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Influence of magnetohydrodynamics on metachronal wave ofparticle-fluid suspension due to cilia motion

M.M. Bhatti a,*, A. Zeeshan b, M.M. Rashidi c,d

a Shanghai Institute of Applied Mathematics and Mechanics, Shanghai University, Shanghai 200072, Chinab Department of Mathematics, International Islamic University Islamabad, Pakistanc Shanghai Key Lab of Vehicle Aerodynamics and Vehicle Thermal Management Systems, Tongji University, Shanghai 201804, Chinad ENN-Tongji Clean Energy Institute of advanced studies, Shanghai 200072, China

A R T I C L E I N F O

Article history:Received 30 December 2015Received in revised form11 March 2016Accepted 14 March 2016Available online

Keywords:Cilia motionMagnetohydrodynamicsParticle fluidMetachronal wave

A B S T R A C T

In this article, the influence of magnetohydrodynamics (MHD) on cilia motion of particle–fluid suspen-sion through a porous planar channel has been investigated. The governing equations of Casson fluidmodelfor fluid phase and particulate phase are solved by taking the assumption of long wavelength and ne-glecting the inertial forces due to laminar flow. The solutions for the resulting differential equations havebeen obtained analytically and a close form of solutions is presented. The expression for pressure risealong the whole length of the channel is evaluated numerically. The influences of all the physical pa-rameters are demonstrated graphically. Trapping mechanism has also been discussed with the help ofstreamlines. It is observed that due to the influence of magnetohydrodynamics and particle volume frac-tion, velocity of the fluid decreases. It is also found that pressure rise shows similar behaviour for particlevolume fraction and Casson fluid parameter.

Copyright © 2016, The Authors. Production and hosting by Elsevier B.V. on behalf of KarabukUniversity. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/

licenses/by-nc-nd/4.0/).

1. Introduction

The term cilia is often used for “eukaryotic cells,” and it is derivedfrom the word “eyelashes,” The term cilia is used when differenttypes of cilia elements are appendages on a single cell. The innerlayer of these cells is described by a cylindrical core which is knownas axoneme. The axoneme contains cylindrical arrangements of mo-lecular motors/dynein and elastic elements (known asmicrotubules).All the cilia elements on the surface of microorganism describe thesame beating pattern. The length range of each cilium element isabout 2 μm to mm and its diameter is about 0.2 μm. The shape ofthe cilia elements is very much similar to hair like motile append-ages which can be observed in the nervous system, digestive system,male and female reproductive system. Cilia motion plays a vital rolein various physiological processes such as reproduction, alimenta-tion, locomotion and respiration. Metachronal wave that is generateddue to the waves of ciliary elements propagates along out of phasedirection in a distensible tube/channel. The main benefit ofmetachronal wave is to control the continuity of the flow and alsoto help us increase the amount of fluid propelled. Ciliary momentsare found in different shapes depending on the ciliary system, suchas oscillatory, excitable, helical, beating and planar. When the

* Corresponding author. Tel.: +8613162146836.E-mail address:muhammad09@shu.edu.cn (M.M. Bhatti).Peer review under responsibility of Karabuk University.

http://dx.doi.org/10.1016/j.jestch.2016.03.0012215-0986/Copyright © 2016, The Authors. Production and hosting by Elsevier B.V. onND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

Please cite this article in press as: M.M. Bhatti, A. Zeeshan, M.M. Rashidi, Influence of magEngineering Science and Technology, an International Journal (2016), doi: 10.1016/j.jestch

metachronal wave propagates along the same path as the effec-tive stroke, then this phenomenon is known as symplecticmetachronism, whereas when they propagate along the oppositedirection, then this phenomenon is known as antiplectic. Severalauthors investigated analytically and numerically the motion of ciliain different situations with various biological fluids [1–5].

On the other hand, magnetohydrodynamics is very helpful tocontrol the flow of fluid. Magnetohydrodynamics (MHD) deals withthe study of electrically conducting fluids such as salt water or elec-trolytes, plasmas and liquidmetals. The application of magnetic fieldcan be found in various engineering process and geophysical studies.Magnetohydrodynamics is also applicable in magnetic drug target-ing for different types of cancer diseases. Magnetohydrodynamicsis useful and applicable in various microchannel design for creat-ing continues and non-pulsating flow. In biomedical engineering,magnetohydrodynamics helps in the regulation of hyperthermia,MRI and magneto-fluid rotary blood pumps, etc. Magnetohydro-dynamics oscillation and waves are very famous tools forastrophysical plasmas and remote diagnostic. Magnetohydrody-namics sensors are often used tomeasure the angular velocity. Akbarand Khan [6–8] analysed the effects of magnetohydrodynamics onmetachronal beating of cilia for Casson fluid model. Ellahi et al. [9]studied the effects of magnetohydrodynamics on peristaltic flow ofJeffery fluid in a rectangular duct through a porous medium.Mekheimer et al. [10] examined the particle fluid suspension of peri-staltic flow in a non-uniform annulus. Akbar and Butt [11] analysed

behalf of Karabuk University. This is an open access article under the CC BY-NC-

netohydrodynamics on metachronal wave of particle-fluid suspension due to cilia motion,.2016.03.001

the effects of heat transfer on metachronal of wave cilia forRabinowitsch fluid model. Later, Akbar and Khan [6–8] character-ize the effects of heat transfer for bi-viscous ciliary motion fluid.Some relevant studies on the present analysis can be found fromthe list of references [6–8,12–29].

With the above analysis in mind, the purpose of this present in-vestigation is to analyse the effects of magnetohydrodynamics onmetachronal wave of particle–fluid suspension due to cilia motionin the uniform porous planar channel. The governing equations ofparticle–fluid suspension have been solved for Casson fluid modelunder the assumption of long wavelength and creeping flow regime.The solution for the resulting equation has been obtained analyti-cally and a closed-form solution is presented for fluid phase andparticulate phase. The impact of all the physical parameters is dis-cussed and plotted. This paper is summarized as follows; Sec. (2)describes the mathematical formulation of the problem, Sec. (3) isdevoted to methodology and solution of the problem and finally,Sec. (4) characterizes the numerical results and discussion.

2. Mathematical formulation

Let us consider the unsteady irrotational, hydromagnetic particle–fluid suspension model, which is incompressible, and electricallyconducting by an external magnetic field is applied through a two-dimensional porous planar channel. Ametachronal wave is travellingwith a constant velocity �c that is generated due to collective beatingof cilia along the walls of the channel whose inner surfaces are cili-ated.We have selected a Cartesian coordinate system for the channelin such a way that is �X − axis taken along the axial direction and�Y − axis is taken along the transverse direction (See Fig. 1).The envelop for cilia tips is assumed to be written as [6–8]

Y F X t a a X ct= ( ) = + −( )� � � � � ��, cos ,ε2πλ

(1)

X G X t X a X ct= ( ) = + −( )� � � � � ��, cos .02

εα πλ

(2)

The vertical and horizontal velocities for cilia motion can bewritten as [26])

�� � � ��

� � � ��U

a c X ct

a c X ct=− −( )− −( )

2 2

12 2

πλ

α πλ

πλ

α πλ

ε

ε

cos

cos, (3)

�� � � ��

� � � ��V

a c X ct

a c X ct=− −( )− −( )

2 2

12 2

πλ

α πλ

πλ

α πλ

ε

ε

sin

sin. (4)

The governing equation of motion, continuity for fluid phase andparticulate phase can be written as

2.1. Fluid phase

∂∂+∂∂=

��

��

UX

VY

f f 0, (5)

1 1

1

−( ) ∂∂+

∂∂+∂∂

⎛⎝⎜

⎞⎠⎟= − −( ) ∂

∂

+ −

CUt

UUX

VUY

CPX

ff

ff

ffρ

��

���

���

��

CCX Y

CS U U J Bk

U

XX XY

p f xs

f

( ) ∂∂

+ ∂∂

⎛⎝⎜

⎞⎠⎟

+ −( ) + × −

� �

� ��

�

� � � �τ τ

μ, (6)

1 1

1

−( ) ∂∂+

∂∂+∂∂

⎛⎝⎜

⎞⎠⎟= − −( ) ∂

∂

+ −

CVt

UVX

VVY

CPY

ff

ff

ffρ

��

���

���

��

CCX Y

CS V V J Bk

V

YX YY

p f ys

f

( ) ∂∂

+ ∂∂

⎛⎝⎜

⎞⎠⎟

+ −( ) + × −

� �

� ���

� � � �τ τ

μ, (7)

2.2. Particulate phase

∂∂+∂∂=

��

��

UX

VY

p p 0, (8)

CUt

UUX

VUY

CPX

CS U Upp

pp

pp

f pρ∂∂+

∂∂+∂∂

⎛⎝⎜

⎞⎠⎟= − ∂

∂+ −(

��

���

���

��

� � )), (9)

CVt

UVX

VVY

CPY

CS V Vpp

pp

pp

f pρ∂∂+

∂∂+∂∂

⎛⎝⎜

⎞⎠⎟= − ∂

∂+ −(

��

���

���

��

� � )). (10)

The mathematical expression for the drag coefficient and the em-pirical relation for the viscosity of the suspension can be described as

Sa

C CC C C

C C

e

s= ( ) ( ) = + − +−( )

=−

=

92

4 3 8 3 32 3 1

0 07

02

2

20μ λ λ μ μχ

χ

ˇ� �, , ,

.22 49

1107 1 69. .

.C

Te C+⎡

⎣⎢⎤⎦⎥

−

(11)

The stress tensor of Casson fluid is defined as

τ τ μ γ101 1n n

sn= + � , (12)

τ μ πi j i j b D yE P, , .= +( )2 2 (13)

In the above equation, π = Ei j, we have consider Py = 0 Now, it isconvenient to define the transformation variable from fixed frameto wave frame

� � �� � � � � � � � � �x X ct y Y u U c v V p Pf p f p f p f p= − = = − = =, , , , ., , , , (14)

Introducing the following non-dimensional quantities

� ��

��

��

��

��

xx

yya

uu

cv

vc

pac

pf pf p

f pf p

s

= = = = = =λ δ λ μ

ρ, , , , , Re,

,,

,2 aac

NSa

Ma

kk

Pa

s

s s s

b D

y

�

� � � �

μ

μσ

μ μζ μ π β

λ

,

, , , , .= = = = √ =2

02 2 2B

(15)

Using Eq. (14) and Eq. (15) and taking the approximation of longwavelength and neglecting the inertial forces, then Eq. (5) to Eq. (10)Fig. 1. Geometry of the problem.

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reduces to the following form. The resulting equations for fluid phaseand particulate phase can be written as

dpdx

uy k

u M uNC

Cu uf

f f p f= +⎛⎝⎜

⎞⎠⎟∂∂− +( )− +( )+

−( )−( )1

1 11 1

1

2

22

ζ, (16)

dpdx

N u uf p= −( ), (17)

and their corresponding non-dimensional boundary conditions are

′ = = = − − ( )− ( )

= = +

u y ux

x

y h

f f0 0 12 2

1 2 21

at and

at

πφαβ ππφαβ π

φ

coscos

,

coos .2πx (18)

3. Solution of the problem

The exact solution of Eq. (15) and Eq. (16) can be written as

uC kM kdp dx C kM N kdp dx N y N

f = −−( ) +( ) + + −( ) +( ) −( )1 1 1 12 2

1 2 2cosh sech hhC kM1 1 2−( ) +( ) , (19)

The volume flow rate is given by

Q C uf f

h

= −( )∫10

dy, (21)

Q C up p

h

= ∫ dy0

, (22)

where

Q Q Qf p= + , (23)

The pressure gradient dp dx is obtained after solving above Eq.(24), we get

dpdx

C kM h Q kM N k N h

h k C C=

−( ) +( ) +( ) + + +( )( )− + −( ) +

1 1 1

1 1

2 21 2ζ ζ ζ tanh

kkM N kM k k N h2 221 1( )( )( ) + + +( )ζ ζ ζ tanh.

(25)

The non-dimensional pressure rise ΔP( ) is evaluated numeri-cally by using the following expression

ΔP dx= ∫ dpdx0

1

. (26)

The expression for dimensionless stream function satisfying equa-tion of continuity is defined as

uy

vx

f pf p

f pf p

,,

,,, ,=

∂∂

= −∂∂

Ψ Ψ(27)

where Nx

xN

kmk

1 2

22 21 2 2 1= ( )− ( )

= ++( )

πφαβ ππφαβ π

ζ ζζ

coscos

, .

4. Numerical results and discussion

In this section, the graphical results of different pertinent pa-rameters are sketched for velocity profile, pressure rise and streamlines. For this purpose, Figs. 2 to 12 have been plotted againsta b, , , ,C k M and z . It is depicted from Fig. 2 that when the param-eters a and b increase, the velocity of fluid shows the oppositebehaviour near the walls. It can be noticed from Fig. 3a that whenthe particle volume fraction C( ) increases, the magnitude of the ve-locity decreases. It can be observed from Fig. 3b that when theporosity parameter k( ) increases, velocity of the fluid increases. From

Fig. 4a, we can observe that Hartmann number M( ) is very helpfulto control the flow because when the Hartmann number in-creases, the velocity of the fluid decreases, whereas the oppositebehaviour has been observed for Casson fluid parameter z( ) asshown in Fig. 4b. Figs. 5 and 6 are sketched for pressure rise versusaverage volume flow rate Q( ) It can be noticed from Fig. 5a thatwhen the particle volume fraction increases, the pressure rise de-creases in retrograde pumping region ΔP > <( )0 0,Q , but its attitudeis opposite in co-pumping region ΔP < >( )0 0,Q . It can be scruti-nized from Fig. 5b that the pressure rise is decreasing in retrogradepumping region and its behaviour seems opposite in free pumpingΔP < <( )0 0,Q region and co-pumping region. It can be observed

from Fig. 6a that when the Hartmann number M( ) increases, thepressure rise decreases in co-pumping and free pumping regionswhile it increases in retrograde pumping region. It can be seen fromFig. 6b that due to Casson fluid parameter, pressure rise decreasesin retrograde pumping region but its behaviour starts changing inthe same region and become opposite in free pumping and co-pumping regions.

Another most interesting part of this section is trapping, whichis taken into account by drawing stream against all the physical pa-rameters. Generally, it is the formulation of internally circulatingbolus that is enclosed by various streamlines. From Fig. 7, we canobserve that when the parameter a increases, the size of trapped

uC kM kdp dx C kM N k

dpdx

N yp = −

−( ) +( ) + + −( ) +( ) −⎛⎝⎜

⎞⎠⎟1 1 1 12 2

1 2cosh secch,

N h N C kM dp dx

C kM

22

2

1 1 1

1 1

− ( ) −( ) +

−( ) +( )( )

(20)

Qh C kM k C C km N dp dx km C kM=

−( ) + + −( ) +( )( )( )( ) + + −( ) +( )1 1 1 1 1 1 12 2 2ζ ζ 221 2

2 2

1

1 1

( ) −( ) +( )− +( ) + +( )

N kdp dx k N h

C kM kM

ζ

ζ ζ

tanh. (24)

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bolus slowly reduces. In Fig. 8, same behaviour can be observedbut the size of the trapping bolus slowly reduces as compared toFig. 7. It can be observed from Fig. 9 that when the particle volumefraction C( ) increases, the number of trapped bolus increases. It

can be noticed from Fig. 10 that when the porosity parameter k( )increases, the magnitude of the bolus slowly reduces, whereasthe number bolus also increases. It can be visualized fromFig. 11 and Fig. 12 that when the Hartmann number M( )and Casson

Fig. 2. Velocity profile for various values of a and b when C M Q= = = = =0 6 1 3 2 0 6. , , , , .z φ .

Fig. 3. Velocity profile for various values of C kand when M Q= = = = = =1 3 0 2 0 5 2 0 6, , . , . , , .z a b φ .

Fig. 4. (a) Velocity profile for various values of M and z when M C k Q= = = = = = =1 0 6 0 5 0 2 0 5 2 0 6, . , . , . , . , , .a b φ .

Fig. 5. Pressure rise vs volume flow rate for various values of C and kwhen M Q= = = = = =1 3 0 2 0 5 2 0 6, , . , . , , .z a b φ .

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fluid parameter z( ) increase then the number of trapping bolusreduces.

5. Conclusion

In this article, the influence of magnetohydrodynamics on ciliamotion of particle–fluid suspension through a porous planar channelhas been investigated. The governing equation of motion and con-tinuity for fluid phase and particulate phase has been obtained byneglecting the inertial forces and taking the long wavelength as-sumption. The solution for the resulting differential equation hasbeen obtained analytically and the closed-form solution is

presented. The major outcomes for the flow problem are: Velocityof the fluid decreases due to the magnetic field while its behaviouris opposite for porosity parameter. Velocity of the fluid behaves asdecreasing function due to particle volume fraction.

• The behaviour of pressure rise is same for Casson fluid param-eter and particle volume fraction.

• The present analysis can also be reduced to Newtonian fluid bytaking z →∞ as a special case of our study.

• The present investigation depicts various interesting behaviourthat warrants further study on cilia motion with particle volumefraction for various biological fluids.

Fig. 6. Pressure rise vs volume flow volume flow rate for various values of M and z when k C Q= = = = = =0 5 0 6 0 2 0 5 2 0 6. , . , . , . , , .a b φ .

Fig. 7. Stream lines for various values of a a a a, . , . , .a b c( ) = ( ) = ( ) =0 25 0 35 0 4when C M k Q= = = = = = =0 6 1 3 0 5 0 5 2 0 6. , , , . , . , , .z b φ .

Fig. 8. Stream lines for various values of b b b b, . , . , .a b c( ) = ( ) = ( ) =0 2 0 4 0 6 when C M k Q= = = = = = =0 6 1 3 0 2 0 5 2 0 6. , , , . , . , , .z a φ .

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Nomenclature

�a Mean radius of the channelB0 Magnetic field�c Wave velocityPy Yield stressC Volume fraction densityM Hartmann number�P Pressure in fixed frameQ Volume flow rateRe Reynolds numberk Porosity parameter

S Drag force�t Time� �U V, Velocity components in fixed frameX0 Reference position of the cilia� �X Y, Cartesian coordinate axis in fixed frame

Greek symbolsσ Electric conductivity of the fluidλ Wavelengthμs Viscosity of the fluidφ Amplitude ratioρ Fluid density

Fig. 9. Stream lines for various values of C a C b C c C, , . , .( ) = ( ) = ( ) =0 0 1 0 2 when k M Q= = = = = = =0 5 1 3 0 2 0 5 2 0 6. , , , . , . , , .z a b φ .

Fig. 10. Stream lines for various values of k a k b k c k, . , . , .( ) = ( ) = ( ) =0 2 0 3 0 5 when C M Q= = = = = = =0 6 1 3 0 2 0 5 2 0 6. , , , . , . , , .z a b φ .

Fig. 11. Stream lines for various values of M a M b M c M, . , ,( ) = ( ) = ( ) =0 1 1 2 when C k Q= = = = = = =0 6 0 5 3 0 2 0 5 2 0 6. , . , , . , . , , .z a b φ .

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τ Stress tensorβ Wave numberε Ratio of the cilia lengthμb Plastic viscosityγ Shear rateζ Casson fluid parameterα Eccentricity of the elliptic path

Subscriptsf Fluid phasep Particulate phase

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Fig. 12. Stream lines for various values of z z z z, . , ,a b c( ) = ( ) = ( ) =0 5 1 3 when C M k Q= = = = = = =0 6 1 0 5 0 2 0 5 2 0 6. , , . , . , . , , .a b φ .

ARTICLE IN PRESS

Please cite this article in press as: M.M. Bhatti, A. Zeeshan, M.M. Rashidi, Influence of magnetohydrodynamics on metachronal wave of particle-fluid suspension due to cilia motion,Engineering Science and Technology, an International Journal (2016), doi: 10.1016/j.jestch.2016.03.001

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