the influence of vibration on poiseuille flow of a non-newtonian fluid. ii

Rheol. Acta 18, 244- 255 (1979) © 1979 Dr. Dietrich Steinkopff Verlag, Darmstadt ISSN 0035-4511 / ASTM-Coden: RHEAAK

Center for the Application of Mathematics, Lehigh University, Bethlehem, PA.

The influence of vibration on Poiseuil le f low of a non-Newtonian fluid. II

J. Y. Kazak ia and R. S. Rivl in

With 1 figure

1. Introduction

Manero and Mena (1) have published ex- periments in which a non-Newtonian fluid flows through a tube of circular cross-section under a uniform time-independent pressure head, the tube being simultaneously vibrated sinusoidally in the longitudinal direction. They found that the effect of the vibration is to produce an increase in the mean rate of discharge of fluid through the tube. This increase in the rate of discharge depends on the frequency and amplitude of the vibration and may be many times that which exists in the absence of a superposed vibration.

In a previous paper (2), we have studied this problem for the situation when the parameters of the problem are such that the change in the rate of discharge is small.

The following two situations were considered: (i) The fluid flows, under the action of a uni-

form time-independent pressure gradient, between two parallel plates which are vibrated sinusoidally and synchronously in their own planes, either in a direction parallel to the pressure gradient or transversely to it.

(il) The fluid flows, under the action of a uniform time-independent pressure gradient through a straight pipe of circular cross-section which is subjected to a sinusoidal vibration either longitudinally or rotationally.

In both cases, the fluid was assumed to satisfy the Rivlin-Ericksen constitutive equation. It was shown how, at any rate in principle, explicit results could be obtained provided that the fluid is assumed to be only slightly non- Newtonian. Explicit results were obtained both for a specific form of the constitutive equation (given in eqs. [2.5] below) and for the case when the amplitude and frequency of vibration are sufficiently small so that a slow flow approximation to the Rivlin-Ericksen equation is valid. 47I

(Received May 29, 1978)

It is seen from the analysis in (2) that in both of the cases (i) and (ii) the rate of discharge may be expected to increase with the frequency and amplitude of the vibration provided that the viscosity of the fluid decreases as the rate of shear increases. Moreover, in the case of flow between parallel plates, such an increase in the rate of discharge will take place both in the case when the direction of vibration is parallel to that of the pressure gradient and when it is transverse to it. Similarly, in the case when the fluid flows through a pipe of circular cross- section, it was found that the increase in the rate of discharge takes place whether or not the tube is vibrated longitudinally or rotationally.

In sections 4 and 7 of the present paper, we consider the effect on the mean rate of discharge of the longitudinal vibration which is the superposition of two sinusoidal vibrations whose frequencies are harmonics of a single frequency. Using the particular Rivlin-Ericksen constitutive equation for a slightly visco-elastic fluid, given in eqs. [2.5] below, as the basis for the calculations, we find that, in addition to a change in the mean rate of discharge which is the sum of the changes which would result from the two sinusoidal vibrations acting independ- ently, and which is present only if the pressure gradient is non-zero, there is a change in the mean rate of discharge, which is independent of the pressure gradient, provided that the frequency of one of the component sinusoidal vibrations is twice that of the other. In section 4 these calculations are carried out for the case of a fluid flowing under a uniform time-independent pressure gradient between parallel plates. In section 7 they are carried out for the case when the fluid flows under a uniform time- independent pressure gradient through a pipe of circular cross-section.

Kazakia and Rivlin, The influence of vibration on Poiseuille flow of a non-Newtonian fluid, H 245

In section 5 of the present paper we consider the effect on the mean rate of discharge, when the fluid flows between parallel plates, of simultaneously acting sinusoidal vibrations in the direction of the pressure gradient and perpendicular to it. We make the same constitutive assumptions as were made in the study of the effect of vibration parallel to the pressure gradient. We again find that in addition to a change in the mean rate of discharge which is the sum of the changes which would result from the two sinusoidal vibrations acting independ- ently, and which is present only if the pressure gradient is non-zero, there is a change in the mean rate of discharge, which is independent of the pressure gradient, provided that the frequency of one of the sinusoidal vibrations is twice that of the other.

Again, in section 8, we consider a pipe of circular cross-section, through which the fluid flows under a uniform time-independent pressure gradient, to be subjected simultaneously to longitudinal and rotational sinusoidal vibrations. The results obtained for the mean rate of discharge of the fluid are qualitatively similar to those obtained in section 5 for the case of flow between parallel plates.

It will be appreciated that the effects of the vibration on the mean rate of discharge, which are studied in this paper, are essentially rectification effects. It will be apparent that many other possible rectification and modulation effects eould be identified by appropriate choices of the constitutive equation and of the vibration which is imposed on the boundary.

2. Constitutive equations

In this paper we shall be concerned with time- dependent flows of an incompressible isotropic non-Newtonian fluid which are superpositions of two shearing flows. For one of these the direction of shear is the 1-direction of a rectangular cartesian coordinate system x and the plane of shear is the 12-plane. For the other the direction of shear is the 3-direction and the plane of shear is the 23-plane. For such flows the velocity field vi, referred to the system x, has the form

1)1 ~--- Vl (X2, t ) , /)3 = /)3(X2, t ) , /)2 = 0 . [ 2 . 1 ]

Let ~c and 2 be the velocity gradients associated with the shearing flows, so that

' 2 = ' [2 2] /£ = /)1~ U3~

where the prime denotes differentiation with respect to x2.

We shall assume that the fluid is slightly non- Newtonian. In this case, we have seen in (2) that for the class of flows defined by eqs. [2.1] the constitutive equation for the components aij of the Cauchy stress referred to the coordinate system x must have the form

G11 = g G Oc~ß, Oo~ßy6, 1£ l~ ) -- p ,

( :'~') 0"33 = g G O«fl, .~~ßT~, - - P ,

0"22 = - - p ,

( :') 0"12 ~- t]/£ -t- g E 0eil, ~«By& ,

a31 = eK 0«p, ~b«~~õ, ~c 2 + . , [2.3]

where F and K are linear in their last arguments, p is an arbitrary hydrostatic pressure, e is a small parameter, q is a constant, 0ca and q~«aeo are defined by

(«)(/~) («)(/~) 0 ,8= ~c~ + 2 2 ,

(«)(/~)(7)(~) («)(/~)(7)(~) ~b«~~~= ~c~~c~c + 2 2 2 2 , [2.43

(«) («) and ~c and 2 are the Œ'th time derivatives of

and 2 respectively. I f e = 0, we obtain the expression for the stress when the fluid is Newtonian and has viscosity t/.

In this paper we shall adopt the special case of eqs.[2.3] (cf. section 8 of (2)) which is obtained by assuming G, F and K to be poly- nomials in their indicated ärguments and dis- carding those terms which involve ~c and 2 with dimensionality in (time)-1 greater than three. We have accordingly

0"11 = - - p -[- 8 [ f l t ( / ¢ 2 -t- 2 2 ) ~- ~2)bJ~ -~ f l3 /£bC] ,

o~» = - p + ~ [ /L(~ = + x =) + / L K k + / h , ~ i ] ,

0"22 = - - p ,

alz = ~~c + e [t/1/c + t/2i + t/3 ~c(~c 2 + 22)],

a2a = n2 + e [t/1,~ + ~2)C + n32(~: 2 + 22)],

0"31 = g [7'1/£)c -I- ~;2(/£.), Aw )~/JC)] , [2.5]

where the il's, q's and ? 'sare constants and the dot denotes the time derivative.

246 Rheolo9ica Acta, Vol. 18, No. 2 (1979)

3. F low between parallel plates - general considerat ions

We consider an incompressible non-Newton- ian fluid to be contained between two infinite parallel rigid plates normal to the 2-axis of a rectangular cartesian coordinate system x and to flow under the action of a uniform time- independent pressure gradient P in the direction of the 1-axis. Simultaneously, the plates are subjected to identical vibrations parallel to the 1 3-plane. We suppose that the components of these vibrations in the 1 and 3 directions consist of the superposition of sinusoidal and cosinusoidal vibrations with angular frequencies co« (« = 1, . . . , /~).

Let the plates be located at x2 = + h and let us denote the velocity at a generic position in the fluid by v~ (i = 1,2,3). Then, the non-slip boundary condition yields

,u #

va = Re ~ U~e '°~~̀ , v3 = Re ~ V~e '°~«' , c t = l ~ = 1

v 2 = 0 on x2 = +_h, [3.1]

where U« and Va are complex constants. Assuming a flow field of the form given in

eqs. [2.1], and denoting the stress components in the system x by a~j, the equations of motion a r e

' ' [3 .2] Ph1 = P + o'12 , ph3 = 0 " 3 2 ,

where p is the density of the fluid, and the prirne and dot denote differentiation with respect to x2 and t respectively.

In order to determine the dependence of v~ and v3 on x2 and t in the case when the flu'id is slightly non-Newtonian and the stress in it is given by eqs. [2.3] or [2.5], we proceed in the following manner. We write

V 1 = g l ( x 2 , t) -}- e Ü l ( x 2 , t ) ,

123 = /~3(X2, t) q- 8~3(x2, t), [3.3]

where vl, v3 are the velocities in the case when the fluid is Newtonian and has viscosity t/(i.e. e = 0 in eqs.[2.3] or [2.5]). We also write correspondingly

0"ij ~ ~i j ~- 8 ~ i j ,

= 17 + et), 2 = ,T + eß. [3A]

We introduce the notation (cf. eqs. [2.3]4.5)

Then, from eqs. [3.2] we obtain, with eqs. [2.3] or [2.5],

• - •

p ~Sa = P + t/vT, p ~3 = t/~~ [3.6]

and

• ~7 . . . . . [3 7] pva = t / + s a , pb3=t /v3 + s 3 .

From eqs. [3.1] and [3.3], it follows that ~t, ~3 taust satisfy the boundary conditions

#

vl = Re ~ U«e '~~~,

#

~3 ~ - Re ~ V~e '~'«~ on X 2 = __+h, [3.8] Œ = l

and ~, and ~3 taust satisfy the boundary conditions

va = v3 = 0 on X 2 = -+h. [3 .9]

With the boundary conditions [3.8], we can obtain the solution of eqs. [3.6] in the form

P 2 & vl = G-7-( h - x 2) + Re Æ N~« '°~«',

zr I c ~ = l

#

= Tee , [3.10] c ¢ = 1

where N« and T~ are complex functions of x2 given by

c o s v « x 2 ( 1 - l) (Ne, Te) = (U«, V«) , [3.11]

cosv«h(l - ~) with

v« = ( ' ~ « ~'~. [3.12]

Let Q1 be the mean volume rate of discharge in the 1-direction per unit length in the 3- direction and let Q3 be the mean volume rate of discharge in the 3-direction per unit length in the 1-direction. By means of an argument similar to that leading to eq. [3.19] in (2), we obtain

2 P h 3 go~ 2~/~ h Q a - - - + - - ~ ~ X 2 S l d x 2 d t ,

3~ rcq o o

2n/m h Q 3 - ~ ( o i ~ x 2 s 3 d x 2 d t . [3.13 3

rot/ o o

Kazakia and Rivlin, The influence o f vibration on Po&euille f low o f a non-Newtonian fluid, H 247

4. Longitudinal vibration superposed on Poiseuille flow between parallel plates

We shall, as an example, consider the case when the vibration of the plates is in the 1- direction and is given by

U 1 = Re { - z ( U l e *m~t + U2e~"ot)},

1) 2 = V 3 = 0 Oll X 2 = __ h, [4.1]

where U1 and U 2 a r e real constants and n > m; i.e. the vibration is the superposition of two sinusoidal vibrations with angular frequencies mco and n~o and amplitudes U~ and U2. We shall also assume that the constitutive eq. [2.5] is valid for the fluid, so that

sl = 17~~ + 17a~ q- 173~ 3. [4.23

Eqs. [3.10] now become

19 -2 fq = -2--~(h - x~) + R e ( N i e *m~'~ + N2e'~'*),

v3 = O, [4.3]

where

N« = - t U « cos v « x 2 ( l - ~)

c o s v« h (1 - z) (a = 1,2), [4.4]

with

B p D'/(D "~ 1/2 ( p h 0 ) "~ 1/2 = , 1/2 = . [4.5] Y1

217 / ,,-Tg-J From eq. [4.3] » we have

P x 2 = - - - + Re (N'~ e 'm~'~ + N~ e '~q . [4.6]

17

Substituting from eq. [4.6] ) in eq. [4.23, we obtain

= - 173-- + 3P (IN ~ x 2 12 + iNil2) $1 173 217

3 - - - Re N1 N2 62m,n

4

+ ReZE«#(x2)#(«m+#")°~t}, [4.7]

where the summation is taken over the following values of the pair (c~,fl):

(1,0), (0,1), (2,0), (0,2), (3,0), (0,3),

(1,-1) , (1,1), (2,1), (2,-1) , (1,2), (-1,2) . [4.8]

IN'il and ]Nil denote the moduli of N'I and N~ respectively,

{10 (2m = n) B2m,n = (2m ~ n)' [4.9]

the tilde denotes the complex conjugate and the E's are given by

E1,0(x2) = z171m~o -- 172m2(D 2 q- 173 172

+ INil 2 4- 7IN' i l2 N'I,

Eo,l(x2) = t171nco - 172 n2co2 + 173 rÜ

+ -~-- lg] I 2 + -~- lg2[ 2 N j ,

Eg,o(X2),Eo,2(x2) 3173 P x 2 . ,2 - - ( N 1 , N 2 2 ) ,

2 17

1 E 3 , o ( X 2 ) , E o , 3 ( x 2 ) = " ,3 ,3 --~173(N1 , N 2 ) ,

El, l (x2) ,El , l (x2)=--3173 P x 2

N1 (N2, N2), 17

3 ~Tt2 ~T t -~Y1 zv2 E2,1 (x2) -~- 173

~ Z - l ( X 2) = 3173NIgN~(1 - 62,ù,n),

3 = N2 (N1,N1). [4.10] E, ,2(x2) ,E-1,2(x2) --~173 ,2 " , - ,

We could, in principle, determine bi by introducing the expression [4.7] for 81 into eq. [3.731 and solving the resulting equation subject to the boundary conditions [3.9]. However, the analysis involved is rather cumbersome and we shall not present these calculations here.

It is, however, evident from eqs. [3.1332 and [4.3]2 that Q» = 0. Also, introducing the expression [4.7] for sl into eq. [3.1331 , we obtain with eq. [4.4]

2Ph3 (1 3ep2h2173.)

Q1- 3--7- 5173

3Ph173 [U~A(v lh ) + U~A(v2h)] - - - e -

U~ U2 v1173 + a g .... g 12~/217 Al(vlh), [4.11]

where A (x) and A l(x) are defined by

248 Rheologica Acta, Vol. 18, No. 2 (1979)

(2x 2 + 1)sinh2x - ( 2 x 2 - 1)sin2x Al(X) = Dl(X) + 12xD2(x) [4.12] A (x) = - 1 + 2x (cosh 2x + cos 2x) ' D3(x)

with

Dl(X) = 2 l-8(1 + cosh 2x cos 2x) + (cosh 2x + cos 2x) 2] (sinh 2 V~x + sin 2 ]/~x)

+ 16 sinh 2x sin 2x (sinh 2 ~/2x - sin 2 ]/~x)

- 41/2 (cosh 2 l//2x + cos 2 ]//2x)(cosh 2x + cos 2x)(sinh 2x + sin 2x),

D2 (x) = ]/2 (cosh 2 ]//2x + cos 2 V~x) [(cosh 2 x - cos 2 x) 2 - 4]

- 2 (cosh 2x + cos 2x) (sinh 2x sin 2 ] /~x - sinh 2 ] /~x sin 2x),

D3(x) = (cosh 2x + cos 2x) 2 (cosh 2 ] /~x + cos 2 V~x). [4.13]

The function A (x) is plotted in figure 1 of (2) and the function A i (x) is plotted in figure I of the present paper.

Allowing for changes in notation, we can recover the expression for Q1 given in eq. [3.27] of (2) for the case when only one sinusoidal vibration of the boundaries is superposed on the Poiseuille flow by taking U1 = 0 or U2 = 0 in eq. [4.11]. The terms in eq. [4.11] due to the vibration are of two kinds - those due to the separate effects of the two sinusoidal components and that due to the interaction of these. The latter is non-zero only if 2m = n and is independent of P, i.e. it is present whether or not an underlying Poiseuille flow exists.

We can obtain from eqs. [4.1211 and [4.13] simple asymptotic formulae for ~A(x), Dl(X), D2 (x) and Da (x) in the cases when x ~ 1 and x ~> 1. When x ~ 1, we obtain (cf. eq. [3.29] of(2))

4 x4 A (x) = T '

( 4 ) Dl(X)=96] / /2x I + T x'~ '

D2(x) = - 8 V~(1 - 4x ' ) ,

Da(x) = 8 (l + 4 x ' ) . [4.14]

From eqs. [4.1212 and [4.14], we see that if x ~ l ,

Al(X) - 288 ]/~ x5 " [4.15] 5

Again, if x » 1, we have (cf. eq. [3.30] of (2))

A (x) = x ,

D 1 (x) = + (1 - 21/~) e (4 + 2 v~) x q .

D 2 (X) = 1 ]//~ e( 4 +2 V2)x

D3 (x) = 1__ e( 4 + 2v~)~. [4.16] 8

From eqs. [4.16] we see that i fx >> 1,

Ai(x) = 12 ] /2x . [4.17]

The approximations [4.15] and [4.17] are shown by broken lines in figures 1 a and 1 b respectively.

5. Simultaneous longitndinal and transverse vibration superposed on Poiseuille flow between parallel plates

We now consider the case when the vibration of the plates has components in both the I and 3-directions and is given by

vl = Re {-zUe'm'° t ) ,

v3 = R e { - t V e '"°''} on x2 = ± h , [5.1]

where U and V are real constants. We again assume that the constitutive eq. [2.5] is valid for the fluid, so that

$3 = t ] l~ ..}_ n2,~ + t/3~(/~2 ..~ /~2). [5.2]

Eqs. [3.10] now become

P .2 vl = -----(h - x 2) + Re(Ne*"~t) ,

v3 = Re (Te'"'~t), [5.3]

T = - I V c°szx2(1 - l) , [5.4] coszh( l - t)

where c o s v x 2 ( 1 - z)

N = - t U cosvh(1 - t )

Kazakia and Rivlin, The influence of vibration on Po&euille f low of a non-Newton&n fluid, II 249

with

v \ 2t/ J ' z \ 217 ] " [5.5]

From eqs. [5.3], we have

F¢ - - P x 2 - - + Re (N'e'm~'t),

17

B = Re (T'e'"~~). [5.6]

Introducing the expressions [5.6] into eqs. [5.2], we obtain

~ p 3 x ~ Px2 (31N,12 + IT,I•)} ~~ = -17~ ( - - ~ - - + 2,1

3 p2 x 2 3 1 ] + 17----r-- + 71Æ'? + 7JT '? . N'«'m~'

2 P x2. (3 N '2 e 2~mc°t + T '2 e 2znc°t)

17

+ N'3e 3,mo,t + T '2 (N,e,(2,+,ù)~,t

+ Kr' e'(2"-"»'t)},

1 { [/171 172 F/2CO2 p2X2 s3 = -~17 3 R e 4 n co - - - + 1 7 ~

173 17

1_34] + -~-IN 12 + [T'l 2 T ' e , ù ~ « + T ' 3 e 3,,,o,~

_ 4P_____x2_~N,(T,e,(,ù+ù)~,~ + ]?'e *"-")~«) 17

+ N '2 (T ' e '(2m+")°'' + T'd(2m-'0~'@. [5.7] )

Introducing the expressions [5.7] into eqs. [3.13], we obtain

2 P h 3 I - - e

Q j = 3 t 1 5173

~ ~17P h- [ 3 -- ~ U a A(vh) + V 2 A(zh)]

+ 62 .... ~ UV2;g173 A l ( zh ) , 3 61/~17

{ 2 P h U 2 Q3 = - e q 3 C~m,ù 172 A(vh)

U 2 Vv } + 62m.ù 30~/-217 Al(vh) • [5.8]

We note that the expression for Q1 consists of a term which arises solely from the presence of a pressure gradient in the 1-direction, terms which arise from the separate interactions of this pressure gradient with the vibrations in the 1 and 3-directions, and a term Which represents an interaction between the vibrations in the 1 and 3-directions and is present only if the frequency of the vibration in the 3-direction is twice that of the vibration in the 1-direction. The latter term is non-zero, whether or not a pressure gradient exists in the fluid.

The expression for Q3 consists of two terms, one of which arises from an interaction of the pressure gradient in the 1-direction with the vibration in the 1-direction and is present only if the frequencies of the vibrations in the 1 and 3-directions are the same. The other arises from an interaction of the vibrations in the 1 and 3-directions and is present only if the frequency of the vibration in the 3-direction is twice that in the 1-direction. This term is present whether or not a pressure gradient exists in the fluid.

The most interesting conclusion of this analysis is that, even in the absence of a pressure gradient, if the frequency of the vibration in one direction is twice that of the vibration in the perpendicular direction, there will result a non-zero mean rate of discharge in the direction corresponding to the higher vibration frequency.

6, Flow through a pille - general considerations

We now consider that the non-Newtonian fluid flows under a uniform time-independent pressure gradient P, say, through a straight pipe with circular cross-section of radius a. Simul- taneously the pipe is vibrated longitudinally and rotationally, the vibrations being superp0sitions of sinusoidal and cosinusoidal vibrations with angular frequeneies co« (~ = 1 . . . . . #).

Let (vr, Vo, V~) be the velocity o f a generic particle of the fluid in a cylindrical polar coordinate system (r,O,z) whose z-axis coincides with the axis of the pipe. We assume a flow field of the form

v~ = ~(~,t), ~o = ro ( r , t ) , v~ = O. [6.1]

250 Rheolo9ica Acta, Vol. 18, No. 2 (1979)

f2(r,t) is the angular velocity of the particle considered about the axis of the pipe.

The no-slip boundary conditions at the wall of the pipe have the form

#

v (a, t) = R e ~ U« e '°'«~ , am1

#

f2 (a, t) = R e Z O« «'~'=', [6.2]

where U« and O« are complex constants. Let x be a local rectangular cartesian coordi-

nate system whose origin is at a generic particle P and which moves so that its 1, 2 and 3 axes are maintained in the longitudinal, radial, and azimuthal directions respectively at P. With respect to this system, the fluid element in the neighborhood of P undergoes simultaneously two shearing motions. For one of these, the direction of shear is the 1-direction and the plane of shear is the 12-plane. Ler te be the velocity gradient for this motion. For the other shearing motion, the direction of shear is the 3-direction and the plane of shear is the 32-plane. Ler 2 be the corresponding velocity gradient. Then,

te = v ' ( r , t ) , 2 = r O ' ( r , t ) , [6.3]

where the prime denotes differentiation with respect to r.

The stress a~j at P referred to the system x is given by eqs. [2.3] and the components of stress a , , a o o , . . . , a ~ o in the cylindrical polar coordinate system r, 0,z are related to 0.~j by

Grr ~- 0"22 ~ 0.00 = 0 " 3 3 , ~zz = 0"11

0.zr = 0 " 1 2 , O'ro = 0 " 2 3 , 0"0z = 0"31" [6 .4]

The equations of motion, referred to the cylindrical polar coordinate system, are

0.;, + l (a~~ - «oo) = - P rf22 , r

0.;o + 28,o p r o r

0.rz a'~~ + + P = ph. [6.5]

We now introduce the notation given in eqs. [3.4] and write

v = ~ + e13, f2 = ~ + eg). [6.6]

Then, from eqs. [2.3], we have with the notation of eqs. [3.5] and [6.6]

~12 = r/~ + s l , a23 = t/,~ + s3. [6.7]

With eqs. [3.4], [6.1], [6.4], [6.6] and [6.7], we obtain from eqs. [6.5]2,3

q {(rOT + 20'} = p r f 2 ,

t/ ~" + + P = pv, [6.8]

and

q {(rOT + 28)'} + s; + ~ - 2 S 3 - - prg), r

ber v« r + I bei v« r

ber v« a + i bei v« a

Io (~b« r) = V« Io((O~a) [6.13]

and v« and 4« are defined by

v« = , qS« = (1 + t). [6.14]

Io( ) denotes the modified Bessel function of the first kind of order zero.

Again, the solution of eq. [6.911 subject to the boundary condition [6.1012 is

,a

Õ R e ~ T~(r)e '°'et, [6.15] c~=l

where

N e = U «

(~) ~1 rl v" + + sl + - - = p v . [6.9]

r

The boundary conditions which must be satis- lied by ~, O, ~ and g) are obtained from eqs. [-6.2] a s

,a

~(a,t) = R e ~ V~e ̀ '°«t , ~ = 1

#

Õ(a, t ) = R e 2 f2« e~°'«~ [6.10]

and

~(a,t) = g)(a,t) = 0. [6.11]

The solution of eq. [6.9]2 subject to the boundary condition [6.10] 1 is

p u = - - - ( a z - r 2) + R e ~ N«(r)e '~'«', [6.12]

4t/ «=1

Kazakia and Rivlin, The influence o f vibration on Po&euille f low o f a non-Newtonian fluM, H 251

where B«a ber1 (ver) + 1 beil (ver)

Te= r berl (v«a) + zbeil (v«a)

= O«a /l(qS«r) [6.16] r I1 (~b«a)

11 ( ) denotes the modified Bessel function of the first kind of order unity and v« and q~Œ are given by eqs. [6.14].

With the neglect of terms of higher degree than the first in 8, we find that the mean volume rate of diseharge Q of the fluid is given by (cf. eq. [4.26] of (2))

a 2~/o rcPa 4 e(o ! 5 r2s~ d t d r . [6.17]

Q - 8tl + 2tl o

7. Longitudinal vibration superposed on Poiseuflle flow through a pipe We consider, as an ex~tmple, the case when

the vibration of the tube is purely longitudinal and is given by

v(a,t) = R e { - t ( U l e 'm°~t + U2em°)t), [7.1]

where Ui and U2 are real constants and n > m. We again assume that the constitutive eq. [2.5] is valid for the fluid,, so that, with % = 0, sl is given by eq. [4.2].

Eq. [6.12] becomes

P = 4t/ (az r2)

+ R e { - , (N 1 e '"~'' + N2 «,ù,ot)}, [7.2]

where, from eqs. [6.13], ber v« r + t bei v« r

N« = ~ U« ber v«a + t bei v«a

I o ( G r ) = z U « - (a = ¤,2) [7.3]

Io( (Ga)

and v» v» qSi and q52 are given by

Y1 ~ ~ Y2 ----- ,

- - ~ 1 # . ~bl = - - ~ (1 + z), q52 = (1 + 0- [7.4]

From eqs. [6.3]i and [7.2], we obtain

- z ( N l e + N2 . 2r/ [7.5]

Introducing the expression [7.5] for ~ into the expression [4.2] for s~, we obtain

JPSr3 3 P r (iNi12 + iNil2) S1 = --£/3 )'--~"--~3 -Jr- at/

?kT '2 ÄT' ~ "~ __3 Rea.vl *'2C'2m, n 4

+ R e • E«~ (r) e ~(«'~ + IJ~)~¢, [7.6]

where the summation is taken over the values of the pair («,fl) listed in [4.8] and the notation [4.9] is used. The E's are given by

{ 3 E1,0(r) = z~lmco -- t/2m2(O 2 -I- 7/'13

\ n2 + 21Nil2 + INil2 N j ,

{ 3 E0,i(r) = t thno~ - t/2n2co 2 + -~t/3

~p2r2 )} k---~--- + 21N'~l= + INŒI2 N j ,

E2,o (r), Eo,2 (r) = 3 t/_____23 P__f_r (Ni2 ' Nj2) ' 4

1

Ea , - i ( r ) ,E~ l(r) 3r l3Pr (N'i ~' ' • , -- - - N 2 , N 1 N ' 2 ) , 2tl

E2,1(r ) = 3 t l 3 N ' i 2 N ' 2 ,

3 v,2~> (1 &2m,,) E~,_~(r) = ~-t/~~,l ~,~ - ,

3 = N2 ( N ~ , N i ) . [7.7] E1,2(r) ,E_i ,2(r) ---~r13 ,2. , ~,

Introducing the expression [7.6] for sl into eq. [6.17] and using eq. [7.3], we obtain the following expression for the mean rate of discharge Q:

(~~a~~~) rcpa4 1 - e - - Q - 8--~ 6tl 3

3 rc P a2 tl 3 - e - [ U ~ A2 (v i a) + U 2 A2 (v2 a)] 4t/2

3 7r U~ U2 r/3 + 62m,ù~-- As(v~a) , [7.8]

4tl

where A2 (x) is defined by (cf. eq. [4.29] in (2))


x (ber z x + bei 2 x) (A2 + 1)

= X 2 (ber x ber' x + bei x bei' x)

+ 2 (ber x bei' x - bei x ber' x), [7.9]

and A3 (x) is defined by

Ei (x)A1 (x) + E2 (x)A2 (x) A3(x ) = , [ 7 . 1 0 ]

E3 (x)

where E 1 (x), E 2 (x) and E3 (x) are defined by

Ei (x) = bei ]/~ x (ber 2 x - bei 2 x)

- 2 ber ] /2x ber x bei x ,

E 2 (x) = ber ] /2x (ber 2 x - bei 2 x)

B 2 bei ] /~x ber x bei x ,

E3 (x) = (ber2 x + bei / x) 2

• (ber 2 ] /~x + bei 2 ]//2x). [7.11]

A1 (x) and A2(x) are defined by

x A~ (x) = ~ y2 {2 ber1 y bei1 y

0

(ber i ] /~y + bei i ]z/2y)

+ (bera 2 y - bei~ y)

(ber~ ]//2y - beii ~2y)} dy, x

A2(x) = ~y2 {2 ber1 ybeix y 0

• (ber 1 ]//2 y - bei1 ]//2 y)

- (ber~ y - bei~ y)

• (ber1 ]//2y + beii ]//2y)} dy. [7.12]

Az(x) is plotted in figure 1 of (2) and A3(x) is plotted in figure i of the present paper.

When x ~ 1, approximate expressions for Az(x) and A3(x) can be obtained by using the series expansions for ber x, beix, berl x and bei1 x (see eqs.[4.30] and [6.34] in (2)). These are

A2(X ) = ~ 1 x4 , A 3 ( x ) = _ _ 1 x6. . [7.13] 24 24

The approximation to A3(x) in eq. [7.1312 is shown by a broken line in figure 1 a.

We note that the expression [7.8] for Q consists of a term which is present whether or not the pipe is vibrated and terms which reflect the effects of the vibration. The latter consist of a term which is non-zero only if a

pressure gradient exists in the fluid and a term which is independent of the existence of a pressure gradient, but is non-zero only if the frequency of one of the component sinusoidal vibrations in the expression [7.1] is twice that of the other.

8. Simultaneous longitudinal and rotational vibration superposed on Poiseuille flow through a pipe

We now consider the case when the vibration of the tube is given by (cf. eqs. [6.2])

v (a, t) = Re ( - z U e'm'°~),

f2(a,t) = Re (-Z(2oe'"'°*), [8.1]

where U and f2o are real constants. We again assume that the constitutive eq. [2.5] is valid for the fluid, so that sl and s3 are given by eqs. [5.2].

With eqs. [8.1], eqs. [6.12] and [6.15] become

= -P---(a 2 - r 2) + Re {N(r)e'm~'t}, 4//

(2 = Re { T(r) e"°t},

where, from eqs. [6.13] and [6.16],

ber vlr + tbei1)lr Io(Óar) N = U - U - -

T = m

ber via + zbeivia Io(4 la) '

£20 a berl (1) 2 F) -q- l beil (Y 2 F)

r beri (v2a) + zbeii (v2a)

f2 o a Io (~b2 r)

r I1(492a)

[8.2]

[ 8 . 3 ]

and (cf. eqs. [6.14]) vl, v2, q51 and q52 are defined by

( Pm~___~ 1/2 ( Pncg ~ i/2 11 ~ Y2 =

\ ~ / ' \ rl I '

Y 1 V2 q51 = -~--~-(1 + z), q52 = --~-(1 + 0. [8.4]

From eqs. [8.2] and [6.3], we have

Pr ; : = - - - + R e ( N ' e"'°t),

2r/

B = r Re (T'e"~'). [8.5]

Introducing these expressions for ~ and 2 into the expression [5.211 for sl, we obtain

Kazakia and Rivlin, The influence of vibration on Poiseuille flow of a non-Newtonian fluid, H 253

4 ( 1 ~P3r32@ P r r2 } si = - - : - t l 3 _ + (3IN'J 2 + IT'[ 2)

1 + --$6zù,ù~tl3r2ReR'T '2 + c>(r,t), [8.6]

where ~(r, t) is the sum of a number of terms of the form Re ~bk(r)e 'k~~ and k is a non-zero integer.

Introducing the expression [8.6] for sl into the expression [6.17] for Q, we obtain

_ _ _ _ ( p2a2t/3 ) gpa2rl3 ~Pa¢ 1 - e - e Q = 8t/ 6t/3 4t/3

[3 U2 A2(v,a) + 2a2(22 A¢(v2a)]

+ 52ù,mS ~Uf22a2t13 A»(v2a), [8.7] 4~

where Az(X ) is defined by eq. [7.9], and Ag(x) is defined by (cf. eq. [6.33] in (2))

2x (ber 2 x + bei 2 x) (A¢ + 1)

= x 2 (ber, x ber] x + bei1 x bei] x)

+ 4 (ber, x bei] x - beil x ber'l x).

As (x) is defined by [8.8]

As(x ) = F , ( x ) B l ( x ) + F2(x)B2(x) , [8.9] F 3 ( x )

where

F t (x) = bei ] / i x (ber 2 x - bei~ x)

- 2 ber ] / i x ber, x bei1 x ,

F2 (x) = ber ~ f i x (ber 2 x - bei 2 x)

+ 2 bei ] / i x ber1 x bei, x ,

F3 (x) = (ber z ]//2x + bei 2 ]//2x)

(ber~ x + bei~ x) 2 , x

B, (x) = S y2 {2 (ber 1 ] / 2y + bei, ] /~y) o

ber2 y bei2 y

+ (ber, ] / / iy - bei, ] ~ y )

" (ber~ y - bei 2 y)} dy,

x

~2(x) = Sy 2 {2 (ber, ~ y - bei~ ~ y ) O

" ber2 y bei2 y

- (ber, ]//2y + bei, ] /2y)

• (ber~ y - bei~ y)} dy. [8.10]

A 2 (x) and A ¢ (x) are plotted in figure I of (2) and A 5 (x) is plotted in figure 1 of the present paper.

40

30 A 1

J3

2C

[

1C

/ /

/ /

A S / -11

A 3 A5

-..---.--'4"----- -~~- I I 0 1 2 3

X

Fig. 1 a. A 1 (x), A 3 (x) and A 5 (x) vs. x. The broken lines represent asymptotic approximations for small x.

103 l o 2

5 xl 02

A1 102

50

20

1 o I II

1 2

Fig. lb. A 1 (x), A 3 (x),

5

represents an asymptotic approximation to A 1 (x) for large x

50

ZX 3 ZX 5

10

/ ' 5 -1 2 Z

I I 5 10 40

X

A s(x)vs.x. The broken line


If x ~ 1, we can approximate A2(x) by eq. [7.13]i and A4(x) and A»(x) by

A4(x) = --1--x4, ~»(x) = - 1 - - ~ 6. [8.113 256 128

This approximate expression for A5 (x) is plotted as a broken line in figure 1 a.

We again see that the expression [8.7] for the mean rate of discharge Q consists of a term which arises solely from the presence of a pressure gradient, terms which arise as interactions between the pressure gradient and the longitudinal or rotational vibrations and a term which arises from the interaction between the longitudinal and rotational vibrations and is non-zero only if the frequency of the longitudinal vibration is twice that of the rotational vibration.

9. Discussion

The calculations in sections 4, 5, 7 and 8 have been carried out on the basis of the particular constitutive equation given in eqs. [2.5]. Anal- ogous calculations can be carried out on the basis of more complicated constitutive equations. However, in general, such calculations will increase in complexity as the constitutive equation employed becomes more complicated. Nevertheless, we can obtain rather easily a qual- itative indication of the manner in which the mean rate of discharge depends on the constitutive equation. This can be done by substituting, in particular terms in the constitutive expressions for Sl and s3, the expressions for and 2 corresponding to the assumed vibration of the boundary, and finding the time-independent term in the resulting expression. We may also consider qualitatively the effect on the mean rate of discharge of more complicated vibrations then those considered in sections 4, 5, 7 and 8.

We return to the discussion in section 3 and consider the fluid to flow under a uniform time- independent pressure gradient P between two parallel plates which are vibrated in accordance with eqs. [3.1], so that the velocity distribution in the case when the fluid is Newtonian is given by eqs. [3.10] - [3 .12] . I f the angular frequencies are such that the ratio between every pair of them is irrational, then whatever particular form the constitutive eq. [2.3] may take, the effect of the vibration on the mean rate of discharge is zero if P is zero. To see this, consider, for

example, the case when the constitutive eq. [2.5] is applicable so that sl is given by eq. [5.2] 1. If the expressions for ~ and 2, obtained from eqs. [3.10] and [2.2], are substituted in eq. [5.2]i and the multinomial theorem is used to expand the expression for (~2 + 72)~ so obtained, then the only terms which are independent of t have P as a factor. Similar considerations apply to terms of higher degree in ~, 2 and their time derivatives, if these occur in the expressions for sl.

It is, however, evident that even if the effect of the vibration on the rate of discharge is zero, when averaged over infinite time, it may be non- zero when averaged over some range of times which is large compared with 2~/c%. For example, consider the case when the vibration of the plates is described by

vl = R e ( U l e ~ol¢ + U2elr°2t),

V2 = /)3 = 0 o n X 2 = _+h, [9.1]

where

0 < 2coi - co 2 ~ co 1 • [9.2]

Suppose also that si is given by eq. [4.2]. Then the expression for Sl will contain a term of the form

const. Re N] 2 ~(r 2 e ~(2~i -o~2)t. [9.3]

This term will make no contribution to the mean rate of discharge calculated over infinite time. It will, however, contribute to the mean rate of discharge calculated over a time small compared with 2n/(2co 1 -~o2) but large compared with 27r/c~1 and this contribution will vary sinusoidally with angular frequency 2coa - o)2.

Acknowledgement

This work was supported by a grant from the National Science Foundation to Lehigh University.

Summary

An incompressible, isotropic, non-Newtonian fluid undergoes plane Poiseuille flow between two parallel plates, or through a pipe of circular cross-section, as a result of a uniform time-independent pressure gradient. The effect is studied of superposed vibrations of the boundaries, which are not necessarily purely sinusoidal, on the mean rate of discharge of the fluid. The calculations are carried out in detail for a particular constitutive equat!on of the Rivlin-Ericksen type, with the assumption that the fluid is slightly non-Newtonian. It is seen that in addition to a change in the mean rate of discharge which arises as an interaction of the

Kazakia and Rivlin, The influence o f vibration on Poiseuille f low o f a non-Newton&n fluid, H 255

vibration with the pressure gradient, there may also occur a change in the mean rate of discharge which arises from the interaction of the harmonic components of the vibration and may be independent of the pressure gradient.

Zusammenfassung

Gegenstand der Untersuchung ist eine inkompres- sible, isotrope, nicht-newtonsche Flüssigkeit entweder in einer ebenen Poiseuille-Strömung zwischen zwei parallelen Platten oder einer solchen durch ein kreis- zylindrisches Rohr unter dem Einfluß eines zeitunab- hängigen Druckgradienten. Es wird der Einfluß von überlagerten Vibrationen der Wände, die nicht rein sinusförmig zu sein brauchen, auf den mittleren Flüssig- keitsdurchsatz untersucht. Die Rechnungen werden im einzelnen für eine spezielle Stoffgleichung vom Rivlin-Ericksen-Typ durchgeführt, wobei die Flüssig-

keit als nur schwach nicht-newtonisch angenommen wird. Man findet, daß außer der Änderung des mittleren Durchsatzes als Folge der Wechselwirkung von Vibra- tion und Druckgradient eine solche Änderung auch als Folge der Wechselwirkung der harmonischen Kom- ponenten der Vibration auftreten kann, die u. U. vom Druckgradienten unabhängig ist.

Reßerences

1) Manero, 0., B. Mena, Rheol. Acta 16, 573 (1977). 2) Kazakia, J. Y., R. S. RivIin, Rheol. Acta 17, 210

(1978).

Authors' address:

J. Y Kazakia, R. S. RivIin Center for the Application of Mathematics, Lehigh University Bethlehem, PA 18015 (USA)

the influence of vibration on poiseuille flow of a non-newtonian fluid. ii

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