Microfiber motion and web formation in a microchannel heat sink: A numerical approach

Computers & Fluids 71 (2013) 28–40

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Computers & Fluids

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Microfiber motion and web formation in a microchannel heat sink:A numerical approach

Alireza Dastan, Omid Abouali ⇑School of Mechanical Engineering, Shiraz University, Shiraz, Iran

a r t i c l e i n f o

Article history:Received 26 August 2011Received in revised form 25 June 2012Accepted 20 September 2012Available online 28 September 2012

Keywords:MicrochannelEllipsoidal particleFiber webNumerical simulation

0045-7930/$ - see front matter � 2012 Elsevier Ltd. Ahttp://dx.doi.org/10.1016/j.compfluid.2012.09.018

⇑ Corresponding author. Address: School of MecUniversity, Mollasadra Street, Shiraz, Iran. Tel.: +98628 7508.

E-mail addresses: [email protected] (A. Da(O. Abouali).

a b s t r a c t

A general computer code which solves the motion equations of non-spherical ellipsoidal particles in thefluid flow was developed and fiber motion and web formation at the channel entrance of a microchannelheat sink were investigated numerically. A circular inlet duct, inlet plenum and 15 parallel channels withhydraulic diameter of 225 lm were considered as the computational domain. Water flow field in themicrochannel was solved with Eulerian approach. Fiber motion equations consisting of translationaland rotational motions were solved by Lagrangian approach assuming a one-way interaction. The numer-ical approach was validated for different aspects of the model and close agreement was obtained in com-parison with experimental and other numerical data. The objective of the present work is to simulate theformation of fiber web at the entrance of the channels, as well as studying the effects of deposited fiberson the flow field and deposition of next fibers. The results show that, the deposited fibers act as a filterthat can lead to deposition of more fibers at the channel entrance. The growth of fiber web in timereported in previous experimental works was also observed in this numerical investigation.

� 2012 Elsevier Ltd. All rights reserved.

1. Introduction

The microchannel heat sink cooling concept was first intro-duced by Tuckerman and Pease [1]. The microchannel heat sinkis usually made up of high thermal conductivity solids such as sil-icon or copper with channels having dimensions ranging from 10to 1000 lm, into its surface. Small mass and volume, large convec-tive heat transfer coefficient, very high surface area to volume ratioand small required coolant volume are some specific characteris-tics of microchannels which make them attractive in both researchand practical areas.

There are other phenomena, however, that adversely affect thefunctionality of microchannels as cooling devices. The term ‘‘foul-ing’’ was originally used in the oil industry and was widely used inliterature to describe any undesirable deposition causing an in-crease in the thermal resistance of heat exchanger [2]. Yiantsiosand Karabelas [3] and Niida et al. [4] studied the particulate foulingin geometries having the hydraulic diameter of 952 lm and727 lm, respectively.

Perry and Kandlikar [5] investigated the fouling in a siliconmicrochannel with hydraulic diameter of 225 lm. The fouling of4 lm silica and 1.25 lm alumina spherical particles dispersed in

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water was investigated experimentally, in their study. They ob-served that no noticeable particle deposition occurs throughoutthe channels because of high flow shear stress at the channel wallscompared to conventional channels, but there is a secondary effectin the particulate fouling when fibrous elements exist. These fibersfound to be from citric acid buffer solution used for preparation ofthe micro-particle suspension. As well, it was observed that nano-particles may scrape the walls of the setup and would produce theundesired fibers [6]. Perry and Kandlikar [5] reported that the fi-brous particles with the length of several hundred micrometersand diameter of about 20 lm are caught at the channel entrance ofthe system and form a ‘‘fiber web’’ which results in trapping of morefibers. The fiber web causes an increase in pressure drop and particledeposition because it acts as a fiber filter. Dastan and Abouali [7]studied numerically the effects of pre-assumed deposited fiberweb at the channel entrance of a microchannel on the pressure drop,spherical particle collection and heat transfer. It was shown that thefiber web has a considerable effect on the pressure drop in the sys-tem and it can act as a fiber filter to collect particles, as well. How-ever, the formation of fiber webs was not discussed in [7] and onlythe effect of pre-defined deposited webs on the microchannel char-acteristics were investigated. Both last mentioned studies depictthat more investigations about the fibrous element motion anddeposition in a microchannel system are required.

One of the major concerns in the microchannels is the high flowpressure drop in the channels. Pressure drop in a microchannel isdue to the viscous friction across the channels, both plenums andalso the pressure losses correspond to the contraction and expansion

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Nomenclature

A rotation matrixa ellipsoid semi minor axis (m)b ellipsoid semi major axis (m)Ct translational Cunningham slip correction factord fluid deformation tensor (1/s)F, f force (N)G fluid velocity gradient tensor (1/s)g gravity acceleration vector (m/s2)I moments of inertia (kg m2)k resistance matrixm mass (kg)p pressure (Pa)T Torque (N m)t time (s)u fluid velocity vector (m/s)v particle mass center velocity vector (m/s)V volume (m3)w fluid spin tensor (1/s)x position vector (m)

Greek symbolsb ellipsoid aspect ratiol dynamic viscosity (kg/m s)

q density (kg/m3)m kinematic viscosity (m2/s)u, h, w Euler angles (rad)ei, g Euler parameterss relaxation time (s)x angular velocity vector (rad/s)

SuperscriptsB Brownianf fluidg buoyancyh hydrodynamicL liftp particlevm virtual mass

Subscriptx with respect to particle coordinate system

A. Dastan, O. Abouali / Computers & Fluids 71 (2013) 28–40 29

at the channel inlet and outlet [8]. For the case with a caught fiberweb at the channel entrance, the pressure losses due to interactionof the flow with the fiber web are added to the mentioned losses.

The concept of fluid with suspended fibrous particles has manyapplications in the industry, e.g. in the composite material produc-tion, environment and chemical engineering and paper productionindustry. The deposition of the inhaled fibrous particles in therespiratory system is significant and also hazardous for health.Therefore, investigation of high aspect ratio particle motion has re-ceived the attention of many researchers. In the case of elongatedparticles, the orientation and rotation of the particles may not beneglected and these lead to more mathematical complexities inthe numerical solution. Fan and Ahmadi [9] studied numericallythe effects of shape, aspect ratio and density of ellipsoidal particleson their dispersion in an isotropic pseudo-turbulent flow field.They also conducted two similar investigations to study the depo-sition of ellipsoidal particles on a wall in turbulent flow [10,11].The Brownian force and Cunningham slip correction factor wereconsidered for the submicron fibers in the latter [11]. Asgharianand Ahmadi [12] studied the motion and deposition of fibers in amodel of small human airways with the diameter of about300 lm. Lin et al. [13] investigated the effects of aspect ratio onthe deposition of cylindrical fibers by the Latice–Boltzman method.Chen and Yu [14] and Högberg et al. [15] studied the motion ofellipsoidal fibers in a laminar flow of a circular duct.

In the most of previous numerical researches, it was assumedthat the elongated particles attach to the wall as soon as they touchthem. In contrast with the spherical particles, performing thisboundary condition in the case of non-spherical particles has muchmore mathematical complexities. Fan and Ahmadi [10] offered anidea to find the touch point of an ellipsoid to a wall which is par-allel to one of the coordinate planes. According to the authors’knowledge, the only research in which rebound of ellipsoidal par-ticles hitting a wall was investigated, is the one performed recentlyby Wynn [16]. Wynn has developed a model to simulate the re-bound of ellipsoidal particles impacting a wall based on the perfectelastic body theory.

In the present paper, the motion of ellipsoidal fibers in a micro-channel is investigated numerically. A procedure for the fiber web

formation is developed and the web formation at the channel en-trance of a microchannel heat sink is simulated.

2. Model description

A microchannel heat sink having 15 parallel channels, shown inFig. 1, is considered as the physical domain of the present work. Eachchannel is 205 lm wide and 251 lm high with the hydraulic diam-eter of 225 lm. The separating fin between the channels is 97 lmwide. These dimensions were chosen to be consistent with thoseof Perry and Kandlikar [5]. All the channels are connected to a ple-num which divides the inlet flow into the channels. The fluid flowenters the plenum through a vertical circular duct located at thetop of the plenum. The other dimensions are specified in Fig. 1. Sincethe goal of this study is the investigation of fiber motion in the inletplenum and the fiber web formation at the channel entrance, theoutlet plenum and some parts of channel length were not consid-ered. The length of modeled channels is 2700 lm to be sure thatflow at the end of the channels is fully developed (according to stud-ied flow rate). The origin of the inertial coordinate system is locatedacross the circular duct center attached to the bottom wall of theplenum. The coordinate system is shown in Fig. 1.

To investigate the fiber web formation at the channel entrance,the governing equations of flow field in the described physical do-main is solved by Eulerian method. After that, based upon the flowfield data, the governing equations of ellipsoidal fiber motion aresolved with Lagrangian approach to find the deposition locationof the fibers and the details of the deposited fiber web. In the fol-lowing sections, the governing equations, numerical solution ap-proach and the required assumptions are discussed.

3. Numerical method

3.1. Flow field governing equations

Some simplifications were assumed before solving the govern-ing equations. Steady incompressible laminar flow with constantfluid properties was considered. The only body force is gravitational

Fig. 1. Different views of the investigated microchannel heat sink. (a) Side view, (b) front view, (c) top view, and (d) schematic isometric view. The inertial coordinate system,different parts of the domain and the channel numbers are shown in the figure. The dimensions are in micrometers.

30 A. Dastan, O. Abouali / Computers & Fluids 71 (2013) 28–40

force acting in the direction of the channel depth (�z direction).Based upon these assumptions the fluid governing equations arecontinuity and momentum as the following:

r � u ¼ 0 ð1Þ

u � ru ¼ � 1qrpþ mr2uþ F ð2Þ

A fixed mass flow rate boundary condition was used at the topplane of the inlet circular duct, while the constant gauge pressureboundary condition was used at the end of the channels. No-slipboundary condition was considered for all walls.

In order to generate the geometry and the computational grid, atechnique extruding a meshed plane to produce a 3-D geometrywas used. Firstly, to make a guess about the needed grid size inthe channel parts of the domain, a single channel was modeledseparately and used to achieve a grid-independent solution ofthe flow field. It was observed that a grid size of 60 � 14 � 26 cellsfor a single channel with dimensions of 2700 � 205 � 251 lm, re-sults in a grid-independent solution. After that, the x–y projectedplane of the domain was divided into different small parts. Asquare was considered around the projected circle of circular ductto connect the grids of the duct to other parts of the domain with-out any interface (see Fig. 2a). The projected plane was meshed bytetrahedron or triangular elements. The element size in the chan-nel parts is consistent with the results of single channel grid study.The other parts were meshed in such a manner that the size ratio oftwo neighbor cells never exceeded 1.1, and the aspect ratio of a cellnever became more than 5. Eventually, the meshed projected planewas uniformly extruded to produce the 3-D geometry. The wholecomputational domain cell number was found to be about1,100,000 prism or tetrahedral elements. In order to carry outthe grid study, a solution-adaptive grid refinement was performed

using the curvature approach [17]. Velocity gradient values werecalculated for the whole domain and 10% of the maximum gradientvalue was selected as a refinement threshold and the cells withgradient value more than the threshold were refined (Fig. 2b andc). The whole domain cell number in the case of refined grid wouldbe 1,300,000 cells. Using this grid, no noticeable change in numer-ical results (less than 1%) was observed, so the initial computa-tional grid was selected for the numerical investigation. For thegrid generation of the model with deposited fibers (see Section 4.3),the approach of multi-zone grid similar to the work of Dastan andAbouali [7] was used and the region around the fiber web was sep-arated from the other parts of domain by two interface planes atx = 1150 and 1750 lm. By this approach, it was possible to increasethe grid resolution around the web to capture the fluid-web inter-action and decrease the overall computational domain size. Due tothe symmetry, just half of the domain was modeled and the wholedomain cell number in this case was found to be about 1,700,000elements.

The governing Eqs. (1) and (2) and corresponding boundaryconditions were solved by an Eulerian approach using OpenFOAMwhich is an open source CFD software package.

The solution is based on finite volume method by employingSIMPLE algorithm. The convective terms are discretized by asecond order upwind scheme and the viscous terms by a centraldifferencing. The continuity scaled residual value of 10�6 isassumed to be the convergence criterion.

3.2. Fiber equations of motion

In comparison to spherical particles, non-spherical particleshave more complex equations of motion because their rotationalmotion cannot be neglected. Three different coordinate systemsused to describe the motion of ellipsoidal particles are shown in

Fig. 2. (a) 2-D grid in a x–y projected plane of the domain used to be extruded and produce the geometry. (b) A selected view of refined cells with high gradient velocity valuein a horizontal plane at x = 175.5 lm. (c) A selected view of refined cells with high gradient velocity value in a vertical plane at y = 151 lm perpendicular to the center of a fin.(d) A selected view of the grid generated in the web region in a plane at x = 1440 lm.

Fig. 3. Three different coordinate systems used in description of fiber motion.

Fig. 4. Transformation between two Cartesian coordinate systems using threesuccessive rotations. Euler angles (u, h, w) are the angle of the rotations. The initialand final coordinate system are (x, y, z) and ðx; y; zÞ, respectively.


Fig. 3. The inertial coordinate system is x = [x, y, z], while x ¼ ½x; y; z�is the particle coordinate system which its origin is on the masscentre of the particle and the z axis is along the major axis of theparticle. The third coordinate system, ^x ¼ ½^x; ^y; ^z�, is co-movingframe coordinate system while its origin is located on the originof the particle coordinate and its axes are parallel to the axes ofinertial coordinate system. The transformation from a given Carte-sian coordinate system to another can be performed by means ofthree successive rotations. The Euler angles (u, h, w) are definedas three successive angles of these rotations. The Euler angles inthis study are as the x-convention of Goldstein [18]. Fig. 4 showsthe definition of the Euler angles.

According to the transformation laws between two Cartesiancoordinate systems, any vector stated in the co-moving coordinatesystem can be stated in the particle coordinate system as thefollowing:

x ¼ A^x ð3Þ


where A is the transformation matrix and may be expressed in theterms of Euler angles [18], i.e.

A ¼cwcu� chsusw cwsuþ chcusw swsh

�swcu� chsucw �swsuþ chcucw cwsh

shsu �shcu ch

264375 ð4Þ

where for instance, s/ ¼ sin ð/Þ and c/ ¼ cos ð/Þ.Due to the singularity happens in the calculation of the time

rate of changes of Euler angles [10], the transformation matrix ofA in terms of four Euler parameters (e1, e2, e3, g) is used in the sim-ulation of ellipsoid motion, i.e.

A ¼1� 2ðe2

2 þ e23Þ 2ðe1e2 þ e3gÞ 2ðe1e3 � e2gÞ

2ðe2e1 � e3gÞ 1� 2ðe23 þ e2

1Þ 2ðe2e3 þ e1gÞ2ðe3e1 þ e2gÞ 2ðe3e2 � e1gÞ 1� 2ðe2

1 þ e22Þ

264375 ð5Þ

The relation between Euler parameters and Euler angles is asfollows [18]:

e1 ¼ cos u�w2 sin h

2

e2 ¼ sin u�w2 sin h

2

e3 ¼ sin uþw2 cos h

2

g ¼ cos uþw2 cos h

2

ð6Þ

The above equations relate a set of four parameters to one ofthree parameters. Therefore, four Euler parameters are subjectedto a constraint given by:

e21 þ e2

2 þ e23 þ g2 ¼ 1 ð7Þ

The translational and rotational equations of motion of a non-spherical particle in a general flow field are as follows:

mp dvdt¼ fg þ fh þ fB þ fvm þ fL ð8Þ

Ixdxx

dt�xyxzðIy � IzÞ ¼ Th

x ð9Þ

Iydxy

dt�xzxxðIz � IxÞ ¼ Th

y ð10Þ

Izdxz

dt�xxxyðIx � IyÞ ¼ Th

z ð11Þ

where the translational motion Eq. (8) is expressed in the inertialcoordinate system and rotational motion Eqs. (9)–(11) are ex-pressed in the particle coordinate system.

The time rate of changes of Euler parameters are given by:

de1dt

de2dt

de3dt

dgdt

2666664

3777775 ¼12

gxx � e3xy þ e2xz

e3xx þ gxy � e1xz

�e2xx þ e1xy þ gxz

�e1xx � e2xy � e3xz

2666437775 ð12Þ

And the particle position can be easily obtained from followingequation:

dxdt¼ v ð13Þ

After solving Eqs. (8)–(11) and obtaining values of translational androtational velocities of the fiber, Eqs. (12) and (13) will be solved todetermine the orientation and position of the particle.

As mentioned, Eqs. (8)–(11) are the general form of governingequations of non-spherical particle motion. In this study the non-spherical particles have been assumed to be in the shape of ellip-soid of revolution and therefore the forces and torque in Eqs.(8)–(11) are those for ellipsoids.

The volume of an ellipsoid is given by:

V ¼ 43pa3b ð14Þ

where a is the semi minor axis of the ellipsoid and b is the ellipsoidaspect ratio (the ratio of semi major axis to semi minor axis,b ¼ b

a > 1).And the principal moments of inertia are given by:

Ix ¼ Iy ¼ 15 mpa2ð1þ b2Þ

Iz ¼ 25 mpa2

ð15Þ

One of the force acting on a suspended particle in a flow is thehydrodynamic drag force, fh. This force for an ellipsoidal particlewas derived by Brenner [19] as follows:

fh ¼ pla^k � ðu� vÞ ð16Þ

where u is the fluid velocity vector at the centre of the fiber andresistance tensor ^k is given by:

^k ¼ A�1 ^kA ð17Þ

A is the transformation matrix given by Eq. (5) and k is a diagonalmatrix which its dimensionless diagonal elements given by Fan andAhmadi [9] are as the following:

kxx ¼ kyy ¼16ðb2 � 1Þ

Ctx

2b2�3ffiffiffiffiffiffiffiffib2�1p lnðbþ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffib2 � 1

qÞ þ b

� � ð18Þ

kzz ¼8ðb2 � 1Þ

Ctz

2b2�1ffiffiffiffiffiffiffiffib2�1p lnðbþ


qÞ � b

� � ð19Þ

where b is the fiber aspect ratio and Ctx and Ct

z are the translationalCunningham slip correction factors [11] which are equal to unity inthis study.

Another force acting on a suspended particle is the buoyancyforce, fg:

fg ¼ Vðqp � qf Þg ð20Þ

where V is the volume of the fiber which is given by Eq. (14).Other forces on the particles in the fluid are the Brownian force,

the virtual mass (added mass) force and lift force. Details of Brown-ian force on ellipsoidal fibers were discussed by Fan and Ahmadi[11]. The results of current study depict that Brownian force on mi-cro-size ellipsoidal fibers in water flow is negligible in comparisonwith drag force. In addition, the effects of virtual mass force [20] onthe fiber motion path in the present investigation is negligible.Therefore both Brownian and added mass forces are excluded fromEq. (8). For the lift force, Harper and Chang [21] offered a correla-tion for lift force due to a simple one-dimensional shear field onany arbitrary shape particle, provided the drag tensor is available.Shear Reynolds number in this study does not satisfy the conditionof using that lift model and the correlation also leads to extra forcecomponent in stream-wise direction which does not have anyphysical reason. Because of these reasons, the lift force was ex-cluded from Eq. (8).

The hydrodynamic torque acting on an ellipsoidal particle in alinear shear flow was derived by Jeffery [22]. The hydrodynamictorque terms in Eqs. (9)–(11) are given by:

Thx ¼

16pla3b

3ðbo þ b2coÞð1� b2Þdzy þ ð1þ b2Þðwzy �xxÞ� �

ð21Þ

Thy ¼

16pla3b

3ðb2co þ aoÞðb2 � 1Þdxz þ ð1þ b2Þðwxz �xyÞ� �

ð22Þ

Thz ¼

32pla3b3ðao þ boÞ

ðwyx �xzÞ ð23Þ

Fig. 5. Ellipsoidal fiber relaxation time with the density of 2400 kg/m3 in water atroom temperature.


where the non-dimensional parameters ao, bo and co were pre-sented by Gallily and Cohen [23], as the following:

ao ¼ bo ¼b2

b2 � 1þ b

2ðb2 � 1Þ3=2 lnb�


qbþ


q264

375 ð24Þ

co ¼ �2

b2 � 1� b

ðb2 � 1Þ3=2 lnb�


qbþ


q264

375 ð25Þ

Additionally, dij and wij are the elements of the fluid deforma-tion rate and spin tensors, respectively:

dij ¼12

@ui

@xj

þ@uj

@xi

!ð26Þ

wij ¼12

@ui

@xj

�@uj

@xi

!ð27Þ

It should be noted that the velocity gradient in the above equa-tions is in the particle coordinate system and it could be obtainedby the following transformation:

bG ¼ AbbGA�1 ð28Þ

where bG and bbG are the fluid velocity gradient tensor with respect toparticle and co-moving frame coordinate systems, respectively andA is the transformation matrix.

As mentioned, the goal of this study is to track the ellipsoidalfibers in the inlet plenum of a microchannel and to determinethe details of the caught fiber web at the entrance of the channels.A general computer code which solves the governing equationswas developed for this aim. After obtaining the flow field, the re-quired data including the fluid velocity vector and fluid velocitygradient tensor elements at each grid node are stored to be usedin the main computer code in which the fiber motion equationsare solved.

The algorithm of fiber tracking in the present investigation issimilar to one performed by Fan and Ahmadi [10]. The fluid veloc-ity vector and fluid velocity gradient tensor are estimated at themass center of the fiber with an interpolation. The solution proce-dure is terminated if the fiber leaves the domain or deposits on thechannel entrance walls.

In this study, a search algorithm benefited from two concepts isdeveloped to find the computational cell in which the particle is lo-cated. In pre-processing part, the concept of auxiliary structuredmesh, ASM, is used to find the neighbor cells and the initial cellin which the particle is released. Two types of neighbors, the firstdegree and the second degree neighbors were defined for signifi-cant reduction of computational effort and time in search algo-rithm. The first degree neighbors have a common face with thecurrent cell, while the second degree ones have a common nodewith the cell. Investigations showed that the neighbor cell searchinstead of a local search, leads to a considerable decrease in com-putational time because the particle is more likely to enter neigh-bor cells than the cells located further.

When the computational cell containing the fiber is determined,the flow parameters should be interpolated at the location of fibermass center based on the data at the nodes of the current cell. Thistask is done by means of an interpolation function. Since differentkinds of computational cells with different node numbers andarrangements may be used in a grid, an interpolation functionbeing able to interpolate among scattered data is required. Oneof these functions is the radial basis function (RBF) interpolation.Press et al. [24] explained different kinds of these functions in de-tails. One of the simplest RBF functions is the Shepard approach asthe following:

uðxÞ ¼XN

i¼1

wiðxÞPNj¼1wjðxÞ

� ui ð29Þ

where

wi ¼1

x� xij jpð30Þ

And jx� xij is the distance between the target point and the ithknown data point (current cell nodes), ui is the value of interpolat-ing parameter (velocity vector or velocity gradient tensor compo-nents) at ith known data point, N is the number of known datapoints (node number of current cell) and p is any arbitrary numberwhich 0 < p 6 3. In this study the p number was selected to bep = 2.

3.3. Numerical solution of the fiber equations of motion

The fiber translational and angular velocities Eqs. (8)–(11) arediscretized by the procedure suggested by Fan and Ahmadi [10].The time derivations in the left hand side are discretized by for-ward differencing. In each equation, the particle translational orangular velocities on the right hand side are evaluated at the nexttime step and the other terms are evaluated at the current timestep. The equations can be restated in the following form:

dXdt¼ AXXþ BX ð31Þ

where X is a translational or angular velocity component of the fi-ber. The discretization procedure of the equation is as follows:

Xnþ1 �Xn

Dt¼ An

XXnþ1 þ BnX ð32Þ

where Dt is the integration time step and the superscripts n andn + 1 depict that the parameter is evaluated in the current or nexttime step, respectively. However, Eqs. (12) and (13) are discretizedby the simple explicit Euler scheme.

Because of non-iterative nature of the solution, the computa-tional error grows in time and causes the Euler parameters notto satisfy Eq. (7) precisely. To prevent the increase of computa-tional error, at the end of each time step, the calculated Eulerparameters are normalized to be sure that Eq. (7) is satisfied.

One criterion for the selection of the numerical integration timestep is the particle relaxation time. The ellipsoidal particle relaxa-tion time is given by:

s ¼ 4a2bðqp � qf Þlðkxx þ kyy þ kzzÞ

ð33Þ


where kii is the element of the resistance tensor given by Eqs. (18)and (19). The offered relaxation time is similar to one presentedby Fan and Ahmadi [9] with the extra term of buoyancy force.Fig. 5 shows the relaxation time of ellipsoidal particles in water atroom temperature. The particle density was assumed to be2400 kg/m3.

3.4. Ellipsoidal fiber boundary conditions

According to experimental observation of Perry and Kandlikar[5], the deposition of fibers on the plenum bottom wall is negligi-ble and the fibers form a web at the inlet of the channels. To sim-ulate this behavior, the reflect boundary condition for the fiberswas considered at all walls of the domain except the channel en-trance walls. A special boundary condition was developed for thefibers which hit the channel entrance walls that will be explainedlater. In both mentioned particle boundary condition (as well as intrap one), it is of great significance to find the precise location oftouch point of the ellipsoid surface to the wall.

The spherical particles are assumed to touch the wall when thedistance from center of particle to wall is smaller than the particleradius. But for ellipsoidal particles the situation is more compli-cated. Based on the idea introduced by Fan and Ahmadi [10], a gen-eral approach was developed to find the touch point of theellipsoid to an arbitrary given wall. The touch point of an ellipsoidto a wall is located on the symmetric plane of the fiber which is anellipse perpendicular to the wall. Two coordinate systems nf and nfare considered at the centre of the ellipse, as shown in Fig. 6. The faxis of nf system is normal to the wall and n axis of nf system isaligned in the direction of particle major axis and a is the angle be-tween two mentioned coordinate system.

The potential touch point of the ellipse to the wall ðn�; f�Þ withrespect to nf coordinate system can be found as the following [10]:

n� ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðb2 � a2Þ cos2 a

a2 þ sin2 ab2

h isin2 a cos2 a

1þ a2b2 1a2 � 1

b2

� �sin2 a cos2 a

vuuut ð34Þ

f� ¼n� sin a cos a 1

a2 � 1b2

� ��

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffifficos2 a

a2 þ sin2 ab2 � n�

2

a2b2

qcos2 a

a2 þ sin2 ab2

ð35Þ

Point ðn�; f�Þ is the location of potential touch point in nf coordinatesystem and it should be transformed to the co-moving coordinatesystem by using a rotation. Subsequently, the touch point coordi-nates in inertial coordinate system can be achieved easily by know-ing the position of the ellipsoid mass center.

Impaction of two bodies is a complicated phenomenon whichdepends on the shape, material, impact velocity, condition of sur-

Fig. 6. An ellipse in contact with a wall and two different coordinate systems to findthe touch point.

faces, orientation, etc. According to authors’ knowledge, the onlyinvestigation about the impaction of non-spherical particles tothe wall has been performed by Wynn [16]. Wynn presented amathematical approach to model the impaction of ellipsoidal par-ticles to walls based on the elastic theory. To simulate the impac-tion of an ellipsoid to a wall it is assumed that the effects of flowfield on the particle motion during the impaction are negligible.In addition, it is assumed that the forces acting on the fiber arein the symmetric plane of the ellipsoid perpendicular to the wall.Therefore for the impaction simulation, the motion of an ellipseis investigated instead of the ellipsoid. The first-order scheme ofWynn model is performed in this study and it will be referred asthe ‘‘reflect boundary condition’’. The restating of the governingequations for the Wynn model is not presented here for the sakeof brevity.

An important issue in the impact simulation is to find the im-pact location of the particle on the wall very precisely. To preventthe divergence of numerical solution of impact simulation, a linearinterpolation approach is used. In the numerical solution, it is as-sumed that the particle touches the wall when it passes throughthe wall (in the present work this length is approximately10 nm). By interpolation all required data between this time stepand the previous time step, the precise impact location can beachieved.

As mentioned, the reflect boundary condition is performed onthe all walls of the domain except the channel entrance verticalwalls (x = +1450 lm). The behavior of the fibers when they formthe web is so complicated and some assumptions and simplifica-tions are needed to model the fiber web formation at the channelinlet. It is assumed that when the particle hits the channel entrancewalls, it attaches to the wall on the contact point, but the solutionprocedure does not terminate and the fiber can rotate around thecontact point with three degrees of rotational freedom. The fiberkeeps its rotation with a constant angular velocity till it hits to an-other point at the area around the channel inlet. In this case, thefiber which has two touching points on the channel entrance wallsor other fibers, is considered as a deposited fiber and the solutionprocess would stop. If the fiber does not find the second supportfor deposition during the rotation process, the fiber detaches fromthe initial contact point and will keep its moving in the domain.This approach for modeling the deposition of fibers at the channelentrance walls will be referred as the ‘‘rotation boundarycondition’’.

The developed computer code should be able to handle thesymmetric plane, as well. In CFD, the symmetric planes may beused to reduce the size of computational domain and therefore re-duce the computational time. When a particle reaches a symmetricplane, it means that another similar particle enters the domainfrom the other part of domain which has not been modeled. Forspherical particles, it is sufficient to change the sign of normalvelocity component of the particle. For non-spherical particles,the angular velocity and orientation of the particle should be no-ticed in a similar manner.

4. Results and discussion

4.1. Validation

Fiber web formation in the channel entrance of a microchannelheat sink is a complicated phenomenon. In addition, there is noquantitative result from the experimental observation that canbe used for validation of the present numerical method in this re-gard. On the other hand the numerical approach which was used topredict the fiber motion and web formation should be checked.Consequently, different parts of the numerical model are validated


by some previous numerical or experimental works to deduce thewhole model is valid.

4.1.1. Flow fieldTo validate the flow field solution, the experimental work of Qu

and Mudawar [8] was considered. The microchannel heat sinkinvestigated in this experimental work consisted of 21 parallelchannels with the width of 231 lm, height of 713 lm (the hydrau-lic diameter of each channel is 349 lm) and length of 4.48 cm. Thewater with temperature of 15 �C enters the inlet plenum of the sys-tem. Constant heat flux of q00 = 100 w/cm2 is dissipated from thesubstrate to the bottom wall of the microchannel heat sink.

A single channel of the system was considered as the computa-tional domain. After the grid study, a computational grid with thesize of 102 � 22 � 43 was selected. Constant fluid properties wereassumed for the numerical investigation. The fluid properties werecomputed on the average temperature of the inlet and outlet fluidpresented in the experimental work.

Fig. 7 shows the pressure drop in the system versus the Rey-nolds number (with respect to channel hydraulic diameter) forboth experimental and the present numerical work. As the figureshows, the numerical results are in good agreement with theexperimental data.

4.1.2. Ellipsoidal fiber motion in a pipe flowValidation of the numerical model for solving the motion equa-

tions of ellipsoidal particle was done by comparing with the workof Chen and Yu [14]. Fully developed pipe flow with the averagevelocity of 0.12 m/s, fluid viscosity of l = 1 � 10�5 kg/m s, densityof q = 1 kg/m3 and the pipe radius of 1 mm were assumed. Ellipsoi-dal fiber with the semi minor axis of a = 5 lm, aspect ratio of b = 20and density of qp = 1000 kg/m3 were injected to the flow field. y-axis of inertial coordinate system is aligned with the pipe axis.The initial location of particle is xo = 0.75R and yo = 0, where R isthe pipe radius. The initial orientation is uo ¼ 45�; ho ¼ �45� andwo ¼ 0�. Location and orientation of the moving ellipsoid are illus-trated in Fig. 8 by showing the major axis of the ellipsoid. As shownthe particle path and orientations are in a very good agreementwith Chen and Yu results [14].

4.1.3. Ellipsoidal fiber reflection from a wallParticle impaction to walls is a very complicated phenomenon

affected by many parameters. Particle orientation and angularvelocity have significant effect on impaction behavior of non-spherical particles. Fig. 9 shows two similar ellipsoidal particleswithout any rotation which hit a wall in different orientationswhile the reflect boundary condition was employed. In one case

Fig. 7. Pressure drop comparison between the present study and the experimentalwork of Qu and Mudawar [8].

the major axis of ellipsoid is in direction of its motion and in theother one the major axis is not. In the former, the ellipsoid hitsthe wall one time and leaves it with a counter clockwise rotation.In the latter case, the particle hits the wall two times. After the firsthitting it experiences a clock wise rotation before hitting the wallfor the second time.

According to authors’ knowledge, no experimental or analyticinvestigation has been done on the ellipsoidal particles impactionto walls, therefore the Wynn reflect boundary condition is validatedby the results of spherical particle impaction. Gorham and Kharaz[25] experimentally investigated the rebound characteristic ofspherical aluminum oxide particles of 5 mm diameter hitting a thicksoda-line glass anvil at 3.9 m/s velocity. Skin friction coefficient fortwo surfaces is 0.092, particle density is 3690 kg/m3, Young’s mod-ulus and Poison’s ratio for the particle are E = 360 GPa, m = 0.23 andfor the wall are E = 70 GPa, m = 0.25, respectively.

The validation is done by three different parameters, rotationspeed, normal and tangential restitution coefficients. Normal andtangential restitution coefficients are defined as the ratio of parti-cle normal and tangential velocity components after impaction tothose of before impaction. Experimental results show that the nor-mal restitution coefficient for overall range of impact angles isaround 0.98. This parameter in the present numerical study isaround 0.99. Fig. 10 shows the tangential restitution coefficientfor different impact angles. The impact angle is defined as the anglebetween the impact and normal directions to the wall. As it can beseen in the figure, the numerical results are in close agreementwith the experiments for the impact angle of larger than 30�. As ex-plained in the experimental work of Gorham and Kharaz [25] forthe impact angle of smaller than 30�, impact involves some degreeof sticking and micro-slip behavior. Wynn impact model cannotpredict these phenomena, and therefore the numerical resultsand the experimental data are different for the impact angles smal-ler than 20�. In the present work, except some fibers entering thestagnation region, the other fibers hit the bottom plenum wall,while they are moving mostly parallel to the bottom wall andthe hitting can be considered far from a normal impact for whichthe Wynn model is not accurate enough. Fig. 11 shows the particlerotation speed after impaction for different impact angles. Thesenumerical results are also in close agreement with the experimen-tal data for the impact angles larger than 30�.

4.2. Flow field results

The fluid flowing in the microchannel heat sink system was as-sumed to be water at 23 �C with the density of qf = 997.6 kg/m3

and viscosity of lf = 9.3958 � 10�4 kg/m s. Average fluid velocityat the channels is 0.84 m/s and the Reynolds number based onchannel hydraulic diameter and this average velocity is 201 andthe fluid total flow rate is 38.6 mL/min. Two other flow rates of29.4 and 20.2 mL/min which lead to Reynolds numbers of 153and 105 were also studied, but the first mentioned flow rate wasmainly investigated.

Fig. 12 shows the velocity contours and vectors at two differentplanes of the domain. The first one is a horizontal plane located atthe middle of channel depth (z = 175.5 lm). The second one is thesymmetric plane of the domain at y = 0. Fig. 12a shows the sym-metric division of flow among the channels and the fully developedflow at their outlets. In Fig. 12b the developing boundary layer onthe inlet circular duct wall and two low speed recirculation zonesare clearly seen. The flow stagnation point occurs on the plenumbottom wall at x � �400 lm. The flow around this point has twodifferent directions to the channels and left recirculation zone.The velocity contours for the other mentioned flow rates are sim-ilar to Fig. 12, but the maximum velocities are 0.91 and 1.33 m/sfor the flow rates of 20.2 and 29.4 mL/min, respectively.

Fig. 8. The motion of an ellipsoid in a fully developed pipe flow in three different views. (a) Present study results. (b) Results of Chen and Yu [14]. x, y and z directions of thepresent work coordinate system are related to �X1, X2 and X3 of the reference work, respectively.

Fig. 9. Hitting of an ellipsoid to a wall in two different orientations.

Fig. 10. Comparison between the present study and the experimental work ofGorham and Kharaz [25] for tangential restitution coefficient.


Due to the constant outlet pressure boundary condition at theend of each channel, the fluid is expected to be divided into thechannels based on the geometric conditions. However, the investi-gation shows that the fluid mass flow rate is approximately equallydivided into all channels. Maximum deviation from the equal divi-sion of fluid into the channels is ±2.7% for the maximum investi-gated Reynolds number, while maximum flow passes throughchannels 2 and 14 (the channels have been numbered in Fig. 1c).

4.3. Ellipsoidal particle motion and the fiber web formation

The ellipsoidal particles with semi minor axis of a = 10 lm andaspect ratio of b = 12.5 were considered as the injected fibers to thedomain. The fiber material is silica with the density of qp = 2400 -kg/m3, Young’s modulus of E = 73 Gpa and Poison’s ratio of m = 0.17.The microchannel is assumed to be made up of copper with theYoung’s modulus of E = 117 Gpa and Poison’s ratio of m = 0.23.

The skin friction coefficient between fiber surface and microchan-nel walls is assumed to be 0.2. Fibers with zero translational andangular velocities are injected to the domain from the top planeof the inlet circular duct (z = +1800 lm). The major axes of all in-jected fibers are parallel to the axis of inlet duct, therefore all Eulerangles are equal to zero. The injection locations of the fibers are thenode points of a uniform grid having equal spacing of 150 lm andconsequently 69 fibers were injected to the domain at the first

Fig. 11. Comparison between the present study and the experimental work ofGorham and Kharaz [25] for rotational speed after hitting the wall.


step. Because of the complexity of the investigation, study was lim-ited to the above specific size and the effects of injected fiber num-ber, size and initial orientation can be studied in future works.

Fig. 13 shows the paths for the center of injected fibers in tophalf of the domain. Because of the symmetry, the fiber motionpaths in the other half of the domain are similar to those ofFig. 13. The results depict that the injected fibers follow the ‘‘flow

Fig. 12. Velocity (m/s) contours and vectors at two different planes in the flow rate of 38vertical plane of the domain (y = 0).

trend’’ in dividing into channels. The fluid which enters the domainthrough the half right of the inlet goes toward the channels in thecenter, while the other half flows to the remaining channels farfrom center. This could be also concluded by observing the velocityvectors shown in Fig. 12. However, expectedly, fiber paths deviatefrom the flow stream lines and about 45% of injected fibers hit theplenum bottom wall at least for one time. The deviation from flowstream lines is due to the large inertia of the fibers and also thesharp change in flow direction when it enters the plenum region(see Fig. 12). Because of boundary layer development at the circu-lar duct wall, fibers move slightly in radial direction during the ver-tical motion along the circular duct.

Fig. 14 shows the paths for the fibers injected at the symmetryline of the domain at y = 0. Figure shows the fibers major axis atdifferent time instances. Fibers follow the behavior of flow trendaround the stagnation point and two of them enter the left recircu-lation zone of the domain. These two fibers hit the wall repeatedlyand do not leave the region till the desired simulation time termi-nates (wall hitting was only drawn once in the figure).

Electronic Annex 1 being available in the online version of thispaper illustrates the animated view of a moving ellipsoidal fiber in-jected at the center line of the domain. The fiber hits the plenumbottom wall and then without any touching to the channel en-trance wall enters to the central channel and finally leaves the do-main. Employing reflect boundary condition at the plenum wall isshown in two views in that video. An animation of another moving

.6 mL/min. (a) Horizontal mid plane of channel depth (z = 175.5 lm). (b) Symmetric

Fig. 13. The paths for the center of injected fibers in the top half of domain. (a) Fibers injected in the right half (x P 0). (b) Fibers injected in the left half (x < 0).

Fig. 14. The path of injected fibers at the symmetric line of domain (y = 0).

Fig. 15. The formed fiber web on the channel entrance walls at the first step of thesimulation. (a) Isometric view. (b) Front view.


fiber is shown in Electronic Annex 2 of the online version of the pa-per. Performing the rotation boundary condition is illustrated inthis video. The fiber hits the channel entrance wall and rotatesaround the touch point till it finds the second support for deposi-tion. This fiber is considered as a deposited fiber in the fiber web.

Among 69 injected fibers in the first step, 26 fibers (38% of in-jected fibers) leave the domain without any hitting the channel en-trance walls and 43 fibers hit them. Seven fibers (16% of hittingfibers) do not find the second support to deposit and therefore de-tach from the wall and leave the region. The second support of 12fibers (28% of hitting fibers) is on the plenum bottom wall andtherefore they lie on the bottom wall in front of channel entrancewalls. And finally, 24 fibers (56% of hitting fibers) deposit on chan-nel entrance walls and form the first step of fiber web. The forma-tion of the fiber web is symmetric with respect to symmetry planeof domain. Fig. 15 illustrates deposited fibers in the first step ofweb formation on the channel entrance walls. The figure showsonly one half of the domain and the fibers were drawn with squarecross section instead of ellipsoid.

At the next step, to investigate how the deposited fibers affectthe flow field and the fiber deposition, a computational modelincluding the fiber web in the first step was prepared. Because ofthe symmetry and the computational domain size reduction, thehalf of domain and web were modeled. In our previous work [7],it was shown that the fiber cross section has no noticeable effecton the fiber web properties and therefore the deposited fiber cross

section was assumed to be square with the width of 20 lm tomake the grid generation process more straightforward.

The existence of fiber web affects the equally division of fluidamong the channels. Channel 3 which has 16% of blockage due tothe fibers has the minimum amount of flow rate which is 3% lessthan that of equal division condition. Channels No. 6 and 7 withoutany blockage have the maximum flow rates. Fig. 16 shows the flowpath lines around the fibers deposited on the entrance walls of thechannels No. 1–3. The flow path lines deviation due to existence ofthe fibers can be clearly observed.

After obtaining the flow field, a new set of fibers are injected tothe domain in a similar manner with the previously discussed step.Because of modeling the half of the domain, 39 fibers are injectedin this step. All injected fibers hit the channel entrance walls ex-cept one which enters the left recirculation zone. It should benoted that the rotation boundary condition is performed on thewalls of previously formed fiber web as well as the channel en-trance walls. Three fibers do not hit any other walls or fibers after

Fig. 16. Flow path lines around the fibers caught in front of channels 1–3. The path lines were colored by velocity magnitude (m/s).

Fig. 17. Different views of the first and second step fiber webs. First step fibers areblue and the second step fibers are red. (a) Front view, (b) top view, (c) side view,and (d) isometric view. (For interpretation of the references to colour in this figurelegend, the reader is referred to the web version of this article.)

Fig. 18. A fiber web observed in the experimental work of Perry [6].


the rotation and detach from the wall and leave the region. 35 fi-bers (90% of injected fibers in the second step) deposit on the chan-nel entrance region and form the second step fiber web.

Due to the large number of deposited fibers in the second step,the impact of fibers and deposition of them on each other are prob-able. Therefore the order of fiber deposition is of a great impor-tance. The fibers interaction with each other has been ignored inthe present numerical simulation, but since the fiber web forma-tion is a random phenomenon, it was decided to inject the secondstep fibers to the domain randomly. The location of each depositedfiber in the second step is saved in the developed code to investi-gate if the next fibers deposit on them or not? Fig. 17 shows thedifferent views of the simulated fiber web in the half of domain.For the qualitative comparison to what happen in experiments, apicture of deposited fiber web in the experimental work of Perry[6] is shown in Fig. 18. Both figures depict the random behaviorof the fiber web formation.

The method used in the present work for the modeling of fiberweb formation in two steps is in agreement with the observation ofPerry and Kandlikar [5] that the fiber web grows in time and addi-tional fibers get caught in the deposited fiber web. First and secondstep fiber webs are the symbols of two webs in different times.Figs. 15 and 17 illustrate the growth of the fiber web in time.


5. Conclusions

In this study, the motion of ellipsoidal fibers and the formationof a fiber web at the channels entrance of a microchannel heat sinkwere investigated numerically. The governing equations of theflow field have been solved with an Eulerian approach. A generalcomputer code was developed to solve the translational and rota-tional motion equations of ellipsoidal fibers with appropriateboundary conditions. Concepts of auxiliary structured mesh andthe neighbor cells led to a reduction in the computational timeand efforts in the particle tracking process.

The outcomess of this study show that, in contrast with spher-ical particles, the orientation of the non-spherical particles has agreat importance and therefore the angular velocities and orienta-tion equations should be solved.

The results showed that fiber web formation at the channels en-trance of a microchannel heat sink can be predicted by a numericalmethod. The effects of the deposited fibers on the flow field and thedeposition of next fibers were modeled and it was shown that thedeposited fiber web grows in time. As discussed, only 35% of in-jected fibers deposits at the channel entrance and makes the firststep fiber web, while 90% of injected fibers in the second stepdeposits on the channel entrance walls and the first step fibers. Itdepicts clearly the growth of the fiber web in time.

Appendix A. Supplementary material

Supplementary data associated with this article can be found, inthe online version, at http://dx.doi.org/10.1016/j.compfluid.2012.09.018.

References

[1] Tuckerman DB, Pease RFW. High-performance heat sinking for VLSI. IEEE,Electron Dev Lett 1981;2:126–9.

[2] Liu R. Study of fouling in a heat exchanger with an application of electronicanti-fouling technology. Philadelphia, PA: Drexel University; 1999.

[3] Yiantsios SG, Karabelas AJ. The effect of gravity on the deposition of micron-sized particles on smooth surfaces. Multiphase Flow 1998;24:283–93.

[4] Niida T, Kousaka Y, Furukawa T. Removal of adhering particles of polystyrenelatex and iron oxide on a wall by shear flow in water. Part Part Syst Charact1989;6:69–73.

[5] Perry J, Kandlikar S. Investigation of fouling and its mitigation in siliconmicrochannel. In; Sixth international conference on nanochannels,microchannels and minichannels. Darmstadt, Germany: ASME; 2008.

[6] Perry J. Fouling in silicon microchannel designs used for IC chip cooling and itsmitigation. Rochester, NY: Rochester Institute of Technology; 2007.

[7] Dastan A, Abouali O. A numerical analysis of the fiber web effects on thepressure drop, particles collection, and heat transfer in a microchannel. HeatTransfer Eng 2011;32:554–65.

[8] Qu W, Mudawar I. Experimental and numerical study of pressure drop andheat transfer in a single-phase micro-channel heat sink. Heat Mass Transfer2002;45:2549–65.

[9] Fan F, Ahmadi G. Dispersion of ellipsoidal particles in an isotropic pseudo-turbulent flow field. Fluid Eng 1995;117:154–61.

[10] Fan F, Ahmadi G. A sublayer model for wall deposition of ellipsoidal particlesin turbulent stream. Aerosol Sci 1995;26:813–40.

[11] Fan F, Ahmadi G. Wall deposition of small ellipsoids from turbulent air flows –a Brownian dynamics simulation. Aerosol Sci 2000;31:1205–29.

[12] Asgharian B, Ahmadi G. Effect of fiber geometry on deposition in small airwaysof the lung. Aerosol Sci Technol 1998;29:459–74.

[13] Lin J, Shi X, You Zh. Effects of the aspect ratio on the sedimentation of a fiber inNewtonian fluids. Aerosol Sci 2003;34:909–21.

[14] Chen Y, Yu C. Sedimentation of fibers from laminar flows in a horizontalcircular duct. Aerosol Sci Technol 1991;14:343–7.

[15] Högberg SM, Akerstedt HO, Lundström TS, Freund JB. Respiratory deposition offibers in the non-inertial regime – development and application of a semi-analytical model. Aerosol Sci Technol 2010;44:847–60.

[16] Wynn EJW. Simulations of rebound of an elastic ellipsoid colliding with aplane. Powder Technol 2009;196:62–73.

[17] Warren GP, Anderson WK, Thomas JL, Krist SL. Grid convergence for adaptivemethods. In: American Institute of Aeronautics and Astronautics (AIAA) 10thcomputational fluid dynamics conference. Honolulu, Hawaii; 1991.

[18] Goldstein H. Classical mechanics. 2nd ed. Reading, MA, USA: Addison-WasleyCompany; 1980.

[19] Brenner H. The stokes resistance of an arbitrary particles-IV. Arbitrary field offlow. Chem Eng Sci 1964;19:703–27.

[20] Brennen CE. Cavitation and bubble dynamics. Oxford, UK: Oxford UniversityPress; 1995.

[21] Harper EY, Chang ID. Maximum dissipation resulting from lift in a slow viscousshear flow. Fluid Mech 1968;33:209–25.

[22] Jeffery GB. The motion of ellipsoidal particles immersed in a viscous fluid. ProcRoy Soc A 1922;102:161–79.

[23] Gallily I, Cohen A. On the orderly nature of the motion of nonsphericalaerosol particles II. Inertial collision between a spherical large droplet and anaxially symmetrical elongated particle. J Colloid Interface Sci 1979;68:338–56.

[24] Press WH, Teukolsky SA, Vetterling WT, Flannery BP. Numerical recepies: theart of scientific computation. 3rd ed. Cambridge, UK: Cambridge UniversityPress; 2007.

[25] Gorham DA, Kharaz AH. The measurement of particle rebound characteristics.Powder Technol 2000;112:193–202.



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Microfiber motion and web formation in a microchannel heat sink: A numerical approach