Couple-stresses in peristaltic transport of fluids

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1994 J. Phys. D: Appl. Phys. 27 1163

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J. Phys. D: Appl. Phys. 27 (1994) ??6$-?170. Prinfed in tha UK

1 Couple-stresses in peristaltic transport 1 I offluids

Elsayed F El Shehaweyt and Kh S Mekheimert t Department of Mathematics, Faculty of Education, Ain Shams University, Heliopolis, Cairo, Egypt

Nasser City, Cairo, Egypt

Received 1 February 1993, in final form 26 October 1993

Department of Mathematics, Faculty of Science (Men), AI-Azhar University,

Abstract. Peristaltic pumping by a sinusoidal travelling wave in the walls of a two-dimensional channel filled with a viscous incompressible couple-stress fluid, is investigated theoretically. A perturbation solution is obtained, which satisfies the momentum equation for the case in which the amplitude ratio (wave amplitude: channel half width) is small. The results show that the mean axial velocity decreases with increasing couple-stress parameter q. The phenomenon of reflux (mean flow reversal) is discussed. A reversal of velocity in the neighboumood of the centre line occurs when the pressure gradient is greater than that of the critical reflux condition. It is found that the critical reflux pressure increases with the couple-stress parameter. Numerical results are reported for various values of the physical parameters of interest.

1. Introduction

The study of the mechanism of peristalsis, in both mechanical and physiological situations, has recently become the object of scientific research. Since the first investigation of Latham [I], several theoretical and experimental attempts have been made to understand peristaltic action in different situations. A review of much of the early literature is presented in an article by Jaffrin and Shapiro [2]. A summary of most of the experimental and theoretical investigations reported, with details of the geometry, fluid, Reynolds number, wavelength parameter, wave amplitude parameter and wave shape has been given in a paper by Srivastava and Srivastava [3]. Most theoretical investigations have been canied out for Newtonian fluids, although it is known that most physiological fluids behave as non-Newtonian fluids. The problem of peristaltic transport of a couple-stress fluid has been investigated by Srivastava 141 under a zero Reynolds number and long-wavelength approximation. The present paper considers the peristaltic transport of a couple-stress fluid for arbitrary value of Reynolds number and wavenumher and wave amplitude small relative to the channel half width. A motivation of the present analysis has been the hope that such a theory of a couple-stress fluid process will be useful in understanding the role of peristaltic muscular contraction in transposing bio-fluids behaving like couple-stress fluids. Also, the theory is important for engineering applications of pumping couple-stress fluids such as fluids of long-chain molecules, animal blood, liquid crystals, polyomeric suspensions and lubrication.

0022-3727/94/061163+08$19.50 0 1994 IOP Publishing Ltd

The couple-stress fluid is a special case of a non- Newtonian fluid, which is intended to take into account particle-size effects.

2. Formulation of t h e problem

We shall consider a two-dimensional channel of uniform thickness filled with a homogeneous Newtonian viscous couple-stress fluid. On the flexible walls of the channel are imposed travelling sinusoidal waves of small amplitude. The equations of motion for a couple-stress fluid are suggested by Stokes [5] to be

a u a u au at ax ay p a x

-=--- * ap + u,v2u - lJ2v4u ( I )

a u a u av I ap -+U- + U- = --- + Y,V~U - u2v4u (2) at ax ay p a y

au au ax ay -+ -=c l (3)

v y = - /*. U1 = -

P P where U and U are the fluid velocities in the x and y directions, q the couple-stress parameter, p is the density and /*. is the viscosity.

The fluid is subjected to boundary conditions imposed by symmetric motion of the flexible walls. Let

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E F El Shehawey and Kh S Mekheimer

. n

a L 5 ( x , t ) = LI cos-(x-Ct) 2U

A

Figure 1. Geometty of the problem.

the vertical displacements of the upper and lower walls be $ and -t, respectively, where

[ ( x , t ) = acos - ( x - ct) (4) (: 1 and U is the amplitude, A the wavelength and c the wave speed; see figure 1. The horizontal displacement will be assumed zero as will the components of the couple- stress tensor (as suggested by Valanis and Sun [61) at the boundaries, thus

at y = &d i t(x, t ) and

In terms of the stream function $(x, y, I) and by selecting the following set of non-dimensional variables and parameters

Reynolds number

couple-stress parameter

R = cd/vl

d = v/(pcd3)

wave number CI = k d / h

amplitude ratio E = a/d

and after eliminating p and dropping the primes, equations (1)-(5) become

a 1 -v2* +*yvz*x - ll.*V2* - -V"l - L 2 V W (7) at Y - R

261 tf I

271

261

261 0.01 0.11 031 0.81 o . ~ 0.61 0.61 o m o.ni 0.91-1.0

' 1

266

260 0.01 0.11 0.21 0.31 0.41 0.63 0.61 0.71 0.61 0.91

? Figure 2. Variation of D, with couple-stress parameter q and wavenumber (Y at (a) R = 1 and (b) R = 10.

eZ is a material constant having the dimensions of length squared and can be identified with a property that depends on the size of the fluid molecule [ I I], and

$ = E cos[cu(x - t ) ]

$y = 0 $x = FCIE sin[ol(x - t)].

at y = f l & t ( x , t )

i(h + @ a y y ) ~ ~ s i n [ 4 x - t ) l + ( h y +@yyy) = 0. ( 8 )

3. Method of solution

Assuming the amplitude ratio E of the wave to be small, we obtain the solution for the stream function as a power series in terms of E . by expanding @ and ap/ax in the form (Fung and Yih [7]).

* = *o + E $ , +E?$* +. . . (9)

2 = (g)o + E (Z), + 2 ($) + I .. . (IO) ax 2

In equation (lo), in which as expected from the peristaltic motion the pressure gradient is a function of the amplitude ratio, the first term on the right-hand side corresponds to the imposed pressure gradient associated

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Couple-stresses in peristaltic transport of fluids

with the primary flow and the higher-order terms correspond to the peristaltic motion, or higher imposed pressure gradient. Substituting (9) into equations (7) and ( 8 ) and collecting terms of like powers of E , we obtain three. sets of coupled linear differential equations with their corresponding boundary conditions in $0, $1

and e2 for the first three powers of E. The first set of differential equations in $0, subject to the steady parallel flow and transverse symmetry assumption for a constant pressure gradient in then direction, yields the following classical Poiseuille flow for a couple-stress fluid

k is the Poiseuille flow parameter, it can be easily shown that

so, lim uo(y) = k(1 - yz)

corresponds to the absence of couple-stress, which leads to the classical Poiseuille flow and

lim uo(y) = 0 Y+O

that is, the effect of couple-stress is quite obvious for small values of y (large values of q).

The second and third sets of differential equations in and +2 with their corresponding boundary conditions

p l ( x , y, t ) = $[@l(y)e'"("-') + @;(y)e-'"("-')] (12)

$2(x, y, t) = i [ b ( y ) + #22(y)e"u(x-') + @;ze-"u(x-')].

Substituting equations (12) and (13) into the differential equations and their corresponding boundary conditions in $1 and $2 leads to the following set of differential equations:

Y"

are satisfied by

(13)

a

GW) e

4

2

0 0 0.1 0.2 0.3 0.4 0.6 0.6 O.? 0.8 0.S y 1

12 I - ..&W

-. -_ ---_ --_ - - - - _ _ 10 -

GM

4 -

2 -

0 0.1 0.2 0.3 0.4 0.6 0.6 0.7 0.1) 0.9 1 Y

Figure 3. Mean velocity perturbation function GO.) for various values of wavenumber a at = 1 with (a) R = 10 and (&) R= 100.

(-$ - 4a2) [ ($ - 4 .2 ) + 2iaR

-qR( -$ -4az ) d 2 ( z - 4 a 2 ) ] & ( y ) dY *

E F El Shehawey and Kh S Mekheimer

I -2

41 I 0 0.1 0.2 0.3 0.4 0.6 0.8 0.7 0.8 0.8 1

8 .... 139.0,

Figure 4. Mean velocity perturbation function G(v) for various values of the couple-stress parameter IJ 2 . u = 1 with (a ) R = 10 and (b) R = 100.

Thus, we obtain a set of differential equations together with the corresponding boundary conditions which are sufficient to determine the solution of the problem up to second order in E . Now, our main intention is to find solutions to differential equations for +l(y), although equation (14) for 41 is a sixth-order ordinary differential equation with variable coefficients, it would perhaps, be impossible to obtain a solution of these differential equations for arbitrary values of R, a and k. This is because of the moving boundary considered in the present problem. The condition of a moving boundary has made the boundary condition inhomogeneous and thus the problem is not an eigenvalue problem, as are all problems of hydrodynamic stability for which solutions are available in the literature. However, we can restrict our investigation to the case of pumping of an initially stagnant fluid, corresponding to no imposed pressure gradient. Thus, in this case (ap/ax)o = 0, which means that the constant k vanishes and we are able to obtain a simple closed form analytical solution for this interesting case of free pumping.

Physically, this assumption means that the fluid is stationary if there are no peristaltic waves. In fact, this assumption is not so restrictive because the maximum pressure gradient that a small-amplitude wave can generate is of the order E' and in the pumping range, the zeroth-order mean pressure gradient must certainly vanish. The solution of equation (14) subject to boundary condition (15) under the assumption that k = 0 may be obtained as

1166

$l(y) = A sinh(h1y) + B sinh(h2y) + Csinh(ay) (19)

where

2 2 112 l j 2 1 hi = - ( ( 2 a 2 + ~ Z ) + [ ( 2 ~ 2 + ~ 2 ) 2 - 4 ( ~ 4 + B Y )I )

J2

[l - E sinh(h2) - C sinh(a)] 1

A=- S W A i )

z1 = XI cosh(hl) sinh(A2) - A2 sinh(A1) cosh(h2)

zz = AI cosh(h1) sinh(a) - orcosh(a) sinh(hl)

2 3 = (A:& - hi) cosh(h2)

24 = (ah: - or3) cosh(a).

Next, in the expansion of @2, we need only concern ourselves with the terms +;o(y) as our aim is to determine the mean flow only. Thus, the solution of coupled differential equation (17) subject to boundary condition (18), under the assumption that k = 0, gives the expression

$&J;o(Y) = g(Y) - g(l) + DI

where

Couple-stresses in peristaltic transport of fluids

Figure 5. Effect of the couple-stress parameter q and wavenumber 01 on critical reflux pressure gradient at R = 10.

0

-1

*.o.o.o,. + “a., -- n.0.s -.“.,.D.

Figure 6. Effect of the couple-stress parameter 7~ on the mean-velocity distribution and reversal flow for R = 10, 01 = 1 and (v/i/a~)~ = -2.0.

Table 1. Effect of the couple-stress parameter and wavenumber on the critical value of (Gpx), at R = 1 .O.

O1 ~~ ~~

q 0.2 0.4 0.6 0.8 1 .o ~~

0.1 4.655 0.2 6.371 0.3 8.130 0.4 9.926 0.5 11.753 0.6 13.604 0.7 15.472 0.8 17.352 0.9 19.242 1.0 21.134

4.822 5.120 6.649 7.146 8.498 9.166

10.355 11.168 12.210 13.151 14.057 15.117 15.896 17.071 17.727 19.017 19.549 20.955 21.364 22.890

5.578 6.239 7.914 9.027

10.212 11.745 12.475 14.414 14.711 17.055 16.932 19.681 19.142 22.299 21.345 24.912 23.544 27.521 25.739 30.128

Thus, we see that one constant CI remains arbitrary in the solution, and CI is found to be proportional to the second-order time-averaged pressure gradient. If we time-average (5 ) for the solution given by equations (1). (2), (9), (10). (12). (13), (19) and (20), we find that

C1= R (2) 2

The constant CI, which is related to the second-order pressure gradient distribution, may be obtained using end conditions of the real physical problem. The mean time- average velocities may be written as

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E F El Shehawey and Kh S Mekheimer

t I

I 0 1 I t

Figure 7. Effect of the mean second-order pressure gradient (%/ax), on the mean-velocity distribution and reversal flow for rl = 0.2, a = 0.6 and R = 10. (a) (%/ax)2 = -2.5, (b) and (d) (@/ax)* = 4.5.

= 0.0, ( c ) ( z / a x ) , = 2.7632

4. Numerical results and discussion

A close look at equation (24) reveals that the mean axial velocity of a couple-stress fluid is dominated by the constants D,, D2, q and the parabolic distribution term

In addition to the terms just mentioned, there is a perturbation term G(y) = g(y) - g(1) = (2/c2)(ipeJ, which represents the perturbation of the velocity across the channel, and its distribution controls the direction of peristaltic mean flow across the cross section.

The constant D,, which initially arose from the non- slip condition of the axial velocity on the wall, is due to the value of & at the boundary and is related to the mean velocity at the boundaries of the channel by i ( & l ) = (c2/Z.)&(&l) = (c2/2)Dl , which shows that the non-slip boundary condition applies to the wavy wall, and not to the mean position of the wall. The variation of D, with q for various values of R and (Y is depicted in figure 2. The numerical results indicate that, for very small values of q, D1 decreases with increasing q , while for large values of q. for which the particles behave like

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a suspension [lo], DI will take a constant value. That is, as the size of the fluid particles increases, the velocity at the boundary decreases and the suspension is solid and no flow can take place.

The variation of G(y) with y for different values of q, a and R is displayed in figures 3 and 4. As expected, G(y) decreases with increasing couple-stress parameter q and, for a fixed value of q, G(y) increases with increasing (Y and R.

It has been observed that urine, bacteria or other material sometimes passes from the bladder to the kidney or from one kidney to the other in the opposite direction to the urine flow. Physiologists term this phenomena 'ureteral reflux'. Two different definitions of reflux exist in the literature. Shapiro et a1 [8] call a flow reflux whenever there is a negative displacement of a particle trajectory, while Yin and Fung [9] define a flow as reflux whenever there is a negative mean velocity in the flow field. In the present analysis the latter definition of reflux is adopted.

Since DI is always a positive quantity, i ( y ) = (E2/2)Dl at y = & 1 shows that mean flow reversal will never occur at the boundaries. Further, from equation (24) it is clear that reflux would occur when the mean pressure gradient ( a p / a x ) z reaches a certain critical value. Thus the critical reflux condition at the centre

Couple-stresses in peristaltic transport of fhids

Yt (.)

I'

~

1

0 I I

~

I

I

Figure 8. Effect of the couple-stress parameter q on the mean-velocity distribution and reversal flow for R = 10, II = 1 and = 2 with (a) q = 1. (b) q = 0.3, (c) q = 0.08 and (d) q = 0.01.

line may be defined as that for which the mean velocity G(y) is equal to zero on the centre line y = 0. Equation (24) yields

1

Y G(0) + DI + --;(Dz - g(1))

(W

For ( F / ~ X ) ~ c ( F / ~ X ) ~ ~ there is no reflux and if ( ; i f ; / a x ) ~ > (%/ax), there will be reflux and backward flow in the neighbourhood of the centre line occurs. The value of (;if;/ax)%" for various values of q. R and (Y are listed in tables 1 and 2 and some values are displayed in figure 5. The results reveal that (V/ax)& increases with increasing couple-stress parameter. Thus, as we take into account the particle size of the fluid, the reversal flow is less favourable. Finally in figures 6-8 the mean velocity distribution and reversal flow are displayed. Figure 6 reveals that the velocity distribution decreases with increasing q and that for large values of q no flow can take place. The effects of q and ( F / a x ) z on mean velocity and reversal are shown in figures 7

Table 2. Effecf of the couple-stress parameter and wavenumber on the critical value of ($ /ax) at R = 100.

U

7) 0.2 0.4 0.6 0.8 1 .o 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 .o -

1.506 1.599 1.766 2.983 3.166 3.499 4.459 4.734 5.232 5.934 6.300 6.963 7.142 7.869 8.697 8.885 9.432 10.425

10.365 11.004 12.163 11.824 12.552 13.873 13.347 14.169 15.661 14.795 15.707 17.361

~~

2.026 2.403 4.015 4.765 6.004 7.126 7.991 9.485 9.981 11.848

11.964 14.202 13.959 16.570 15.922 18.901 17.974 21.337 19.924 23.652

and 8. Figure 8 reveals that the reversal flow decreases with increasing couple-stress parameter (increasing mean diameter of fluid particles). Interpreted physiologically, this means that urine in which particles are suspended (namely urine from a diseased kidney) is less susceptible to reversal flow in the ureter, as the mean diameter of the particles increases.

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E F El Shehawey and Kh S Mekheimer

References

Latham T W 1966 Fluid motion in a peristaltic pump M S Thesis MIT Cambridge, Massachusetts

laffrin M Y and Shapiro A H 1971 Peristaltic pumping Annual Review of Fluid Mechunics vol 3 (Palo Alto. California: Palo Alto Publications) pp 13-36

transport of blood. Casson model41 J. Biomech. 17 821-9

Srivastava L M 1986 Peristaltic transport of a couple-stress fluid Rheof. Acra 25 63841

Stokes V K 1965 Couple-stresses in fluids Phys. Fluids 9 1709-15

Valanis K C and Sun C T 1969 Poiseuille flow of a fluid

Srivastava L M and Srivastava V P 1984 Peristaltic

with couple-stress with applications to blood flow Biorheofogy 6 85-97

Fung Y C and Yih C S 1968 Peristaltic transport ASME J. Appl. Mech. 33 669-75

[SI Shapiro A H, Jaffrin M Y and Weinberg S L 1969 Peristaltic pumping with long wave length at low Reynolds number J. FlKid Mech. 37 799-825

I91 Yin F C and Fung Y C 1971 Comparison of theory and experiment in peristaltic transport J. Fluid Mech. 47 93-112

transport of a patticle-fluid suspension J. Bwmech. Eng. 111 15745

11 11 Chandan S 1982 Lubrication theory for couple-stress fluids and its applications to sholt bearing Wear 80 281-90

[IO] Srivastava L M and Srivastava V P 1989 Peristaltic

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