-200

200

400

Im@L1D

(d)

Figure 5.9: Real and imaginary components of the no-slip solution Λ0 as well as the first-ordercorrection Λ1

As we have seen previously, the incorporation of slip serves to remove the square-

root singularities exhibited by the no-slip solution. From Figs. 5.9(c) and 5.9(d) above,

we note that the first-order correction to the vorticity possesses a strong singularity

at the edges which opposes that of the zeroth-order solution. Attempting to integrate

these two curves poses some numerical problems. In particular, the singularity in Λ1

is only captured by the very last point before the edge of the blade. As such, any

numerical integration which we may undertake is susceptible to considerable error. We

find that the best method by which to determine the hydrodynamic function is to solve

Eq. (5.40) in one single swoop. This avoids the problems associated with competing

singularities.



Recall Eq. (2.6) which is used to calculate the quality factors:

Q =4λπρb2

+ Γr(ωR)

Γi(ωR). (5.44)

As alluded to earlier, the quality factor is indeed independent of the displacement of

the beam, W (x|ω). This is of great significance, since we can predict the quality factors

associated with the in-plane oscillations of the infinitely thin beam without knowing

anything about the actual mode shape. Note that Eq. (5.44) represents the quality

factor associated with the fluid only. In stark contrast to the infinitely thin boundary

layer model presented in Chapter 4, a closed expression for the quality factor does not

exist. Indeed, the calculation of such values requires us to construct and solve a vorticity

matrix-vector equation each time we need to evaluate the hydrodynamic function. This

takes a considerable amount of time, even using a powerful software package such as

mathematica R© . Since we know how to solve for the hydrodynamic functions in both

the no-slip and slip scenarios, we can proceed to calculate the quality factors. They

are presented in Table 5.1. The first column represents the fluid contribution to the

experimental quality factors.

Experiment Theoryβ Qfluid only Qno-slip Qfirst-order slip Qsecond-order slip

0.3527 136 128 197 2290.9223 339 270 446 5281.7477 535 435 761 9172.8268 804 616 1133 13914.1315 1160 804 1550 19420.4346 157 151 236 2750.8115 311 245 401 472

Table 5.1: Quality factors for the beam of zero thickness

62

Notice that all three regimes presented in Table 5.1 have associated with them

quite different quality factors. In particular, note that the two different slip models

yield very different results. This is very different to the thin boundary layer solution

in Chapter 4, where the first and second-order slip models yielded extremely similar

results (see Fig. 4.2).

10−2

10−1

10010

0

101

102

103

β

Q

No−slip

First−order slip

Second−order slip

Figure 5.10: Beam of zero thickness: Predicted and experimental quality factors associatedwith the first five in-plane modes of the 16µm-long device (dots) and the second and thirdin-plane modes of the 24µm-long device (crosses). No-slip (solid), first-order slip (dashed) andsecond-order slip (dotted) predictions are depicted alongside experiment values.

To this point we have looked at the effects associated with both the first and

second-order slip conditions. Since the correction arising from first-order slip is quite

large, there is very little point solving to second-order. As such, we will focus on the

no-slip and first-order slip solutions in the proceeding chapter.

Chapter 6

Beam of Non-Zero Thickness

Until now, we have treated the nanoelectromechanical resonator as infinitely thin.

Consequently, we have ignored all effects associated with the leading face of the beam.

Since the thickness of the beam is by no means negligible compared to its width (aspect

ratio 1:5), we anticipate that the quality factors associated with this model will be

markedly different to the infinitely thin scenario. Consider the schematic diagram of

the cross-section of the beam.

−a ay

z

d

−d

Figure 6.1: Figure showing cross-section of beam of non-zero thickness. The origin of thecoordinate axes is located at the geometric center of the cross-section.

Recall the two-dimensional streamfunction presented in Eq. (5.6). In the case

of the infinitely thin blade undergoing in-plane oscillations, this expression simplified

substantially with three of the four terms vanishing. In the case of non-zero thickness

63

64

however, very little simplification is possible. We have

ψ =∫C


1µpΨl

]dl, (6.1)

where the subscripts n and l denote derivatives with respect to the normal and tan-

gential coordinates to the surface respectively. Recall that the contour C is taken to

enclose the entire fluid and so the normal vector points out of the fluid.


Let us first apply the no-slip boundary condition traditionally used in continuum

mechanics. As previously discussed, this is not strictly valid given the relatively large

Knudsen number. However, this will serve as a good first approximation and indeed

will allow us to set up the machinery necessary for application of the slip model in the

proceeding section. For two-dimensional flow in the y-z plane, the fluid velocity is given

by

u = ∇× (ψx) =∂ψ

∂zy − ∂ψ

∂yz.

If we demand that the fluid velocity on Cbeam is precisely equal to the velocity of

the beam itself, we see that for unidirectional motion of the beam (Ubeam = Ubeamy),

the following equations must hold at the surface of the beam:

∂ψ

∂z= Ubeam and

∂ψ

∂y= 0. (6.2)

We require the precise form of the streamfunction on Cbeam. Unfortunately, Eq. (6.2)

cannot provide this. We thus proceed by taking a very different approach to that of

the previous chapters. We know that for a stationary beam, the surface constitutes a

single streamline on which the streamfunction is constant. It follows that if we pose the

problem in the reference frame of the oscillating beam, we can resolve the dilemma of

the unknown streamfunction.

Chapter 6. Beam of Non-Zero Thickness 65

Let us consider a stationary beam surrounded by an infinite fluid oscillating in the

y-direction at angular frequency ω. We write the fluid velocity in the following way:

u = ubeam + u,

where ubeam is the velocity of the beam. The corresponding boundary conditions are

given by

u = 0 at beam surface (no-slip condition),

u = −ubeam at infinity.

The linearized Fourier transformed governing equation for the fluid is given by

Eq. (2.3). Substitution of the above form of u yields

−iρω(ubeam + u) = −∇P + µ∇2(ubeam + u).

Noting that ubeam = Ubeamy and ∇2ubeam = 0, we see that

−iρωu = −∇(P − iρωUbeamy

)+ µ∇2u. (6.3)

The governing equation remains unchanged except for the inclusion of an extra

body force term. This is a direct consequence of transforming into the non-inertial

reference frame associated with the oscillating beam. We can proceed with the boundary

integral formulation working in this new reference frame. Since the fluid infinitely far

from the beam is no longer stationary, we must treat the contour integral around C

with caution. Remember that this contour encloses the entire fluid and so we write

C = Cbeam + C∞ as depicted in Fig. 6.2. Strictly speaking C∞ is an infinitely large

contour that encloses the entire system, however we must first treat it as finite and

proceed to take the limit as r →∞.

66

dl

n

(-a,-d)

(-a,d) (a,d)

(a,-d)

(-r1,r2)

fluid

beam

fluid

fluid

(r1,r2)

(r1,-r2)(-r1,-r2)

(a)

C1

C2

C3

C4

dl

n

dl

n

(-r1,r2)

fluid

beam

(r1,r2)

(r1,-r2)(-r1,-r2)

(b)

Figure 6.2: Diagrams of the contour enclosing the entire fluid. We see that C = Cbeam + C∞as depicted in Figure 6.2(b).


Equation (6.1) can be written in terms of the two constituent contours of C:

ψ =∫Cbeam

[ψGn−ψnΩ−ωΨn+

1µpΨl

]dl+

∫C∞

[ψGn−ψnΩ−ωΨn+

1µpΨl

]dl. (6.4)

We note that infinitely far from the beam, the flow is uniform. Thus, both the

vorticity and pressure are zero and the streamfunction can be written as ψ = −Ubeamz.

Furthermore, the streamfunction is constant on Cbeam since the beam is stationary in

the reference frame we are considering. Without loss of generality we can thus take

ψ = ψn = 0 on Cbeam. Equation (6.4) thus becomes

ψ =∫Cbeam

[− ωΨn +

1µpΨl

]dl +

∫C∞

[ψGn − ψnΩ

]dl. (6.5)

The Green’s functions G(y, z; y′, z′) and Ω(y, z; y′, z′) presented earlier are

G =12π

log(αR), (6.6)

Ω = − 12πK0(αR), (6.7)

where R =√

(y − y′)2 + (z − z′)2. Since Ω is exponentially small at infinity, we note

that the corresponding contribution to the integral in Eq. (6.5) vanishes. Furthermore,

we can evaluate the following integral:∫C∞

ψGndl = −Ubeamz′,

where we have made use of the fact that ∇2G = 0 on C∞. The streamfunction for the

fluid can now be written primarily in terms of the unknown quantities ω and p on the

surface of the beam. We find that

ψ(y′, z′) = −Ubeamz′ +

∫Cbeam

[− ωΨn +

1µpΨl

]dl. (6.8)

The integral in the above equation can be interpreted as the perturbation to the

uniform flow streamfunction, attributed to the presence of the beam. In the absence

of the beam, the integral in Eq. (6.8) vanishes and we are left with the streamfunction

corresponding to uniform flow. We consider four distinct components of the contour

Cbeam as depicted in Fig. 6.3.

68

C1

C2

C3

C4

dl

n

Figure 6.3: Contour around the face of the beam. Cbeam = C1 + C2 + C3 + C4.

We consider motion of the beam to be strictly in the y-direction. Since each path

Ci represents a straight line parallel to either the y or z-axes, we can replace the line

integrals with integrals over y and z. For example, on C1 we have dl = dz, ∂/ ∂l = ∂/ ∂z

and ∂/ ∂n = − ∂/ ∂y. Similar expressions follow for the other three faces. We make

use of the fact that for pure in-plane motion, the pressure on the surface of the beam is

symmetric about the y-axis and antisymmetric about the z-axis. Thus, p2(y) = p4(y)

and p1(z) = −p3(z). Conversely, the vorticity at the surface is antisymmetric about

the y-axis and symmetric about the z-axis. Hence, ω2(y) = −ω4(y) and ω1(z) = ω3(z).

With this in mind Eq. (6.8) becomes

ψ(y′, z′) = −Ubeamz′

+∫ d

−d

[ω1(z)

(Ψy

∣∣y=a

−Ψy

∣∣y=−a

)+

1µp1(z)

(Ψz

∣∣y=a

+ Ψz

∣∣y=−a

)]dz

+∫ a

−a

[ω2(y)

(Ψz

∣∣z=d

+ Ψz

∣∣z=−d

)− 1µp2(y)

(Ψy

∣∣z=d

−Ψy

∣∣z=−d

)]dy. (6.9)

We have posed this problem in the reference frame of the beam itself and so the fluid

is considered stationary on Cbeam. Equation (6.2) is thus an inappropriate boundary

condition. Instead we have

∂ψ(y′, z′)∂z′

∣∣∣∣∣beam

= 0 and∂ψ(y′, z′)

∂y′

∣∣∣∣∣beam

= 0. (6.10)

Combining Eqs. (6.9) and (6.10), we obtain the following pair of coupled integral equa-


tions:

Ubeam =∫ d

−d

[ω1(z)

(Ψyz′

∣∣y=a

−Ψyz′∣∣y=−a

)+

1µp1(z)

(Ψzz′

∣∣y=a

+ Ψzz′∣∣y=−a

)]dz

+∫ a

−a

[ω2(y)

(Ψzz′

∣∣z=d

+ Ψzz′∣∣z=−d

)− 1µp2(y)

(Ψyz′

∣∣z=d

−Ψyz′∣∣z=−d

)]dy,

(6.11)

and

0 =∫ d

−d

[ω1(z)

(Ψyy′

∣∣y=a

−Ψyy′∣∣y=−a

)+

1µp1(z)

(Ψzy′

∣∣y=a

+ Ψzy′∣∣y=−a

)]dz

+∫ a

−a

[ω2(y)

(Ψzy′

∣∣z=d

+ Ψzy′∣∣z=−d

)− 1µp2(y)

(Ψyy′

∣∣z=d

−Ψyy′∣∣z=−d

)]dy, (6.12)

where the right hand sides of the above equations are evaluated on the face of the beam.

We now proceed to scale these equations in the following way. Since the dominant length

scale is the width of the beam (y-direction), the spatial variables are scaled according

to

ξ =y

a, χ =

z

a. (6.13)

The pressure and vorticity are scaled as

P1(χ) =a

µUbeamp1(z), Λ1(χ) =

a

Ubeamω1(z),

P2(ξ) =a

µUbeamp2(y), Λ2(ξ) =

a

Ubeamω2(y). (6.14)

The beam for which we possess experimental data has an aspect ratioA = widththickness =

2a2d = 5, however we continue the formulation of this model in terms of an arbitrary as-

pect ratio A. This will enable us to check the model against the infinitely thin results

obtained in Chapter 5 by setting A→∞. From this scaling, it follows that ξ ∈ [−1, 1]

and χ ∈ [−1/A, 1/A]. Adopting the scaled Green’s function presented in Eq. (5.34), the

integral equations in (6.11) and (6.12) can be written as

70

1 =∫ 1/A

−1/A

[Λ1(χ)

(Ψξχ′

∣∣ξ=1

− Ψξχ′∣∣ξ=−1

)+ P1(χ)

(Ψχχ′

∣∣ξ=1

+ Ψχχ′∣∣ξ=−1

)]dχ

+∫ 1

−1

[Λ2(ξ)

(Ψχχ′

∣∣χ=1/A

+ Ψχχ′∣∣χ=−1/A

)+ P2(ξ)

(Ψξχ′

∣∣χ=−1/A

− Ψξχ′∣∣χ=1/A

)]dξ,

(6.15)

0 =∫ 1/A

−1/A

[Λ1(χ)

(Ψξξ′

∣∣ξ=1

− Ψξξ′∣∣ξ=−1

)+ P1(χ)

(Ψχξ′

∣∣ξ=1

+ Ψχξ′∣∣ξ=−1

)]dχ

+∫ 1

−1

[Λ2(ξ)

(Ψχξ′

∣∣χ=1/A

+ Ψχξ′∣∣χ=−1/A

)+ P2(ξ)

(Ψξξ′

∣∣χ=−1/A

− Ψξξ′∣∣χ=1/A

)]dξ.

(6.16)

Notice that in the limit when A → ∞, Eq. (6.15) collapses to become equivalent

to Eq. (5.18), the only exception being that in Chapter 5 we considered the vorticity

jump rather than the actual vorticity. Henceforth, we will drop the ‘bar’ notation for

Ψ and consider it to be scaled.


We now implement a similar numerical method to that used in Chapter 5 in order

to solve the above integral equations. This is significantly harder than for the beam of

zero thickness since we now have two coupled equations. We must transform the integral

equations into a system of matrix equations using an appropriate quadrature method. In

discretizing the integrals over the faces of the beam, we adopt the unequal quadrature

points with χm = − 1A cos(πm/M) for m = 0, 1, . . . ,M and ξn = − cos(πn/N) for

n = 0, 1, . . . , N . This will account for the presence of the square-root singularities

in the pressure and vorticity at the corners of the beam cross-section [5]. Provided

we discretize the integration intervals using a sufficiently large number of segments,

the pressure and vorticity can be approximated as constant in each interval. That is,

P1(χ) = P1m and Λ1(χ) = Λ1m for χm < χ < χm+1 and P2(ξ) = P2n and Λ2(ξ) = Λ2n

for ξn < ξ < ξn+1. Equations (6.15) and (6.16) thus become


M−1∑m=0

[Λ1mI1 + P1mI2

]+N−1∑n=0

[Λ2nI3 + P2nI4

]= 1, (6.17)

M−1∑m=0

[Λ1mJ1 + P1mJ2

]+N−1∑n=0

[Λ2nJ3 + P2nJ4

]= 0, (6.18)

where

I1 =∫ χm+1

χm

(Ψξχ′

∣∣ξ=1

−Ψξχ′∣∣ξ=−1

)dχ,

I2 =∫ χm+1

χm

(Ψχχ′

∣∣ξ=1

+ Ψχχ′∣∣ξ=−1

)dχ,

I3 =∫ ξn+1

ξn

(Ψχχ′

∣∣χ=1/A

+ Ψχχ′∣∣χ=−1/A

)dξ,

I4 =∫ ξn+1

ξn

(Ψξχ′

∣∣χ=−1/A

−Ψξχ′∣∣χ=1/A

)dξ,

(6.19)

J1 =∫ χm+1

χm

(Ψξξ′

∣∣ξ=1

−Ψξξ′∣∣ξ=−1

)dχ,

J2 =∫ χm+1

χm

(Ψχξ′

∣∣ξ=1

+ Ψχξ′∣∣ξ=−1

)dχ,

J3 =∫ ξn+1

ξn

(Ψχξ′

∣∣χ=1/A

+ Ψχξ′∣∣χ=−1/A

)dξ,

J4 =∫ ξn+1

ξn

(Ψξξ′

∣∣χ=−1/A

−Ψξξ′∣∣χ=1/A

)dξ.

The above integrals still have both ξ′ and χ′ unspecified. We demand that Eqs. (6.17)

and (6.18) hold at the midpoint of each segment on the discretized intervals. In other

words, we force these equations to hold at (ξ′, χ′) = (ξ′k,1A) = (1

2(ξk + ξk+1), 1A) for

k = 0, . . . , N − 1 and (ξ′, χ′) = (1, χ′j) = (1, 12(χj + χj+1)) for j = 0, . . . ,M − 1.

Applying these conditions to Eq. (6.17) yields the following matrix-vector equation:

[I1(1, χ′j)

]jm

[I2(1, χ′j)

]jm

[I3(1, χ′j)

]jn

[I4(1, χ′j)

]jn[

I1(ξ′k,1A)

]km

[I2(ξ′k,

1A)

]km

[I3(ξ′k,

1A)

]kn

[I4(ξ′k,

1A)

]kn

Λ1

P1

Λ2

P2

=

1

1...

1

.

72

If x is the vector containing the unknowns, then we write this in the following form:

Ax = a. (6.20)

Similarly, Eq. (6.18) becomes

[J1(1, χ′j)

]jm

[J2(1, χ′j)

]jm

[J3(1, χ′j)

]jn

[J4(1, χ′j)

]jn[

J1(ξ′k,1A)

]km

[J2(ξ′k,

1A)

]km

[J3(ξ′k,

1A)

]kn

[J4(ξ′k,

1A)

]kn

Λ1

P1

Λ2

P2

=

0

0...

0

,(6.21)

or alternatively

Bx = b. (6.22)

In the above equations, A and B are matrices of dimension (M +N)× (2M +2N)

while x, a and b are vectors of length 2M+2N . We want to solve Eqs. (6.20) and (6.22)

simultaneously. With the dimensions of these systems in mind, it becomes apparent

that we can construct one large system to be solved,[A

B

]x =

[a

b

],

or

Cx = d. (6.23)

We have now posed the problem in such a way as to have 2M +2N unknowns, and

the same number of corresponding equations. Construction of the matrix C can be ex-

tremely slow if we fail to make several simplifications first. We note that the majority of

the integrals in Eq. (6.19) can be evaluated exactly without resorting to numerical meth-

ods. Noting that Ψξχ′(ξ, χ; ξ′, χ′) = Ψχξ′(ξ, χ; ξ′, χ′), Ψξ′(ξ, χ; ξ′, χ′) = −Ψξ(ξ, χ; ξ′, χ′)

and Gχξ′(ξ, χ; ξ′, χ′) = Gξχ′(ξ, χ; ξ′, χ′), we see that that all but two of the integrals

can be evaluated directly by the Fundamental Theorem of Calculus. The integrals I3and J1 are more stubborn and we must resort to numerical integration. The matrix C

and vector d can be constructed using a software package such as mathematica R© .

Consider Figs. 6.4 and 6.5 on the following pages. These display the computed vorticity

and pressure on faces 1 and 2 of the beam for β = 0.3527 and A = 5.


As anticipated, we see that the vorticity and pressure on C1 are odd and even

functions of χ respectively. Conversely, the vorticity and pressure on C2 are even and

odd functions of ξ respectively. We need not display the computed results for C3 and

C4 since these are trivially found using the symmetry properties that we exploited to

simplify the problem from the outset.

74

Re HL1L

Im HL1L

-0.2 -0.1 0.1 0.2 Χ

-6

-4

-2

2

4

6

L1

(a)

Re HP1L

Im HP1L

-0.2 -0.1 0.1 0.2 Χ

-10

-5

5

10

15

P1

(b)

Figure 6.4: (a) Vorticity and (b) pressure distribution on the leading face of the beam (C1; seeFig. 6.3) for dimensionless frequency β = 0.3527 and aspect ratio A = 5.


Re HL2L

Im HL2L

-1.0 -0.5 0.5 1.0 Ξ

-4

-2

2

4

L2

(a)

Im HP2L

Re HP2L

-1.0 -0.5 0.5 1.0 Ξ

-1.0

-0.5

0.5

1.0

P2

(b)

Figure 6.5: (a) Vorticity and (b) pressure distribution on the top face of the beam (C2; seeFig. 6.3) for dimensionless frequency β = 0.3527 and aspect ratio A = 5.

76


We now endeavour to calculate the force on the beam as it oscillates. There are

a few subtleties which we must bear in mind when performing this calculation. Recall

that in undertaking this problem, we have considered the non-inertial reference frame

associated with the beam. In the rest frame of the laboratory, we have an additional

body force term which is a direct consequence of this transformation. From Eq. (6.3),

we see that the pressure in the reference frame of the laboratory is given by

p = pcalculated + iρωUbeamy,

where pcalculated is the (unscaled) pressure arising from our preceding numerical analysis.

Noting that Ubeam = −iωW , we see that

p = pcalculated + ρω2yW . (6.24)

The vector force F acting on any three-dimensional body with surface S, moving in any

manner in an incompressible fluid is given by [5]

F =∫S

[− pdS + µω × dS

].

The component of this force in the y−direction is then

Fy = j · F =∫Cbeam

[− pdz + µωdy

].

Since we know the pressure and vorticity on all four faces of the beam, we can readily

evaluate the total force in the y−direction,

F = −∫ d

−d

[2p1(z) + 2ρω2aW

]dz −

∫ a

−a

[2µω2(y)

]dy.

We must take care not to confuse the angular frequency ω with the fluid vortic-

ity ω2(y). Making use of the scaled variables from Eqs. (6.13) and (6.14), the above

expression can be rewritten as follows:

F = 2µiωW[Jpres(β) + Jvort(β)

]− 4ρadω2W , (6.25)


where

Jpres =∫ 1/A

−1/AP1(χ)dχ and Jvort =

∫ 1

−1Λ2(ξ)dξ. (6.26)

Since we have solved the problem in terms of the scaled pressure and vorticity, the

integrals Jpres and Jvort can be readily computed using mathematica R© . Equating

(2.4) and (6.25) yields the following expression for the hydrodynamic function:

Γ(ω) =2iπβ

[Jpres(β) + Jvort(β)

]− 4πA

. (6.27)

It is particularly interesting to see that the contribution to Γ(ω) due to the change

of reference frame depends only on A. Furthermore, 4/(πA) → 0 as A → ∞ and so

there is no additional body force for an infinitely thin blade. In order to evaluate the

hydrodynamic function for a particular value of β, we must construct and solve the

matrix system mentioned earlier numerically and proceed to integrate the calculated

pressure and vorticity over the faces of the beam. From a computational point of view,

this procedure is extremely demanding. Indeed, if we use M = N = 20; a reasonably

modest number of subdivisions on each face, we are required to construct an 80 × 80

matrix, of which 1,600 entries are numerically computed integrals! This process needs

to be undertaken every single time we want to evaluate Γ for some aspect ratio A

and frequency β. One can appreciate that plotting the hydrodynamic function as a

continuous curve is very difficult. Nevertheless, it is possible to evaluate Γ at sufficiently

many points to enable us to construct an accurate representation of the behaviour of

Γ over the appropriate frequency domain. As an aside, we note that production of

the following results in Fig. 6.6 required approximately 10 hours of CPU time using a

3.2GHz Pentium 4 processor.

78

10−1

100

10110

−1

100

101

102

β

Γ

Figure 6.6: Real (solid) and imaginary (dotted) components of the hydrodynamic function asa function of dimensionless frequency β. The above results correspond to aspect ratio A = 5and have been computed using M = N = 30 subdivisions for small β and M = N = 40 forlarger values of β. This value of A was chosen since it coincides exactly with the dimensions ofthe beams used by Bargatin et al. [2].


6.1.3 Alternative Scaling For Out-of-Plane Modes

So far we have been primarily concerned with in-plane motion of the beam. This

corresponds to motion in the y-direction for A > 1. The hydrodynamic function for

out-of-plane motion of an infinitely thin beam is known [3]. We would ultimately like to

check our model in this particular limiting case. The situation of out-of-plane motion

can be adopted be setting A < 1. Care must be taken when choosing such values of A.

Recall that the dimensionless frequency, spatial dimensions, vorticity and pressure have

all been scaled according to the dominant length scale. For in-plane motion (A > 1),

the characteristic length scale is the width of the beam, and so we chose to scale using

a. Conversely, for A < 1 the thickness of the beam becomes the characteristic length

scale and so it becomes appropriate to scale the aforementioned quantities in terms of

d. This is particularly easy to do. We see that the spatial discretization of the beam

faces is done with χm = − cos(πm/M) for m = 0, 1, . . . ,M and ξn = −A cos(πn/N) for

n = 0, 1, . . . , N . It turns out that the expression for the hydrodynamic function is very

similar to Eq. (6.27) and is given by

Γout(ω) =2iπβ


]− 4A

π. (6.28)

Notice that 4A/π → 0 as A→ 0. That is, the body force contribution arising from

solving the problem in a non-inertial reference frame vanishes for an infinitely thin blade.

This is perhaps slightly counter-intuitive since one would expect that for out-of-plane

motion, there would be a body force. The consequences of changing reference frame de-

pend on the volume of fluid displaced by the beam and not on the exposed surface area.

It is now possible to check the hydrodynamic function calculated by the present

model against the values obtained by Sader [3] simply by choosing A→ 0. Furthermore,

the present model should yield the same results as obtained in Chapter 5 in the case

as A → ∞ since this scenario represents in-plane motion of an infinitely thin blade.

Consider Fig. 6.7, which shows the computed hydrodynamic function as a function

of aspect ratio A. The dotted lines represent the infinitely thin solutions previously

mentioned. One can see that there is very good agreement between the present model

and the two aforementioned limiting cases.

80

10−1

100

101

102

100

101

A

Re(

Γ)

Increasing β

(a)

10−1

100

101

102

100

101

A

Im(Γ

)

Increasing β

(b)

Figure 6.7: (a) Real and (b) imaginary components of the hydrodynamic function plotted as afunction of aspect ratio A. Shown as dotted lines are the infinitely thin results derived by Sader[3] (left) and in Chapter 5 (right). A = 0 corresponds to out-of-plane motion of an infinitelythin blade, while A → ∞ corresponds to in-plane motion of an infinitely thin blade. Resultshave been computed for β = 0.3527, 1, 4 and 15.


One may wonder why there exists a sharp peak at A = 1 in Fig. 6.7. We must bear

in mind that in constructing these plots, we are really amalgamating two different aspect

ratio regimes corresponding to the respective scalings for the in-plane and out-of-plane

oscillations. Thus, we can actually think of this “peak” as the point where the length

scale of the problem switches. It is important to note that starting from A = 1, as A→ 0

and A → ∞ , both the real and imaginary components of the hydrodynamic function

are monotonic decreasing. That is, for a particular scaling, Γ decreases monotonically

as the blade becomes thinner. This is in perfect accordance with our physical intuition.

Recall that the real and imaginary components of the hydrodynamic function represent

the added mass and viscous losses respectively of the beam as it vibrates. We expect

that these would decrease as the blade thickness decreases and this is indeed what is

observed.

82

6.2 First-Order Slip Boundary Condition

We now seek to apply the slip boundary condition at the surface of the beam. As

discussed in preceding chapters, it is necessary to consider a boundary condition of this

form since the mean free path of air molecules is comparable to the thickness of the

blades used by Bargatin et al. [2]. Indeed, at this scale (Kn ≈ 0.17), the no-slip bound-

ary condition traditionally used in continuum mechanics is certainly not applicable.

We will see that the subsequent correction to the no-slip results is extremely strong,

and as such, seeking anything more than a first-order correction is unreasonable. The

cross-section of the beam is still given as in Fig. 6.1, the only thing changing in our

analysis being the boundary condition at the beam surface.

Recall that in Section 6.1 we posed the problem in terms of a stationary beam

surrounded by a fluid oscillating at infinity so that the streamfunction could be set to

zero at the beam surface. This analysis follows in exactly the same way, except that in

moving from Eq. (6.4) to Eq. (6.5), we can no longer set ψn = 0. Differentiation of the

streamfunction with respect to the normal coordinate yields the tangential velocity at

the beam surface which is certainly no longer zero! Instead we obtain

ψ =∫Cbeam

[− ψnΩ− ωΨn +

1µpΨl

]dl +

∫C∞

[ψGn − ψnΩ

]dl, (6.29)

where G and Ω are the Green’s function defined in Eqs. (6.6) and (6.7). Since the

pressure and vorticity infinitely far from the beam will be unaffected by the presence

of slip, this equation reduces to give

ψ(y′, z′) = −Ubeamz′ +

∫Cbeam

[− ψnΩ− ωΨn +

1µpΨl

]dl. (6.30)

This is precisely the same as in Eq. (6.8) with the addition of one extra term. We can

actually write the above equation as

ψ(y′, z′) = ψno-slip(y′, z′)−∫Cbeam

[ψnΩ

]dl, (6.31)

where ψno-slip is given by Eq. (6.8). In order to remove the dependence of Eq. (6.31) on


ψn, we examine the first-order slip boundary condition. This is given by

U∣∣wall

− uw = −γη ∂U∂n

∣∣∣∣wall

, (6.32)

where the subscript n denotes differentiation in the direction of the normal vector,

which points out of the fluid. We again consider the four faces of the beam as distinct

components of the contour Cbeam as in Fig. 6.3. Since the fluid velocity and vorticity

can be written as u = ∇×(ψx) and ω = −∇2ψ respectively, the slip boundary condition

becomes

ψz′ = 0 on C1 (no penetration condition),

ψz′ + γηω2 = 0 on C2 (slip condition), (6.33)

ψy′ + γηω1 = 0 on C1 (slip condition),

ψy′ = 0 on C2 (no penetration condition),

and we have∫Cbeam

(ψnΩ

)dl = γη

[ ∫ d

−dω1(z)

(Ω

∣∣y=a

+Ω∣∣y=−a

)dz+

∫ a

−aω2(y)

(Ω

∣∣z=d−Ω

∣∣z=−d

)dy

].

Combining this with Eq. (6.9), we see that the streamfunction can be written as


+∫ d

−d

[ω1(z)

(Ψy

∣∣y=a

−Ψy

∣∣y=−a

)+

1µp1(z)

(Ψz

∣∣y=a

+ Ψz

∣∣y=−a

)]dz

+∫ a

−a

[ω2(y)

(Ψz

∣∣z=d

+ Ψz

∣∣z=−d

)− 1µp2(y)

(Ψy

∣∣z=d

−Ψy

∣∣z=−d

)]dy

− γη

[ ∫ d

−dω1(z)

(Ω

∣∣y=a

+ Ω∣∣y=−a

)dz +

∫ a

−aω2(y)

(Ω

∣∣z=d

− Ω∣∣z=−d

)dy

].

(6.34)

84

Regrouping the terms slightly yields


+∫ d

−d

[ω1(z)

(Ψy

∣∣y=a

−Ψy

∣∣y=−a − γη(Ω

∣∣y=a

+ Ω∣∣y=−a)

)+

1µp1(z)

(Ψz

∣∣y=a

+ Ψz

∣∣y=−a

)]dz

+∫ a

−a

[ω2(y)

(Ψz

∣∣z=d

+ Ψz

∣∣z=−d − γη(Ω

∣∣z=d

− Ω∣∣z=−d)

)− 1µp2(y)

(Ψy

∣∣z=d

−Ψy

∣∣z=−d

)]dy.

(6.35)

The incorporation of slip has altered the streamfunction slightly by introduction of

additional terms. If we force Eq. (6.35) to satisfy the boundary conditions in Eq. (6.33)

we obtain the following four equations:

Ubeam =∫ d

−d

[ω1(z)

(Ψyz′

∣∣y=a

−Ψyz′∣∣y=−a − γη(Ωz′

∣∣y=a

+ Ωz′∣∣y=−a)

)+

1µp1(z)

(Ψzz′

∣∣y=a


)]dz

+∫ a

−a

[ω2(y)

(Ψzz′

∣∣z=d

+ Ψzz′∣∣z=−d − γη(Ωz′

∣∣z=d

− Ωz′∣∣z=−d)

)− 1µp2(y)

(Ψyz′

∣∣z=d


)]dy on C1,

(6.36)

Ubeam =∫ d

−d

[ω1(z)

(Ψyz′

∣∣y=a

−Ψyz′∣∣y=−a − γη(Ωz′

∣∣y=a

+ Ωz′∣∣y=−a)

)+

1µp1(z)

(Ψzz′

∣∣y=a


)]dz

+∫ a

−a

[ω2(y)

(Ψzz′

∣∣z=d

+ Ψzz′∣∣z=−d − γη(Ωz′

∣∣z=d

− Ωz′∣∣z=−d)

)− 1µp2(y)

(Ψyz′

∣∣z=d


)]dy + γηω2(y′) on C2,

(6.37)


0 =∫ d

−d

[ω1(z)

(Ψyy′

∣∣y=a

−Ψyy′∣∣y=−a − γη(Ωy′

∣∣y=a

+ Ωy′∣∣y=−a)

)+

1µp1(z)

(Ψzy′

∣∣y=a


)]dz

+∫ a

−a

[ω2(y)

(Ψzy′

∣∣z=d

+ Ψzy′∣∣z=−d − γη(Ωy′

∣∣z=d

− Ωy′∣∣z=−d)

)− 1µp2(y)

(Ψyy′

∣∣z=d


)]dy + γηω1(z′) on C1,

(6.38)

0 =∫ d

−d

[ω1(z)

(Ψyy′

∣∣y=a

−Ψyy′∣∣y=−a − γη(Ωy′

∣∣y=a

+ Ωy′∣∣y=−a)

)+

1µp1(z)

(Ψzy′

∣∣y=a


)]dz

+∫ a

−a

[ω2(y)

(Ψzy′

∣∣z=d

+ Ψzy′∣∣z=−d − γη(Ωy′

∣∣z=d

− Ωy′∣∣z=−d)

)− 1µp2(y)

(Ψyy′

∣∣z=d


)]dy on C2. (6.39)

The terms arising from the incorporation of slip are coloured red above. Notice

that if we remove them by setting η = 0, Eqs. (6.11) and (6.12) corresponding to the

no-slip scenario are recovered. Proceeding to scale the above four integral equations,

we obtain the following:

86

1 =∫ 1/A

−1/A

[Λ1(χ)

(Ψξχ′

∣∣ξ=1

− Ψξχ′∣∣ξ=−1

− γη(Ωχ′∣∣ξ=1

+ Ωχ′∣∣ξ=−1

))

+ P1(χ)(

Ψχχ′∣∣ξ=1

+ Ψχχ′∣∣ξ=−1

)]dχ

+∫ 1

−1

[Λ2(ξ)

(Ψχχ′

∣∣χ=1/A

+ Ψχχ′∣∣χ=−1/A

− γη(Ωχ′∣∣χ=1/A

− Ωχ′∣∣χ=−1/A

))

+ P2(ξ)(

Ψξχ′∣∣χ=−1/A

− Ψξχ′∣∣χ=1/A

)]dξ on C1, (6.40)

1 =∫ 1/A

−1/A

[Λ1(χ)

(Ψξχ′

∣∣ξ=1

− Ψξχ′∣∣ξ=−1

− γη(Ωχ′∣∣ξ=1

+ Ωχ′∣∣ξ=−1

))

+ P1(χ)(

Ψχχ′∣∣ξ=1

+ Ψχχ′∣∣ξ=−1

)]dχ

+∫ 1

−1

[Λ2(ξ)

(Ψχχ′

∣∣χ=1/A

+ Ψχχ′∣∣χ=−1/A


− Ωχ′∣∣χ=−1/A

))

+ P2(ξ)(

Ψξχ′∣∣χ=−1/A

− Ψξχ′∣∣χ=1/A

)]dξ + γηΛ2(ξ′) on C2,

(6.41)

0 =∫ 1/A

−1/A

[Λ1(χ)

(Ψξξ′

∣∣ξ=1

− Ψξξ′∣∣ξ=−1

− γη(Ωξ′∣∣ξ=1

+ Ωξ′∣∣ξ=−1

))

+ P1(χ)(

Ψχξ′∣∣ξ=1

+ Ψχξ′∣∣ξ=−1

)]dχ

+∫ 1

−1

[Λ2(ξ)

(Ψχξ′

∣∣χ=1/A

+ Ψχξ′∣∣χ=−1/A

− γη(Ωξ′∣∣χ=1/A

− Ωξ′∣∣χ=−1/A

))

+ P2(ξ)(

Ψξξ′∣∣χ=−1/A

− Ψξξ′∣∣χ=1/A

)]dξ + γηΛ1(χ′) on C1, (6.42)

0 =∫ 1/A

−1/A

[Λ1(χ)

(Ψξξ′

∣∣ξ=1

− Ψξξ′∣∣ξ=−1

− γη(Ωξ′∣∣ξ=1

+ Ωξ′∣∣ξ=−1

))

+ P1(χ)(

Ψχξ′∣∣ξ=1

+ Ψχξ′∣∣ξ=−1

)]dχ

+∫ 1

−1

[Λ2(ξ)

(Ψχξ′

∣∣χ=1/A

+ Ψχξ′∣∣χ=−1/A


− Ωξ′∣∣χ=−1/A

))

+ P2(ξ)(

Ψξξ′∣∣χ=−1/A

− Ψξξ′∣∣χ=1/A

)]dξ on C2. (6.43)



In order to solve the integral equations (6.40) - (6.43), we will use precisely the

same numerical solution method as in Section 6.1. We discretize the integration inter-

vals using Chebyshev points and proceed to construct a giant matrix-vector equation.

Following the same procedure as before, we see that Eqs. (6.40) - (6.43) become

M−1∑m=0

[Λ1mI1 + P1mI2

]+N−1∑n=0

[Λ2nI3 + P2nI4

]= 1 on C1, (6.44)

M−1∑m=0

[Λ1mI1 + P1mI2

]+N−1∑n=0

[Λ2nI3 + P2nI4

]+ γηΛ2(ξ′) = 1 on C2, (6.45)

M−1∑m=0

[Λ1mJ1 + P1mJ2

]+N−1∑n=0

[Λ2nJ3 + P2nJ4

]+ γηΛ1(χ′) = 0 on C1, (6.46)

M−1∑m=0

[Λ1mJ1 + P1mJ2

]+N−1∑n=0

[Λ2nJ3 + P2nJ4

]= 0 on C2, (6.47)

where

I1 =∫ χm+1

χm

(Ψξχ′

∣∣ξ=1

−Ψξχ′∣∣ξ=−1

− γη(Ωχ′∣∣ξ=1

+ Ωχ′∣∣ξ=−1

))dχ,

I2 =∫ χm+1

χm

(Ψχχ′

∣∣ξ=1

+ Ψχχ′∣∣ξ=−1

)dχ,

I3 =∫ ξn+1

ξn

(Ψχχ′

∣∣χ=1/A

+ Ψχχ′∣∣χ=−1/A


− Ωχ′∣∣χ=−1/A

))dξ,

I4 =∫ ξn+1

ξn

(Ψξχ′

∣∣χ=−1/A

−Ψξχ′∣∣χ=1/A

)dξ,

(6.48)

J1 =∫ χm+1

χm

(Ψξξ′

∣∣ξ=1

−Ψξξ′∣∣ξ=−1

− γη(Ωξ′∣∣ξ=1

+ Ωξ′∣∣ξ=−1

))dχ,

J2 =∫ χm+1

χm

(Ψχξ′

∣∣ξ=1

+ Ψχξ′∣∣ξ=−1

)dχ,

J3 =∫ ξn+1

ξn

(Ψχξ′

∣∣χ=1/A

+ Ψχξ′∣∣χ=−1/A


− Ωξ′∣∣χ=−1/A

))dξ,

J4 =∫ ξn+1

ξn

(Ψξξ′

∣∣χ=−1/A

−Ψξξ′∣∣χ=1/A

)dξ.

88

Using the above equations, we can assemble the matrix-vector system. Only a few

small modifications need be made to the mathematica R© code corresponding to the

no-slip case. In addition, we note that half of the integrals involving Ωχ′ and Ωξ′ are

directly integrable whilst for the other half we must resort to numerical integration.

The matrix-vector equation can be written in the following way:

Mx = r,

where x is the global vector containing all of the unknown values of Λ1, P1, Λ2 and P2.

By observation of Eqs. (6.44)-(6.48), it is evident that the matrix-vector equation can

also be written as (M0 + KnM1

)x = r, (6.49)

where the matrices M0 and M1 are independent of Kn = η/b. The vast majority of

computational time is spent constructing M0 and M1. We need only compute these

once for given β and A since solutions corresponding to all Kn can then be quickly

evaluated using the above equation.

We now examine one set of solutions. The plots in Figs. 6.8 and 6.9 show the real

and imaginary components of the vorticity and pressure on the front (C1) and top (C2)

faces of the beam. Although the Knudsen number corresponding to the experiments

undertaken by Bargatin et al. [2] is Kn = η/b = η/2 ≈ 0.17, the results presented are

for Kn = 0.01. We will deal with Kn rather than η from now on since it is more widely

used in the literature. Note that these two quantities differ by a factor of 1/2.

Notice from Figs. 6.8 and 6.9 that the incorporation of slip has the effect of removing

the square-root singularities in the vorticity near the corners. This phenomenon was

observed upon introduction of slip for the infinitely thin blade (see Fig. 5.4). Notice

also that the square-root singularities in the pressure remain intact. Although the

vorticity and pressure are coupled through the global matrix-vector system, we note

that incorporating first-order slip affects elements of the matrix corresponding to the

vorticity only.


Re HL1L

Im HL1L

-0.2 -0.1 0.1 0.2Χ

-2

-1

1

2

L1

(a)

Re HP1L

Im HP1L

-0.2 -0.1 0.1 0.2Χ

-15

-10

-5

5

10

15

P1

(b)

Figure 6.8: (a) Vorticity and (b) pressure distribution on the leading face of the beam (C1;see Fig. 6.3) for dimensionless frequency β = 0.3527, aspect ratio A = 5 and Knudsen numberKn = 0.01.

90

Re HL2L

Im HL2L

-1.0 -0.5 0.5 1.0Ξ

-1.0

-0.5

0.5

1.0

1.5

L2

(a)

Im HP2L

Re HP2L

-1.0 -0.5 0.5 1.0Ξ

-1.0

-0.5

0.5

1.0

P2

(b)

Figure 6.9: (a) Vorticity and (b) pressure distribution on the top face of the beam (C2; seeFig. 6.3) for dimensionless frequency β = 0.3527, aspect ratio A = 5 and Knudsen numberKn = 0.01.



Once the vorticity and pressure are known on each face of the beam, the sub-

sequent calculation of the hydrodynamic function is relatively straightforward. Recall

Eqs. (6.27) and (6.28) which are expressions for the hydrodynamic function correspond-

ing to in-plane and out-of-plane modes respectively. We note that as well as being de-

pendent on ω (or equivalently β) and A, Γ will also have a dependence on the Knudsen

number Kn = η/b = η/2. As such, we write

Γin(ω,Kn) =2iπβ


]− 4πA

for A > 1, (6.50)

Γout(ω,Kn) =2iπβ


]− 4A

πfor A < 1. (6.51)

Recall that the above terms involving A arise due to the fact that we have solved

the problem in the non-inertial reference frame of the beam itself. At this stage we

must exercise caution in evaluating the solution as there are several subtleties which, if

ignored, will be catastrophic to our results. In the limit as Kn → ∞, we require that

the hydrodynamic loading on the beam approach zero, since this represents a vacuum.

Consider the following plots which show the computed hydrodynamic function for a

range of Knudsen numbers.

10-5 0.001 0.1 10 1000

2.6

2.8

3.

3.2

3.4

Kn

ReHGL

(a)

10-5 0.001 0.1 10 10006.

6.2

6.4

6.6

6.8

7.

Kn

ImHGL

(b)

Figure 6.10: Figure showing real and imaginary components of the calculated hydrodynamicfunction for β = 0.3527 and A = 5. These correspond to the raw solutions obtained from thematrix-vector equation.

92

Linear Regression of Hydrodynamic Function

The real and imaginary components of the hydrodynamic function level-off as Kn

increases, and fail to approach zero. Upon first glance this is quite concerning, since

it means that we cannot evaluate Γ directly for Kn = 0.17. As Kn → ∞, inspection

of the first-order slip condition yields ∂U/ ∂z → 0. In this limit, the variation in

the velocity diminishes and the pressure contribution on the front face of the beam

prevents the loading from approaching zero. Whilst the numerical process employed

fails to converge to the required solution as Kn increases, we note that Γ is monotonic

decreasing in Kn as required. We remedy the problem by considering very small values

of Kn so as to identify the linear region of the plots in Fig. 6.10. Remember that we

have applied the first-order slip condition and as such will be looking for a correction

to Γ to first-order in Kn. That is, we seek a solution of the following form:

Γ(ω,Kn) = Γ0 + KnΓ1 for Kn 1. (6.52)

We note that the above equation is valid only for small Kn and that Γ0 = Γno-slip

which was found in the preceding section. We proceed to perform a linear regression

for small values of Knudsen number. Consider the following plots, which are the same

curves as in Fig. 6.10, but have been plotted for small Kn only. In order to identify

the linear component of the curve, we consider Kn ∼ 10−7 − 10−6. For larger values

of Kn, the linearity exhibited by the curves in Fig. 6.11 is disrupted. We perform an

interpolation of the points depicted in Fig. 6.11 to degree two, and subsequently extract

the linear component. This is more accurate than simply undertaking a linear regression

since we allow for the curve to be ever so slightly non-linear and correctly capture this

behaviour. We force the interpolated function to go through the point corresponding

to Kn = 0 since this represents the no-slip solution which is well-known and can be

computed to high accuracy. In performing the calculations using mathematica R© , we

have set the working precision to 20 significant figures. We can thus be confident in

the accuracy of the following plots despite the fact that Kn is extremely small. For

the purposes of plotting, we have subtracted the constant Γ0 from the hydrodynamic

function.


æ

æ

æ

æ

æ

æ

æ

æ

æ

æ

æ

0 2.´10-7 4.´10-7 6.´10-7 8.´10-7 1.´10-6

-0.000015

-0.00001

-5.´10-6

0

Kn

Re@G - G0D

(a)

æ

æ

æ

æ

æ

æ

æ

æ

æ

æ

æ

0 2.´10-7 4.´10-7 6.´10-7 8.´10-7 1.´10-6

-0.000014

-0.000012

-0.00001

-8.´10-6

-6.´10-6

-4.´10-6

-2.´10-6

0

Kn

Im@G - G0D

(b)

Figure 6.11: Figure showing real and imaginary components of the hydrodynamic function forβ = 0.3527 and A = 5.

94

Improving Accuracy of Γ for Kn 1

The expression in Eq. (6.52) is linear and monotonic decreasing in Kn. We know

that it is valid for Kn 1, but for sufficiently large Knudsen number, it gives both

Re(Γ) < 0 and Im(Γ) < 0. This is unphysical since it represents a situation where

viscous losses are negative! We seek to improve the convergence of the asymptotic

expansion of the hydrodynamic function in the large Kn regime. Moreover, we require

that Γ → 0 as Kn → ∞. This is done using a Pade approximant and is outlined by

Hinch [12]. We write

Γ = Γ0 + KnΓ1

= Γ0

(1 +

Γ1

Γ0Kn

),

and since 1 + x ≈ (1− x)−1 for small x,

ΓPade(ω,Kn) = Γ0

(1− Γ1

Γ0Kn

)−1

. (6.53)

The expression in Eq. (6.53) has the beauty of being valid in both the low and

high Knudsen number regimes. Thus, the curves in Fig. 6.11 will remain intact. The

Pade approximant approaches zero as Kn → ∞ as it should. In fact, for Kn 1,

ΓPade(ω,Kn) ∼ −Γ20/(Γ1Kn).

GPadeG0+ G1Kn

0.0 0.2 0.4 0.6 0.8 1.00

2

4

6

8

Kn

ReHGL

(a)

GPade

G0+ G1Kn

0 1 2 3 40

5

10

15

20

Kn

ImHGL

(b)

Figure 6.12: Figure showing the real and imaginary components of both the linear constructand the Pade approximant.


Convergence

As with any numerical procedure, understanding the convergence of the computed

solutions is paramount. We now examine the behaviour of the solutions obtained as we

vary the number of subdivisions used in the discretization of the beam faces. We have

already seen how to calculate Γ0 and Γ1 given frequency β and aspect ratio A. Consider

the plots in Fig. 6.13 which show how both the real and imaginary components of Γ0

and Γ1 converge as M and N are increased. For simplicity we have taken M = N in

each case. Values of β = 0.3527 and A = 5 have been adopted for this example.

æ

æ

æ

0 10 20 30 40 50 60

3.145

3.146

3.147

3.148

3.149

3.150

3.151

N

Re@G0D

(a)

æ

æ

æ

0 10 20 30 40 50 606.702

6.703

6.704

6.705

6.706

6.707

N

Im@G0D

(b)

æ

æ

æ

æ

0 10 20 30 40 50 60-17

-16

-15

-14

-13

-12

N

Re@G1D

(c)

æ

æ

æ

æ

0 10 20 30 40 50 60-16

-15

-14

-13

-12

-11

N

Im@G1D

(d)

Figure 6.13: Real and imaginary components of the no-slip hydrodynamic function Γ0 as wellas the first-order correction Γ1 . The solid curve represents the interpolated solution whilst thedotted lines depict the limiting values obtained using Richardson extrapolation.

96

The interpolated functions given by the solid curves in Fig. 6.13 are of the form

a + b/N c and adopt the exponents c = 2 and c = 1/2 for Γ0 and Γ1 respectively.

The values of Γ0 converge fast enough that we can directly evaluate the solution by

taking a sufficiently large value of N . In contrast, the first-order correction converges

much slower, and as such, we are required to extrapolate in order to find the N → ∞solution. We know the exponents of the dominant error terms associated with both

Γ0 and Γ1 respectively. The process of Richardson extrapolation can thus be employed

to accurately predict Γ0 and Γ1 for N → ∞. We emphasise that in order to use this

process, we need to use a series of meshes in which the number of points doubles each

time. For example, we consider N = 5, 10, 20, 40, 80. The problem of slow convergence

of the first-order correction has been remedied by calculating the solution for several

modest values of N , and extrapolating to find the limiting value. We can adopt this

method for any values of β and A.

6.2.3 Comparison with Free Molecular Solution

The solution presented in Eq. (6.53) possesses the correct asymptotic behaviour

for Kn 1 since the solution is constructed using points in this range. We know that

ΓPade(ω,Kn) ≈ −Γ20/(Γ1Kn) → 0 as Kn → ∞. That is, the hydrodynamic function

Γ ∼ Kn−1 as Kn → ∞. We know that this solution exhibits the correct qualitative

behaviour in converging to zero, but cannot be sure that its asymptotic form is indeed

correct. As such, we examine the free molecular solution which we know to be true in

the limit as Kn →∞.

Consider the stationary beam surrounded by a fluid, infinite in extent, and oscil-

lating back and forth with velocity −Ubeam = iωW . The force per unit length exerted

on the beam is given by [10]

F = −µUbeam

Kn

[(1− ε) +

[2 +

π

2

√T +

(2− π

2

√T

)ε

]1A

]. (6.54)

In the above expression, the ratio between the temperature of the beam and the

temperature at infinity is given by T = Tbeam/T∞. The value of ε relates to the extent

of interactions which are specular in nature. Scenarios involving purely diffuse and

specular reflection are represented by ε = 0 and ε = 1 respectively. For simplicity


we will consider T = 1, representing the case where the temperature is uniform. The

hydrodynamic function can be readily found and is given by

Γfree mol(β,Kn) =i

Knπβ

[(1− ε) +

[2 +

π

2+

(2− π

2

)ε

]1A

]. (6.55)

Compare Eqs. (6.53) and (6.55). Notice that both expressions scale as ∼ Kn−1 in

the large Knudsen number regime. Up to a constant, the Pade approximant constructed

earlier possesses the correct asymptotic form as Kn →∞. In fact, for β = 0.3527, A = 5

and ε = 0, ΓPade/Γfree mol → 1.53 as Kn →∞. In constructing the Pade approximant,

we have managed to dramatically improve the convergence of the calculated hydrody-

namic function. For the beams used by Bargatin et al. [2], the Knudsen number is

Kn = 0.17. Although the hydrodynamic function does not agree perfectly with the free

molecular solution for very large Kn, we expect that for Kn ∼ 0.1−0.2, the accuracy will

be quite high. Consider the following plot, which shows the linear components of the

hydrodynamic function as well as the Pade approximant, alongside the free molecular

solution. We have plotted the molecular case for ε = 0 and ε = 0.25.

G0+ G1Kn

GPade

Gfree mol HΕ = 0LGfree mol

HΕ = 0.25L

1.000.50 5.000.10 10.000.050.01

0.1

1

10

Kn

Im@GD

Figure 6.14: Figure showing the linear component of the hydrodynamic function (dotted),the Pade approximant (solid) and the free molecular solution for ε = 0 (dashed) and ε = 0.25(dot-dashed).

98


We are now in a position to compare the results predicted by the non-zero thickness

models with experimental values. The principal method of doing so is to evaluate and

compare the quality factors associated with the various in-plane modes. The hydrody-

namic function associated with such modes are given by Eqs. (6.27) and (6.53) for the

cases involving no-slip and slip respectively. These have already been computed, and

so we can readily calculate the associated quality factors using Eq. (2.6). The following

table contains the experimental quality factors, as well as the values predicted by both

the no-slip and first-order slip models. The first five rows correspond to the first five in-

plane modes of the 16µm-long device. The last two rows correspond to the second and

third in-plane resonances of the 24µm-long device. As you can see, there is very good

agreement between the experimentally determined quality factors and those predicted

by the full theoretical model including slip.

Experiment Theoryβ Qfluid only Qfirst-order slip Qno-slip

0.3527 136 144 1010.9223 339 313 2061.7477 535 516 3242.8268 804 749 4504.1315 1160 1002 5790.4346 157 171 1190.8115 311 283 188

Table 6.1: Predicted and experimental quality factors associated with the beam of non-zerothickness


6.4 Determination of Surface Properties

In making the predictions displayed in Table 6.1, we have implicitly assumed cer-

tain properties of the surface of the beam. We have derived the results assuming that

all fluid-beam interactions are diffuse in nature. That is, all fluid molecules temporar-

ily adhere to the surface of the beam and are subsequently re-emitted at a range of

velocities. Another mechanism by which the fluid particles can interact with the beam

is through specular reflection. Quite simply, this consists of the particles reflecting off

the surface of the beam, much like a mirror. The extent to which the fluid particles

undergo specular reflection is related to the surface properties of the device. For the

beams used by Bargatin et al. [2], the precise nature of the surface was unknown, and

so, to begin with we have assumed pure diffuse reflection. We now seek to determine

the relative proportions of diffuse and specular reflection. Recall the first-order slip

condition given by

U∣∣wall

− uw = −2γKn∂U

∂n

∣∣∣∣wall

,

where the unit normal n points out of the fluid. To this point we have considered γ to

be fixed and equal to 1.11. If we allow for the presence of specular reflection, the above

slip condition becomes [11]

U∣∣wall

− uw = −2γ(

1.79σ

− 0.65− 0.14σ)

Kn∂U

∂n

∣∣∣∣wall

. (6.56)

In Eq. (6.56), σ is the thermal accommodation coefficient and represents the propor-

tional of fluid particles undergoing diffuse reflection. Solutions for any Kn are available

to us using the Pade approximant discussed earlier. As such, we can easily find an

expression that gives the quality factors as a function of Knudsen number. Thus, we

need not construct and solve the entire matrix-vector equation for each value of σ. The

quality factors presented Table 6.1 are evaluated for Kn = 0.17. Our goal is to vary

σ (or equivalently Kn) in order to achieve the closest agreement between experiment

and theory. We do so by using the method of least squares to minimise the percentage

error associated with the quality factors. The following plot shows the predicted and

experimental quality factors. We have included the experimental results for two beams

of different length.

100

No-Slip

Slip

0.5 1 2 3 4

100

200

400

1000

Β

Q

Figure 6.15: Figure showing experimental and predicted quality factors for beam of non-zero thickness. The experimental values for the 16µm-long (blue) and 24µm-long (red) beamsare depicted by dots. The two first-order slip solutions correspond to purely diffuse reflection(dashed) and mixed diffuse/specular reflection (solid).

β Qexperiment QFittedfirst-order slip % error

0.3527 136 149 10.00.9223 339 324 4.11.7477 535 539 0.62.8268 804 784 2.54.1315 1160 1051 9.40.4346 157 177 12.70.8115 311 293 5.7

Table 6.2: Predicted and experimental quality factors associated with the beam of non-zerothickness. The sum of the percentage errors has been minimised by setting σ = 0.95.


Recall that the actual Knudsen number for the beams is 0.17. The Knudsen number

which minimises the percentage error committed by the predicted quality factors is

Kn = 0.1879. By observation of Eq. (6.56), we see that varying Kn is equivalent to

altering σ. The fitted value of Kn corresponds to a thermal accommodation coefficient of

σ = 0.95. This means that fluid-beam interactions are predominantly diffuse in nature,

with a small degree of specular reflection. This is a particularly interesting result,

since the model we have developed is able to determine the nature of the gas-surface

interaction.

Chapter 7

Concluding Remarks

When a clamped-clamped silicon carbide beam is excited thermoelastically, it ex-

hibits a wide range of in-plane and out-of-plane resonances. Recent experiments con-

ducted by Bargatin et al. [2] involved immersing such beams in a viscous fluid and

measuring the quality factors associated with the displayed resonances. Whilst the ex-

perimental data could be fitted, a comprehensive mathematical model accounting for

the underlying physical principles remained elusive. This was the motivation that led

to the formulation of the theory presented in this thesis.

In order to understand the behaviour of the nanoelectromechanical devices used,

we have tackled the problem from two distinct angles. The first involved analysing the

devices using beam theory in order to predict the resonant frequencies. There was very

good agreement between experimental results and the predicted values, both for the

in-plane and out-of-plane modes. The second, and considerably larger component of

this thesis was to formulate a model able to predict the quality factors observed by

Bargatin et al.. On our journey towards the full theoretical model we have adopted

many “simple” models which capture key elements of the underlying physics one by one.

Since the beams are extremely small (width 400nm and thickness 80nm), it is necessary

to move away from a description which relies entirely on classical continuum mechanics.

Indeed, we have found that the effects associated with slip are quite significant.

Chapters 5 and 6 constitute original research. Recall in Chapter 5, we found that

the beam of zero thickness, subject to the no-slip condition yielded quality factors which

103

104

were reasonably accurate. The leading face of the beam and the inclusion of slip serve

to diminish and enhance the quality factors, respectively. Consequently, we conclude

that these two mechanics provide competing effects for the infinitely thin blade subject

to no-slip.

The full theoretical model accounting for the non-zero thickness of the beam as well

as the effects of slip produced results which agreed remarkably well with experimental

values. Interestingly, the predicted values are independent of the amplitude of vibration

and the boundary conditions at the ends of the beam. In addition, we have also been

able to draw some conclusions about the nature of the surface of the beams. We have

found that the coefficient of thermal accommodation at the surface is approximately

σ = 0.95 representing predominantly diffuse reflection.

Further Work

There are several aspects of this problem which could be pursued in the future. As

with any model attempting to account for physical observations, rigorous testing and

repetition of experiments is required. The data available to us is limited to beams of

only one aspect ratio. Further testing could be undertaken for various aspect ratios

since the model presented allows for arbitrary rectangular cross-section.

As mentioned in Appendix B, the corners of the beam represent a flow regime

distinct from all other regions. A rigorous asymptotic analysis of this region could be

undertaken, accounting for the infinitesimal length scale. Having said this, the Navier

slip condition employed by us has proved very successful in capturing the behaviour at

the corners.

Beams of different cross-sectional geometry could be explored without too much

difficulty. The construction of the present model has been done using a general boundary

integral formulation. As such, it would be quite achievable to look at triangular or

trapezoidal cross-sections.

In order to improve the rate of convergence of the slip solutions in Chapter 6,

we could employ a different method of discretization in order to capture the strong

singularities at the edges more effectively.

Appendix A

Richardson Extrapolation

Richardson extrapolation is a method used to accelerate the rate of convergence of

a sequence. This is of particular interest to us since it enables us to obtain extremely

accurate results for the computed vorticity and pressure, whilst only having to solve the

matrix-vector equations a few times with modest mesh sizes. The process of Richardson

extrapolation is outlined by Hornbeck [13].

Suppose we have some value I which we wish to evaluate. Suppose also that the

outcome of some numerical method yields an approximation I to the desired value I.

We write

I = I − C(∆x)p −D(∆x)2p − E(∆x)3p − . . . , (A.1)

where ∆x is the mesh size used for the quadrature method and p is the exponent of the

dominant error term. The functions C, D and E are independent of ∆x. If we take

two different mesh sizes, ∆x1 and ∆x2, then the corresponding estimates of I are given

by I1 and I2 respectively. The above equation then becomes

I1 = I − C(∆x1)p −D(∆x1)2p − E(∆x1)3p − . . . , (A.2)

I2 = I − C(∆x2)p −D(∆x2)2p − E(∆x2)3p − . . . . (A.3)

Upon setting ∆x1 = 2∆x2, we find that Eq. (A.2) can be written as

I1 = I − 2pC(∆x2)p − 22pD(∆x2)2p − 23pE(∆x2)3p − . . . . (A.4)

105

106

It follows that

2pI2 − I12p − 1

= I + 2pD(∆x2)2p +2p(22p − 1)

2p − 1E(∆x2)3p + . . . . (A.5)

We now have a new estimate for I obtained using I1 and I2 in which the error

term involving (∆x)p has vanished. This estimate will thus be more accurate than

either of the two initial guesses. The above procedure can be repeated provided we

have estimates for I on additional mesh spacings. We can readily generalise to the

case with l starting guesses for I, evaluated on meshes which differ by a factor of two.

That is, ∆x1 = 2∆x2 = 4∆x3 = 8∆x4 = . . .. If each starting value is given by I1,i for

i = 1, . . . , l, then we can perform an iteration according to the following rule:

Il,k =2p(l−1)Il−1,k+1 − Il−1,k

2p(l−1) − 1. (A.6)

This will yield l − 1 new estimates for I, each with the dominant error term re-

moved. We can iterate using Eq. (A.6) until we have only one value left. This will be a

very good approximation to the actual value of I since each iteration involves removing

the dominant error term.

Appendix B

Matched Asymptotic Solutions

B.1 Beam of Zero Thickness

We now attempt a matched asymptotic expansion for the beam of zero thickness.

We will only consider the first-order correction arising due to the slip. In a similar

fashion to Chapter 4, we perform a perturbation expansion of the appropriate variables.

These can be written as

ψ = ψ(0) + ηψ(1) +O(η2), (B.1)

ω = ω(0) + ηω(1) +O(η2). (B.2)

The first-order slip condition is given by

U (0) + ηU (1) +O(η2) = Ubeam + aγη

(∂U (0)

∂z′+ η

∂U (1)

∂z′+O(η2)

),

where both sides of the above equation are evaluated at z′ = 0. Upon substitution of

Eqs. (B.1) and (B.2) into the above equation, we can equate the coefficients of η to

obtain

O(1) :∂ψ(0)

∂z′

∣∣∣∣z=0

= U (0) = Ubeam,

O(η) :∂ψ(1)

∂z′

∣∣∣∣z=0

= U (1) = aγ∂U (0)

∂z′= −aγω(0).

107

108

Recall that the streamfunction of the fluid is given by Eq. (5.7) where the term

involving the pressure vanishes by symmetry for in-plane modes. Upon differentiation

with respect to the normal coordinate z′, we obtain

∂ψ(y′, z′)∂z′

=∫ a

−a∆ω(y)Ψzz′(y, 0; y′, 0)dy.

Upon substitution of Eqs. (B.1) and (B.2) into the above integral equation, and

equating different orders of η, we obtain the following coupled integral equations:

O(1) :∫ a

−a∆ω(0)(y)Ψzz′(y, 0; y′, 0)dy =

∂ψ(0)

∂z′

∣∣∣∣z=0

= Ubeam, (B.3)

O(η) :∫ a

−a∆ω(1)(y)Ψzz′(y, 0; y′, 0)dy =

∂ψ(1)

∂z′

∣∣∣∣z=0

= −aγω(0)(y). (B.4)

Notice that the zeroth-order vorticity can be solved directly using Eq. (B.3). This

gives the vorticity associated with the no-slip condition and has already been found

in Chapter 5. The first-order correction to the vorticity can then be found recursively

using Eq. (B.4). The fluid vorticity jump across the blade to first-order in η is then

given by

∆ω = ∆ω(0) + η∆ω(1).

From a practical point of view, this procedure is not particularly difficult. The

necessary mathematical machinery has already been set up in Chapter 5. We are

required to solve the matrix system in Eq. (5.21) again but with a new right-hand side.

The kernel function inside the integrand remains the same and so the kernel matrix

remains unchanged. We must solve the equation

MΛ(1) = −γΛ(0), (B.5)

where M is precisely the same matrix as in Eq. (5.21). Several problems arise when

we implement the solution method just outlined, both in qualitative and quantitative

manners.

Let us consider what happens to the calculated vorticity jump upon inclusion of

the first-order correction term. Fig. B.1 contains plots showing Λ(0)(ξ) and Λ(0)(ξ) +

ηΛ(1)(ξ), the scaled vorticity jumps to zeroth and first-order respectively. Note that

Chapter B. Matched Asymptotic Solutions 109

the scaling has been performed in the same way as in Chapter 5. There are two main

issues with these plots. Note that the first-order correction actually serves to increase

the vorticity over the bulk of the blade face. This is problematic since it would suggest

that viscous losses actually increase upon inclusion of slip! Furthermore, the vorticity

exhibits a sharp jump at the edges from ±∞ to ∓∞. Suppose we were to put these

concerns aside briefly and proceed to integrate the dotted curves across the face of the

blade. The plots in Fig. B.1 represent the solution corresponding to N = 50 subdi-

visions. As we increase N , we actually find that convergence is not achieved. The

singularities near the edges of the blade are so strong that as the solutions approach

ξ = ±1, the area enclosed beneath the curves does not converge. This suggests that

there is some fundamental problem with the formulation of the solution since we have

obtained a singularity which is not integrable.

Let us consider the nature of the singularity in the first-order vorticity. Recall

that solutions arising from inversion of the matrix in Eq. (5.21) exhibit a square-root

singularity at the edges of the blade. If we are to re-use the matrix equation with Λ(0)

on the right-hand side, then the solution obtained will have have a stronger singularity

at the edges since the square-root singularity will have been counted twice. Indeed, a

singularity will be observed which is not integrable. Thus, there exists some fundamen-

tal flaw in the entire procedure of performing a matched asymptotic expansion in this

way.

110

First order

Zeroth order

-1.0 -0.5 0.5 1.0Ξ

-20

-10

10

20

30

40ReHLL

(a)

First order

Zeroth order

-1.0 -0.5 0.5 1.0Ξ

-40

-30

-20

-10

10

20ImHLL

(b)

Figure B.1: Real and imaginary components of the vorticity jump across the infinitely thinblade. The bold curves represent the zeroth-order solution obtained in Chapter 5 while thedashed curves show matched asymptotic expansions correct to first-order. These results areplotted for dimensionless frequency β = 0.3527, Kn = 0.17 and have been solved with N = 50subdivisions.

Chapter B. Matched Asymptotic Solutions 111

So far we have matched two solutions corresponding to inner and outer components

of the flow, for small Knudsen number Kn. This worked perfectly for the infinite plate

solution obtained in Chapter 4, but presents difficulties for the beam of zero thickness.

In performing the matched asymptotic expansion, we have treated the fluid as con-

sisting of two distinct flow regimes; close to the blade and far away from the blade.

Consider however, the region at the leading edge of the blade. In this region, the effec-

tive length scale is δ ≈ 0, depicted in Fig. B.2, while the thickness of the kinetic layer is

still O(η). Thus, the Knudsen number at the leading edge is Kn →∞ which represents

free molecular flow. Obviously an expansion in small Kn will not be valid here. We

certainly would not expect the vorticity to exhibit a non-integrable singularity in such

circumstances. To handle the leading edge of the blade appropriately, we would need

to perform a matched asymptotic expansion for the three distinct flow regions depicted

in Fig. B.2.

η δ

Continuum

Knudsen layer Free

molecular

Figure B.2: Figure showing the three distinct flow regions

We do not proceed with this analysis since it is incredibly complex. Instead, we

will stand by our solution obtained in Chapter 5 where the Navier slip condition was

used. As previously mentioned, this slip condition automatically accounts for both the

Kn → 0 and Kn →∞ limits, enabling its use across the entire face of the blade.

112

B.2 Beam of Non-Zero Thickness

The process of searching for a matched asymptotic expansion across two distinct

flow regimes proved to be successful for the infinite plate model but fruitless for the

beam of zero thickness. We may also wonder whether this approach is likely to work for

the beam of non-zero thickness. Suppose we expand the relevant variables presented in

Chapter 6 in terms of small η (or equivalently Kn) to obtain

ψ = ψ(0) + ηψ(1) +O(η2), (B.6)

ω = ω(0) + ηω(1) +O(η2), (B.7)

p = p(0) + ηp(1) +O(η2). (B.8)

Combining the above equations with Eq. (6.30) and equating the different orders

of η, we obtain the following:

O(1) : ψ(0) = −Ubeamz′ +

∫Cbeam

[− ω(0)Ψn +

1µp(0)Ψl

]dl, (B.9)

O(η) : ψ(1) =∫Cbeam

[− aγω(0)Ω− ω(1)Ψn +

1µp(1)Ψl

]dl. (B.10)

In the latter of the above equations, we have made use of the slip condition so

that ψ(1)n = aγω(0). Equation (B.9) corresponds to the no-slip boundary condition. We

have already solved this problem in Chapter 6 to obtain the vorticity ω(0) and pressure

p(0), correct to O(1). With ω(0), we can use Eq. (B.10) subject to the appropriate slip

boundary condition to recursively determine the first-order corrections to the vorticity

and pressure. This appears to be a feasible way of incorporating slip into the model

of the beam of non-zero thickness. However, the difficulties that became evident in

Section B.1 also arise here. Although we do not explicitly present the results, we find

that through using the zeroth-order results to compute the first-order correction, the

singularities in the vorticity and pressure become stronger. As in Section B.1, we find

that the computed results are in fact non-integrable. While this solution method may

appear appealing at first, we find that it is fruitless. Its failure can be attributed to the

fact that we have ignored the fluid near the corners of the beam as a third and distinct

flow regime.

Bibliography

[1] Y.T. Yang, C. Callegari, X.L. Feng, K.L. Ekinci, and M. L. Roukes. Zeptogram-

scale nanomechanical mass sensing. Nano Letters, 6(4), 2006.

[2] I. Bargatin, I. Kozinsky, and M. L. Roukes. Efficient electrothermal actuation

of multiple modes of high-frequency nanoelectromechanical resonators. Applied

Physics Letters, 90(9), Feb 26 2007.

[3] J.E. Sader. Frequency response of cantilever beams immersed in viscous fluids

with applications to the atomic force microscope. Journal of Applied Physics,

84(1):64–76, Jul 1 1998.

[4] L.F. Crabtree, G.E. Gadd, N. Gregory, C.R. Illingworth, C.W. Jones, D. Kuche-

mann, M.J. Lighthill, R.C. Pankhurst, L. Rosenhead, L. Sowerby, J.T. Stuart, E.J.

Watson, and G.B. Whitham. Laminar Boundary Layers. Oxford University Press,

1963.

[5] E.O. Tuck. Calculation of unsteady flows due to small motions of cylinders in a

viscous fluid. Journal of Engineering Mathematics, 3(1):29–44, 1969.

[6] C.P. Green and J.E. Sader. Small amplitude oscillations of a thin beam immersed

in a viscous fluid near a solid surface. Physics of Fluids, 17(7), Jul 2005.

[7] C.P. Green and J.E. Sader. Torsional frequency response of cantilever beams im-

mersed in viscous fluids with applications to the aomic force microscope. Journal

of Applied Physics, 92(10), Nov 2002.

[8] J.E. Sader, J.W.M. Chon, and P. Mulvaney. Calibration of rectangular atomic

force microscope cantilevers. Review of Scientific Instruments, 70(10):3967–3969,

Oct 1999.

113

114

[9] N.G. Hadjiconstantinou. Oscillatory shear-driven gas flows in the transition and

free-molecular-flow regimes. Physics of Fluids, 17(10), Oct 2005.

[10] G.A. Bird. Molecular Gas Dynamics and the Direct Simulation of Gas Flows.

Oxford University Press, Melbourne, 2003.

[11] C.R. Lilley and J.E. Sader. Velocity profile in the knudsen layer according to the

boltzmann equation. Proceedings of The Royal Society A, 464(2096), 2008.

[12] E.J. Hinch. Perturbation Methods. Cambridge University Press, 1995.

[13] R.W. Hornbeck. Numerical Methods. Quantum Publishers, 1975.

[14] S. Wolfram. Mathematica: a system for doing mathematics by computer. Addison-

Wesley Publishing Co., 1991.

[15] M. Abramowitz and I.A. Stegun. Handbook of mathematical functions. Dover

Publications, New York, 1965.

[16] D. Halliday, R. Resnick, and J. Walker. Fundamentals of physics, extened, with

modern physics. John Wiley and Sons. Inc., New York, 4th edition, 1993.

[17] J.H. Park, P. Bahukudumbi, and A. Beskok. Rarefaction effects on shear driven

oscillatory gas flows: A direct simulation monte carlo study in the entire knudsen

regime. Physics of Fluids, 16(2):317–330, Feb 2004.

[18] S Basak, A Raman, and S.V. Garimella. Hydrodynamic loading of microcantilevers

vibrating in viscous fluids. Journal of Applied Physics, 99(11), Jun 1 2006.

[19] R.D. Knight. Physics for Scientists and Engineers: A Strategic Approach. Pearson

Addison-Wesley, 2008.

[20] H.L. Evans. Laminar Boundary-Layer Theory. Addison-Wesley Publishing Co.,

London, 1968.

[21] C. Pozrikidis. Boundary integral and singularity methods for linearized viscous

flow. Cambridge University Press, Melbourne, 1992.

[22] H. Power and L.C. Wrobel. Boundary Integral Methods in Fluid Mechanics. Com-

putational Mechanics Publications, Southampton, 1995.

BIBLIOGRAPHY 115

[23] L.D. Landau and E.M. Lifshitz. Fluid Mechanics. Pergamon Press Ltd., London,

1959.

[24] R.L. Daugherty, J.B. Franzini, and E.J. Finnemore. Fluid Mechanics with Engi-

neering Applications. John Wiley and Sons, Inc., 2003.

Documents

The Dynamics of High-Frequency Nanoelectromechanical