Peristaltic transport of a two-layered model of physiological fluid

J. Biomvchanics

Vol.

IS. No. 4. pp. 257

PnntedinGreatBritain.

265. 1982

W?I -92908? 040257 09 w3.00/0 0 1982 Perpamon Press Ltd

PERISTALTIC TRANSPORT OF A TWO-LAYERED MODEL OF PHYSIOLOGICAL FLUID* L. M.

SRlVAsTAVAt

and V. P. SRIVASTAVA

Department of Applied Mathematics, Motilal Nehru Regional Engineering College, Allahabad, India Abetract-The problem of peristaltic transport of a two-fluid (peripheral and core fluid) model in a nonuniform tube and channel has been investigated under long wavelength approximation. A comparison of these rqsults with those for without peripheral layer fluid shows that the magnitude of the pressure rise under a given spt of conditions is smaller in the case of with peripheral layer fluid. For a given non zero pressure drop, the flow rate increases as the viscosity of the peripheral layer fluid decreases. However, for zero pressure drop, the flow rate is independent of the presence of peripheral layer fluid. Pressure rise in the case of non-uniform geometry is found to he much smaller than the corresponding values in the case of uniform geometry. The analy$s has been applied and compared with the observed flow rates of spermatic fluid (semen) in vas deferas of rhesus monkeys and to the experimental results of Weinberg et al. (1971).

INTRODUCTION

Peristalsis is now well known to the physiologists to be one of the major mechanisms for fluid transport in many biological systems. In particular, peristaltic mechanism may be involved in swallowing food through the esophagus, urine transport from kidney to bladder through ureter, movement of the thyme in the gastro-intestisal tract, in ductus efferentes of the male reproductive tracts, transport of spermatozoa in cervical canal, movement of ovum in fallopian tubes, transport of @mph in lymphatic vessels and in the vasomotion of small blood vessels such as arterioles, venules and cgpillaries. In addition, peristaltic pumping occurs in many practical applications involving bio-mechanical systems. The study of the mechanism of peristalsis, in both mechanical and physiological situations, has only recently become the object of scientific research. Since the first inv@stigation of Latham (1966), several theoretical andexperimental attempts have been made to understand’peristaltic action in different situations. A review of much of the early literature is presented by Jaffrin and Shapiro (1971). Bums and Parkes (1967), Barton and Raynor (1968), Shapiro et al. (1969), Lykoudis and iRoos ( 1970), Roos and Lykoudis ( 197 1), Lykoudis (19?1), Lew et al. (1971) and Gupta and Seshadri (1978) studied the case of a vanishingly small Reynolds nuniber. Non-linear analysis of two dimensional peristaltic flow has been investigated : (i), assuming a small arhplitude ratio, by Fung and Yin (1968), Yin and Fung (1969, 1970), Yin and Fung (1971). Chow (1970). Raju and Devanathan (1972), Mittra and Prasad (1973,1974), Devi and Devanathan (1975), Wilson et al. (1977), Wilson and Peral (1979) and lReceiwd 19~Nouember 1980. t-1 Address: Department of Mathematics, University of Liberia, Monrovia, Liberia.

Wilson (1979); (ii), assuming a small wave number and Reynolds number, by Zien and Ostrach (1970), Li (1970X Jaffrin (1971X Manton (1975), Liron (1976). Kaimal (1978) and Wilson and Panton (1979). Tong and Vawter (1972) used finite element method to study peristaltic flow. Their work was further extended by Negrin (1972) and Negrin et al. (1974). Hung and Brown (1977) and Brown and Hung (1974) obtained the velocity and pressure fields in orthogonal curvilinear coordinates for two-dimensional peristaltic flow, using an implicit finite difference method. Most studies until about 1970, dealt with sinusoidally varying channels. Lew et al. (1971) considered a nonsinusoidal wave to explain the motion of thyme in the small intestine. In order to provide a better understanding of the peristaltic action in ureter, Lykoudis and Roos (1970), Manton (1975) and Liron (1976) considered the peristaltic wave of arbitrary shape. Peristaltic pumping by a lateral bending wave is studied by Hanin (1968), Wilson et al. (1977) and Wilson and Panton (1979). Further, Mank (1976), Zimmermann (1977) and Mahrenholtz and Zimmermann (1978) investigated the case of unsymmetric wave forms. All these above investigations on peristaltic motion are restricted to Newtonian fluids. However, Raju and Devanathan (1972), and Devi and Devanathan (1975). studied with power law and micropolar fluids respectively. Hung and Brown (1976), and Kaimal(1978), studied the peristaltic flow of solid-particle mixture. Peristaltic flow through elastic and viscoelastic channel is investigated by Mittra and Prasad (1973) and in elastic circular cylindrical tube by Rath (1978) and in non-uniform tube by Gupta and Seshadri (1976). Some new ideas on peristaltic transport are contained in the review article by Fung (1971) and in the book by Liron (1978). Most of the studies on peristaltic motion referred to above have been carried out by assuming physiological fluids to behave like Newtonian fluids of constant

257

L. M.

258

SRIVASTAVA

viscosity. Although this approach provides satisfactory understanding of the peristaltic mechanism in ureter, it fails to give a better understanding when peristaltic mechanism involves small blood vessels, lymphatic vessels, intestine, the ductus efferentes of the male reproductive tract, and the transport of spermatozoa in the cervical canal. In these body organs, viscosity of fluids varies across the thickness of the ducts, and viscosity of the fluid near the wall isdifferent from its viscosity in the centre of the duct [Hynes (1960), Bugliarello and &villa (1970), Goldsmith et al. (1975), and Bugliarallo and Sevilla (1970). have shown experimentally that blood flow through small diameter tubes has blunted velocity profiles and a red cell free plasma layer - Newtonian fluid near the tube wall]. In addition, these organs are generally observed to be non-uniform ducts (We&l and Zotterman, 1926; Wiedeman, 1963 ; Widerhielm, 1967 ; Lee and Fund, 1971). In particular, this is true of vas deferens (a thick-walled tube which connects the epididymis, an elongated organ on the posterior surface of the testis, close to the ejaculatory duct). Spermatic fluid (semen), consisting of spermatozoa and their fluid medium, flows through the vas deferens to the ejaculatory duct and is finally expelled during sexual intercourse from the penis by a series of rapid muscular contractions. Movement through vas deferens is accomplished by means of peristaltic action of contractile cells in the duct wall, the sperm themselves are non-motile at this time (Semens and Longworthy, 1938; Vander et al., 1975). The vas deferens in rhesus monkeys is in the form of a diverging tube with a ratio of exit to inlet dimensions of approximately 4 : 1 (Guha et al., 1975). Hence peristaltic analysis of fluid of single layered constant viscosity in uniform channels cannot explain the mechanisms of transport of fluid in these organs. Recently Shukla et al. (1980) studied the effects of viscosity variation in a uniform tube. With the above discussion in mind we propose to study the effects of viscosity variation of the fluid on the mechanism of peristaltic transport in a nonuniform tube and channel considering two-layered fluid model with particular reference to the flow of spermatic fluid (semen) in vas deferens. The application of the results for flow rates to the flow rates observed in vas deferens in rhesus monkeys has been considered. PERlSTALTlC TRANSPORT IN AN AXISYMMETRIC UNIFORM TUBE

and V. P.

SRIVASTAVA

with a(x’) = a, + k(x’),

(2)

where a(x’) is the half width of the tube at any axial distance x’ from the inlet, a, is the half width at inlet, k (CC1) is a constant whose magnitude depends on the length of the tube, exit and inkt dimensions, b is the amplitude of the wave, 1 is the wave length, and c is the wave propagation velocity. It is observed by Guha et al. (1975). and Gupta and Sheshadri (1976), that the Reynolds number for vas deferens in rhesus monkeys is of the or&r 10m3, and thus the inertia term is neglected in the present problem. In view of this and using long wavelength approximation (Barton and Raynor, 1968; Shapiro et a/., 1969 ; and Lardner and Shack, 1972) the approp riate equations in dimensionless form, describing the flow in the laboratory frame of reference arc

aP -= ax

i a _ au ;g VW~

ap

-= dr

( >

(3)

9

0,

(4)

where r’ r = -, a,

x = -, 1

t=$,

P(r)=,I

U’

X’

u = -,

C

P(J)

p=Pi, J%

with p = p(x,t) pressure and u = u(x,r,t) axial velocity. The non-dimensional boundary conditions are

at4 ar

=

at

r

0,

u=O,

NON.

Formulation and analysis

We consider the flow of an incompressible Newtonian fluid moving in axisymmetric form in a nonuniform tube (Fig. 1). The viscosity of the fluid varies across the tube. We assume an infinite sinusoidal wave train moving with velocity c along the walls of the tube. The geometry of the wall surface is described as H(x’,t’) = a(~‘) + b sin F (x’ - ct’h

(1)

(5)

Fig. 1 Peristaltic transport in a non-uniform tube peripheral layer adjacent to the wall.

(6) (7)

259

Peristaltic transport of a two-layered model of physiological fluid The no-slip condition (7) implies that there is no axial movement of the wall, and so the wall material is necessarily extensible. Integrating equation (3) and using boundary conditions (6), (7), we get the velocity profile as ldp u(x, r. t) = - Idx The instantaneous Qk

0 =

(8)

volume flow rate is given by 2nrudr

-n* 2dx

=

s h r3 .adr*

where

s h

r3

(11)

The pressure rise APL(t) and friction force F,(t) (at the wall) in the tube of length L.,in their nondimensional form are given by

&L(t) =

so “I

0

dp

ii!+

where p, and ~1~are the viscosities of the central and the peripheral layers respectively (pz < p,). The geometry of the interface h, may be given as,

where or= 5 is the dimensionless radius at the inlet a, b, of the central layer and 4, = a, b, is the amplitude of 0 the interface wave. In view of the viscosity variation given in (16). equations (14) and (15) assume the form Ap L(t)

FL(t)

(12)

and

F,(t,_S.“‘^h’(+ix.

(17)

(9)

(10)

0 jYodr.

(16)

h, =?=a+$+$,sin2n(r-r)

Or

I, =

h, < r < h,

(13)

=

_

s s

8p2“’ =

=

D

“I

gii2 7c

h4 -

h2

h*-(1

0

Q(w)

(18)

- /i,)h’:

QW -/i,)h’:

dx.

(19)

Following Shukla et al. (1980), the relation for 4, and h, may be written as 41 = ad,

hl = ah.

(20)

Use of equation (10) in equations (12) and (13), yields 2

APL(t)= - ;

L’LQ(x,t) s * -Yx1

(14)

and FL(C)= f

h2 Q(w) Tdx.

(1%

1

Theetkcts of vilscosityvariation on peristaltic flow can

be investigated ithrough equations (14)and (15)for any given viscosity function p(r). For the present investigation, we, study the viscosity variation through two-layered (psripheral and core) fluid model. Ejkcts

of peripheral

layer

viscosity

To see the effects of peripheral layer viscosity on the flow characterintics of spermatic fluid through vas deferens, we assume the viscosity variation in the dimensionless form as

Equation (20) is obtained by using the fact that the total flux is equal to the sum of the fluxes across both the regions, 0 6 r < h, and h, < r < h. Further, equation (20) implies that the ratio of the amplitude of the interface wave to the amplitude of the wave on the wall is the same as that of the radial displacement in the interface to the radial displacement in the wall. Further, following Gupta and Seshadri (1976), assume the form of Q(x,t) as Q(x,r)

Q

4’

It II -=---z-+-

22x a,

C$sin 2x(x - 1)

+ 24 sin 2x(x --If) + 4’ sin2 271(x- t).

(21)

where Q is the time mean flow over one period of the Nave. The above form of Q(x,t) has been assumed at constant values of Q(x, t) giving APL(t) (equation 18) always negative, and hence there will be no pumping action. After using equations (20) and (2 1) in equations (18) and (191 the final expressions for APL(t) and FL(r) can be obtained as

- t) + 24 sin 2n(x - t) + #‘sin* 2n(x - t) dx

APL(t) = - 8/i, + #sin2n(x

- f)

(22)

L. M.

260

s ’

L./l ;

FL(?) = 8ji2

-

-

0

SRIVASTAVA

and V. P.

results

.-

b2+2ikx4sin2x(xt) + 24 sin 2x(x 2

4, 1 1 (1 P2)a4 I_

When k = 0 in equations (22) and (23), we can get the expressions for a uniform tube which correspond to Shukla et al. (1980) results for pressure drop and frictional force. In addition with fi2 = 1, the present problem reduces to that of Shapiro er al. (1969). Further, equation (22) reduces to the same as that obtained by Gupta and Seshadri (1976) for the single layered fluid of constant viscosity, if P2 = 1. Numerical

SRIVASTAVA

and discussion

I[

1

+

t) + 42 sin2 2x(x - t) 2

+ $ 0

4 sin 27r(x

f)

b

1

(23)

pressure rise and thus maximum flow rate is achieved at zero pressure rise and maximum pressure occurs at zero flow rate. Also the effect of increasing the amplitude ratio is to increase the pressure rise. Furthermore, the distinction between the two values of pressure rise for one and two fluid models increases with increasing amplitude ratio at zero flow rate, but as flow rate increases both the values come closer and closer and ultimately coincide and after that difference increases with increasing flow rate. Figure 6 shows that

To discuss the results obtained above quantitatively, we shall compute the dimensionless pressure rise ApL(t) and friction force FL(t) over the tube length for different given values of the dimensionless timemean flow 0, with amplitude ratio 4 and ji2. The average rise in pressure L\pL and friction force F, are then evaluated by averaging ApL(t) and FL(t) over one period of the wave. As the integrals in equations (22) and (23) are not integrable in the closed form, they are evaluated numerically, using a digital computer. We use the Collowing values of the various parameters in equations (22) and (23) for flow in vas deferens of rhesus monkeys (Gupta and Sheshadri, 1976). a,, =

O.O12cm,

a = 0.92,

L = i = 2Ocm,

k = ?

=

0.0018,

c = 1Ocmfs.

Figures 2 and 3 represent the variation of dimensionless pressure rise with dimensionless time for 4 = 0.8 and for different values of ji2, in the case of nonuniform and uniform tube respectively. A comparison of the results with those for without peripheral layer of the pressure rise (F2 = 1) shows that the magnitude under a given set of conditions is smaller in the case of with peripheral layer (ji2 < I). As p2 approaches one, the values for the two layered model in both uniform and non-uniform geometry cases approach the corresponding values of the single layered model. Comparison of Figs. 2 and 3 shows further that the pressure rise in the case of the non-uniform tube is much smaller than the corresponding values in the case of the uniform tube. This happens because complete occlusion occurs only at the entry in a diverging tube, whereas in a uniform tube it occurs at all points along the tube. It can also be seen that the effect of increasing the flow rate is to reduce the pressure rise for different values of I#Jand &. The average pressure riseversus time mean flow rate is plotted for different values of 4 and ii2 in Figs. 4 and 5, which shows a linear relation between them. It is clear that an increase in flow rate decreases the

Fig. 2. Variation of pressure rise over the non-uniform tube length.

__ _-________

_____

____

Fig. 3. Variation of pressure rise over the uniform tube length.

Peristaltic transport of a two-layered model of physiological flui,

at a given flow rate an increase in the value of wavelength leads to a decrease in pressure rise. Further, p2 has more effect on pressure rise at zero flow rate. Finally the friction forces FL(t) and F, have been plotted in Figs. 7 to 10.

PERISTALTICTRASNSPORTIN A NON-UNIFORM TWODIMENSIONALCHANNEL

Following the same procedure as in the case of a pressure rise and friction force are obtained as tube, the expressions for non-dimensional L,i

AM)

= - 3&

i I)

The variations are qualitatively similar to those in the axisymmetric uniform and non-uniform tube as discussed earlier. The results of the present analysis have been compared to those of the theoretical results of Jatfrin and Shapiro (1971) and the experimental results of Weinberg et al. (1971) in Fig. 13. APPLICATION

Our analysis of an axisymmetric non-uniform tube may be applied in the study of flow rates observed in vas deferens. According to the experimental observations of flow in the vas deferens of rhesus monkeys

Q + f$ sin 27r(x - t) [ 1 - (1 - &)a3

I[

1+ 5

261

3 dx

+ 4 sin 2x(x - t)

1

(24)

and

s L/i

FL(f) = 3&

Q + 4 sin 2rr(x - t)

0

When k = 0 in equation (24) and (25), we can get the expressions for a uniform channel which correspond to the Shukla et al. (1980) results for pressure drop and frictional force. In addition, with & = 1, the present problem reduce$ to the same as that of Shapiro et al. (1969) and Lardner and Shack (1972), when the eccentricity of the elliptical motion of cilia tips is zero in their analysis Further, equation (24) reduces to the same as that obtained by Gupta and Seshadri (1976) for the single layered fluid of constant viscosity, if & = 1. Numerical

results and discussions

Variations of APL(t), FL(r), & and FL with t and 0 for uniform and non-uniform two dimensional channels with a = 0.833 have been shown in Figs. 11 to 15.

-0I

.2

4

,dx. t)

-6

.6

10

1.2

14

(25)

1

(8-10 kg body wt) made by Guha et al. (1975k the period of ejaculation was about 2 s and the average flow rate was 0.02 ml/s for 30 mm fig pressure rise. To compare our analysis with their observations, we assume the following approximate values of the various parameters for flow in the vas deferens of rhesus monkeys (Gupta and Sheshadri, 1976); a, = O.O12cm, k = 0.0018, L = i. = 2Ocm, c = lOcm/s. The viscosity of rhesus monkey semen is of the order 4 c.P. (Gupta and Sheshadri, 1976) and hence viscosity of the fluid in the core region may be taken as of the same order. Assuming the peripheral layer viscosity as 1 c.P. (of water), p1 may be found as 0.25. Considering the values of peripheral layer thickness as 0.00096 cm

16

ii/n Fig. 4. Pressure-tlow rate relationship for a non-uniform tube.

Fig. 5. Pressure-flow relationship for a non-uniform tube.

L. M. SRIVASTAVA and V. P. SRIVASTAVA

262

10.25

P/W80 ____ __--_----_____

--.* ‘L,’

I

-50

I

I

t

.2

.I

.6

0

.o 6pl

ln

1.2

1.A

ls

Fig. 6. Effect of wave length on the prcssurc-fiow relationship in a non-tmiform tube.

rate

and b = O.OlO!? cm, the values of 4 and a are obtained as 0.9 and 0.92 respectively. With the above values, the measured pressure rise of 30 mm Hg corresponds to the dimensionless pressure rise (4 G) of 0.184. The corresponding dimensionless flow rate is then obtained from Fig. 5 as 1.475which gives the flow rate as 0.0066ml/s. This value differs from the observed value by 67%. CONCLUSIONS The problem of peristaltic transport of a two-fluid (peripheral and core fluid) model in a uniform and non-uniform tube and channel have been discussed under long wavelength approximation. A comparison of the results with those for without peripheral layer shows that the magnitude of the pressure rise under a given set of conditions is smalkr in the case with peripheral layer. Pressure rise in the case of non-

Fig. 8. Variation of friction force over the uniform tube length.

uniform geometry is found to be much smaller than the corresponding values in the case of uniform geometry. Friction force possesses opposite character to the pressure rise. Further, it can also be concluded that the flow rate increases as viscosity of the peripheral layer fluid decreases for any non-zero value of pressure drop. However for zero pressure drop (or pressure rise), it has been observed that the flow rate is independent of the presence of peripheral layer fluid, but the same is not true in the case of friction force. The results of the present analysis have been compared to those of the theoretical results of JatTrinand Shapiro (1971)and the experimental results of Weinberg er al. (1971).Finally, -.7 -

-6

e.7

-+=o -.5

Fig. 7.

Variation

of friction force over the non-uniform length.

tube

Fig.9. Friction

forc+flow

rate relationship for a non-uniform tube.

263

Peristaltic transport of a two-layered model of physiological fluid

6

Q=7

-6-O

5 ‘ .3 2 -1 n’ Ia

6 -1 -2 -3 4

-.6

A0

1

.2

1

.I.

1 .6

1 .6

a/n

1 so

0

1.2

1 14

0

1

2

3

.L

5

6

7

6

9

B

J 1.6

Fig. 12. Pressure-flow rate relationshiu for a non-uniform

Fig. 10. Effect of wave length on the friction force-flow rate relationship in a non-uniform tube.

the analysis has been applied and compared with those observed flow rates in the vas deferens of rhesus monkeys. Our calculated value of flow rate differs from the observed value by 67%. This error may be due to the fact that in our calculation of flow rate we had to use guessed vahres of various parameters such as 4, & and c, as data are not available. Also, it seems that there are other thctors, including cilia, responsible for semen transport. Moreover, the assumption of a sinusoidal wave’shape in the present analysis may not be true in the case of the actual vas deferens tube. The authors feel that icomplex wave motion as suggested by Wilson et al. (1977) may provide better results for the vas deferens. We conclude that considerably more

Fig. 13. PressurcRow rate relationship for a dimensional uniform channel. two-dimensional theory for R = 0,

# 1.6 -

tip25

----

PI'1

K =

%

=

0; - .- .- two-dimensional theory

corrected for end walls (JaKrinand Shapiro, 1971); --- twodimensional theory corrected for end walls and inactive pumping regions (Jaffrin and Shapiro, 1971). Data points, experimental results of Weinberg et al (1971).

6-

51 0 Fig. 11. Variation of pressure rise over the non-uniform channel length.

I

.I

’

.2

’

.3

’

4

’

.5

’

.6

’

.7

’

.6

’

.6

Fig. 14. Variation of friction force over the non-uniform channel length.

L. M.

264

SRIVASTAVA

-.5

-.L -.3

-d -.l 0 .l

16 .I .3

.4 .5 .6 .7 0

.l

.7,

.3

.L

.5

.6

.7

.6

.)

6 Fig. 15. Friction force-tlow rate relationship for a nonuniform channel.

theoretical and experimental investigations are necessary to understand adequately the flow in the vas deferens. Acknowledgement--The

authors gratefully acknowledge sup port of this work by a project grant from the Ministry of Education and Social Welfare, Government of India.

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radius or half width at inlet of the central

al,

layer t(

( 1 =-

aI,

a,

b

amplitude of the peristaltic wave amplitude of the interface wave wave speed constant length of tube. or channel dimensionless pressure rise in the tube or channel of length L

b, c k L &#I

dimensionless average pressure rise dimensionless instantaneous volume flow rate dimensionless time average flow rate dimensionless friction force at the wall dimensionless average friction force time dimensionless time wave length of the peristaltic wave coordinate in circular tube viscosity function viscosities of the fluids of the central and the peripheral layers

APL

Q

Q FAt)

FL t’ i

(r’flr’)

dr’) ihG2

dimensionless layer fluid

b

a

4 a,

radius or half width of tube or channel radius or half width of the central layer radius or half width at inlet of the tube or channel

viscosity of the peripheral

density of the fluid

P NOMENCLATURE

dimensionless radius or half width at inlet of the central layer

4(=-)

amplitude ratio

a,

amplitude ratio at the interface dimensionless dimensionless

pressure axial velocity