Numerical Simulation of Biomagnetic Fluid Flow in an Oscillating Lid Driven Cavity · · 2016-12-16technique based on simple algorithm. Results are presented for wide ranges of

International Journal of Applied Engineering Research ISSN 0973-4562 Volume 11, Number 23 (2016) pp. 11259-11265

© Research India Publications. http://www.ripublication.com

11259

Numerical Simulation of Biomagnetic Fluid Flow in an Oscillating Lid

Driven Cavity

Abdullah A.A.A. Al-Rashed1, Abdulwahab Ali Alnaqi1 and M. A. Hossain2

1Dept. of Automotive and Marine Engineering Technology, College of Technological Studies, The Public Authority for Applied Education and Training, Kuwait.

2Department of Mathematics, University of Dhaka, Dhaka-1000, Bangladesh. *Corresponding author

Abstract

The fundamental problem of biomagnetic fluid dynamics in a

lid driven cavity is numerically studied. The flows are driven

by the top sliding wall, which executes sinusoidal oscillations.

Numerical solutions are acquired by solving a coupled and non

linear system of PDEs, with their appropriate boundary

conditions and by using a finite differences numerical

technique based on simple algorithm. Results are presented for

wide ranges of principal physical parameters, i.e., Re, the

Reynolds number, MnF, the magnetic number and , the non-

dimensional frequency of the lid oscillation. Comprehensive

details of the flow structure are presented. Results concerning

the velocity and skin friction indicate that the presence of the

magnetic field influence the flow field considerably.

Keywords: Biomagnetic fluid, oscillating lid, magnetic fluid,

driven cavity

Nomenclature

,x y

Cartesian coordinates

,x y

non dimensional Cartesian coordinates , /x y L

M

magnetization

H

magnetic field strength

Re Reynolds number = L U

FMn magnetic number 2

0 0HU

t

time

u

velocity in x-direction

u

dimensionless horizontal velocity

v

velocity in y-direction

v

dimensionless vertical velocity

Greek symbols

biomagnetic fluid density

dynamic viscosity

kinematic viscosity

magnetic susceptibility

ω lid oscillation frequency

D dimensionless average drag force

INTRODUCTION

Biological fluid is a fluid that exists in a living creature. The

fluid dynamics of biological fluids in the presence of applied

magnetic fields is biomagnetic fluid dynamics (BFD). Due to

its bioengineering and medical applications, an extensive

research work has been done on this relatively new area, during

the last decades [1-3]. Development of magnetic devices for

cell separation, magnetic wound or cancer tumor treatment

causing magnetic hyperthermia, targeted transport of drugs

using magnetic particles as drug carriers, reduction of bleeding

during surgeries or provocation of occlusion of the feeding

vessels of cancer tumors and development of magnetic tracers

[4-7].

Biomagnetic fluid flow is analogous to the principals of

FerroHydroDynamics (FHD) and Magneto Hydro Dynamics

(MHD) where the magnetization and the Lorentz force are

dominating force in the flow field. Magnetization is the

measure of how much the magnetic fluid is affected by the

magnetic field and a function of the magnetic field intensity. In

FHD, the flow is affected by the magnetization of the fluid in

the magnetic field [8]. MHD deals with conducting fluids and

the mathematical model ignores the effect of polarization and

magnetization.

Blood can be considered as a magnetic fluid [1]. Moreover, it

can also be considered as diamagnetic, paramagnetic or

ferromagnetic fluid depending upon certain conditions. Mature

red blood cells contain the hemoglobin molecule, in the form

of iron oxides at a exclusively high concentration, so it can be

considered as the most characteristic biomagnetic fluid [9]. The

orientation of the erythrocyte disk plane is parallel to the

magnetic field [10]. Oxygenated blood behaves like a

diamagnetic material whereas paramagnetic when

deoxygenated [11].

Incompressible viscous flow in a closed container constitutes

an important subject, from the standpoints of both theoretical

analyses and technological applications. Fluid flow of viscous

fluid in a driven cavity has been considered as a benchmark

configuration for numerical and experimental model validation

[12–15]. Concerning the practical applications, researchers are

interested in understanding the unsteady cavity flow

phenomena. Time periodic flows in cavity can be viewed as

prototypes for studying mixing processes. A typical

configuration of unsteady flow in cavity is constructed by a

moving top lid with a sinusoidal or cosinusoidal motion [16–

18]. Soh and Goodrich [17] studied the fluid flow within a

closed finite square cavity, driven by a sliding wall that

executes sinusoidal oscillation. They found that the variation of

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the fluid flow structure due to the oscillating motion of the lid

in one complete cycle. Later, Iwatsu et al. [18] conducted a

numerical study of flow driven by a torsionally oscillating lid

in a square cavity for a wide range of Reynolds numbers and

frequencies of the oscillating lid. They reported that the effect

of the lid motion penetrates a larger depth into the cavity at low

frequencies, and flow is similar to the steady driven cavity flow

at the maximum plate velocity. However, the flow was

confined within a thin layer near the oscillating lid, at high

frequencies.

The fundamental problem of the biomagnetic fluid flow in a

steady lid driven cavity under the influence of localized

magnetic field has been conducted by Tzirtzilakis and Xenos

[19]. They concluded that the flow field influences

considerably in the presence of the magnetic field. Also the

influence of the magnetic field on the flow is local and is

confined close to the area of application of the magnetic field.

From the above discussion and an exhaustive survey of

literature, it reveals that there has not been much work on the

periodic nature of fluid flow in a cavity. Furthermore, such

studies have been limited to pure lid driven cavity flow

problem. To the best of the authors’ knowledge, the problem of

biomagnetic fluid flow in a square cavity driven by a

periodically oscillating lid, has not yet been studied. The

objective of the present work is to obtain extensive

computational results of the biomagnetic fluid flow in the

driven cavity with an oscillating lid. Complete flow details

have been acquired over wide ranges of some principal

parameters, i.e., Reynolds number, magnetic number and

oscillation frequency. The solution of the problem is obtained

numerically by the development of an efficient numerical

methodology based on the SIMPLE algorithm.

MATHEMATICAL FORMULATION

The problem under consideration is the viscous, steady, two–

dimensional, incompressible, laminar biomagnetic fluid

(blood) flow inside a square duct under the influence of an

applied magnetic field. The length and height of the duct is L

and the axes intersect at the bottom left corner of the square

duct with coordinates (0, 0) i.e the origin of the Cartesian

system is at the bottom left corner of the cavity. The flow is

subject to a magnetic source, which is placed very close to the

lower plate of the cavity and below it. A schematic

representation of the flow field as well as the magnetic field

strength contours are given in Figure 1.

Figure 1: Contour lines of the dimensionless magnetic field

strength

Blood is assumed as a homogeneous Newtonian fluid,

according to the mathematical model of BFD [1, 20-22] and the

principles of FHD [23-26]. The apparent viscosity due to the

application of the magnetic field is considered to be negligible.

The rotational forces acting on the erythrocytes, when entering

and exiting the magnetic field are discarded (equilibrium

magnetization). Moreover, The Lorentz force due to the

electrical conductivity of blood is considered negligible

compared to the magnetization force. Thus, blood is also

considered electrically non-conducting fluid.

The governing equations of the fluid flow, under the action of

the applied magnetic field, are similar to those derived in FHD.

Hence at the cavity flow the dimensional velocity components

(bar above the quantities) u , v and pressure p are governed

by the mass conservation and the fluid momentum equations at

the x , y directions, which are given respectively by

Continuity equation:

0u vx y

(1)

Momentum equations:

2 2

02 2

1u u u p u u Hu v Mt x y x x y x

(2)

2 2

02 2

1v v v p v v Hu v Mt x y y x y y

(3)

The boundary conditions are

Upper wall ( , 0 ): sin , 0

Lower wall ( 0, 0 ): 0, 0

Left wall ( 0, 0 ): 0, 0

Right wall ( , 0 ): 0, 0

y L x L u U t v

y x L u vx y L u v

x L y L u v

(4)

In the above equations , is the biomagnetic fluid density,

is the dynamic viscosity, 0 is the magnetic permeability,

/ is the kinematic viscosity, M is the magnetization

and H is the magnetic field strength.

In equations (3) and (4) equation terms 0 /M H x and

0 /M H y represents, respectively, the component of the

magnetic force per unit volume, and depends on the existence

of the magnetic gradient.

For the variation of the magnetization M , with the magnetic

field intensity H experiments which has been carried out in [2]

showed that it can be fairly approximated by the linear relation

[11]

M H (5)

where, is a constant called magnetic susceptibility. The

components of the magnetic field intensity xH and yH along

the x and y coordinates, ,x yH H H is given respectively

by



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2 22 2

,2 2

x yy b x aH H

x a y b x a y b

(6)

where ,a b is the point where the magnetic source is placed

and γ is the magnetic field strength at this point ,x a y b .

The magnitude H of the magnetic field intensity, is given by

1/ 2

22

1, ,

2x yH x y H H

x a y b

(7)

The conservation equations of mass and momentum are non-

dimensionalized by introducing the following non-dimensional

variables and numbers into the conservation equations:

2

0 0 0

2

0 0

, , , , ,

, , , ,

Re (Reynolds number),

(Magnetic nmber)

yxx y

F

x y u v px y u v pL L U U U

HHtU L Ht H H HL U H H H

L U

HMnU

(8)

where 0 ,0H H a .

Applying these non-dimensional variables and numbers the

non-dimensional conservation equation of mass and

momentum together with the boundary conditions may be

written as follows:

0u vx y

(9)

2 2

2 2

1

ReF

u u u p u u Hu v Mn Ht x y x x y x

(10)

2 2

2 2

1

ReF

v v v p v v Hu v Mn Ht x y x x y y

(11)

Upper wall ( 1, 0 1): sin , 0

Lower wall ( 0, 0 1): 0, 0

Left wall ( 0, 0 1): 0, 0

Right wall ( 1 0 1): 0, 0

y x u t vy x u v

x y u vx y u v

(12)

The magnitude H of the magnetic field intensity is also derived

from the relations (8) and (9) is given by:

2 2,

bH x y

x a y b

(13)

NUMERICAL METHODOLOGY

The governing equations are discretized using the finite

difference method while the coupling between velocity and

pressure fields is done using the SIMPLE algorithm. The

diffusion terms in the equations are discretized by a second

order central difference scheme, while a hybrid scheme (a

combination of the central difference scheme and the upwind

scheme) is employed to approximate the convection terms. The

convergence of solutions is assumed when the relative error for

each variable between consecutive iterations is recorded below

the convergence criterion ε such that

1

1

n n

n

U UU

(14)

where n denotes the number of iterations and the convergence

criterion was set to 10−6.

RESULTS AND DISCUSSION

The verification of the present numerical code is carried out in

two folds. First, since there is no public data available for

biomagnetic flow inside an oscillating lid-driven cavity, the

present numerical code was validated with the previous

benchmark solutions of pure lid-driven flow in a square cavity

reported by Iwatsu et. al. [18]. By considering the magnetic

parameter 0FMn , our present numerical simulation for

biomagnetic flow becomes pure lid-driven cavity flow

problem. Typical time history of the drag force for Re = 400 at

different frequencies (low and moderate) are presented in

Figure 2 together with the previous result of lid driven flow

with oscillating lid [18]. It is found that there is a very good

agreement between the present results and the previous study

performed by Iwatsu et. al. [18].

Figure 2: A representative time history of force coefficient,

Cf, for Re = 400 for frequency = 0.1 and 1.0.



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(a)

(b)

Figure 3: Streamlines for Re = 400 and various values

of MnF = 1 and 8 (a) Tzirtzilakis and Xenos [19] and (b) Present

The second fold is a comparison between the predicted stream

function contours for Re = 400 and various values of MnF (MnF

= 1, 4 and 8), under steady state condition of the present work

to that of Tzirtzilakis and Xenos [19]. As displayed in Figure 2,

the comparison strikes an excellent agreement between both

studies. These validation cases enhance the confidence in the

numerical outcome of the present work.

After validation of the present code for pure lid-driven cavity

flow with an oscillating motion of the top lid, simulation of

biomagnetic fluid dynamics for the oscillating lid case was

carried out. The flow field becomes periodic with an identical

frequency to the oscillating lid. To ensure the periodic steady

state of the fluid motion, results after a sufficient number of lid

oscillation cycles are considered.

t = T/8 t = T/4 t = 3T/8 t = T/2

t = 5T/8 t = 3T/4 t = 7T/8 t = T

Figure 3: Streamline plots during a complete period of the cycle at Re = 400, ω = 1.0 and MnF = 0

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

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1



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The contour lines of stream function have been constructed to

visualize the overall flow patterns inside the cavity at different

instants, caused by the oscillation of the lid motion and some

typical pictures are shown in Figure 3 for Re = 400, ω = 1.0 and

MnF = 0.0. Temporal variations of the streamline plots during

a complete period are displayed in Figure 3. Each period was

divided into eight intervals. The first four intervals represent

first half of the cycle. The next four intervals represent the

second half of the cycle which is exhibits a perfect temporal

reversal of the first half of the cycle. Initially, a leading primary

vortex appeared which occupies the whole region of the cavity

at t = T/8. A secondary vortex at the upper portion of the right

side wall and two small counter-rotating vortices at the two

corners of the bottom wall is also observed along with leading

vortex. As a result of the no-slip condition between the fluid

and solid wall, a clockwise vortex forms just below the lid at t = T/4. This clockwise vortex amplifies and merges with the top

right secondary vortex and eventually engages the maximum

space of the cavity. Accordingly, in the last interval of the first

half cycle, the central counterclockwise vortex that emerge at

beginning of the period, gradually reduces in size and

eventually becomes a secondary vortex at the upper portion of

the left wall. The phenomenon observed in the first half of the

cycle reverses in the second half as the fluid flow is forced by

a sinusoidal oscillation.

t = T/8 t = T/4 t = 3T/8 t = T/2

t = 5T/8 t = 3T/4 t = 7T/8 t = T

Figure 4: Streamline plots during the complete cycle at Re = 400, ω = 1.0 and MnF = 1

The flow field revealed significant changes in the location and

number of vortices with the variation of magnetic force which

is depicted in Figure 4. As MnF was increased from 0 to 1,

together with the vortices that created for the pure lid driven

flow, a single prominent minor vortex at the center of the

bottom wall is created because of the influence of the magnetic

field.

MnF = 0 MnF = 1 MnF = 4 MnF = 8

Figure 5: Streamline plots for different values of MnF at a instant t = 3T/8, Re = 400 and ω = 1.0

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

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1

0 0.2 0.4 0.6 0.8 10

0.2

0.4

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0 0.2 0.4 0.6 0.8 10

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0 0.2 0.4 0.6 0.8 10

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0 0.2 0.4 0.6 0.8 10

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0.4

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0 0.2 0.4 0.6 0.8 10

0.2

0.4

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0 0.2 0.4 0.6 0.8 10

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0 0.2 0.4 0.6 0.8 10

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0 0.2 0.4 0.6 0.8 10

0.2

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1



11264

At the beginning of the period, the bottom wall centre vortex

merge with right side corner vortex and ultimately merge with

the vortex that is created under the lid. At the end of the first

half cycle, a vortex due to magnetic force appeared again at the

center of the bottom wall and in the opposite direction.

Furthermore, as MnF increased from 1 to 4, a similar trend for

vortex formation is observed. The centre vortex of the bottom

wall increased with increasing values of magnetic force.

Fluid flow inside the cavity with various values of magnetic

number at the instant t = 3T/8 is shown in Figure 5, where Re

and ω is considered to be 400 and 1.0 respectively. A

remarkable variation of flow occurs for the implementation of

the magnetic field. An increase in MnF at a given oscillation

frequency and at a given instant promotes the development

vortices. Therefore, fluid is well mixed at higher values of

magnetic number.

Figure 6: Variation of the drag force predictions, for

Re = 400 and MnF = 4 at different frequencies

( = 0.1, 1.0 and 5.0)

Finally, to cover a wide range of lid frequency, ω = 0.1, 1 and

5 were used as the input variables. The effect of lid frequency

on the predicted drag force is displayed in Figure 6. This figure

illustrates how quickly the steady periodic solution is reached

for various values of . From Figure 6 it is observed that steady

periodic solution reaches quicker for large and for small

frequency, = 0.1 more cycles are required to reach a steady

periodic solution. It is observed that the drag force follows the

same sinusoidal behavior of the external excitation offered by

the sliding lid. Furthermore, it is noticed that magnitude of the

drag force increase with an elevation of frequency value.

CONCLUSION

Bio-magnetic fluid flow within a square cavity with an

oscillating upper lid has been investigated numerically. The

investigation was carried out for the magnetic number and the

oscillation frequency of the sliding lid. The steady-periodic

natures of the solutions have been captured. A profound effect

of magnetic numbers on the flow field structure has been

recognized. Formation of vortices within the cavity amplifies

with the increasing values of MnF which accelerates the mixing

of the fluid. As the oscillation frequency increases at a fixed

MnF, the number of vortices within the cavity decreases. Steady

periodic solution is reached faster for small values of the

normalized frequency. Moreover, the results indicate that drag

force increases with an increasing values of the lid frequency.

The numerical results of this study could be useful in

biomedical applications and analyzing the behavior of the fluid

driven by oscillatory motion of the lid and the influence of the

magnetic field.

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Documents

Numerical Simulation of Biomagnetic Fluid Flow in an Oscillating Lid Driven Cavity · · 2016-12-16technique based on simple algorithm. Results are presented for wide ranges of