Cilia-a s sis t e d hyd ro m a g n e tic p u m pin g of bio r h eological cou ple

s t r e s s fluidsRa m e s h, K, Trip a t hi, D a n d Beg, OA

h t t p://dx.doi.o r g/10.1 0 1 6/j.jppr.20 1 8.0 6.00 2

Tit l e Cilia-a s sis t e d hyd ro m a g n e tic p u m pin g of bio r h eological cou ple s t r e s s fluids

Aut h or s R a m e s h, K, Trip a t hi, D a n d Be g, OA

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Q5

Q4Q1

+ MODEL

Propulsion and Power Research 2019;xxx(xxx):1e13

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ORGINAL ARTICLE 727374757677

Cilia-assisted hydromagnetic pumping ofbiorheological couple stress fluids

787980

K. Ramesha, D. Tripathib,*, O. Anwar Bégc 81828384858687

aDepartment of Mathematics, Symbiosis Institute of Technology, Symbiosis International University, Pune, 412115,IndiabDepartment of Mechanical Engineering, Manipal University Jaipur, Rajasthan, 303007, IndiacFluid Mechanics/Propulsion, Aeronautical/Mechanical Engineering Department, Salford University, Manchester,UK

88899091

Received 19 December 2017; accepted 17 June 2018Available online XXXX

9293949596979899100101102103

KEYWORDS

Magnetohydrodynamics;Metachronal waves;Cilia;Synchronous beating;Polar couple stress fluid;Physiological transport;Magnetic blood pumps

*Corresponding author.;

E-mail addresses: dharmtri@gmail.com

Peer review under responsibility of Be

Production and Hosting by

https://doi.org/10.1016/j.jppr.2018.06.0022212-540X/ª 2019 Beihang University. Plicense ().

104105106107108109110

Abstract A theoretical study is conducted for magnetohydrodynamic pumping of electro-conductive couple stress physiological liquids (e.g. blood) through a two-dimensional ciliatedchannel. A geometric model is employed for the cilia which are distributed at equal intervalsand produce a whip-like motion under fluid interaction which obeys an elliptic trajectory. Ametachronal wave is mobilized by the synchronous beating of cilia and the direction of wavepropagation is parallel to the direction of fluid flow. A transverse static magnetic field isimposed transverse to the channel length. The Stokes’ couple stress (polar) rheological modelis utilized to characterize the liquid. The normalized two-dimensional conservation equationsfor mass, longitudinal and transverse momentum are reduced with lubrication approximations(long wavelength and low Reynolds number assumptions) and feature a fourth order linear de-rivative in axial velocity representing couple stress contribution. A coordinate transformation isemployed to map the unsteady problem from the wave laboratory frame to a steady problem inthe wave frame. No slip conditions are imposed at the channel walls. The emerging linearizedboundary value problem is solved analytically and expressions presented for axial (longitudi-nal) velocity, volumetric flow rate, shear stress function and pressure rise. The flow is effec-tively controlled by three geometric parameters, viz cilia eccentricity parameter, wave

, dtripathi@nituk.ac.in (D. Tripathi).

ihang University.

Elsevier on behalf of KeAi

roduction and hosting by Elsevier B.V. on behalf of KeAi. This is an open access article under the CC BY-C-D

111112113114115116117118119120121122

2 K. Ramesh et al.

+ MODEL

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636465

Nomenclature

c wave speed (unit: m$s�1

a mean width of the channl wavelength (unit: m)t time (unit: s)r density (unit: kg$m�3)s electrical conductivity (uB0 magnetic field (unit: T)m dynamic viscosity (unit:Q volume flow rate (unit: mP pressure (unit: Pa)ε cilia length parameter (uU longitudinal velocity (unV transverse velocity (unit:a measure of the eccentricx non-dimensional axial coy dimensionless transverseu non-dimensional axial vev dimensionless transverset dimensionless timep dimensionless pressureb wave numberRe Reynolds numberg couple stress (polar) fluidM Hartmann magnetic numQ dimensionless time-mean

frameF dimensionless time-mean

666768697071727374757677

number and cilia length and two physical parameters, namely magnetohydrodynamic (MHD)body force parameter and couple stress non-Newtonian parameter. Analytical solutions arenumerically evaluated with MATLAB software. Axial velocity is observed to be enhanced inthe core region with greater wave number whereas it is suppressed markedly with increasingcilia length, couple stress and magnetic parameters, with significant flattening of profiles withthe latter two parameters. Axial pressure gradient is decreased with eccentricity parameterwhereas it is elevated with cilia length, in the channel core region. Increasing couple stressand magnetic field parameter respectively enhance and suppress pressure gradient across theentire channel width. The pressure-flow rate relationship is confirmed to be inversely linearand pumping, free pumping and augmented pumping zones are all examined. Bolus trappingis also analyzed. The study is relevant to MHD biomimetic blood pumps.ª 2019 Beihang University. Production and hosting by Elsevier B.V. on behalf of KeAi. This is an open

access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

)el (unit: m)

nit: S$m�1)

kg$m�1$s�1)3$s�1)

nit: m)it: m$s�1)m$s�1)ityordinatecoordinatelocityvelocity

parameterberflow in the laboratory

flow in the wave frame

78798081828384858687888990919293949596979899100101102103104105106107108109110111112113114115116117118119120121122123124

1. Introduction

Ciliated propulsion arises in numerous physiological andbiological systems. Cilia constitute small but intricateappendage structures which protrude from vessel walls.Cilia which average 10 mm in length can flex easily andassist therefore in many sophisticated transport mechanisms.They usually occur in high density arrays unlike flagellawhich usually appear in nature as single structures or pairs.Cilia beating mechanisms, which control the direction ofinduced thrust, therefore differ significantly from flagellar

beating. They exhibit whip-like- motions and appear oncells, plants, physiological organs, marine organisms. Theyexert a substantial role across the biological spectrum andfeature in for example physiology [1], respiration [2], em-bryonic mechano-transduction processes [3], coral reefsystems [4], ventricular cerebrospinal fluid dynamics [5].The beating mechanisms of cilia tend to be dominated bymetachronal waves (these also dictate oscillations offlagella). The highly efficient energy usage of cilia has beendocumented in several studies including Gueron and Levit-Gurevich [6], Guirao and Joanny [7] and Vilfan and Julicher[8] in the latter, it has been shown that optimized perfor-mance is achieved by hydrodynamic interactions whichresult in synchronization of the cilia as nonlinear oscillators.The importance of cilia in sensory reception, mucociliaryclearance and renal physiology disease has been addressedby Inés Ibañez-Tallon [9]. Brueckner [10] has examined thefunction of cilia in embryonic navigation.

The ciliated propulsion problem is attractive to fluiddynamics researchers since it is characterized by manygeometric and viscous hydrodynamic features which areamenable to mathematical modelling. Historically the fieldof biological propulsion, specifically the fluid dynamics of abeating flagellum, was initiated in the early 1950s by Taylor(as lucidly reviewed in Lighthill [11]) who approximated thelocomotion induced on a body by a tail beating in a regularmanner in a Newtonian viscous medium. He derived anapproximate relation connecting the organism velocity tothe tail wave propagation speed. More recently there hasbeen a considerable resurgence in interest in ciliated hy-drodynamics, largely motivated by the emerging areas ofbiomimetics and bio-inspired engineering systems. Gener-ally the approaches for simulating ciliated flows fall into twobroad categories. The first has become known as the enve-lope model (followed in the current article) and the secondis the sub-layer model. In the envelope model the cilia aredensely packed and the engulfing fluid medium interactswith a waving material surface enveloping the top of thelayer. Further sub-categories of the envelop approach featuremodels of an actively driven semi-flexible filament (this

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aims to recreate actual cilia beat mechanisms) and beatshapes optimized based on pumping efficiency. A goodperspective of many different approaches to ciliated hy-drodynamics has been provided by Elgeti and Gompper[12]. Modern efforts have included both theoretical andcomputational analyses. Dauptain et al. [13] presented two-dimensional direct numerical simulation (DNS) results forhydrodynamics of regular oscillating flexible cilia arrays(ctenophore Pleurobrachia pileus), noting that power uti-lized is enhanced with cilia beating frequency, in consis-tency with actual behaviour reported in laboratory testing.Vilfan and Jülicher [14] presented computational simula-tions of flow fields produced by single and dual beating ciliaconfigurations, studying carefully the development of syn-chronized states occur as a function of distance of cilia andrelative beat orientation. Yang et al. [15] presented a nu-merical analysis of an unsteady viscous hydrodynamics foran individual elastic cilium by linking the internal forcegeneration with the surrounding fluid as a simulation ofmulti-ciliary interaction in ovum transport in the oviduct andalso mucus migration in the trachea. They showed thathydrodynamic coupling does lead to both synchrony andmetachrony. Further investigations include the works ofDillon et al. [16] and Dresdner et al. [17].

The above investigations have invariably considered thebiological fluid medium to be Newtonian i.e. they haveignored rheological effects. It is now firmly established formany decades that the vast majority of physiological mediaare characterized by substantial non-Newtonian properties.Accurate simulations therefore require non-Newtonianmodels. These models may be classified as biorheologicalmodels [18] and include viscoplastic fluids (e.g. Binghammodel, Vocadlo model etc), thixotropic (time-dependentshear-thinning) fluids, rheopetic (anti-thixotropic) fluids,viscoelastic fluids (e.g. Oldroyd-B, FENE-P, PPT, Wil-liamson, Maxwell etc) fluids and so on. Unfortunately thesemodels although they include extra terms in the amendedNavier-Stokes equations, the non-Newtonian models do notintroduce length dependent effects which are associated withparticles suspended in biofluids. These give rise to couplestresses which can exert a non-trivial influence on shear (andother) characteristics of physiological liquids. Stokes [19]introduced the “couple stress” fluid model in the mid-1960s to provide a framework for simulating biologicaland other industrial suspensions more precisely, by extend-ing the conventional Cauchy stress. Couple stress or “polar”fluid models therefore are more elegant than the classicalnon-polar models. They also have the advantage thatalthough they lead to boundary value problems with higherorder than the Navier-Stokes, the supplementary terms arelinear. This feature has stimulated considerable interestamong engineers and applied physicists. Stokes [20] pre-sented closed-form solutions for thermal conduction andconvection in couple stress flows. Soundalgekar [21]investigated hydrodynamic solute dispersion in channelcouple stress flows. Devakar et al. [22] considered couplestress fluid flows in several classical cases (Couette,

Poiseuille and generalized Couette flow) observing thatcouple stresses decelerate the flow. Rammkissoon [23]conducted a detailed analytical study of drag effects incouple stress flow past a sphere using a Galerkin-typeformulation and stream function due to a concentratedpoint force. Ramanaiah and Sarkar [24] used the polar modelto analyse lubrication performance of infinitely long sliderbearings, deriving expressions for load capacity, frictionalforce and centre of pressure and noting that greater couplestress effect enhances load capacity but decreases shearstress (wall frictional coefficient) i.e. induces deceleration. Anumber of biological couple stress flow models have alsobeen communicated. These studies have considered peri-staltic pumping in porous media [25], pulsatile blood flows[26], synovial tribology [27], stenotic hemodynamics withcatheter and wall slip [28], micro-organism swimming dy-namics [29], capillary transport with wall permeability[30,31] and mildly stenosed bifurcated arterial blood flow[32]. Excellent details of polar fluid dynamics are providedin Cowin [33] and more recently by Eringen [34].

Many physiological liquids are electrically-conductingowing to the presence of ions and iron content (e.g. hae-moglobin in red blood cells). Such fluids are responsive toelectrical and magnetic fields. Magnetohydrodynamic(MHD) pumping [35] has emerged as a robust mechanismfor transporting such fluids in clinical applications (extra-corporeal surgical blood flow control). MHD pumps utilizethe Lorentz effect which involves the injection of an electriccurrent into two electrodes positioned at sidewalls facingeach other in a micro-channel. This charge injection gen-erates a transversal ionic current in the micro-channel whichcan be regulated by applied magnetic fields. The Lorentzforce acting on the ionic current in the solution mobilizes afluid flow in the micro-channel direction. Bi-directionalpumping is also possible since flow reversal is induced bya reversal of the electric current or the magnetic field vector.Gastric flows may also be doped with magnetic particles toenable MHD endoscopy treatments. Several excellentstudies of MHD pumping in medical engineering have beenmade which have included both theoretical and fabricationaspects. These include Zhong et al. [36] and Lim and Choo[37]. Recently Ponalagusamy and Selvi [38] simulated themagnetohydrodynamic blood flow in stenotic arterial flowwith viscosity and peripheral plasma layer effects. Couplestress magnetohydrodynamic pumping flows have alsoreceived some attention. Tripathi and Bég [39] derivedapproximate solutions for velocity, pressure gradient, localwall shear stress and volumetric flow rate in hydromagneticperistaltic pumping of couple stress fluids, considering bothnon-integral and integral numbers of waves. They high-lighted the retarding effects of both couple stresses andmagnetic field. Srinivasacharya and Rao [40] used a finitedifference numerical method to compute the magnetizedpulsating arterial couple stress blood flow, presentingextensive solutions for shear stress, flow rate and impedancenear the apex, noting the significant deceleration by couplestresses in streaming blood flow near the lateral junction and

4 K. Ramesh et al.

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secondary flow damping by magnetic field near the apex inthe daughter artery. Magnetic polar physiological hydrody-namic include periodic spinning blood magnetohydrody-namic separation device modelling was studied by Bég et al.[41]. Slip hydromagnetic couple stress flow in distensiblechannels was analyzed by El Shennawy and Elkhair [42].Couple stress magnetic biological bearing tribology wassimulated by Naduvinamani et al. [43] in which a Reynoldspolar lubrication equation and the Christensen stochastictheory for roughness were used. Further studies include bio-thermodynamics of two-fluid couple stress liquids withthermal radiation for ablation therapy as examined byMurthy et al. [44] and thermo-magnetic Hall couple stressperistaltic pumping as reported by Hayat et al. [45]. Severalresearchers have also attempted to simulate ciliated mag-netohydrodynamic flows. Siddiqui et al. [46] consideredNewtonian channel flows and derived analytically the ve-locities, stream function and axial pressure gradient as afunction of the cilia and metachronal wave velocity and alsocomputed numerically the pressure rise per wavelength.They showed that trapping is reduced substantially withstronger magnetic field. Akbar et al. [47] used a viscoplasticfluid model to investigate hydromagnetic metachronal cili-ated flow and heat transfer under an inclined magnetic fieldwith wall slip and thermal jump effects. Akbar et al. [48]further investigated the ciliated propulsion of magnetizednanofluids using an elliptic beating cilia model. Bhatti et al.[49] have presented a theoretical study on the effects ofmagnetohydrodynamics on the peristaltic flow of Jeffreyfluid in a rectangular duct. Hayat et al. [50] have discussedthe magnetohydrodynamic peristaltic motion based on theconstitutive equations of a Carreau fluid in a channel.Saleem et al. [51] have constructed a mathematical modelfor MHD blood flow through an artery with mild stenosis.Saleem and Munawar [52] have studied the flow of bloodthrough a stenotic artery in the presence of a uniformmagnetic field. Ellahi et al. [53] have studied the peristalticflow of a viscous fluid in a non-uniform rectangular duct.Bhatti et al. [54] have discussed the simultaneous effects ofcoagulation (blood clot) and variable magnetic field onperistaltically induced motion of non-Newtonian Jeffreynanofluid containing gyrotactic microorganisms through anannulus. Bhatti et al. [55] have investigated the effects ofvariable magnetic field on peristaltic flow of Jeffrey fluid ina non-uniform rectangular duct having compliant walls.

In the current work a mathematical model is developedfor hydromagnetic pumping of electrically-conductingcouple stress physiological liquids (e.g. blood) through atwo-dimensional ciliated channel under with metachronalwaves generated by beating of the cilia. The Stokes’ couplestress (polar) rheological model is employed. Closed-formsolutions are developed for the problem under lubricationapproximations and extensive computations are presentedusing MATLAB symbolic software for axial pressuregradient, axial (longitudinal) velocity, volumetric flow rate,wall shear stress function and pressure rise. Streamlinevisualization is also included to scrutinize bolus (trapping)

phenomena. The study is relevant to MHD biomimeticblood pumps [36] and also may be of interest in simulatingthe hydrodynamics of micro-/nano-scale robots in biomed-ical devices [56].

2. Mathematical model

Consider the flow of an incompressible electrically-conducting couple stress non-Newtonian fluid through asymmetric channel with inner surfaces that are ciliated. Weselect Cartesian co-ordinates (X, Y ), where the X-axis liesalong the centre of the body and Y-axis is transverse to thefluid flow. The flow is generated due to a metachronal wavepropagation which is produced by the collective, rhythmicbeating of the cilia with constant wave speed c, along thechannel walls. The schematic structure of the problem underconsideration is depicted in Fig. 1. The geometry of themetachronal wave form proposes that the whip-like dy-namics of the cilia may be simulated according to:

Y Z f ðX ; tÞZ��aþ aε cos

�2p

lðX � ctÞ

��Z�H ð1Þ

The experimental studies of Sleigh [57] confirm that thecilia tips move in elliptical paths. Therefore, the verticalposition of the cilia tips can be written as:

X ZgðX ; tÞZX0 þ aεa sin

�2p

lðX � ctÞ

�ð2Þ

Here a is the mean width, H is the semi-width of the channel, l isthe wavelength, t is time, a is the measure of the eccentricity, ε isthe cilia length parameter and X0 is indicated position of the par-ticle. If the no slip hydrodynamic condition is imposed at thechannel walls, then the velocities of the transporting fluid aremerely those produced by the cilia tips, which can be given as:

UZvX

vt

����X0

Zvg

vtþ vg

vX

vX

vtZ

vg

vtþ vg

vXU ð3Þ

V ZvY

vt

����X0

Zvf

vtþ vf

vX

vX

vtZ

vf

vtþ vf

vXU ð4Þ

Invoking Eqs. (1) and (2) in Eqs. (3) and (4), readilyyields the following expressions for the longitudinal ve-locity (U ) and transverse velocity (V) components at theboundaries:

UZ��

2pl

�acεa cos

�2plðX � ctÞ�

1� �2pl

�aεa cos

�2plðX � ctÞ� ð5Þ

VZ

�2pl

�acεa sin

�2plðX � ctÞ�

1� �2pl

�aεa cos

�2plðX � ctÞ� ð6Þ

Fig. 1 Symmetric MHD metachronal wave propagation in a channel with ciliated walls.

Biorheological couple stress fluids 5

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In view of this formulation of the boundary conditionswe are able to distinguish between the effective stroke of thecilia and the effective recovery stroke by approximatelyaccounting for the shortening of the cilia. The governingequations for two-dimensional unsteady flow of an incom-pressible electrically-conducting couple stress (polar) fluidsunder the effect of a magnetic field applied transverse to theXeY plane (which generates two Lorentzian drag compo-nents in the X and Y directions respectively, viz �sB2

0U and� sB2

0V ) may be written as follows:

vU

vXþvV

vYZ0 ð7Þ

r

�vU

vtþU

vU

vXþV

vU

vY

�Z� vP

vXþ m

�v2U

vX 2 þv2U

vY 2

�

�h

�v4U

vX 4 þ 2v4U

vX 2vY 2 þv4U

vY 4

�� sB2

0U

ð8Þ

r

�vV

vtþU

vV

vXþV

vV

vY

�Z� vP

vYþ m

�v2V

vX 2 þv2V

vY 2

�

�h

�v4V

vX 4 þ 2v4V

vX 2vY 2 þv4V

vY 4

�� sB2

0U

ð9Þ

Here vPvX is axial pressure gradient and vP

vY is transverse pressuregradient. To determine analytical solutions, it is advantageous todefine the transformation between the laboratory (unsteady) frameand wave (steady) frame, for the present moving boundary valueproblem. The appropriate coordinate transformations are definedas:

xZX � ct; yZY ;uðx; yÞZUðX ;Y ; tÞ � cvðx; yÞZV ðX ;Y ; tÞ;pðx; yÞZpðX ;Y ; tÞ ð10Þ

Furthermore, we introduce the following non-dimensional parameters and hydrodynamic parameters:

xZx

l; yZ

y

a;uZ

u

c; vZ

v

c;HZ

H

a; tZ

ct

l

pZa2p

lmc;bZ

a

l;ReZ

rca

m

gZ

ffiffiffim

h

ra;MZ

ffiffiffis

m

rB0a

ð11Þ

Where x; y; u; v; H ; t; p ; b; Re; g; M ;denote non-dimensional axial coordinate, dimensionless transversecoordinate, non-dimensional axial velocity, dimensionless trans-verse velocity, dimensionless channel semi-width, dimensionlesstime, dimensionless pressure, wave number, Reynolds number,couple stress (polar) fluid parameter and Hartmann magneticnumber, respectively. In the limit g/N, polar effects areneglected and the governing equations reduce to the classicalNavier-Stokes equations for magnetohydrodynamic ciliated pro-pulsion of a Newtonian fluid. Applying lubrication theory ap-proximations i.e. long wavelength (b << 1) and low Reynoldsnumber (Re/0) (i.e. Stokes flow) assumptions, implementingEqs. (10) and (11) in Eqs. (1)e(9), after dropping the bars, thefollowing linearized equations are generated:

vp

vxZ

v2u

vy2� 1

g2

v4u

vy4�M 2ðuþ1Þ ð12Þ

vp

vyZ0 ð13Þ

vu

vyZ0;

v3u

vy3Z0 at yZ0 ð14Þ

The non-dimensional forms of the physical boundaryconditions emerge as:

6 K. Ramesh et al.

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uZ� 1� 2pεab cosð2pxÞ1� 2pεab cosð2pxÞ

v2u

vy2Z0 at yZhZ1þ ε cosð2pxÞ:

ð15Þ

It is important to note here that couple stress fluids aremuch simpler than another group of microstructural non-Newtonian fluids i.e. micropolar fluids [58], since theypossess no microstructure at the kinematic level and there-fore the kinematics of such fluids is totally described usingthe velocity field i.e. angular velocity (micro-rotation) fieldis not required. The higher order gradient boundary condi-tions for axial velocity in Eqs. (14) and (15) are the so-called“stress-free conditions” which physically imply that couplestresses vanish at the channel wall inner surfaces.

The instantaneous volume flow rate in the fixed frame isgiven by

QZ

ZH0

UðX ;Y ; tÞdY ð16Þ

The above expression in the wave frame becomes

qZ

Zh

0

uðx; yÞdy ð17Þ

From Eqs. (10), (16) and (17), the expression for thevolume flow rate can be written as

QZqþ ch ð18Þ

The time-mean flow over time period T at fixed positionX is defined as

QZ1

T

ZT

0

Qdt ð19Þ

Using Eq. (18) into Eq. (19) and then integrating, yields

QZqþ ac ð20Þ

Defining the dimensionless time-mean flows, Q and F, inthe laboratory and wave frame, respectively, as

QZQ

acand FZ

q

acð21Þ

From Eq. (20), the flow rate in the non-dimensional formcan be written as follows:

SfZm1m2

ðm22 �m2

1

�� 1

1� 2pεab cosð2pxÞ �1

M 2

dp

dxþ 1

g2

��m1 sinhðm2yÞcoshðm2hÞ �m2 sinhðm1yÞ

coshðm1hÞ �m1m2

g2

�m2 sinhðm2yÞcoshðm2hÞ �m1 sinh

coshð

QZF þ 1 ð22Þ

where. FZR h0 udy

3. Analytical solutions

The normalized boundary value problem is well-posed andadmits exact solutions. The closed-form solutions of Eqs.(12) and (13) subject to the dimensionless boundary con-ditions Eqs. (14) and (15) are obtained as:

uZ1

ðm22 �m2

1

�� 1

1� 2pεab cosð2pxÞ �1

M 2

dp

dxþ 1

g2

��m2

1 coshðm2yÞcoshðm2hÞ �m2

2 coshðm1yÞcoshðm1hÞ

�

� 1

M 2

dp

dxþ 1

g2

ð23Þdp

dxZ

F �A1

A2ð24Þ

where,

m1Z

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffig2 þ g

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffig2 � 4M 2

p2

sð25aÞ

m2Z

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffig2 � g

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffig2 � 4M 2

p2

sð25bÞ

A1Z1

ðm22 �m2

1

�� 1

1� 2pεab cosð2pxÞ þ1

g2

��m2

1 sinhðm2hÞm2 coshðm2hÞ �

m22 sinhðm1hÞ

m1 coshðm1hÞ�þ 1

g2

ð25cÞ

A2Z � 1

M 2

�hþ 1

ðm22 �m2

1Þ�m2

1 sinhðm2hÞm2 coshðm2hÞ

� m22 sinhðm1hÞ

m1 coshðm1hÞ��

ð25dÞ

The dimensionless wall shear stress distribution for thepresent problem is given by

Sf Zvu

vy� 1

g2

v3u

vy3ð26Þ

ðm1yÞm1hÞ

�� ð27Þ

116117118119120121122123124

Biorheological couple stress fluids 7

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Upon substituting the Eq. (18) into Eq. (20), we obtainthe solution for shear stress distribution as follows:

The non-dimensional pressure rise ðDpÞ is computed bythe following expression:

DpZ

Z1

0

�dp

dx

�dx ð28Þ

The stream function in the wave frame obeying the

Cauchy-Riemann equations

�uZ vj

vy and vZ � vjvx

�may

be computed by using Eq. (23). Numerical evaluation ofthe exact solutions is executed in MATLAB which permitsa parametric assessment of the flow characteristics. Alsovisualization of streamlines is achieved with MATLABsoftware which allows an examination of bolus formationand trapping.

Fig. 2 Velocity profiles for different flow parameters for fixed valuesof. ðaÞ εZ0:5; aZ0:2; xZ1; QZ2; gZ3; MZ2:5; ðbÞ aZ0:1;

bZ0:1; xZ1; QZ2; gZ2; MZ3ðcÞ εZ0:5; aZ0:1; bZ0:1;

xZ1; QZ2; MZ3; ðdÞ εZ0:5; aZ0:1; bZ0:1; xZ1; QZ2;

gZ3.

84858687888990919293949596979899100101102103104105106107108109110111112113114115116117118119120121122123124

4. Computational results and interpretation

Selected computations for velocity, axial pressure gradient,shear stress and pressure rise characteristics, and alsostreamline distributions are illustrated in Figs. 2e7.Generally flow rate (Q) is prescribed as 2 and a relativelystrong magnetic field is imposed (M is at least set to 2 in allthe graphs, unless otherwise indicated, in consistency withactual MHD blood pump devices e see Zhong et al. [36]and Lim and Choi [37] wherein generally double the mag-netic Lorentz force relative to the viscous hydrodynamicforce is recommended). All data is provided in the captions.

Table 1 gives the comparison of velocity distribution forthe present and earlier literature (Newtonian fluid). In thebenchmark study of Ramesh [59], cilia effects were notconsidered, therefore in our model we have set ε (cilialength parameter) Z 0, a (measure of the eccentricity) Z 0,b (wave number) Z 0, Q (ddimensionless time-mean flowin the laboratory frame) Z 1, g (couple stress (polar) fluidparameter)/N (i.e. Newtonian case) and M (Hartmannmagnetic number)/0 (electrically non-conducting case i.e.vanishing magnetic field). Furthermore, porous medium andslip effects do not appear in our model and therefore datafrom Ramesh [59] which corresponds to the simplest case inthat study (i.e. where the porous medium vanishes [infinitepermeability] and there is no channel wall slip) is selected toensure that the exact same conditions are enforced in bothmodels. The legend used in Table 1 therefore is ε Z 0:5;aZ 0; bZ 0; QZ 1; g/N; M/0:It is evident

that our results are in close agreement with that of Ramesh[59]. Confidence in the present analysis is therefore justifi-ably high.

Fig. 2(a)e(d) illustrate the evolution in axial velocityacross the entire channel span (�1.5 � y�þ1.5) forrespectively a variation in metachronal wave number (b),cilia length (ε), couple stress parameter (g) and magnetic

field parameter (M ). Generally with the exception ofFig. 2(c), different responses are induced in the core zoneand the edge zones (near the ciliated walls). Increasingwave number (Fig. 2(a)) is found to accelerate the axial

Fig. 3 Pressure gradient profiles for different flow parameters forfixed values of. ðaÞ εZ0:5; bZ0:1; yZ0; QZ1; gZ7; MZ7; ðbÞaZ0:1; bZ0:1; yZ0; QZ1; gZ7; MZ7ðcÞ εZ0:5; aZ0:1; bZ0:1; yZ0; QZ1; MZ7; ðdÞ εZ0:5; aZ0:1; bZ0:1; yZ0;QZ1;

gZ7.

Fig. 4 Pressure rise profiles for different flow parameters for fixedvalues of. ðaÞ εZ0:5; aZ0:1; yZ0; gZ2; MZ2; ðbÞ aZ0:1; bZ0:1; yZ0; gZ2; MZ2ðcÞ εZ0:5; aZ0:1; bZ0:1; yZ0; MZ2;

ðdÞ εZ0:5; aZ0:1; bZ0:1; yZ0; gZ2.

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flow in the core zone but to decelerate it in the end zones.Peak axial velocity is computed logically at the channelcentre line.

The parameter bZ al, where a is the mean width and l

is the wavelength of the metachronal wave. Clearly greaterwavelengths reduce the wave number and vice versa forshorter wavelengths. In the channel core region, shorter

Fig. 5 Shear stress distribution for different flow parameters forfixed values of ðaÞ εZ0:5; bZ0:1; QZ2;gZ7; MZ7;

ðbÞ aZ0:1; bZ0:1; QZ2;gZ3; MZ3ðcÞ εZ0:5; aZ0:1; bZ0:1; QZ2;MZ3; ðdÞ εZ0:5; aZ0:1; bZ0:1; QZ2;gZ5.

Fig. 6 Streamline patterns for different values of wave number forfixed values of. ε Z 0:5; aZ 0:1; QZ 1;gZ 2; MZ 2:5

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wavelengths therefore result in acceleration whereas theyinduce deceleration in the proximity of the ciliated walls.The converse response is associated with larger wavelengths(smaller wave numbers). A similar observation for power-

Fig. 7 Streamline patterns for different values of Hartmann numberfor fixed values of. ε Z 0:5; aZ 0:1; QZ 1;gZ 2; bZ 0:1.

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law and Newtonian fluids in ciliated channel wave propa-gation has been reported by Siddiqui et al. [60]. Fig. 2(b)shows that increasing cilia length induces a strong decel-eration in the core region and also causes a substantial

acceleration in the end zones (near wall regions). Clearlygreater cilia length will offer more resistance to the axialflow which will impede pumping in the core region. Theimpact of cilia length is minimal at the walls since theprotuberances exert a greater effect deeper into the channel.This manifests in a boost in axial velocity at the boundaries.Several studies have highlighted this including Sleigh [61]and more recently Khaderi et al. [62]. Elasticity of thecilia will also inevitably influence axial flow although thisaspect has not been considered in the present work. It couldbe simulated in future by using a ratio between the viscousforce and the elastic force in the cilia [62]. An increase incouple stress (polar) parameter, g, as depicted in Fig. 2(c), isfound to consistently decrease the axial velocity. The rise ing, clearly leads to a reduction in the couple stress term inEq. (12) � 1

g2v4uvy4. The retarding nature of couple stresses as

demonstrated in many other investigations [18e25] isconfirmed by the present computations. Symmetry is sus-tained across the channel span in consistency with the wallboundary conditions. However very little flow reversal isinduced compared with the near-wall axial flow fieldresponse in Fig. 2(a) and (b). Fig. 2(d) shows that an in-crease in magnetic body force parameter, M, stifles the bulkaxial flow in the core zone but accelerates it towards thewall zones. This is a classical result from magnetohydro-dynamics, confirmed in many standard works includingSutton and Sherman [63]. Significant backflow is inducedclose to both the lower and upper channel walls (approxi-mately one third of the plots are below the zero line).Magnetic field is generally found to inhibit bulk flow of thecouple stress fluid which is desirable in MHD pumpingoperations for enhanced flow control [36].

Fig. 3(a)e(d) present the response in axial pressuregradient (dp/dx) to alteration in cilia eccentricity parameter(a), cilia length (ε), couple stress parameter (g) and mag-netic field parameter (M ) plotted against axial coordinate(channel length coordinate). An increase in eccentricity ofthe beating cilia induces a weak deceleration in the coreflow and a corresponding acceleration in the edge flow, asillustrated in Fig. 3(a). The modifications in pressuregradient are more pronounced at the walls than in the coreregion. An increase in cilia length has a much more sig-nificant impact than cilia eccentricity, as plotted in Fig. 3(b).Considerable elevation is produced in the axial pressuregradient in the core zone with even greater depletion in axialpressure gradient in the end zones. Clearly the presence oflonger elastic appendages (cilia) emanating from the wallsinteracts more intensively with the flow field than shortercilia. There is never any skewness in the profiles which areall perfectly symmetric about the channel centre line. Anincrease in axial pressure gradient is generated with greatercouple stress parameter (Fig. 3(c)) and once again this trendis enforced across the entire channel span. This enhance-ment in pressure gradient is physically consistent with thedeceleration in axial flow computed earlier. Increasingmagnetic field is also observed to diminish the axial pres-sure gradient, a characteristic feature of MHD pumps

Table 1 Comparison of velocity profile when.ε Z 0:5; aZ 0; bZ 0; QZ 1; g/N; M/0.

y Newtonian fluid Ramesh [59] Present study

�1.5 �1.000000000000000 �0.999999999998181 �1.000000000000000�1.2 �0.460000008344650 �0.460057371537914 �0.460063751198660�0.9 �0.040000021457672 �0.040003307696679 �0.040008760610394�0.6 0.260000005364418 0.260032609010523 0.260030306976534�0.3 0.439999967813492 0.440053078913479 0.4400534515621230 0.499999970197678 0.500059722193328 0.5000621529792110.3 0.439999967813492 0.440053078913479 0.4400534515621230.6 0.260000005364418 0.260032609010523 0.2600303069765340.9 �0.040000021457672 �0.040003307696679 �0.0400087606103941.2 �0.460000008344650 �0.460057371537914 �0.4600637511986601.5 �1.000000000000000 �0.999999999998181 �1.000000000000000

Biorheological couple stress fluids 11

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[36,37]. This reduction is sustained across the entire channeli.e. there is no switch-over in profiles at the walls.

Fig. 4(a)e(d) illustrate the evolution of pressure rise (Dp)with volumetric flow rate (Q) with variation in respectivelymetachronal wave number (b), cilia length (ε), couple stressparameter (g) and magnetic field parameter (M ). Theclassical inverse relationship between pressure rise andvolumetric flow rate (known for Newtonian fluids-see e.g.Wu [1] is also obtained for couple stress (polar) fluids for allplots. With increasing wave number (Fig. 4(a)), pressure riseis consistently decreased for all flow rates. However, withincreasing cilia length (Fig. 4(b)), pressure rise is elevated inthe pumping region ðDp > 0Þ whereas it is depressed in theaugmented pumping region Dp < 0, with increasing flowrate. It is known that cilia spacing and length influences theviscous resistance per cilium and thereby the axial flow. Thelatter is assisted with greater cilia length and this aids inpressure rise in the lower channel half space (Fig. 4(b)). Theintroduction of extra energy to the flow at the lower wallhowever must be compensated for by an extraction at theupper wall, and these features are also related to synchro-nicity of beating cilia [2e10]. The pressure rise is thereforefound to decrease with greater cilia length in the upperchannel half space. The special case ε Z 0 although notplotted implies vanishing cilia and absence of a metachronalwave - in this scenario the flow is a purely peristalticmechanism due to flexibility of the walls. Maximum pres-sure rise therefore occurs for εZ 0.6, in the pumping regionand the minimum pressure rise also corresponds to ε Z 0.6,but in the augmented region. There is a significantly greaterspread of linear decay profiles in Fig. 4(b) compared withthe other plots. In the pumping region ðDp > 0Þ, the pres-sure rise decreases with greater couple stress parameter (g),whereas the opposite effect is induced in the augmentedpumping region, as observed in Fig. 4(c). An increase inmagnetic field parameter is conversely found to enhancepressure rise in the pumping region whereas it reducespressure rise in the augmented pumping region, as shown inFig. 4(d). In all the plots there exist also free pumping zones(Dp Z 0) although this is essentially a cross-over point inFig. 4(b)e(d). The impact of wave number on pressuredifference is generally weak at all flow rates; however the

cilia length, couple stress parameter and magnetic fieldimpart a tangible modification to the pressure difference inthe channel.

Fig. 5(a)e(d) present the shear stress distributions forvarious values of cilia eccentricity (a), cilia length (ε),couple stress parameter (g) and magnetic field parameter(M ). The oscillatory nature of shear (frictional wall force) isclearly captured and this is in synchrony with the meta-chronal beating of the cilia. Peaks and troughs are observedalternately in each of the plots. Increasing eccentricity ratiois found in Fig. 5(a) to considerably elevate the peakmagnitudes of shearing stress but only weakly decrease theminimum values. More oblate topologies are also observedat the peaks compared with the troughs which are sharper.The shear stress direction is however never changed with avariation in eccentricity ratio (values are invariably nega-tive) i.e. there is no flow reversal in the MHD pump chan-nel. Fig. 5(b) indicates that peak shear stress values aremildly elevated with increasing cilia length whereas thetroughs are very sharply decreased. The influence of cilialength is markedly greater than cilia eccentricity. Fig. 5(c)shows that with increasing couple stress parameter both thepeaks and troughs are strongly reduced which is consistentwith flow deceleration in polar fluids. In this plot there isalso a weak flow reversal computed at low values of couplestress parameter, although this is quickly eliminated withhigher values. Fig. 5(d) illustrates that increasing magneticfield similarly damps the shear stress strongly. The greaterLorentzian magnetic drag generated with higher M valuesretards the ciliated propulsion and the couple stress fluidshears therefore more slowly against the inner surfaces ofthe walls.

Finally in Fig. 6(a)e(d) and 7(a)e(d), MATLAB softwarehas been used to plot streamline visualizations for the influ-ence of two parameters, namely metachronal wave number(b) and magnetic field parameter (M ), respectively. In bothsets of plots the doubly symmetric dual bolus structure pat-terns are captured.With increasingmetachronal wave number(b) i.e. decreasingwave length the central streamlines becomeincreasingly distorted generating new bolus structures whichgrow symmetrically outwards into the channel space.Conversely with increasing magnetic parameter the newly

Q2

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emerging bolus structures are contracted and vortex intensityis decreased in the channel. Trapping phenomena are there-fore non-trivially influenced by both the metachronal wavelength and applied magnetic field.

5. Conclusions

A theoretical investigation is presented for metachronal-wave generated propulsion of electro-conductive couplestress fluids in a channel with inner ciliated walls, motivatedby simulating bio-inspired MHD pumps. The metachronalwave is aligned to the axial pumping direction and issimulated via an elliptical geometric model. Closed-formsolutions for the transformed linearized boundary valueproblem are derived. Visualization of the analytical solu-tions is evaluated with symbolic software, MATLAB. Thepresent computations have shown that:

� There is an inverse relationship between pressure rise andflow rate for variation in all geometric, material andmagnetic control parameters.

� Increasing metachronal wave number elevates axial ve-locity in the core zone (whereas it induces axial flowretardation at the channel walls), generates decreasingpressure difference in the pumping and augmentedpumping regions and encourages the growth of newsecondary bolus structures in the channel.

� Increasing cilia eccentricity generates decreases axialpressure gradient in the core region (and increases it inthe near-wall zones),

� Increasing cilia length decelerates the axial flow, in-creases the axial pressure gradient (in the core region),boosts pressure difference in the pumping region (posi-tive pressures) and reduces it in the augmented pumpingregion (negative pressures) and furthermore weakly in-creases peak shear stresses and strongly reduces troughmagnitudes.

� Increasing couple stress parameter retards the axial flow,elevates the axial pressure gradient across the full span ofthe channel, reduces pressure difference in the pumpingzone, increases pressure difference in the augmentedzone and enhances both peak and trough shear stressvalues.

� Increasing magnetic body force parameter decelerates theaxial flow, reduces axial pressure gradient across theentire channel span, enhances pressure difference in thepumping zone, suppresses pressure difference in theaugmented pumping zone, depresses both peak andtrough shear stress values and mitigates the developmentof new bolus structures in the channel space.

The present study has presented a simple but interestingmodel for MHD physiological ciliated pumping systems. Ithas provided some useful benchmarks for further analysiswhich is presently being explored with ANSYS FLUENTcomputational fluid dynamics software. This will allowgeneralization to three-dimensional geometries and will also

incorporation of elasticity of the ciliated structures. Effortsin this regard will be communicated in the near future.

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