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A long wave approximation to peristaltic motion
J. i3rwwclrun~c~
Vol. 3. pp. 6?-75.
Pergamon
Press.
1970.
Printed
inGreat
Brttain
A LONG WAVE APPROXIMATION PERISTALTIC MOTION*
TO
T.-F. ZIEN and S. OSTRACH Case Western Reserve University, Cleveland, Ohio 44106, U.S.A. Abstract-The problem of motion of a viscous incompressible fluid induced by travelling wave motions of the conflning walls has been studied analytically for the two-dimensional geometry. The analysis is aimed at the possible application to urine flow in human ureters. The wave length, A, of the peristaltic waves is assumed to be large compared to the half channel width, d, whereas the amplitude of the wave, a, need not be small compared to d. A systematic approach baaed on an asymptotic expansion of the solution in terms of the small parameter d/A has been used and solutions up to OW/A~) have been presented in closed forms. The limiting solution has been studied including the effects of externally applied pressure gradients, and the criteria for backward flow have been established and discussed in detail. Higher order solutions which include the effects of non-linear inertial terms of the Navier-Stokes equations, have been studied for the case of zero mean volume flow. The results indicate that the peristaltic waves on the wall do not contribute to the time-averaged flow quantities to O(cf/A), but they do give rise to a mean axial velocity of OW/A*), and a mean pressure gradient of O(d*/A*) is associated with the motion. Comparisons with some earlier results have been made and certain agreements are brought out. 1. INI’RODLJCTION
metric, travelling transverse waves are imposed. The channel has a mean half width d, and the travelling wave has,an amplitude a, a wavelength A and a phase velocity c. The motion of the fluid is studied with and without externally imposed axial pressure gradients. The assumptions are that the wavelength is long compared to the mean half width of the channel, i.e. 6 = d/h + 1, and that the wave propagation speed is moderate so that a Reynolds number defined by R = cd/v is of order unity. This latter assumption is equivalent to requiring the frequency of the peristatic wave to be small compared to the reciprocal of a characteristic time for the vorticity diffusion. Bums and Parkes ( 1967) studied peristaltic motions in both two dimensional and axisymmetric configurations using Stokes equation and assuming that the wavelength is large compared to the wave amplitude. The inertial terms in the Navier-Stokes equations are entirely neglected. Therefore, their results may apply only LOthe case of creeping motion
of fluid transport by peristaltic motion of the confining walls has received careful study in the recent literature, (see the list of references of the present paper). The need for peristaltic pumping may arise in circumstances where it is desirable to avoid using any internal moving parts, such as pistons, in the pumping process. Moreover, peristalsis is also well known to the physiologists to be one of the main mechanisms for fluid transport in a biological system. Specifically, the passage of urine through a human ureter is an example of such a process. It is for the purpose of understanding this specific process of urine flow that the present investigation is undertaken. A simple two dimensional model has been chosen in the following analysis, although the more realistic case of axisymmetric configuration presents no essential difficulties. A viscous incompressible fluid with kinematic viscosity v is assumed to be confined in a channel with flexible walls upon which symTHE DYNAMICS
*Received
1April 1969. 63
B.M.Vol.3 No. I- E
64
T.-F. ZIEN and S. OSTRACH
of the fluid, which corresponds to very slow motion with very low frequency. Hanin ( 1968) dealt with the case of two dimensional motion with zero mean axial pressure gradients under the assumptions of small afd and large X/d, while imposing no restriction on the frequency of the peristaltic waves. Fung and Yih ( 1968) treated the motion also in two dimensional geometry, assuming a/d cg 1 but allowing arbitrary wavelength and frequency. They presented results which apply only to the case where the externally imposed axial pressure gradient is either zero or of the order (a/d)2. Furthermore, Shapiro, JatIiin and Weinberg (1968) reported some theoretical and experimental studies of the problem for both two dimensional and axisymmetric geometries. They assumed that d/A4 1 and low frequency, leaving the amplitude ratio a/d arbitrary. Only the limiting solution for infinite wavelength and zero frequency has been presented, and the flow is under some externally imposed pressure gradient in the axial direction. Most recently, an analysis based on the assumptions similar to those used by Shapiro et al. was presented by Barton and Raymore ( 1968) for axisymmetric geometry. They included a discussion for short wavelength case also. Our analysis is based on the same assumptions as those used by Shapiro et al. (1968), but the method of solution is different. In addition, we have carried out the calculations to the order (d/Q2 and obtained closed form solutions. Thus the nonlinear streaming effects on the fluid. motion are clearly demonstrated and the effects of large but finite wavelength are made obvious. While the higher-order solutions are mainly concerned with the case of zero mean volume flux, pumping under arbitrarily imposed pressure gradients with the consequence of finite mean flows is considered for the zeroth-order solution and criteria for reflux are established to this limiting order. Our results agree with those of Shapiro er al. ( 1968) to the zeroth order, and reduce to those due to Fung and Yih (1968) in higher orders when taken to their limits.
Physiologically speaking, our results as well as those of Shapiro et al. ( 1968) relate to the urine flow in human ureters under healthy conditions whereas the results of Fung and Yih f 1968) apply to the pathologic conditions of ureteral obstructions. 2. MATHEMATICAL FORMULATION We consider the flow of a viscous incompressible fluid with density p and kinematic viscosity Y through a two dimensional channel of width 2d with flexible walls. A rectangular Cartesian coordinate system (X, Y) is chosen with the X-axis aligned with the centerline of the channel and in the direction of wave propagation on the walls. A velocity vector (U, v) is defined accordingly. The travelling waves are then represented by Y =+q*(X,
T) =& d+acos [
t(X-cT)], (1)
where a, A and c’ represent the amplitudk, wavelength and the phase speed of the waves respectively (see Fig. I). 2.1 Boundary
conditions
and equation
of
motion
The boundary conditions that must be satisfied by the fluid on the waving walls are the no-slip and impermeability conditions. The fact that the equation of the waving walls is given determines the velocity component normal to the moving walls in the following way. Consider the general case where the surface of a moving wall is represented by the equation F(X,Y,T)=Y-Y,(X,T)=O.
(2)
Then the fact that fluid particles originally on the wall must remain on the wall implies that ETF=O
(3.1)
A LONG
WAVE APPROXIMATION
65
Fig. I. Geometry of the problem.
which gives explicitly V- UY,,--
Y1r= 0 on Y = Y,(X,T)
(3.2)
where, as usual, D/DT is the substantial derivative and subscripts denote partial differentiation. Since the velocity component normal to the moving surface, qn, is equal to equation l/+(1 + YL)(V- UYIX), (3.2) determines qn as (4) The tangential component of the fluid velocity on the wall is usually determined from the mechanical property of the wall. For example, if the wall is assumed to be inextensible, then another relation between U and V can be derived from the no-slip condition. The fluid velocity vector q is thus completely specified on the moving wall. However, we shall choose the simple condition here that the wall has only transverse displacements at all times so that U = OonY=YI(X,T). It can general, no-slip surface velocity
(5)
be shown that this assumption, in implies an extensible wall. The exact boundary conditions on a moving can also be used to determine the components (U, V) on the surface
which is flexible but inextensible. Taylor ( 195 1) has given explicitly the boundary conditions on U and V for a flexible but inextensible waving plate for small values of a/A. He showed that U/c on the boundary starts with 0( az/h2). Since the mechanical properties of the actual ureter wall are quite complicated and it is highly unlikely that the wall is inextensible, we propose to adopt the simpler condition of U = 0 on the wall. We note here that all the previous work (cited in the reference list) used the same assumption. Substituting YI(X, T) = A [d+a cos 2rr/X (X-CT)] into equations (3.2) and (5) for the particular wall motion under consideration, we find the following set of boundary conditions on (U, V) for our problem:
u=o
(6.1)
(X-CT).
(6.2)
Finally an axial pressure gradient must be prescribed at some X station. This boundary condition will be discussed later in connection with the distinction between zero and finite mean volume flows. The equations of two dimensional motion of a viscous incompressible Newtonian fluid can be written in terms of a stream function 9 as
66
T.-F. ZIEN and S. OSTRACH
(see, for example, Schlichting,
1960)
VW’Ti- **WIfx - *\IIxvz*), = UVQQ, (7) where V2 denotes the Laplacian operator (a2@X2+ P/aY2) and * is the stream function defined by *x=--c/ Yy= 2.2 Dimensionless sions
made and we shall seek an asymptotic expansion of Ji in the limit of 6 -+ 0. The following form of asymptotic expansion is postulated:
4.4%Y, t; 8%f, RI - Jtok Y, t; e, R) +Grfi,(x,y,t;e,R)+62Jr2(x,y,t;e,R)+...
(13)
(8.1) u.
(8.2)
formulation
and
expan-
The characteristic lengths in the X and Y directions, respectively, are A and d, which will be chosen as the length scales for X and Y. A natural time scale for the problem is the period of the peristaltic wave, A/c. Therefore, we introduce the following dimensionless quantities:
Equations ( IO) and ( 11) will be studied by using (13) together with the following limit process k Y, t; e, R)
fixed as 6 -+ 0.
(14)
The leading term of the expansion (13) should be interpreted as the limiting solution as 6 + 0. It is to be noted here that the limit process (14) has the property of leaving the complete travelling wave structure unchanged and to be of order unity in the imposed boundary conditions. Hence & should represent x=- X k,y=$, t=;T; the meaningful limiting solution of very long (9) waves. Also implied in this limit process is the JI(x,y,t) =$,*=$v=~. fact that the transverse and axial coordinates are scaled differently and, as a consequence, that the axial velocity component U is an Substitution of (9) into (6), (7) and (8) yields order of magnitude larger than the transverse the folowing partial differential equation for $I velocity component V. This type of limiting together with the boundary conditions: process is the same as that used in studying the propagation of long waves on shallow water (see Stoker, 1957). Therefore, the corresponding expansion procedure will be referred to as the long-wave approximation. .
(IO) JI, = T 2ne sin 27r(x - t) (11.1)
y=+(1+ECOS27r(X-t)): I& = 0.
(11.2)
Three dimensionless parameters present themselves in this formulation, namely E = a/d,
S = d/h,
R = cd/v.
A long wave approximation
(12)
will now be
2.3 Axial pressure gradients The axial pressure gradient in the flow field will play an important role in the subsequent analysis, particularly in the discussion of volume flux, and hence, deserves special attention. In terms of the stream function, the axial pressure gradient can be expressed as
A LONG WAVE APPROXIMATION
The dimensionless
67
3.1 Zeroth order solution
form of it is
The general solution of Jl,,is $0 =X3(x, t)y3+A2(x, t)y’ +
In the limit of 6 + 0, we can obtain an asymptotic expansion for the pressure gradient by substituting the asymptotic expansion for tj~.equation (13), into equation (16). We have P= Bo+6F1+P82+.
* *,
t)y+&x.
+&x.
JI,h, - WG,, 1. ( 16)
t).
(26)
Boundary conditions (23) and symmetry conditions on JiOindicate that +,, must be odd in Y. hence /jO 3 A’, = 0.
(17)
F&(X, 1) and A, (x, t) are found to satisfy the following two equations
(18)
s3(x.t)~+n(x,t)~=-2~~sinp
where
(27.1)
3772(x. t)A, +A,
= 0,
(27.2)
(19.1) ( 19.2)
where, for brevity, we have introduced and p(x. t) defined by ?)(X,t)
=
l+ECOShr(X-t)
/3(x-t) = 2n(x - t). - 91,90x,, - JlOI/!LU- +1,1.
(20) (21) 2 aVo R
aX2ay
i a%
atay2
(28.1)
(28.2)
(19.3)
3. SOLUTIONS Applying the limit process to the equations ( 10) and ( II), and substituting equation ( 13) into (IO) we obtain the following system of equations for Jlo, &, JI2etc. -$JI,=O
11(x, t)
I a+, a3Jio
ay axay*
(22) The boundary conditions are
Equations (27) are easily solved to give A (x t) _ C(t) -3ecosP 39
-
a,(,,
t) =-3
c(t) -3e cos p (29.2) r) *
It is worth mentioning here that we must realize the obvious restriction on the value of E, i.e. l < 1. The case E = 1 would lead to the appearance of a singularity because the denominators in equations (29) would become zero for x-t=n/2, n=l,3,5 ,.... This is due to the fact that the channel would be completely ‘blocked’ at these points. and continuity of flow then demands infinite velocity.
Jlot=~2~csin2rr(x-t).JI,,=0 $lI = JI1, = 0 $22
(29.1)
q3
=
dhY
=
0.
(23) (24) (25)
68
T.-F. ZIEN
and S. OSTRACH
The zeroth-order solutions then take the form
_; PO=;
$Jo= ~C(f)-~ecoS~l~-3[C(t)-facos~l~
go=-
l Tdt-+ej+;=$@dt] C(t)
[I 0
77
2 ‘C(t)dt,
(35)
(36)
I0
(30)
with bar denoting the time average. In terms of these expressions, the meaning (3 1) of the function C(t) is more obvious. It is seen to be directly related to the mean volume flux. (32) The relation between the function C(t) and the pressure gradient is evident from equation The arbitrary function C(t) arises during (35). For convenience, we shall consider only the the integration of equations (27), and is related simple case C(t) = Co = const. in the folto the volume flow in the following manner. lowing. This corresponds to the case of imThe instantaneous volume flow through the posing a constant volume flux on the flow. We channel is given by (actually one-half of the note the following results of elementary intevolume flow) gral calculus: ,,3
.Y?-I [c(fj-hOSp] rl ( r12 >
120= &A,--,= -2C(t)+Ecos&
(33)
From equation (33), it is clear that the volume flow consists of two parts, one being due to the peristaltic waves and the other being represented by the arbitrary function C(t). This arbitrary function is, of course, related to the as yet unspecified axial pressure gradient through equation (32). Note that the case of C(t) = 0 gives U, = 0 for c = 0, i.e. the fluid is completely at rest if the peristaltic waves do not exist. It will be shown later that this case corresponds to zero mean volume flux. The case of non-vanishing C(t) may be referred to as forced pumping, because here the fluid velocity exists even if E = 0. It will be revealed later that this case corresponds to a mean fluid transport of finite magnitude. (1) Time-aueraged jhw variables. Let us investigate the flow variables averaged over one period of the wave motion. Thus
I I
:; -=
(1 _
E2)
;=@$dt=$-
I
-I/2
(1-,2)-1’2]
(37.1) (37.2)
+(2+8)(1-2)-5~z
‘cosp dt = -Qe( 1 - @)-=. I 0 .n3
(37.4)
It follows then _=
P ,=~~~-FP)-j’2[(1+~)C,+~~~]
(38)
ao=~[1-(1-EZ)-1’*+QE~(1-eZ)--5’~y*] -i-3c, [(l+f)(l--F*)-S’S’-(l-~*)--ll~] (39) d o=-2co.
(40)
Hence ~o=~~*-n-5f2[tel-(l+~)ao]
(41)
A LONG WAVE APPROXIMATION
7i,==${l-(1 +&
_ l2) -112+ SE2( 1 - E’) -5/zy2
(~_E2)-1/2[
I+; (
(I--2)-512y2 >
II
.
(42) Equation (41) suggests that a finite mean volume flow would imply the existence of a ‘favorable’ mean pressure gradient in addition to the pressure gradient associated with peristaltic motion of the walls. To be more explicit, we denote the pressuregradient corresponding to zero mean flow by ( P,do,. then equation (4 1) gives
with 91 (~~)o=z~E2(l--2)-5/2.
(44)
We note here that equations (43) and (44) are in exact agreement with the results of Shapiro et al. (1968). Also implied in the above results is the fact that peristaltic waves can overcome anzdverse pressure gradient of the magnitude ( PO)p A mean volume transport in the direction of wave propagation will be possible fez any pressure gradients less adverse than ( PO),,. In the absence of peristalsis (E = 0), equations (41) and (42) combine to reduce to the well-known plane Poiseuille flow results with Q,, directly proportional to the externally imposed pressure gradient. However with peristalsis present, the required pressure gradient for producing a given mean volume flow is not directly proportional to the mean volume flow, but depends on the amplitude parameter E of the peristalsis also, see equation (41). It is because of this interaction that the additional velocity profile due to Q0 in equation (42) is not quite Poiseuillean. In fact, this additional profile tends to give a forward velocity near axis and backward velocity near the walls for any finite E.
69
(2) Criteria for r&u. The phenomenon of reflux is known as the backward flow of urine from bladder to kidney and is regarded as a pathological condition in human physiology. Since it is revealed in the previous section that the case of zero mean volume flux corresponds to an adverse mean pressure gradient, it appears likely that reflux might occur in this case. Indeed, we see from equation (42) that in the absence of QO, the mean axial velocity along the axis (y = 0) is always negative for all E, 0 < l < 1. The question of importance is then under what conditions of pressure gradient the reflux can occur. In the following, the reflux criteria are discussed in terms of QO, which is obviously related to the mean pressure gradient. We note here first that since the imposed velocity profile due to Q0 tends to contribute a negative velocity near the mean location of the wall, large Q0 is likely to incur reflux. Rewrite equation (42) as
+g
(
1+;
>
(l-E2)-5$$-Q0)y2.
Then the following easily established: (a)
criteria
0 G Q,, < l-
(45) for reflux are
(l-e2)1’2.
This range of Q0 will always result in negative ii0 near the axis of symmetry, and hence reflux will always occur near the axis. The corresponding velocity profile is sketched in Fig. 2(a). (b)
do=
1-(l---2)1/2
I Q,,.
This is the critical value of QO, below which reflux always occurs. Corresponding to &, there is a critical mean pressure gradient P,, obtained from equation (4 1) BC+_-E2)-2
1 >
0.
(46)
T.-F. ZEN
and S. OSTRACH
(a)
5
(b)
Cc)
0
.l_(l_E+
l_$)!i<;
I-(
<+
0 2+c
Cd)
(4
(0
_----I
-.3c2 :+E
2
_zIYjsig.___. _._
To > j$+
k2
Fig. 2. Velocity profiles for various ranges of mean volume flow; cases (a) and (f) correspond to reflux.
The velocity profile is sketched in Fig. 2(b).
(c)
&>
&, = iioP+
(1-3)-*‘2 +Qe2 l-I-2E2 Yl ( >
Q. > 1-(l-E2)1’2.
- 8) -W
I
> 0, (47)
This range of & will never result in reflux. The velocity profile is sketched in Fig. 2(c).
and a zero mean resultant pressure gradient. (e)
I?+$> Qo>&, p = flE-2
This particuiar velocity profile
value of eO yields a uniform
E
( 1-
$) 512
1-(1-8)~~2(1+~)-1](1-~~~)-1.
A LONG
WAVE
As 0, gets larger and falls in this range, the velocity profile shows an opposite curvature, although reflux does not occur. (See Fig. 2(e) for velocity profile.)
07
Q,, >&+k2.
When o,, gets exceedingly large and falls into this range, then reflux occurs near the I.0
71
APPROXIMATION
Hence reflux will never occur and the upper bound for 0, given above is meaningless. It is clear from the above discussions that reflux will occur, in the presence of peristalsis, for the range of values of 0, indicated in cases (a) and (f). Based on these results, a region for no reflux is sketched on a go - E diagram in Fig. 3._It is to be emphasized that the whole axis of Q. also belongs to the region of no reflux. Since the critical mean pressure gradient,
c
0.0
0.6
la” 0.4
0.2
0
0.2
o-4
O-6
0.6
Fig. 3. Criteria for reflux (zeroth-order results). The entire positive &axis plus the bounded area indicates the region for no reflux.
wall. The velocity profile for this case is sketched in Fig. 2(f). It must be borne in mind that the above criteria apply to the case of finite amplitude ratio (i.e. E > 0). For e = 0, that is, in the absence of peristalsis, any finite positive 9, will result in an exact Poiseuillean profile, (ii, - 1 -y2), as is evident from equation (42).
pw, is positive, it is seen that reflux will not necessarily occur even when pumping against an adverse pressure gradient. On the other hand, it is shown in case (d) that the case of peristalsis under zero mean resultant pressure gradient corresponds to a uniform profile of I&,.It is positive and ensures the condition of no reflux.
72
T.-F. ZIEN and S. OSTRACH
3.2 First-order solution Substitution of the solution of Jlo into equation (2 1) gives
and the pressure equation ( 19.2)
gradient
follows
P, = -A,,--A&+6&.
if C(t) is taken to be C,. The solution of +I satisfying ment &(y) = -h(--~) is f!h
the require-
from
(54)
In the following, higher-order solutions will be investigated only for the case of zero mean flow. Thus we shall let C(t) = K(t) = 0. Let
(-GY, t) = ~~A,y?+~(A,A,-A,A,,-K,)y’ +&(x,
t)y3+I;,(x,
t)y.
(49)
The set of homogeneous boundary conditions on I& (equation (24)) determines the functions k, (x, t) and k3(x, t) up to an arbitrary function K(t) , with Mx, t) --l(A,A,-~,A,,--A,,)?” lo
1 -
Then velocity and pressure gradient reduce to, respectively,
-
1 -u1 R = +&&Y~+~
(&AI
--AI,A,-&)Y~
+ 3ksy*+ k, As is in the case of zeroth order solution, the function K(t) relates to the value of +I along the waving wall through
Q, = +~l,-, = RK(tl,
(52)
and hence represents the volume flow due to external causes, e.g. an external pressure gradient of O(6). The velocity, &, is immediately found to be
1’1= -AIt -A,Als+6kp
(55) (56)
It is easily verified that G, = 0
and
F1 = 0.
(57)
Therefore we conclude that the peristaltic motion of the walls contributes nothing to the mean axial velocity and mean pressure gradient of O(6). Also, with some algebraic manipulation, we can write down the solution up to O(6) for this case, and then pass it to the limit of small wave amplitude ratio with x, y, t held fixed.
73
A LONG WAVE APPROXIMATION
The result is lii$
1
a5(x, t) = jj -~A3z~+6R(k,A3z-k,J3
- ky(l-iy7)
cosp-$leR6y(l-y7)2
&+o
+A&ss--Ww-M
sinp+0(c2,ijz).
(58)
Note that equation (58) is in exact agreement (to the orders considered, of course) with Fung and Yih’s (1968) result when their free pumping solutions up to Ok) are expanded in the limit of 6 * 0, holding our variables (x, y, t) fixed. It is worth noting that a finite limiting solution exists in Fung and Yih’s problem of small amplitude waves when the wavelength is allowed to become very large, with independent variables properly scaled, as was done in our analysis. 3.3 Second-order solutions Since the peristaltic waves on the walls do not produce any effects on the flow field to order S, we propose to carry out the secondorder solutions. Again we consider the zero mean flow case. Using previous results for &, and $,, we have, from equation (22)
1 av2 --=+&,,(x, Ray2 .
t)y”
t)y7+$us(x,
7! $-3!a,(x,
tw+
f)Y
(59&b,
(59)
1.
The general solution of I~J~, with the required symmetry properties, is easily obtained as j$2=ully*l+uOY9+a7y7+a5y5+s3y3+S1y (60) The functions s3(x, t) and sI (x, t) are again determined from the homogeneous boundary conditions, equation (25). For zero mean volume flow, we find s, (x, t) = (a,+ Z!q2u7+ 3~7~u~+4$u,,)-q~ (61.1) s3(x, t) = - (2u5 + 3u7v2+ 4u&!4+ 5u&)772. (61.2) Therefore, the second-order solution has also been obtained in closed form. In principle, the mea+ velocity ii2 and mean pressure gradient P, can be calculated explicitly as was done for the zeroth- and fist-order solutions. However, the algebra becomes quite extensive and tedious. In the following, the mean quantities are investigated in the limit E --* 0. This simplifies the algebra somewhat, and gives the essential information needed. We note that
where
. [ 1 4x,t> =$6 64, - 1)W&)t -;&z
a,,(~, t) = j$lZR
(AA1
3s
$43(&43J,-$43A~z
-AJ
(AI&-&AI~ a.;(~,t) = +
1
3
-A
3E
)+zA10 3
-A3,L--3Aw43,-43
;RNA- l)(A~&-A3A~r
1
Au2 = 1l&y’o+
9dgy8+ 7a7y6 + 56,~~ + 3s3y2 + sl.
As E -+ 0, we find, with considerable of algebraic manipulation, d,l = O(E2)
fro=
&
Rr2e2 + o(e2)
--A,),
12(k,A,),--A,,(A,A,,--A,A,,--A,,)l
a7 = -&-R&l
+ o(e2)
(62)
amount
74
T.-F. ZIEN and S. OSTRACH
and hence
&
=
( -& (
s* = -
Therefore, ii*=
$2
>
RA*
$R-*+&)R~~E*
F2= RZRe*yh-&)
+ ok*) +o(c*)
we have
ly*_L&+_Ly4 xZR*e* 16o [ +(;j&-&)Y*-(&j+f~)]+ok*). (6%
Similarly, we have for the second-order sure gradient
pres-
= const.
(64)
Notice here that_there are no terms of O(E) existing in ti2 and P, as E + 0. This is also in agreement with the findings of Fung and Yih ‘( 1968). However, it is somewhat surprising to find that p2 is constant across the channel (and generally positive), with all the coefficients appearing in each y-dependent terms cancelled with each other. This conclusion is also consistent with the results of Fung and Yih for E --* 0 with 6 fixed. 4. SUMMARY AND CONCLUSIONS
Peristaltic motion has been studied for two dimensional geometry in the limit of long wavelength and low frequency. Asymptotic expansions in terms of a dimensionless wave number, 6, have been constructed, and solutions are obtained in closed form to O(S*). The limiting solution (i.e. the zeroth-order solution) has been studied including the effects of arbitrarily imposed pressure gradients. It is found that the peristaltic waves for the case of zero mean volume flux correspond to an ‘adverse’ mean pressure gradient and a backward mean axial velocity near the plane of symmetry. For this back-flow (reflux) not to occur, there must exist an additional pressure gradient which gives rise to a mean volume flux. The strength of this additional pressure gradient can be measured in terms of the mean volume flow, and the criteria for reflux are thus established in terms of this mean volume flux. It is shown that reflux will not occur even in the presence of a resultant adverse mean pressure gradient of certain magnitude. Higher-order solutions have been studied to reveal the effects of peristaltic waves with long but finite wavelengths on the mean flow variables. The results indicate that, for the case of zero mean volume flux, the peristaltic motion of the walls does not produce any mean axial velocity to order 6, nor does it corre-
A LONG
WAVE APPROXIMATION
spond to any pressure gradient of order 6, however it does give rise to a mean axial velocity and a mean axial pressure gradient to order a*. The results of the higher-order solutions suggest that a Reynolds number more relevant than the one introduced earlier for this problem is perhaps
75
results of Fung and Yih ( 1968) in higher-order solutions when both results are reduced to a common limit. i.e. E + 0,6 + 0. research was carried out with the financial support from National Aeronautics and Space Administration under Grant NGR-36-003-088.
Acknowledgement-The
REFERENCES
Barton, C. and Raymor. S. (1968) Peristaltic flow in tubes. Bull. math. Biophys.
30.663-680.
Bums, J. C. and Parkes, T. (1967) Peristaltic motion. J. Fluid Mech. 29.73 I-743.
Fung, Y. C. and Yih, C. S. (1968) Peristaltic transport.
where o is the frequency of the peristaltic wave, because of the group appearance of 6R in the solutions. Thus the limitation in the validity of our solutions to low frequencies is
made evident, as SR is assumed to be small throughout our analysis. Comparisons with previous relevant works have been made, and agreement has been found with the results of Shapiro et al. (1968) in the zeroth-order solution, and with the
J. appl. Mech. 35.669-675.
Hanin, M. (19681 The flow through a channel due to transversally
oscillating walls. fsrael J. Technol. 6, 67-
71.
H. 11960) Boundary Layer Theory. (Trans. by J. Kestin), McGraw-Hill, Nrw York. 4th Edn. p. 59.
Schlichting.
Shapiro, A. S., Jaffrin, M. and Weinberg, S. L. (1968) Peristaltic pumping with long wave length at low Reynolds numbers. M.I.T. Fluid Mechanics Lab. Rep. No. 68-5.
Stoker, J. J. (1957) Water Waves. Interscience, New York. Taylor, G. 1. (195 1) Analysis of the swimming of microscopic organisms. Proc. R. Sot. A209.447-461.
Vol. 3. pp. 6?-75.
Pergamon
Press.
1970.
Printed
inGreat
Brttain
A LONG WAVE APPROXIMATION PERISTALTIC MOTION*
TO
T.-F. ZIEN and S. OSTRACH Case Western Reserve University, Cleveland, Ohio 44106, U.S.A. Abstract-The problem of motion of a viscous incompressible fluid induced by travelling wave motions of the conflning walls has been studied analytically for the two-dimensional geometry. The analysis is aimed at the possible application to urine flow in human ureters. The wave length, A, of the peristaltic waves is assumed to be large compared to the half channel width, d, whereas the amplitude of the wave, a, need not be small compared to d. A systematic approach baaed on an asymptotic expansion of the solution in terms of the small parameter d/A has been used and solutions up to OW/A~) have been presented in closed forms. The limiting solution has been studied including the effects of externally applied pressure gradients, and the criteria for backward flow have been established and discussed in detail. Higher order solutions which include the effects of non-linear inertial terms of the Navier-Stokes equations, have been studied for the case of zero mean volume flow. The results indicate that the peristaltic waves on the wall do not contribute to the time-averaged flow quantities to O(cf/A), but they do give rise to a mean axial velocity of OW/A*), and a mean pressure gradient of O(d*/A*) is associated with the motion. Comparisons with some earlier results have been made and certain agreements are brought out. 1. INI’RODLJCTION
metric, travelling transverse waves are imposed. The channel has a mean half width d, and the travelling wave has,an amplitude a, a wavelength A and a phase velocity c. The motion of the fluid is studied with and without externally imposed axial pressure gradients. The assumptions are that the wavelength is long compared to the mean half width of the channel, i.e. 6 = d/h + 1, and that the wave propagation speed is moderate so that a Reynolds number defined by R = cd/v is of order unity. This latter assumption is equivalent to requiring the frequency of the peristatic wave to be small compared to the reciprocal of a characteristic time for the vorticity diffusion. Bums and Parkes ( 1967) studied peristaltic motions in both two dimensional and axisymmetric configurations using Stokes equation and assuming that the wavelength is large compared to the wave amplitude. The inertial terms in the Navier-Stokes equations are entirely neglected. Therefore, their results may apply only LOthe case of creeping motion
of fluid transport by peristaltic motion of the confining walls has received careful study in the recent literature, (see the list of references of the present paper). The need for peristaltic pumping may arise in circumstances where it is desirable to avoid using any internal moving parts, such as pistons, in the pumping process. Moreover, peristalsis is also well known to the physiologists to be one of the main mechanisms for fluid transport in a biological system. Specifically, the passage of urine through a human ureter is an example of such a process. It is for the purpose of understanding this specific process of urine flow that the present investigation is undertaken. A simple two dimensional model has been chosen in the following analysis, although the more realistic case of axisymmetric configuration presents no essential difficulties. A viscous incompressible fluid with kinematic viscosity v is assumed to be confined in a channel with flexible walls upon which symTHE DYNAMICS
*Received
1April 1969. 63
B.M.Vol.3 No. I- E
64
T.-F. ZIEN and S. OSTRACH
of the fluid, which corresponds to very slow motion with very low frequency. Hanin ( 1968) dealt with the case of two dimensional motion with zero mean axial pressure gradients under the assumptions of small afd and large X/d, while imposing no restriction on the frequency of the peristaltic waves. Fung and Yih ( 1968) treated the motion also in two dimensional geometry, assuming a/d cg 1 but allowing arbitrary wavelength and frequency. They presented results which apply only to the case where the externally imposed axial pressure gradient is either zero or of the order (a/d)2. Furthermore, Shapiro, JatIiin and Weinberg (1968) reported some theoretical and experimental studies of the problem for both two dimensional and axisymmetric geometries. They assumed that d/A4 1 and low frequency, leaving the amplitude ratio a/d arbitrary. Only the limiting solution for infinite wavelength and zero frequency has been presented, and the flow is under some externally imposed pressure gradient in the axial direction. Most recently, an analysis based on the assumptions similar to those used by Shapiro et al. was presented by Barton and Raymore ( 1968) for axisymmetric geometry. They included a discussion for short wavelength case also. Our analysis is based on the same assumptions as those used by Shapiro et al. (1968), but the method of solution is different. In addition, we have carried out the calculations to the order (d/Q2 and obtained closed form solutions. Thus the nonlinear streaming effects on the fluid. motion are clearly demonstrated and the effects of large but finite wavelength are made obvious. While the higher-order solutions are mainly concerned with the case of zero mean volume flux, pumping under arbitrarily imposed pressure gradients with the consequence of finite mean flows is considered for the zeroth-order solution and criteria for reflux are established to this limiting order. Our results agree with those of Shapiro er al. ( 1968) to the zeroth order, and reduce to those due to Fung and Yih (1968) in higher orders when taken to their limits.
Physiologically speaking, our results as well as those of Shapiro et al. ( 1968) relate to the urine flow in human ureters under healthy conditions whereas the results of Fung and Yih f 1968) apply to the pathologic conditions of ureteral obstructions. 2. MATHEMATICAL FORMULATION We consider the flow of a viscous incompressible fluid with density p and kinematic viscosity Y through a two dimensional channel of width 2d with flexible walls. A rectangular Cartesian coordinate system (X, Y) is chosen with the X-axis aligned with the centerline of the channel and in the direction of wave propagation on the walls. A velocity vector (U, v) is defined accordingly. The travelling waves are then represented by Y =+q*(X,
T) =& d+acos [
t(X-cT)], (1)
where a, A and c’ represent the amplitudk, wavelength and the phase speed of the waves respectively (see Fig. I). 2.1 Boundary
conditions
and equation
of
motion
The boundary conditions that must be satisfied by the fluid on the waving walls are the no-slip and impermeability conditions. The fact that the equation of the waving walls is given determines the velocity component normal to the moving walls in the following way. Consider the general case where the surface of a moving wall is represented by the equation F(X,Y,T)=Y-Y,(X,T)=O.
(2)
Then the fact that fluid particles originally on the wall must remain on the wall implies that ETF=O
(3.1)
A LONG
WAVE APPROXIMATION
65
Fig. I. Geometry of the problem.
which gives explicitly V- UY,,--
Y1r= 0 on Y = Y,(X,T)
(3.2)
where, as usual, D/DT is the substantial derivative and subscripts denote partial differentiation. Since the velocity component normal to the moving surface, qn, is equal to equation l/+(1 + YL)(V- UYIX), (3.2) determines qn as (4) The tangential component of the fluid velocity on the wall is usually determined from the mechanical property of the wall. For example, if the wall is assumed to be inextensible, then another relation between U and V can be derived from the no-slip condition. The fluid velocity vector q is thus completely specified on the moving wall. However, we shall choose the simple condition here that the wall has only transverse displacements at all times so that U = OonY=YI(X,T). It can general, no-slip surface velocity
(5)
be shown that this assumption, in implies an extensible wall. The exact boundary conditions on a moving can also be used to determine the components (U, V) on the surface
which is flexible but inextensible. Taylor ( 195 1) has given explicitly the boundary conditions on U and V for a flexible but inextensible waving plate for small values of a/A. He showed that U/c on the boundary starts with 0( az/h2). Since the mechanical properties of the actual ureter wall are quite complicated and it is highly unlikely that the wall is inextensible, we propose to adopt the simpler condition of U = 0 on the wall. We note here that all the previous work (cited in the reference list) used the same assumption. Substituting YI(X, T) = A [d+a cos 2rr/X (X-CT)] into equations (3.2) and (5) for the particular wall motion under consideration, we find the following set of boundary conditions on (U, V) for our problem:
u=o
(6.1)
(X-CT).
(6.2)
Finally an axial pressure gradient must be prescribed at some X station. This boundary condition will be discussed later in connection with the distinction between zero and finite mean volume flows. The equations of two dimensional motion of a viscous incompressible Newtonian fluid can be written in terms of a stream function 9 as
66
T.-F. ZIEN and S. OSTRACH
(see, for example, Schlichting,
1960)
VW’Ti- **WIfx - *\IIxvz*), = UVQQ, (7) where V2 denotes the Laplacian operator (a2@X2+ P/aY2) and * is the stream function defined by *x=--c/ Yy= 2.2 Dimensionless sions
made and we shall seek an asymptotic expansion of Ji in the limit of 6 -+ 0. The following form of asymptotic expansion is postulated:
4.4%Y, t; 8%f, RI - Jtok Y, t; e, R) +Grfi,(x,y,t;e,R)+62Jr2(x,y,t;e,R)+...
(13)
(8.1) u.
(8.2)
formulation
and
expan-
The characteristic lengths in the X and Y directions, respectively, are A and d, which will be chosen as the length scales for X and Y. A natural time scale for the problem is the period of the peristaltic wave, A/c. Therefore, we introduce the following dimensionless quantities:
Equations ( IO) and ( 11) will be studied by using (13) together with the following limit process k Y, t; e, R)
fixed as 6 -+ 0.
(14)
The leading term of the expansion (13) should be interpreted as the limiting solution as 6 + 0. It is to be noted here that the limit process (14) has the property of leaving the complete travelling wave structure unchanged and to be of order unity in the imposed boundary conditions. Hence & should represent x=- X k,y=$, t=;T; the meaningful limiting solution of very long (9) waves. Also implied in this limit process is the JI(x,y,t) =$,*=$v=~. fact that the transverse and axial coordinates are scaled differently and, as a consequence, that the axial velocity component U is an Substitution of (9) into (6), (7) and (8) yields order of magnitude larger than the transverse the folowing partial differential equation for $I velocity component V. This type of limiting together with the boundary conditions: process is the same as that used in studying the propagation of long waves on shallow water (see Stoker, 1957). Therefore, the corresponding expansion procedure will be referred to as the long-wave approximation. .
(IO) JI, = T 2ne sin 27r(x - t) (11.1)
y=+(1+ECOS27r(X-t)): I& = 0.
(11.2)
Three dimensionless parameters present themselves in this formulation, namely E = a/d,
S = d/h,
R = cd/v.
A long wave approximation
(12)
will now be
2.3 Axial pressure gradients The axial pressure gradient in the flow field will play an important role in the subsequent analysis, particularly in the discussion of volume flux, and hence, deserves special attention. In terms of the stream function, the axial pressure gradient can be expressed as
A LONG WAVE APPROXIMATION
The dimensionless
67
3.1 Zeroth order solution
form of it is
The general solution of Jl,,is $0 =X3(x, t)y3+A2(x, t)y’ +
In the limit of 6 + 0, we can obtain an asymptotic expansion for the pressure gradient by substituting the asymptotic expansion for tj~.equation (13), into equation (16). We have P= Bo+6F1+P82+.
* *,
t)y+&x.
+&x.
JI,h, - WG,, 1. ( 16)
t).
(26)
Boundary conditions (23) and symmetry conditions on JiOindicate that +,, must be odd in Y. hence /jO 3 A’, = 0.
(17)
F&(X, 1) and A, (x, t) are found to satisfy the following two equations
(18)
s3(x.t)~+n(x,t)~=-2~~sinp
where
(27.1)
3772(x. t)A, +A,
= 0,
(27.2)
(19.1) ( 19.2)
where, for brevity, we have introduced and p(x. t) defined by ?)(X,t)
=
l+ECOShr(X-t)
/3(x-t) = 2n(x - t). - 91,90x,, - JlOI/!LU- +1,1.
(20) (21) 2 aVo R
aX2ay
i a%
atay2
(28.1)
(28.2)
(19.3)
3. SOLUTIONS Applying the limit process to the equations ( 10) and ( II), and substituting equation ( 13) into (IO) we obtain the following system of equations for Jlo, &, JI2etc. -$JI,=O
11(x, t)
I a+, a3Jio
ay axay*
(22) The boundary conditions are
Equations (27) are easily solved to give A (x t) _ C(t) -3ecosP 39
-
a,(,,
t) =-3
c(t) -3e cos p (29.2) r) *
It is worth mentioning here that we must realize the obvious restriction on the value of E, i.e. l < 1. The case E = 1 would lead to the appearance of a singularity because the denominators in equations (29) would become zero for x-t=n/2, n=l,3,5 ,.... This is due to the fact that the channel would be completely ‘blocked’ at these points. and continuity of flow then demands infinite velocity.
Jlot=~2~csin2rr(x-t).JI,,=0 $lI = JI1, = 0 $22
(29.1)
q3
=
dhY
=
0.
(23) (24) (25)
68
T.-F. ZIEN
and S. OSTRACH
The zeroth-order solutions then take the form
_; PO=;
$Jo= ~C(f)-~ecoS~l~-3[C(t)-facos~l~
go=-
l Tdt-+ej+;=$@dt] C(t)
[I 0
77
2 ‘C(t)dt,
(35)
(36)
I0
(30)
with bar denoting the time average. In terms of these expressions, the meaning (3 1) of the function C(t) is more obvious. It is seen to be directly related to the mean volume flux. (32) The relation between the function C(t) and the pressure gradient is evident from equation The arbitrary function C(t) arises during (35). For convenience, we shall consider only the the integration of equations (27), and is related simple case C(t) = Co = const. in the folto the volume flow in the following manner. lowing. This corresponds to the case of imThe instantaneous volume flow through the posing a constant volume flux on the flow. We channel is given by (actually one-half of the note the following results of elementary intevolume flow) gral calculus: ,,3
.Y?-I [c(fj-hOSp] rl ( r12 >
120= &A,--,= -2C(t)+Ecos&
(33)
From equation (33), it is clear that the volume flow consists of two parts, one being due to the peristaltic waves and the other being represented by the arbitrary function C(t). This arbitrary function is, of course, related to the as yet unspecified axial pressure gradient through equation (32). Note that the case of C(t) = 0 gives U, = 0 for c = 0, i.e. the fluid is completely at rest if the peristaltic waves do not exist. It will be shown later that this case corresponds to zero mean volume flux. The case of non-vanishing C(t) may be referred to as forced pumping, because here the fluid velocity exists even if E = 0. It will be revealed later that this case corresponds to a mean fluid transport of finite magnitude. (1) Time-aueraged jhw variables. Let us investigate the flow variables averaged over one period of the wave motion. Thus
I I
:; -=
(1 _
E2)
;=@$dt=$-
I
-I/2
(1-,2)-1’2]
(37.1) (37.2)
+(2+8)(1-2)-5~z
‘cosp dt = -Qe( 1 - @)-=. I 0 .n3
(37.4)
It follows then _=
P ,=~~~-FP)-j’2[(1+~)C,+~~~]
(38)
ao=~[1-(1-EZ)-1’*+QE~(1-eZ)--5’~y*] -i-3c, [(l+f)(l--F*)-S’S’-(l-~*)--ll~] (39) d o=-2co.
(40)
Hence ~o=~~*-n-5f2[tel-(l+~)ao]
(41)
A LONG WAVE APPROXIMATION
7i,==${l-(1 +&
_ l2) -112+ SE2( 1 - E’) -5/zy2
(~_E2)-1/2[
I+; (
(I--2)-512y2 >
II
.
(42) Equation (41) suggests that a finite mean volume flow would imply the existence of a ‘favorable’ mean pressure gradient in addition to the pressure gradient associated with peristaltic motion of the walls. To be more explicit, we denote the pressuregradient corresponding to zero mean flow by ( P,do,. then equation (4 1) gives
with 91 (~~)o=z~E2(l--2)-5/2.
(44)
We note here that equations (43) and (44) are in exact agreement with the results of Shapiro et al. (1968). Also implied in the above results is the fact that peristaltic waves can overcome anzdverse pressure gradient of the magnitude ( PO)p A mean volume transport in the direction of wave propagation will be possible fez any pressure gradients less adverse than ( PO),,. In the absence of peristalsis (E = 0), equations (41) and (42) combine to reduce to the well-known plane Poiseuille flow results with Q,, directly proportional to the externally imposed pressure gradient. However with peristalsis present, the required pressure gradient for producing a given mean volume flow is not directly proportional to the mean volume flow, but depends on the amplitude parameter E of the peristalsis also, see equation (41). It is because of this interaction that the additional velocity profile due to Q0 in equation (42) is not quite Poiseuillean. In fact, this additional profile tends to give a forward velocity near axis and backward velocity near the walls for any finite E.
69
(2) Criteria for r&u. The phenomenon of reflux is known as the backward flow of urine from bladder to kidney and is regarded as a pathological condition in human physiology. Since it is revealed in the previous section that the case of zero mean volume flux corresponds to an adverse mean pressure gradient, it appears likely that reflux might occur in this case. Indeed, we see from equation (42) that in the absence of QO, the mean axial velocity along the axis (y = 0) is always negative for all E, 0 < l < 1. The question of importance is then under what conditions of pressure gradient the reflux can occur. In the following, the reflux criteria are discussed in terms of QO, which is obviously related to the mean pressure gradient. We note here first that since the imposed velocity profile due to Q0 tends to contribute a negative velocity near the mean location of the wall, large Q0 is likely to incur reflux. Rewrite equation (42) as
+g
(
1+;
>
(l-E2)-5$$-Q0)y2.
Then the following easily established: (a)
criteria
0 G Q,, < l-
(45) for reflux are
(l-e2)1’2.
This range of Q0 will always result in negative ii0 near the axis of symmetry, and hence reflux will always occur near the axis. The corresponding velocity profile is sketched in Fig. 2(a). (b)
do=
1-(l---2)1/2
I Q,,.
This is the critical value of QO, below which reflux always occurs. Corresponding to &, there is a critical mean pressure gradient P,, obtained from equation (4 1) BC+_-E2)-2
1 >
0.
(46)
T.-F. ZEN
and S. OSTRACH
(a)
5
(b)
Cc)
0
.l_(l_E+
l_$)!i<;
I-(
<+
0 2+c
Cd)
(4
(0
_----I
-.3c2 :+E
2
_zIYjsig.___. _._
To > j$+
k2
Fig. 2. Velocity profiles for various ranges of mean volume flow; cases (a) and (f) correspond to reflux.
The velocity profile is sketched in Fig. 2(b).
(c)
&>
&, = iioP+
(1-3)-*‘2 +Qe2 l-I-2E2 Yl ( >
Q. > 1-(l-E2)1’2.
- 8) -W
I
> 0, (47)
This range of & will never result in reflux. The velocity profile is sketched in Fig. 2(c).
and a zero mean resultant pressure gradient. (e)
I?+$> Qo>&, p = flE-2
This particuiar velocity profile
value of eO yields a uniform
E
( 1-
$) 512
1-(1-8)~~2(1+~)-1](1-~~~)-1.
A LONG
WAVE
As 0, gets larger and falls in this range, the velocity profile shows an opposite curvature, although reflux does not occur. (See Fig. 2(e) for velocity profile.)
07
Q,, >&+k2.
When o,, gets exceedingly large and falls into this range, then reflux occurs near the I.0
71
APPROXIMATION
Hence reflux will never occur and the upper bound for 0, given above is meaningless. It is clear from the above discussions that reflux will occur, in the presence of peristalsis, for the range of values of 0, indicated in cases (a) and (f). Based on these results, a region for no reflux is sketched on a go - E diagram in Fig. 3._It is to be emphasized that the whole axis of Q. also belongs to the region of no reflux. Since the critical mean pressure gradient,
c
0.0
0.6
la” 0.4
0.2
0
0.2
o-4
O-6
0.6
Fig. 3. Criteria for reflux (zeroth-order results). The entire positive &axis plus the bounded area indicates the region for no reflux.
wall. The velocity profile for this case is sketched in Fig. 2(f). It must be borne in mind that the above criteria apply to the case of finite amplitude ratio (i.e. E > 0). For e = 0, that is, in the absence of peristalsis, any finite positive 9, will result in an exact Poiseuillean profile, (ii, - 1 -y2), as is evident from equation (42).
pw, is positive, it is seen that reflux will not necessarily occur even when pumping against an adverse pressure gradient. On the other hand, it is shown in case (d) that the case of peristalsis under zero mean resultant pressure gradient corresponds to a uniform profile of I&,.It is positive and ensures the condition of no reflux.
72
T.-F. ZIEN and S. OSTRACH
3.2 First-order solution Substitution of the solution of Jlo into equation (2 1) gives
and the pressure equation ( 19.2)
gradient
follows
P, = -A,,--A&+6&.
if C(t) is taken to be C,. The solution of +I satisfying ment &(y) = -h(--~) is f!h
the require-
from
(54)
In the following, higher-order solutions will be investigated only for the case of zero mean flow. Thus we shall let C(t) = K(t) = 0. Let
(-GY, t) = ~~A,y?+~(A,A,-A,A,,-K,)y’ +&(x,
t)y3+I;,(x,
t)y.
(49)
The set of homogeneous boundary conditions on I& (equation (24)) determines the functions k, (x, t) and k3(x, t) up to an arbitrary function K(t) , with Mx, t) --l(A,A,-~,A,,--A,,)?” lo
1 -
Then velocity and pressure gradient reduce to, respectively,
-
1 -u1 R = +&&Y~+~
(&AI
--AI,A,-&)Y~
+ 3ksy*+ k, As is in the case of zeroth order solution, the function K(t) relates to the value of +I along the waving wall through
Q, = +~l,-, = RK(tl,
(52)
and hence represents the volume flow due to external causes, e.g. an external pressure gradient of O(6). The velocity, &, is immediately found to be
1’1= -AIt -A,Als+6kp
(55) (56)
It is easily verified that G, = 0
and
F1 = 0.
(57)
Therefore we conclude that the peristaltic motion of the walls contributes nothing to the mean axial velocity and mean pressure gradient of O(6). Also, with some algebraic manipulation, we can write down the solution up to O(6) for this case, and then pass it to the limit of small wave amplitude ratio with x, y, t held fixed.
73
A LONG WAVE APPROXIMATION
The result is lii$
1
a5(x, t) = jj -~A3z~+6R(k,A3z-k,J3
- ky(l-iy7)
cosp-$leR6y(l-y7)2
&+o
+A&ss--Ww-M
sinp+0(c2,ijz).
(58)
Note that equation (58) is in exact agreement (to the orders considered, of course) with Fung and Yih’s (1968) result when their free pumping solutions up to Ok) are expanded in the limit of 6 * 0, holding our variables (x, y, t) fixed. It is worth noting that a finite limiting solution exists in Fung and Yih’s problem of small amplitude waves when the wavelength is allowed to become very large, with independent variables properly scaled, as was done in our analysis. 3.3 Second-order solutions Since the peristaltic waves on the walls do not produce any effects on the flow field to order S, we propose to carry out the secondorder solutions. Again we consider the zero mean flow case. Using previous results for &, and $,, we have, from equation (22)
1 av2 --=+&,,(x, Ray2 .
t)y”
t)y7+$us(x,
7! $-3!a,(x,
tw+
f)Y
(59&b,
(59)
1.
The general solution of I~J~, with the required symmetry properties, is easily obtained as j$2=ully*l+uOY9+a7y7+a5y5+s3y3+S1y (60) The functions s3(x, t) and sI (x, t) are again determined from the homogeneous boundary conditions, equation (25). For zero mean volume flow, we find s, (x, t) = (a,+ Z!q2u7+ 3~7~u~+4$u,,)-q~ (61.1) s3(x, t) = - (2u5 + 3u7v2+ 4u&!4+ 5u&)772. (61.2) Therefore, the second-order solution has also been obtained in closed form. In principle, the mea+ velocity ii2 and mean pressure gradient P, can be calculated explicitly as was done for the zeroth- and fist-order solutions. However, the algebra becomes quite extensive and tedious. In the following, the mean quantities are investigated in the limit E --* 0. This simplifies the algebra somewhat, and gives the essential information needed. We note that
where
. [ 1 4x,t> =$6 64, - 1)W&)t -;&z
a,,(~, t) = j$lZR
(AA1
3s
$43(&43J,-$43A~z
-AJ
(AI&-&AI~ a.;(~,t) = +
1
3
-A
3E
)+zA10 3
-A3,L--3Aw43,-43
;RNA- l)(A~&-A3A~r
1
Au2 = 1l&y’o+
9dgy8+ 7a7y6 + 56,~~ + 3s3y2 + sl.
As E -+ 0, we find, with considerable of algebraic manipulation, d,l = O(E2)
fro=
&
Rr2e2 + o(e2)
--A,),
12(k,A,),--A,,(A,A,,--A,A,,--A,,)l
a7 = -&-R&l
+ o(e2)
(62)
amount
74
T.-F. ZIEN and S. OSTRACH
and hence
&
=
( -& (
s* = -
Therefore, ii*=
$2
>
RA*
$R-*+&)R~~E*
F2= RZRe*yh-&)
+ ok*) +o(c*)
we have
ly*_L&+_Ly4 xZR*e* 16o [ +(;j&-&)Y*-(&j+f~)]+ok*). (6%
Similarly, we have for the second-order sure gradient
pres-
= const.
(64)
Notice here that_there are no terms of O(E) existing in ti2 and P, as E + 0. This is also in agreement with the findings of Fung and Yih ‘( 1968). However, it is somewhat surprising to find that p2 is constant across the channel (and generally positive), with all the coefficients appearing in each y-dependent terms cancelled with each other. This conclusion is also consistent with the results of Fung and Yih for E --* 0 with 6 fixed. 4. SUMMARY AND CONCLUSIONS
Peristaltic motion has been studied for two dimensional geometry in the limit of long wavelength and low frequency. Asymptotic expansions in terms of a dimensionless wave number, 6, have been constructed, and solutions are obtained in closed form to O(S*). The limiting solution (i.e. the zeroth-order solution) has been studied including the effects of arbitrarily imposed pressure gradients. It is found that the peristaltic waves for the case of zero mean volume flux correspond to an ‘adverse’ mean pressure gradient and a backward mean axial velocity near the plane of symmetry. For this back-flow (reflux) not to occur, there must exist an additional pressure gradient which gives rise to a mean volume flux. The strength of this additional pressure gradient can be measured in terms of the mean volume flow, and the criteria for reflux are thus established in terms of this mean volume flux. It is shown that reflux will not occur even in the presence of a resultant adverse mean pressure gradient of certain magnitude. Higher-order solutions have been studied to reveal the effects of peristaltic waves with long but finite wavelengths on the mean flow variables. The results indicate that, for the case of zero mean volume flux, the peristaltic motion of the walls does not produce any mean axial velocity to order 6, nor does it corre-
A LONG
WAVE APPROXIMATION
spond to any pressure gradient of order 6, however it does give rise to a mean axial velocity and a mean axial pressure gradient to order a*. The results of the higher-order solutions suggest that a Reynolds number more relevant than the one introduced earlier for this problem is perhaps
75
results of Fung and Yih ( 1968) in higher-order solutions when both results are reduced to a common limit. i.e. E + 0,6 + 0. research was carried out with the financial support from National Aeronautics and Space Administration under Grant NGR-36-003-088.
Acknowledgement-The
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Fung, Y. C. and Yih, C. S. (1968) Peristaltic transport.
where o is the frequency of the peristaltic wave, because of the group appearance of 6R in the solutions. Thus the limitation in the validity of our solutions to low frequencies is
made evident, as SR is assumed to be small throughout our analysis. Comparisons with previous relevant works have been made, and agreement has been found with the results of Shapiro et al. (1968) in the zeroth-order solution, and with the
J. appl. Mech. 35.669-675.
Hanin, M. (19681 The flow through a channel due to transversally
oscillating walls. fsrael J. Technol. 6, 67-
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Shapiro, A. S., Jaffrin, M. and Weinberg, S. L. (1968) Peristaltic pumping with long wave length at low Reynolds numbers. M.I.T. Fluid Mechanics Lab. Rep. No. 68-5.
Stoker, J. J. (1957) Water Waves. Interscience, New York. Taylor, G. 1. (195 1) Analysis of the swimming of microscopic organisms. Proc. R. Sot. A209.447-461.