Mathematical modelling of peristaltic propulsion of viscoplastic bio-fluids.

Proceedings of the Institution of Mechanical Engineers, Part H: Journal of Engineering in Medicine http://pih.sagepub.com/

Mathematical modelling of peristaltic propulsion of viscoplastic bio-fluids D Tripathi and Osman A Be´g Proceedings of the Institution of Mechanical Engineers, Part H: Journal of Engineering in Medicine published online 29 November 2013 DOI: 10.1177/0954411913511584 The online version of this article can be found at: http://pih.sagepub.com/content/early/2013/11/29/0954411913511584

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Original Article

Mathematical modelling of peristaltic propulsion of viscoplastic bio-fluids

Proc IMechE Part H: J Engineering in Medicine 0(0) 1–22 Ó IMechE 2013 Reprints and permissions: sagepub.co.uk/journalsPermissions.nav DOI: 10.1177/0954411913511584 pih.sagepub.com

D Tripathi1 and Osman A Be´g2

Abstract This article studies theoretically the transportation of rheological viscoplastic fluids through physiological vessels by continuous muscle contraction and relaxation, that is, peristalsis. Both cases of planar and cylindrical physiological vessels are considered. A mathematical model is developed under long wavelength and low Reynolds number approximations. Expressions for axial velocity in core region, axial velocity in plug flow region, volume flow rate and pressure gradient in non-dimensional form are obtained. A comparative study of velocity profiles, pressure distribution, friction force and mechanical efficiency for different viscoplastic liquids is conducted. The influence of width of plug flow region, shear rate strain index and yield stress index on the pressure distribution, friction force and mechanical efficiency is elaborated. The study is relevant to gastric fluid mechanics and also non-Newtonian biomimetic pump hazardous waste systems exploiting peristaltic mechanisms.

Keywords Peristalsis, generalized viscoplastic fluids, pressure gradient, fiction force, mechanical efficiency, yield stress, mathematical modelling, gastric hydrodynamics, biomimetic chemical waste pumps

Date received: 1 February 2013; accepted: 15 October 2013

Introduction Physiological fluids are transported into the human body by a continuous process of muscle contraction and relaxation. This process is known as peristaltic transport. The most recognizable example is human swallowing,1 in which the oesophagus conveys masticated food peristaltically from the mouth to stomach. This is achieved even when the human body is inverted. Numerous other systems in the human body feature peristaltic flows, including uretral hydrodynamics,2,3 cardiovascular flows,4 intestinal chyme movement,5 bile migration6,7 and spermatozoa propulsion in the cervical canal.8 The periodic lateral movements of these vessels are controlled by intricate electrochemical reactions taking place in the body. In medical device technology, peristaltic flow mechanisms have also been exploited. Significant developments in this regard include peristaltic pumps for insulin in diabetes treatments,9 antithrombogenic peristaltic micropumps for haematological testing10 and electrostatic pumps for pharmacological delivery.11 The complex nature of peristaltic flow has stimulated significant attention in terms of mathematical and numerical modelling. Aranda et al.12 utilized a three-dimensional computational model based on the method of regularized

Stokeslets to simulate peristaltic pumping of a viscous fluid in non-axisymmetric tube as a model of ovum transport in the oviduct, identifying conditions for maximum mean flow rate. Xiao and Damodaran13 used a finite volume computational method to study the moderate to high Reynolds number peristaltic flow in a circular tube, presenting new visualizations for the effect of the wave amplitude, wavelength and Reynolds number on the ‘flow trapping’ mechanism. JimenezLozano and Sen14 investigated streamline patterns and their local and global bifurcations in a two-dimensional planar and axisymmetric peristaltic flow for an incompressible Newtonian fluid using methods of dynamical systems. They identified three different flow scenarios, namely, backwards flow, trapping and augmented flow each associated with different bifurcations. Tripathi and Be´g15 investigated analytically the transient

1

Department of Mathematics, National Institute of Technology Delhi, Delhi, India 2 Gort Engovation Research (Propulsion and Biomechanics), Bradford, UK Corresponding author: Osman A Be´g, Gort Engovation Research (Propulsion and Biomechanics), 15 Southmere Avenue, Great Horton, Bradford, BD7 3NU, UK. Email: [email protected]


2

Proc IMechE Part H: J Engineering in Medicine 0(0)

hydromagnetic peristaltic flow and heat transport in a finite length physiological vessel, showing that greater pressure is needed to propel the fluid by peristaltic pumping in comparison to a non-conducting Newtonian fluid, whereas a lower pressure is required if heat transfer is effective. The above studies have been confined to Newtonian fluids. However, many physiological liquids are known to be non-Newtonian in nature, including bile, blood, mucus, digestive fluids and so on.16 These fluids do not obey the linear relationship between the shear stress and shear rate as do classical Newtonian fluids. An important subclass of non-Newtonian fluids is the viscoplastic or ‘yield stress’ fluids, which exhibit both viscous and plastic properties. Such liquids do not flow under the imposition of low shear stress. The shear stress must exceed a critical value known as the yield stress for the fluid to flow. Viscoplastic fluids arise in many diverse areas of medical and biotechnological engineering, including pharmacological creams,17 food stuffs,18 white blood cell suspensions,19 capillary fluids20 and biomembranes.21 Popular viscoplastic models include the Casson model,18,20,22,23 Herschel–Bulkley (H-B) model24 and Bingham plastic (B-P) model.25 Gnoevoi et al.26 investigated the viscoplastic Bingham flow in deformable conduits using a thin-layer approximation. Vishnyakov et al.27 used the ‘narrowband’ asymptotic method to study peristaltic flow of a viscoplastic H-B liquid in a slot channel, demonstrating that the mode of flow deviates significantly from that in a channel with rigid walls when the axial pressure gradient is low. Muravleva and Muravleva28 studied peristaltic flow of a Bingham–Il’yushin viscoplastic liquid evaluating the effects of geometry and the yield point on flow characteristics. Sankar29 obtained analytical solutions for peristaltic transport of Casson fluid in a channel and also in an inclined tube is analysed under long wavelength and low Reynolds number assumptions, showing that pressure rise decreases with the increase of the time mean flow rate and angle of inclination and is enhanced with the increase of stress ratio and amplitude ratio. He further observed that the frictional force is elevated with stress ratio, whereas it is depressed with increase in inclination angle. Further studies of viscoplastic biological flows in undulating conduits have been reported by Samy,30 Misra and Pandey31 and Mernone et al.32 using the Casson model and by Vajravelu et al.33 with the B-P model. In these studies, solutions were derived to study the effects of plug flow region (i.e. yield stress) and index of shear rate strain (i.e. shear thickening and shear thinning) in channel and cylindrical tube geometries. Viscoplastic flows are also of considerable interest in the analysis of biotechnological transport phenomena. Food engineers have experimentally categorized the viscoplastic rheological characteristics of numerous

consumable materials. Recent studies include Prakash et al.34 who studied deep-fat fried coated products, Sai Manohar et al.35 who obtained the rheological properties of a semi-liquid breakfast foods and Sopade et al.36 who studied the rheological behaviours of food thickeners. A further important viscoplastic model in biotechnology and also liquid swallowing is the Mizrahi–Berk (M-B) model,37 originally developed to simulate non-Newtonian behaviour of orange juice. Yet, another popular viscoplastic model is the Vocˇadlo model,38 that is, Robertson–Stiff model,39,40 which has become popular in simulating both biophysics and also biopolymer pumping liquids for petroleum engineering. These rheological models are also of relevance in modelling waste sludge41 and alkali halide suspension flows in biochemical engineering.42 The previous investigations of peristaltic transport of viscoplastic fluids only considered either B-P model, Casson model or H-B models or subsequent modifications of these models. These previous studies do not examine and compare peristaltic flows of a full range of viscoplastic models. This is the objective of the present study, that is, a comparative analysis of the peristaltic flow of generalized viscoplastic fluids, which spans the principal robust mathematical models available for viscoplastic models (H-B, M-B, Casson, Vocˇadlo and B-P models) and furthermore includes the generalized Newtonian model and also the power law model. Emphasis is focused on evaluating the influence of yield stress index on the peristaltic propulsion, an aspect which is neglected in the existing literature. We study the flows in channel and axisymmetric geometries. Our model is mainly applicable in the biophysics of viscoplastic food bolus conveyance through the human alimentary canal, chyme migration in the small intestine and haemodynamics in narrow vessels. It is also useful in simulating peristaltic pump flow characteristics for medical and hazardous material disposal applications.43 The empirical data available for certain foods are also studied in relation to swallowing. Graphical results are included for pressure and friction force distributions.

Viscoplastic bio-fluid constitutive models We consider the peristaltic transport of rheological fluids modelled by a generalized viscoplastic model. The constitutive equations for such fluids are as follows ~t m = ~t0m + mg_ n , g_ = 0,

for ~t 5 ~t0

for ~t 4 ~t0

ð1Þ ð2Þ

where ~t is the shear stress, ~t0 is the yield stress, g_ is the rate of shear strain, m is the coefficient of dynamic viscosity, m is the index for yield stress and n is index of shear rate. The limiting cases of equations (1) and (2) are as follows


Tripathi and Be´g

3

(i) m = 1,

~t = ~t 0 + mg_ n

(HerschelBulkley model)

(ii) m = 1=2,

1=2 ~t = ~t0 + mg_ n 1=2 ~t 1=2 = ~t0 + mg_ 1=2 ~t m = ~t0m + mg_ 1=m ~t 1=m = ~t0 + mg_

(MizrahiBerk model)

1=2

(iii) m = 1=2, n = 1=2, (iv) n = 1, (v) m ! 1=m, n = 1, (vi) m = 1, n = 1, (vii) m = 1, ~t 0 = 0,

~t = ~t 0 + mg_ ~t = mg_ n (viii) m = 1, n = 1, ~t 0 = 0, ~t = mg_

(Casson model) (Generalized Bingham plastic model) (Bingham plastic model) (Power-law model) (Newtonian model)

Non-dimensional analysis and flow models Herein, two conduit geometries are studied – twodimensional channel flow and axisymmetric pipe flow. To facilitate analytical solutions, the conservation equations in primitive independent spatial and temporal variables (x, y, t) are transformed to non-dimensional form. For the channel flow case, the non-dimensional parameters are as follows ~

j = lj , h = h~a , t = t=

~ta mc ,

t0 =

~t0 a mc ,

ct~ l,

u = uc~ , v =

f=

~ f a,

p=

2

v~ ca ,

p~a mcl ,

h~ a ~ Q ac

h=

Q=

= 1 f cos2 p(j t),

)

ð4Þ

For the axisymmetric flow case, the parameters are taken as j ! x, h ! r and Q =

~ Q pa2 c

ð5Þ

where j, l, h, a, t, c, u, v, a, h, f, p, Q, x and r indicate the non-dimensional axial distance, the wavelength, transverse distance, channel semi-width (or conduit radius), time, wave velocity, axial velocity, transverse (radial) velocity, wave number, transverse (radial) displacement from centre line, wave amplitude, pressure, volume flow rate, axial distance and radial distance, respectively. In dimensionless form, equations (1) and (2) become n ∂u tm = tm + for t 5 t 0 ð6Þ 0 ∂h ∂u = 0 for t 4 t 0 ð7Þ ∂h

∂p =0 ∂h

ð9Þ

The prescribed boundary conditions are ∂u (j, hpl , t) = 0 ∂h

that is, the regularity condition ð10Þ

where hpl is width of the plug flow region, defined as follows ∂p ð11Þ hpl = t 0 ∂j and ð12Þ

u(j, h, t) = 0

that is, the ‘no-slip’ condition on inner surface of the vessels. Integration of equation (8) leads to t=h

∂p +A ∂j

ð13Þ

with A as function of integration. Invoking equation (13) into equation (6) and then using the regularity boundary condition (10) and equation (11), we arrive at the following expression for velocity gradient mn n o1n ∂u ∂p = hm h m pl ∂h ∂j

ð14Þ

Integrating equation (14) from h to h and employing the no-slip boundary condition (10), the axial velocity is obtained as

Channel flow Under the assumption of long wavelength and low Reynolds number approximation, we have a ! 0 and Re ! 0. We note that a ! 0 removes the curvature effects and Re ! 0 makes the inertial forces negligibly small in comparison with the viscous forces. The conservation equations governing the flow are as follows ∂p ∂t = ∂j ∂h

ð3Þ

(Vocadlo model)

ð8Þ

mn ðh n o1n ∂p u= s m hm pl ds for hpl 4 h 4 h ∂j

ð15Þ

h

The axial velocity in the plug flow region is given by mn hðpl n o1n ∂p upl = s m hm pl ds for 0 4 h 4 hpl ∂j h


ð16Þ

4


The volumetric flow rate24 for hpl 4 h 4 h can also be defined as ðh Q=

ð17aÞ

udh hpl

Using equation (15), we arrive at 9 8 > mn > ðh ðh = < ∂p m m 1n Q= (s hpl ) dsdh > ∂j > ; :

ð17bÞ

the analytical solutions for axial pressure gradient (a key variable in peristaltic transport) for all the rheological fluid models studied here, this has been achieved for a number of cases, as described in the following. Case I. For m = 1, equation (19) after integration gives the pressure gradient for H-B model " )# n ( Q + h + hpl 1 + f2 ∂p 2n + 1 = 2n + 1 ∂j n (h hpl ) n

hpl h

ð20-IaÞ

We note that the expression for Q is only defined in region for hpl 4 h 4 h, which can be studied without plugging (hpl ! 0). Therefore, there is no additional term necessary for simulating the core region. The transformations between a wave frame (X, Y) moving with velocity c and the fixed frame (j, h) are given by X = j t, Y = h, U = u 1, V = v

ð17cÞ

where (U, V) and (u, v) are the velocity components in the wave and fixed frame, respectively. Using the transformations of equation (17c) in equation (17a), we get

For n = 1, it reduces to pressure gradient for B-P model, and for hpl = 0, it also reduces to the pressure gradient of power law model. When n = 1 and hpl = 0, then we have the case of Newtonian model. These reduced expressions are given by + h + hpl 1 + f 3 Q ∂p 2 = ð20-IbÞ ∂j (h hpl )3 " )# n ( Q + h + hpl 1 + f2 ∂p 2n + 1 = 2n + 1 ∂j n h n ð20-IcÞ

ðh Q=

ð17dÞ

(U + 1)dY

+ h + hpl 1 + 3 Q ∂p = ∂j h3

f 2

ð20-IdÞ

hpl

which, on integration, yields Q = q + h hpl

ð17eÞ

Averaging volumetric flow rate along one time period, we have = Q

ð1

31=m f Q + h + hpl 1 + 2 ∂p 7 6 = 4 hm n o5 +2 hm (m + 2)hpl hm + 1 ∂j 2 pl pl 1 m + 2 + m+1 2 (h hpl ) m + 2 h m+1 2

ð21-IIaÞ

ð17fÞ

Qdt

When m ! 1=m, the pressure gradient for the Vocˇadlo model is obtained, that is

0

From equations (17e) and (17f), we get = q hpl + 1 f = Q h hpl + 1 f Q 2 2

ð18Þ

∂p = ∂j 2

From equations (17b) and (18), the axial pressure gradient is derived as 2 3mn 6 7 6 Q + h + hpl 1 + f2 7 ∂p 6 7 ( ) =6 7 ∂j 1 4 Ðh Ðh m 5 m n (s hpl ) dsdh

Case II. For n = 1, equation (19) reduces to the pressure gradient for the generalized B-P model

ð19Þ

hpl h

Since the plug region is very small and parallel to x-axis. The variation in hpl with axial position is negligible so that hpl is taken constant in this analysis. This approach is consistent with the earlier methodology of Vajravelu et al.24 Although it is complicated to extract

3m

6 6 4h1=m pl

2

+ h + hpl 1 + f 7 Q 2 7 m=(m + 2) 5 m=(m + 1) mh (2m + 1)hpl h (h hpl )2 2mm+ 1 hm=(m + 2) + mpl + 1 m+1

ð21-IIbÞ

and for m = 1, it reduces to pressure gradient for the B-P model, which is presented in equation (20-Ib). For m = 1 and hpl = 0, we have the expression corresponding to the Newtonian model presented in equation (20-Id). Case III. For m = 1=2, equation (19) represents the pressure gradient for the M-B model, that is


Tripathi and Be´g

5

∂p (n + 1)(2n + 1)(3n + 1)(4n + 1) = ∂j 2n 8 932n ð22-IIIaÞ < = + h + hpl 1 + f Q 2 5 3 pffiffiffi pffiffiffiffiffiffi2n + 1 pffiffiffiffiffiffiffiffi ; : h hpl n (3n + 1)(2n2 + 3n + 1)h + n(6n2 + 7n + 1)hpl + (12n3 + 20n2 + 9n + 1) hhpl

When n = 1=2, equation (19) becomes the pressure gradient for the Casson model, namely 2 3 + h + hpl 1 + f 15 Q ∂p 2 = 4pffiffiffi pffiffiffiffiffiffi4 pffiffiffiffiffiffiffiffi5 ∂j h hpl 5h + 2hpl + 8 hhpl ð22-IIIbÞ

For n = 1=2 and hpl = 0, equation (19) further reduces to the Newtonian model, defined earlier in equation (20-Id). A number of other definitions are also very useful for characterizing the peristaltic flow. Pressure difference across one wavelength is written as ð1 Dp1 =

∂p dj ∂j

ð23Þ

hpl = 2t 0

∂p ∂x

and

The solution of equation (26) is written as u=

m ðr o1n 1 ∂p n n m s hm pl ds for hpl 4 r 4 h 2 ∂x

h

ð31Þ

h

Plug flow velocity is expressed as follows m hðpl o1n 1 ∂p n n m upl = s hm pl ds for 0 4 r 4 hpl 2 ∂x h

ð32Þ

Friction force is defined by

F=

ð30Þ

u(x, h, t) = 0

0

ð1

ð29Þ

Volumetric flow rate is defined for hpl 4 h 4 h as

∂p dj ∂j

ð24Þ

0

ðh Q=

ð33Þ

2rudr hpl

Mechanical efficiency is defined as the ratio between the average rate per wavelength at which work is done by the moving fluid against a pressure head and average rate at which the walls do work on the fluid. For rheological fluids, the appropriate expression is 1 QDp E= f(I1 Dp1 ) Ð1 where I1 = 0 (∂p=∂j) cos 2pjdj

ð25Þ

1 ∂p Q=2 2 ∂x

mn ðh ðr 1 n r (sm hm pl ) dsdr

ð34Þ

h

The averaged flow rate in this case is 2 = q h2 + 1 f + 3f Q pl 8

= Q h2 h2pl + 1 f +

Employing the same methodology as with channel flow, the relevant conservation equations for this scenario assume the form ∂p 1 ∂(rt) = ∂x r ∂h

ð26Þ

∂p =0 ∂r

ð27Þ

3f2 8

ð35Þ

Pressure gradient is computed with 2

3mn

27 6 6Q + h2 + h2pl 1 + f 3f8 7 ∂p 7 = 26 6 7 ∂x Ðh Ðr 1 4 5 m m n 2 r (s hpl ) dsdr

hpl

ð36Þ

h

The analytical solutions of pressure gradient for various rheological cases are now elaborated.

with the following boundary conditions

where

hpl

Axisymmetric flow

∂u (x, hpl , t) = finite ∂h

Introducing equation (31) into above equation yields

ð28Þ

Case I. For m = 1, equation (36) reduces to the pressure gradient for the H-B model


6

Proc IMechE Part H: J Engineering in Medicine 0(0) n o3n + h2 + h2 1 + f 3f2 (3n + 1) Q pl 8 ∂p 5 = 24 3n + 1 ∂x nh n 2

∂p = ∂x

n o3n + h2 + h2 1 + f 3f2 (n + 1)(2n + 1)(3n + 1) Q pl 8 24 5 2n + 1 n(h hpl ) n (2n2 + 3n + 1)h + (4n2 + 5n + 1)hpl 2

ð37-IcÞ

ð37-IaÞ

When n = 1, this reduces to the pressure gradient for the B-P model and for hpl = 0, it also reduces to pressure gradient for the power law model. For n = 1 and hpl = 0, we can extract the case of the Newtonian model. The relevant expressions for each model are given by o3 2 n + h2 + h2 1 + f 3f2 24 Q pl 8 ∂p 5 = 4 ∂x (3h + 5hpl )(h hpl )3

∂p = ∂x

n o + h2 + h2 1 + f 3f2 8 Q pl 8 h4

ð37-IdÞ

Case II. For n = 1, equation (2.2.10) reduces to the pressure gradient for generalized B-P model, that is

ð37-IbÞ

2

31=m

6 7 + h2 + h2 1 + f 3f2 Q ∂p 6 7 pl 8 = 26 7 n o 3m + 1 3m + 1 4 h 5 ∂x hpl m + 1 h pl h 2 2 2 (m + 1)(3m + 1) + (h hpl ) 2(m + 1) (h + hpl ) + 6 (h + hhpl 2hpl )

When m ! 1=m, the pressure gradient is obtained for the Vocˇadlo model 2 6 6 + h2 + h2 1 + f 3f2 6 Q ∂p pl 8 6 = 26 2 3m + 1 3m + 1 ∂x n 6 m2 h m hpl m m+1 4 24 mh m + (h hpl ) 2(m (m + 1)(3m + 1) + 1) (h + hpl ) +

ð38-IIaÞ

3m 7 7 7 37 7 o 7 hpl 2 5 2 5 6 (h + hhpl 2hpl )

ð38-IIbÞ

and for m = 1, it reduces to pressure gradient for B-P model, that is, equation (37-Ib), and for m = 1 and hpl = 0, it further reduces to the classical Newtonian model, namely equation (37-Id). Case III. When m = 1=2, equation (36) gives the pressure gradient for the M-B model, which is expressed as n o32n 2 + h2 + h2 1 + f 3f2 (n + 1)(2n + 1) Q pl 8 ∂p 5 = 24 2n2 F(x, t) ∂x

ð39-IIIaÞ

hence " F(x, t) = (h

1=2

1=2 2n + 1 hpl ) n

1=2 2

5h2

1=2 3

120n3 h1=2 (h1=2 hpl )

(3n + 1)(4n + 1)(5n + 1)

60n2 h(h1=2 hpl ) 20n 3=2 1=2 1=2 h (h hpl ) + : 3n + 1 (3n + 1)(4n + 1) 1=2 4

+

120n4 (h1=2 hpl )

ð39-IIIbÞ

(3n + 1)(4n + 1)(5n + 1)(6n + 1) #

2n + 1 1=2 1=2 1=2 h (h + hpl )(h + hpl ) h2pl n

And for n = 1=2, it reduces to pressure gradient for Casson model given in the following 2 3 + h2 + h2 1 + f 3f2 12 Q pl 8 ∂p n oi5 = 4 4 h 2 1=2 3=2 ∂x h1=2 hpl 3h2 17 23hhpl + 18h3=2 hpl + 26h1=2 hpl


ð39-IIIcÞ

Tripathi and Be´g

7

η

h

u c

h pl

Core region

upl

ξ

Plug flow region

Figure 1. Geometry of peristaltic flow regime.

and if n = 1=2 and hpl = 0, it further reduces to the Newtonian model in equation (37-Id). Pressure difference for the axisymmetric flow is given by ð1 Dp1 =

∂p dx ∂x

ð40Þ

0

Friction force is evaluated with ð1 F=

h2

∂p dx ∂x

ð41Þ

0

Mechanical efficiency is obtained with E=

1 QDp 2f½I11 + fI12 (1 + f)Dp1

ð42Þ

where ð1 I11 = 0

∂p cos (2px)dx, ∂x

ð1 I12 =

∂p cos4 (px)dx ∂x

ð43Þ

0

Numerical results and discussion Figure 1 shows the geometry of peristaltic channel flow and axisymmetric flow. Comprehensive computations have been performed using MATHEMATICA software44 to explore the influence of rheological

characteristics on velocity profile (u), pressure rise per wavelength (Dp), friction force (F) and mechanical efficiency (E) in the peristaltic transport of a variety of viscoplastic fluids. These are illustrated in Figures 2–21. Figures 2–11 correspond to the channel flow scenario. Figures 12–21 are associated with the axisymmetric flow geometry. The effects of yield stress index (m), shear rate index (n) and plug flow region width (hpl ) are particularly emphasized in the simulations. In these figures, H-B denotes Herschel–Bulkley, M-B designates Mizrahi–Berk and B-P represents the Bingham plastic model. In all these figures plotted, pressure rise is observed to be a maximum for zero flow rate. While the pressure rise is observed to decay with increasing averaged flow rate, for most fluids, the relationship is linear (e.g. Vocˇadlo, power law, H-B, B-P and Casson), whereas for others it is clearly non-linear (e.g. M-B). The viscoplastic models are frequently referred to as ‘discontinuous’.45 H-B fluids exhibit an infinite shear stress: shear rate gradient at zero shear rate. The Vocˇadlo model, however, attains a finite value at zero shear rate.40 In the case of the B-P model, below a certain stress yield, the fluid assumes ‘rigidity’; above this yield, the medium flows as an incompressible viscous fluid.46 A similar pattern is evident for the variation of friction force (in opposite direction, that is, negative direction) with averaged flow rate, that is, friction force peaks at zero averaged flow rate and decays as averaged flow rate increases. Figure 2(a)–(d) illustrates to see the effects of plug flow width (hpl), index of shear rate (n) and index for


8

Proc IMechE Part H: J Engineering in Medicine 0(0) 0.4 0.3

Axisymmetric flow Channel flow

0.3 0.2

u 0.2

u

0.1

0.1 -1

hpl = 0.0 hpl = 0.025 hpl = 0.05 hpl = 0.1

-0.5

0.5

1

η, r

-0.1

0.2

0.4

0.2

1

-0.1

(b)

n= 0.8 n= 0.9 n= 1.0 n = 1.1

u

0.8

η

(a) 0.3

0.6

m= 0.5 m= 1.0 m= 1.5 m = 2.0

0.3 0.2

u

0.1

0.1 0.2

0.4

0.6

0.8

1

η

-0.1

(c)

0.2

0.4

0.6

0.8

1

η

-0.1 -0.2

(d)

Figure 2. Velocity profiles (axial velocity vs transverse displacement) at f = 0:5, j = 1:0, ∂p=∂j = 1:0, t = 1:0: (a) m = 1:0, n = 1:0 and hpl = 0:01; (b) m = 1:0, n = 1:0 and hpl = 0:0, 0:025, 0:05, 0:1; (c) m = 1:0, hpl = 0:01, n = 0:8, 0:9, 1:0, 1:1 and (d) n = 1:0, hpl = 0:01 and m = 0:5, 1:0, 1:5, 2:0.

yield stress (m) on axial velocity distributions across the conduit width. Figure 2(a) shows the comparison between channel flow and axisymmetric flow pattern under peristaltic propulsion, and it is evident that in the central region, axial velocity for axisymmetric flow is greater than that for channel flow. However, as we approach the periphery of the conduit, the channel flow velocity is dominant. Both profiles exhibit a clear symmetry about the centre line. In the central zone, velocity is in fact negative indicating the presence of flow reversal, a characteristic that vanishes towards the conduit walls. Figure 2(b) indicates that with increasing plug flow width, for small values of radial (transverse) distance, the axial velocity is accentuated; however, with further progression towards the conduit walls, a prominent decrease in axial velocity is observed. This trend is consistent with the fact that plug flow effects will benefit the core zone regime and will be counterproductive external to this zone. Figure 2(c) demonstrates that with a rise in shear rate index (n), initially, the axial flow is strongly decelerated (values become increasingly negative). However, as we move towards the boundaries of the conduit, this effect is reversed, and a slight enhancement in axial flow velocity is induced, that is, acceleration in the axial flow is witnessed. Figure 2(d) reveals that the yield stress index (m) results in the contrary response in axial velocity, compared with index of shear stress. In the core region, increasing m values serve to accelerate the axial flow, whereas further from the centre line, a significant retardation arises in the axial flow with increasing m values.

In Figure 3(a)–(e), the pressure rise per wavelength In these Dp is plotted against the average flow rate Q. figures, the cases of different bio-rheological fluids are when n . 1 considered. Figure 3(a) shows Dp versus Q (dilatant fluids), n = 1 (Newtonian fluids) and n \ 1 (pseudo-plastic fluids), and it is apparent that Dp is maximized for dilatant fluids and minimized for pseudoplastic fluids. It is also noticed here that Dp for the Newtonian fluid is greater than for the pseudo-plastic fluid. However, the opposite trend is observed for the dilatant fluid. Figure 3(b) depicts the distribution of Dp for H-B and B-P fluids, revealing that pressure with Q rise, Dp, for H-B fluids exceeds that for B-P fluid when n . 1. However, for n \ 1, the converse is observed. The influence of plug flow region width (hpl ) on Dp is presented in Figure 3(c)–(e). Clearly, for all rheological fluids and even Newtonian fluid, Dp increases when hpl is increased. A larger plug flow length is therefore beneficial to peristaltic propulsion. The maximum pressure rise is associated with the H-B fluid with n = 1.5 (Figure 3(e)) and the lowest pressure rise corresponds to the pseudo-plastic fluid (n = 0.5) in Figure 3(c); both cases are achieved with the zero plug width (hpl = 0). In Figure 4(a)–(c), the variations of m and hpl on Dp in B-P, Newtonian and Vocˇadlo fluids are illustrated. Figure 4(a) demonstrates the effects of m on Dp in Newtonian and Vocˇadlo fluids. Here, it is found that Dp is also decreasing function of yield stress index, m. Figure 4(b) depicts that Dp for B-P fluid is of greater magnitude than for Vocˇadlo fluids. Figure 4(c)


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Figure 3. Pressure versus averaged flow rate for channel flow at f = 0:4, m = 1:0 and different values of n and hpl : (a) hpl = 0:0 and n = 0:5, 1:0, 1:5; (b) hpl = 0:05 and n = 0:5, 1:0, 1:5; (c) n = 0:5 and hpl = 0:0, 0:05, 0:1; (d) n = 1:0 and hpl = 0:0, 0:05, 0:1 and (e) n = 1:5 and hpl = 0:0, 0:05, 0:1. H-B: Herschel–Bulkley; B-P: Bingham plastic. Different lines represent corresponding models.

indicates that for the Vocˇadlo fluid case, Dp is enhanced as hpl is increased. Inspection of Figure 4(a)– (c) also shows that even for high values of the yield stress index, the Vocˇadlo model produces a linear decay in pressure rise with averaged flow rate. This linear behaviour is also witnessed for the Newtonian model and the B-P model. distributions for M-B, Figure 5 presents the Dp Q Casson and pseudo-plastic fluids. Here, it is seen that Dp is substantially elevated when n is large, that is, n . 1. For n = 1.5, a non-linear profile is observed for the M-B fluid, for which the highest pressure rise is computed. Pressure rise is lower for the Casson fluid but higher than for the pseudo-plastic fluid. The M-B fluid37 is in fact a modification of the Casson model with variable flow index. Clearly, fluid rheology

contributes to increasing pressure in the peristaltic flow, a characteristic which is not captured with the pseudo-plastic flow model. Figure 6(a)–(e) shows the variation of friction force F with averaged flow rate for different shear rate index values (n) and plug flow widths (hpl), with yield stress index fixed at unity. It is evident that the H-B model with n = 1.5 (Figure 6(b)) achieves the greatest friction force (a measure of wall shear); for this fluid, the resistance to peristaltic propulsion in the channel will therefore be greatest. The lowest friction force is obtained for pseudo-plastic fluids in Figures 6(a) (for n = 0.5) and (c) (for hpl = 0). It is therefore expected that pseudo-plastic fluids will be propelled with greater ease in the channel compared with all other rheological (or Newtonian) fluids. Figure 6(d) and (e) shows that


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Figure 4. Pressure versus averaged flow rate for channel flow at f = 0:4, n = 1:0 and different values of m and hpl : (a) hpl = 0:0 and m = 1:0, 2:0, 3:0; (b) hpl = 0:05 and m = 1:0, 2:0, 3:0 and (c) m = 2:0 and hpl = 0:0, 0:05, 0:1. B-P: Bingham plastic. Different lines represent corresponding models.

Figure 5. Pressure versus averaged flow rate for channel flow at f = 0:4, m = 0:5 and different values of n = 0:5, 1:0, 1:5 and hpl = 0:0, 0:1. M-B: Mizrahi–Berk. Different lines represent corresponding models.

increasing the plug flow length consistently boosts the friction force. The highest friction forces correspond to the B-P model and H-B model with hpl = 0.1; the lowest friction force magnitudes (again arising at 0 average flow rate) are produced with both Newtonian and dilatant fluids for zero plug flow length. Figure 7(a)–(c) shows the variation of friction force F with averaged flow rate for different yield stress index values (m) and plug flow widths (hpl), with shear rate index (n) of unity. In Figure 7(a), it is apparent that the

Vocˇadlo model, even with high-yield stress index, generates lower friction forces for all flow rates as compared with the Newtonian model. Similarly, in Figure 7(b), the Vocˇadlo model corresponds to substantially lower friction forces than the B-P model. As computed in earlier plots, an increase in plug flow width markedly accentuates fiction force for the Vocˇadlo model (Figure 7(c)) implying a deceleration in the peristaltic axial flow. The larger the plug flow width, the greater the extent of homogenous velocity across the channel cross section. Figure 8 presents the friction force evolution for the M-B model, Casson model and pseudo-plastic model. Non-linearity in the decay profile for the M-B model is again present, whereas it is absent in the Casson model. In both these cases, there is a non-zero plug length, although shear rate index (n) for the former is higher. The friction force for the pseudo-plastic fluid is almost invariant with change in the averaged flow rate and of very small magnitude. Similar trends have been reported by Samy30 and Medhavi.47 Figure 9(a)–(e) illustrates the variation of mechani in the chancal efficiency E with averaged flow rate Q nel for all the models studied and the effects of n and hpl . The curves for E are non-linear. These curves show that an optimum efficiency for peristaltic propulsion is attained at relatively low averaged flow rates in the


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Figure 6. Friction force versus averaged flow rate for channel flow at f = 0:4, m = 1:0 and different values of n and hpl : (a) hpl = 0:0 and n = 0:5, 1:0, 1:5; (b) hpl = 0:05 and n = 0:5, 1:0, 1:5; (c) n = 0:5 and hpl = 0:0, 0:05, 0:1; (d) n = 1:0 and hpl = 0:0, 0:05, 0:1 and (e) n = 1:5 and hpl = 0:0, 0:05, 0:1. H-B: Herschel–Bulkley; B-P: Bingham plastic. Different lines represent corresponding models.

= 0:02. The maximum optimum effivicinity of Q ciency corresponds to the H-B model with hpl = 0.1 (Figure 9(e)) and n = 1.5 (Figure 9(b)). Very low efficiencies are produced with the B-P (Figure 9(b)) and pseudo-plastic models (Figure 9(c)). Increasing shear rate index strongly enhances the efficiency for the H-B model (Figure 9(b)). Increasing plug flow width also increases the efficiency for the B-P model (Figure 9(d)), although values are much lower than for the H-B model. Figure 10(a)–(c) depicts the efficiency curves for the effect of yield stress index and plug flow width. The efficiency, E, is reduced in the Vocˇadlo model as yield stress index (m) increases. Conversely, an increase in

plug flow width enhances the efficiency for the Vocˇadlo model with a high m value (Figure 10(c)). Both the B-P model and Newtonian model achieve higher optimum efficiencies for all flow rates as compared with the Vocˇadlo model (Figure 10(a) and (b)). Figure 11 presents efficiency E plots for the M-B, Casson and pseudo-plastic models. The M-B model produces the highest efficiency, considerably in excess of any other model. Very low efficiency is obtained with the pseudo-plastic fluid for zero plug flow width, with a slightly better performance for the Casson model. In Figures 12–21, the axisymmetric peristaltic flow results are depicted. It is immediately evident that pressure rise and friction force magnitudes are at least one


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Figure 7. Friction force versus averaged flow rate for channel flow at f = 0:4, n = 1:0 and different values of m and hpl : (a) hpl = 0:0 and m = 1:0, 2:0, 3:0; (b) hpl = 0:05 and m = 1:0, 2:0, 3:0 and (c) m = 2:0 and hpl = 0:0, 0:05, 0:1. B-P: Bingham plastic. Different lines represent corresponding models.

Figure 8. Friction force versus averaged flow rate for channel flow at f = 0:4, m = 0:5 and different values of n = 0:5, 1:0, 1:5 and hpl = 0:0, 0:1. M-B: Mizrahi–Berk. Different lines represent corresponding models.

order of magnitude higher than in the corresponding cases for channel flow. In some instances, values are several orders of magnitude in excess of the channel flow results, in particular for the M-B model and the Casson model (Figure 15). Figure 12(a)–(c) shows the influence of plug flow width (hpl), index of shear rate (n) and index for yield stress (m) on axial velocity profiles for peristaltic

axisymmetric flow. In these figures, magnitudes are observed generally to be lower than the channel flow case (Figure 2(b)–(d)). Inspection of Figure 12(a) indicates that increasing plug flow width initially enhances the axial flow (axial flow velocity is minimized with the zero plug flow width case). As we migrate towards the conduit walls, the contrary behaviour is demonstrated and there is a deceleration in axial flow with increasing plug flow width. Figure 12(b) shows that with increasing index of shear rate (n), in the core region, the axial flow is initially decelerated, whereas further from the core, that is, for larger radial distances, the axial flow is consistently accelerated. With increasing index for yield stress (m), Figure 12(c) demonstrates the opposite response is induced by increasing yield stress index (m) compared with that for increasing shear rate index (n) in Figure 12(b). As m values increase, the axial flow in the core region undergoes a significant acceleration, whereas as radial coordinate increases, the rise in yield stress index generates a substantial deceleration in axial flow. Figure 13(a)–(e), in which m is fixed at 1.0, shows that the pressure rise per wavelength Dp is a maximum for the H-B model with hpl = 0.1 for zero average flow (Figure 13(e)) and a minimum for the pseudorate Q plastic fluid model with zero plug flow width (Figure 13(c)). The dilatant fluid model with shear rate index


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Figure 9. Mechanical efficiency versus averaged flow rate for channel flow at f = 0:4, m = 1:0 and different values of n and hpl : (a) hpl = 0:0 and n = 0:5, 1:0, 1:5; (b) hpl = 0:05 and n = 0:5, 1:0, 1:5; (c) n = 0:5 and hpl = 0:0, 0:05, 0:1; (d) n = 1:0 and hpl = 0:0, 0:05, 0:1 and (e) n = 1:5 and hpl = 0:0, 0:05, 0:1. H-B: Herschel–Bulkley; B-P: Bingham plastic. Different lines represent corresponding models.

(n = 1.5) generates significantly greater pressure rise than both the Newtonian model and the pseudo-plastic model (Figure 13(a)). The B-P model with a non-zero plug flow width also produces greater pressure rise than the Newtonian model (Figure 13(d)), but lesser values than the H-B model (Figure 13(b)). It is important, however, to note that the magnitudes of the average are also much greater than in the channel flow rate Q flow case. Therefore, in axisymmetric propulsion flow, all the pseudo-plastic models may create greater pressure rise but require higher flow rates to attain this. Figure 14(a)–(c), in which n is fixed at unity, confirms that the Vocˇadlo model generates very low pressure rise relative to the Newtonian model (Figure 14(a)) and the B-P model (Figure 14(b)). Also, the pressure

rise in the Vocˇadlo model is minimized for all flow rates with zero plug flow width (Figure 14(c)). Pressure rise in the Vocˇadlo model also vanishes at a lower flow rate than for the Newtonian and B-P models as intimated axis. by the earlier intersection of the profile with the Q Figure 15 shows that the pressure rise in the M-B model is massively in excess of the Casson or pseudoplastic models. Again, the non-linear nature of the decay in the M-B profile is emphasized. The other models produce a linear depletion in pressure rise with increasing flow rate. Figure 16(a)–(e) shows the friction force (wall shear) distributions with averaged flow rate for various shear rate index (n) values with m fixed at unity. The largest friction force corresponds to the H-B model


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Figure 10. Mechanical efficiency versus averaged flow rate for channel flow at f = 0:4, n = 1:0 and different values of m and hpl : (a) hpl = 0:0 and m = 1:0, 2:0, 3:0; (b) hpl = 0:05 and m = 1:0, 2:0, 3:0 and (c) m = 2:0 and hpl = 0:0, 0:05, 0:1. B-P: Bingham plastic. Different lines represent corresponding models.

Figure 11. Mechanical efficiency versus averaged flow rate for channel flow at f = 0:4, m = 0:5 and different values of n = 0:5, 1:0 and hpl = 0:0, 0:1. M-B: Mizrahi–Berk. Different lines represent corresponding models.

(Figure 16(e)). The B-P model48 also produces considerable resistance to the peristaltic flow (Figure 16(d)) exceeding the Newtonian model. Furthermore, the B-P model with n = 1.0 creates greater friction force at the conduit walls than the H-B model with n = 0.5 (Figure 16(b)). The lowest friction forces are associated at any flow rate with the pseudo-plastic model (Figure 16(a)

and (c)). The dilatant model, however, generates substantially greater friction force values with zero plug flow width (Figure 16(a)). Figure 17(a)–(c) demonstrates that the friction force is much lower for the Vocˇadlo model with a high-yield stress index than the Newtonian (Figure 17(a)) or B-P models (Figure 17(b)). Increasing plug flow width is observed, however, to significantly elevate the friction force magnitudes in the Vocˇadlo model (Figure 17(c)). Friction force is also reduced to 0 much faster for the Vocˇadlo model than the other material models. Figure 18 again highlights the non-linear decay in friction force profile for the M-B model and indicates that a significantly greater wall resistance is experienced by the M-B fluid (with non-zero plug flow width) compared with the Casson or pseudo-plastic fluids. The maximum friction force is 22 compared with a value less than 5 for the Casson and a value of approximately 1 for the pseudo-plastic fluid, all at zero averaged flow rate. Figure 19(a)–(e) illustrates the variation of mechani in the axical efficiency E with averaged flow rate Q symmetric flow for several models with the influence of n and hpl . Peak efficiency is observed for the H-B model (Figure 19(e)), and the lowest efficiency corresponds to the pseudo-plastic model with non-zero plug flow width (Figure 19(c)). The B-P model produces greater


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(c) Figure 12. Velocity profiles (axial velocity vs radial displacement) at f = 0:5, x = 1:0, ∂p=∂x = 1:0, t = 1:0: (a) m = 1:0, n = 1:0, hpl = 0:0, 0:025, 0:05, 0:1; (b) m = 1:0, hpl = 0:01, n = 0:8, 0:9, 1:0, 1:1 and (c) n = 1:0, hpl = 0:01, m = 0:5, 1:0, 1:5, 2:0.

Figure 13. Pressure versus averaged flow rate for axisymmetric flow at f = 0:4, m = 1:0 and different values of n and hpl : (a) hpl = 0:0 and n = 0:5, 1:0, 1:5; (b) hpl = 0:05 and n = 0:5, 1:0, 1:5; (c) n = 0:5 and hpl = 0:0, 0:05, 0:1; (d) n = 1:0 and hpl = 0:0, 0:05, 0:1 and (e) n = 1:5 and hpl = 0:0, 0:05, 0:1. H-B: Herschel–Bulkley; B-P: Bingham plastic. Different lines represent corresponding models. Downloaded from pih.sagepub.com at Universitats-Landesbibliothek on December 16, 2013

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Figure 14. Pressure versus averaged flow rate for axisymmetric flow at f = 0:4, n = 1:0 and different values of m and hpl : (a) hpl = 0:0 and m = 1:0, 2:0, 3:0; (b) hpl = 0:05 and m = 1:0, 2:0, 3:0 and (c) m = 2:0 and hpl = 0:0, 0:05, 0:1. B-P: Bingham plastic. Different lines represent corresponding models.

Figure 15. Pressure versus averaged flow rate for axisymmetric flow at f = 0:4, m = 0:5 and different values of n = 0:5, 1:0, 1:5 and hpl = 0:0, 0:1. M-B: Mizrahi–Berk. Different lines represent corresponding models.

efficiency for all flow rates compared with the Newtonian model (Figure 19(d)). The dilatant flow model achieves quite large efficiency relative to the Newtonian and pseudo-plastic models (Figure 19(a)). However, the H-B values for peristaltic mechanical efficiency strongly exceed the B-P (Figure 19(b)), the pseudo-plastic (Figure 19(c)) and the dilatant fluid

(Figure 19(e)) models. Generally, the optimum efficiency is attained at lower averaged flow rates for all the fluid value is required for the models. However, a greater Q H-B and B-P models at lower shear rate index (n = 0.5) than at higher shear rate index (n = 1.5), as testified to by comparing Figure 19(c) and (e), respectively. Figure 20(a)–(c) shows that a much lower efficiency is produced by the Vocˇadlo model than any other fluid model, including the Newtonian (Figure 20(a)) and the B-P model (Figure 20(b)). Increasing plug flow width, however, induces a marked increase in mechanical efficiency of the Vocˇadlo model (Figure 20(c)), although the peak efficiency is still sustained at much lower magnitudes than for the other models studied. Finally, Figure 21 illustrates the variation of for mechanical efficiency E with averaged flow rate Q the M-B, Casson and pseudo-plastic models. While the M-B model generates a much greater efficiency than the other two models, the peak efficiency does not exceed the B-P model (Figure 20(b)), although the wave amplitude is double that in Figure 18 (u = 0.4 in Figure 20(b) and u = 0.2 in Figure 21). The pseudoplastic fluid model is associated with the lowest mechanical efficiency (with zero plug flow width). Increasing shear rate index is found to distinctly increase the M-B model efficiency.


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Figure 16. Friction force versus averaged flow rate for axisymmetric flow at f = 0:4, m = 1:0 and different values of n and hpl : (a) hpl = 0:0 and n = 0:5, 1:0, 1:5; (b) hpl = 0:05 and n = 0:5, 1:0, 1:5; (c) n = 0:5 and hpl = 0:0, 0:05, 0:1; (d) n = 1:0 and hpl = 0:0, 0:05, 0:1 and (e) n = 1:5 and hpl = 0:0, 0:05, 0:1. H-B: Herschel–Bulkley; B-P: Bingham plastic. Different lines represent corresponding models.

Conclusion A mathematical study has been conducted for the peristaltic propulsion of generalized viscoplastic fluids in channels and axisymmetric conduits. Expressions have been derived for axial velocity, pressure rise, friction force and peristaltic mechanical efficiency. The H-B, M-B, Casson, Vocˇadlo and B-P models have been examined in addition to the Newtonian model and also the power law models (pseudo-plastic and dilatant fluids). Extensive graphical results have been included. The present analysis has shown that

Apparently, based on the non-dimensionalization approach implemented in the present study,

non-dimensional pressure rise (Dp) is maximized for dilatant fluids and minimized for pseudo-plastic fluids. In channel flow, increasing plug flow width (hpl ) initially accelerates the axial flow but with greater transverse distance, it decelerates the flow. Increasing yield stress index (m) exhibits a similar influence to plug flow width. However, increasing shear rate index (n) tends to initially decelerate the flow and with greater transverse distance acts to accelerate it. In axisymmetric flow, a similar response is computed to that for channel flow; however, magnitudes are markedly lower in axisymmetric flow. Therefore, responses for axial flow in channel and


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Figure 17. Friction force versus averaged flow rate for axisymmetric flow at f = 0:4, n = 1:0 and different values of m and hpl : (a) hpl = 0:0 and m = 1:0, 2:0, 3:0; (b) hpl = 0:05 and m = 1:0, 2:0, 3:0 and (c) m = 2:0 and hpl = 0:0, 0:05, 0:1. B-P: Bingham plastic. Different lines represent corresponding models.

Figure 18. Friction force versus averaged flow rate for axisymmetric flow at f = 0:4, m = 0:5 and different values of n = 0:5, 1:0 and hpl = 0:0, 0:1. M-B: Mizrahi–Berk. Different lines represent corresponding models.

axisymmetric cases are similar in a qualitative sense but differ in quantitative sense. In channel flow, the influence of plug flow width (hpl ) on pressure rise (Dp) is greater for H-B fluids than pseudo-plastic or dilatant fluids. In channel flow, Dp for the B-P fluid considerably exceeds that computed for the Newtonian fluid.

For channel flow, with larger values of yield stress index (m) in the Vocˇadlo fluid, Dp is significantly lower than for the Newtonian fluid. In channel flow, qualitatively, the effects of plug flow width (hpl ) on pressure rise (Dp) are similar for the H-B, B-P and Vocˇadlo viscoplastic models. Pressure rise (Dp) in the M-B fluid is significantly greater than for the Casson or Newtonian fluids at in channel flow. all flow rates (Q) Pressure rise (Dp) in the Casson fluid is larger than in for the Newtonian fluid at any flow rate (Q) channel flow. An optimum peristaltic mechanical efficiency (E) is observed at generally lower flow rates (Q). Magnitudes for the channel flow are lower than for axisymmetric flow. Generally, for both channel and axisymmetric flows, mechanical efficiency is an increasing function of plug flow width (hpl ). Mechanical efficiency, as defined by equation (25), is a decreasing function of yield stress index (m) for both channel flow and axisymmetric flow. In channel flow, friction force (F), that is, wall shear, is a maximum for the H-B model with a high shear rate index, whereas the lowest friction force is observed for pseudo-plastic fluids.


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Figure 19. Mechanical efficiency versus averaged flow rate for axisymmetric flow at f = 0:4, m = 1:0 and different values of n and hpl : (a) hpl = 0:0 and n = 0:5, 1:0, 1:5; (b) hpl = 0:05 and n = 0:5, 1:0, 1:5; (c) n = 0:5 and hpl = 0:0, 0:05, 0:1; (d) n = 1:0 and hpl = 0:0, 0:05, 0:1 and (e) n = 1:5 and hpl = 0:0, 0:05, 0:1. H-B: Herschel–Bulkley; B-P: Bingham plastic. Different lines represent corresponding models.

In channel flow, there is a non-linear decrease in pressure rise (Dp) and friction force (F) for the M-B model. In axisymmetric flow, a significantly greater pressure rise (Dp) occurs for the M-B model than for any other model (pressure profile is also a nonlinear function of flow rate for the M-B model). In axisymmetric flow, friction force is maximized for the H-B model (and minimized for the pseudoplastic model and is also very low for the Vocˇadlo model). Maximum efficiency in axisymmetric flow corresponds to the H-B model and the lowest efficiency is attained with the pseudo-plastic model (power law model, with shear rate index, n \ 1).

A greater mechanical efficiency is computed in axisymmetric flow, for the B-P model compared with the Newtonian model. Significantly greater magnitudes of pressure rise, friction force and mechanical efficiency are observed for axisymmetric flow compared with channel flow. The pattern of profiles for both cases is, however, similar. In axisymmetric flow, the dilatant flow model achieves a greater mechanical efficiency than the Newtonian or pseudo-plastic models.

The present study has revealed some interesting characteristics of viscoplastic peristaltic flows. Future investigations will consider non-Newtonian nanofluids49


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Figure 20. Mechanical efficiency versus averaged flow rate for axisymmetric flow at f = 0:4, n = 1:0 and different values of m and hpl : (a) hpl = 0:0 and m = 1:0, 2:0, 3:0, (b) hpl = 0:05 and m = 1:0, 2:0, 3:0 and (c) m = 2:0 and hpl = 0:0, 0:05, 0:1. B-P: Bingham plastic. Different lines represent corresponding models.

Acknowledgements The authors are extremely grateful to the reviewer for his or her comments, which have served to significantly improve the quality and clarity of the present study. They would also like to express their gratitude to the Assistant Managerial Editor, Miss Anita Treso. Declaration of conflicting interests The authors declare that there is no conflict of interest. Figure 21. Mechanical efficiency versus averaged flow rate for axisymmetric flow at f = 0:2, m = 0:5 and different values of n = 0:5, 1:0, 1:5 and hpl = 0:0, 0:1. M-B: Mizrahi–Berk. Different lines represent corresponding models.

Funding This research received no specific grant from any funding agency in the public, commercial or not-for-profit sectors. References

and also heat transfer and will be reported imminently. Such studies are also relevant to gastric transport and novel peristaltic pumps in medical engineering. It is also hoped that that experimental researchers in particular will be motivated on reading the article, to conduct investigations on viscoplastic propulsion flows, which would further develop this interesting area of medical engineering sciences.

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Appendix 1 Notation a c E F

non-dimensional channel semi-width (or conduit radius) non-dimensional wave velocity mechanical efficiency for peristaltic propulsion friction force (internal wall shear)

h hpl m n p Dp Dp1 Q Q ∂p=∂j R T (u, v) (U, V) x, y a g_ h l m v j ~t ~t0 f

non-dimensional transverse (radial) displacement from conduit centre line width of the plug flow region index for yield stress index of shear rate non-dimensional pressure pressure rise per wavelength pressure difference for the axisymmetric flow non-dimensional volumetric flow rate averaged volumetric flow rate along one time period non-dimensional axial pressure gradient non-dimensional radial distance non-dimensional time velocity components in the fixed frame velocity components in the wave frame coordinates along the conduit centre line and transverse to it non-dimensional wave number rate of shear strain non-dimensional transverse distance non-dimensional wavelength coefficient of dynamic viscosity non-dimensional transverse (radial) velocity non-dimensional axial distance shear stress yield stress non-dimensional wave amplitude


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