Time dependent compressible flow about a spherically symmetric polymer in solution

Physica 94A (1978) 271-286 © North-Holland Publishing Co.

TIME DEPENDENT COMPRESSIBLE FLOW ABOUT A SPHERICALLY SYMMETRIC POLYMER IN SOLUTION

R.B. JONES

Department of Physics, Queen Mary College, Mile End Road, London E1 4NS, England

Received 6 July 1978

We obtain the general solution of the linear Navier-Stokes equation for time dependent compressible viscous flow about a spherically symmetric polymer molecule. The solution is presented in a covariant form valid in a general cartesian coordinate frame. In the course of deriving the solution we obtain a general decomposition of the unperturbed flow in the absence of the polymer. Our solution generalizes the earlier solution derived by Schmitz and Felderhof for the case of creeping flow.

1. Introduction

In this article we present a general solution of the linear Navier-Stokes equation for time dependent compressible viscous flow in the presence of a polymer molecule described as a spherically symmetric permeable object. The method of solution includes also the case of a hard sphere with stick or mixed stick-slip boundary conditions. The solution we present can be used to discuss the interaction of a sound wave with a polymer and to describe the frequency dependent intrinsic viscosity of a dilute polymer solution subject to a time dependent shear flow. The flow through the polymer is described by the Debye-Bueche-Brinkman equationsl"2'~). We obtain a general solution of these equations by extending the method of cartesian ansatz already used to solve the time independent case4'5). Our solution has the advantage of being valid in a general cartesian coordinate frame in contrast to methods of solution using vector spherical harmonics which require the choice of a preferred axis.

In section 2 we gather together the equations describing the fluid flow. In sections 3 and 4 we show how an arbitrary unperturbed flow in the absence of the polymer can be decomposed into a series of three distinct types of irreducible cartesian tensors. In section 5 we show how the perturbed flow in

271

272 R.B. JONES

the presence of the po lymer can be described by a simple ansatz utilizing the already defined irreducible tensors. The ansatz reduces the partial differential equations to a set of ordinary differential equations which depend only on the unper turbed flow and on the inverse permeabil i ty of the polymer molecule.

2. Time dependent Debye--Bueche-Brinkman equations

Following Landau and Lifshitz 6) we write the equations of motion for a

compress ible viscous fluid as

V . or = O((OvlOt) + (v • V)v), (2.1a)

OplOt + V- (Or) = 0. (2.1b)

In these equations p and v are density and velocity while the pressure tensor or is given by

trii = n ( ( Ovi/ Oxi) + ( Ovi/ Oxi)) - (2r//3 - O ( V . v )6i i - paq, (2.2)

where r/ and s r are shear and bulk viscosity respectively, and p is the

hydrostat ic pressure. Assuming that the density deviations f rom the equilibrium density are small and of the same order as the pressure and velocity fields we can write the usual linearized form of these equations as

• /V2v + (~t/3 + O V ( V . v ) - V p = po(OvlOt) , (2.3a)

O/sIOt + p o ( V . v) = 0. (2.3b)

Here /5 is the deviation of the density f rom its equilibrium value p0,

p = p0 + t~. (2.4)

We relate 15 and the pressure p in the usual way6),

p = c~]/5, (2.5)

where Co is the adiabatic sound velocity in the fluid. Now let us fourier t ransform all time dependent fields in the same manner,

e.g.

v ( r , t) = e '~'v(r, to)dto.

Hencefor th all fields are to be regarded as functions of position and frequency. Using (2.5) to eliminate the density ~ we can write the equations

TIME DEPENDENT FLOW ABOUT A POLYMER IN SOLUTION 273

of motion entirely in terms of the fields v and p as

V2v - 1]'rll V p = q2v, (2.6a)

V' . v = ( d ~ l ) P . (2.6b)

In these equations we have made the following definitions:

l l , h = ( l l , l ) [ 1 - io~co ( ~ql3 + ~) ] , (2.7a)

qZ = -(itopo/~), Re q > 0, (2.7b)

itoa9~ = po-~Co" (2 .7c )

From eqs. (2.6a) and (2.6b) we can derive an equation for the pressure field

alone,

V2p -- b'2p = 0, (2.8)

where

V 2 = - -O)2 /C 2, Re u > 0, (2.9a)

with

c 2 = c 2 - (ito[0o)(4~1]3 + ~). (2.9b)

There are various identities which relate the quantities defined in eqs. (2.7)

and (2.9). Two useful identities are these:

~hcZ (2.10a) = 1 - nC2o ,

/ ,2=_ Eq 2 (2.10b) 1 - ~ "

Assume now that an unperturbed flow (v0, P0) is given in an infinite unbounded region satisfying eqs. (2.6a) and (2.6b). Into this flow at r = 0 we place a spherically symmetr ic polymer molecule which per turbs the flow to

(v, p). The polymer exerts a force density on the fluid of the form

- ; t ( r ) ( v - U - g~ x r ) ,

where A(r), the inverse permeability1'2), is assumed to be a spherically symmetr ic function, while U(to), g/(to) are the fourier components of the rigid body translational and rotational velocities of the po lymer which are assumed to be specified. In the presence of this extra force on the fluid, the flow is

274 R.B. JONES

described by the equations

V 2 v - ( 1 / ~ I O V p - q 2 v = A(r) ( v - U - / ~ x r), (2.11a) 7/

V. v = (e/r/1)p, (2.11b)

which we will call the time dependent Debye-Bueche-Br inkman equations. By taking the limit Co--> ~ we obtain the time dependent equations appropriate to incompressible flow.

Finally, let us express the perturbed flow (v, p) in terms of the deviation from the unperturbed flow:

/3 ~ V 0 - ~ - V l , (2.12)

p = p o + p l .

Remembering that the flow (v0, p0) satisfies eqs. (2.6a) and (2.6b) we obtain:

V2vl_ ( 1 / r l l ) l r p 1 _ (q2 + k 2 ( r ) ) v t = k2 ( r ) ( vo - U - 1~ × r ) , (2.13a)

V . v l = ( e [ r l l ) p l , (2.13b)

where

k2 ( r ) = A(r)/rl. (2.14)

The problem now is to find the solution (v~,pl) of eqs. (2.13a) and (2.13b) which vanishes at infinity, given a general incident unperturbed flow (v0, p0) satisfying eqs. (2.6a) and (2.6b).

3. Tensor analysis of the unperturbed flow

The basic idea for solving (2.13a) and (2.13b) is to find a decomposit ion of the unperturbed flow (v0, p0) into a sum of simpler fields and then to use the linearity of the equations to obtain a similar decomposit ion of (vl, pt). One might at tempt to do this by using vector spherical harmonicsT), but at the price of specifying a preferred direction in advance. If we wish instead to obtain a solution valid in an arbitrary cartesian frame of reference, we must first obtain a decomposit ion of the unperturbed flow valid in such an arbitrary frame. This aim has already been achieved for the static D e b y e - B u e c h e - Brinkman equations4"5). The technique rests on the observation that the vector spherical harmonics give a basis set for representations of the rotation group in three dimensions. Hence we require a basis set for the same group but expressed in covariant form valid in a general frame. Such a complete set can be written in terms of irreducible cartesian tensorsS). The irreducible tensors must be chosen to characterize the given unperturbed flow (v0, p0).


As already seen in the static case4S), one generates the irreducible tensors f rom the tensor coefficients of the Taylor 's series expansion of the fields

(vo, p~). We write

= . . , , , . . . i , x i , . • • x~,, (3.1) n~O n~O

with

h(.) . 1 0"v0i I (3.2)

and

P0 = 2 P~"÷'>= 2 k'~.-) . , . . a~ Ii.,. t n ~ l I • • • Xin~ r t ~ O r l = ( }

with

(3.3)

K~, ) _ 1 O"po [ " ~ " - n ! axe,-. ~ _ a & ,=o' (3.4)

It is obvious that the tensor K ~") is completely symmetric in all its indices while the tensor h ~") is symmetric only in the indices i ~ . . . i,.

The tensors h ~") and K ~") are not independent. To see this it is convenient to introduce the tensors (<") instead of K ¢"~ by the defining equation

{•,,• = 1 K~m+, (3.5a) ii~...im ~(m +2) i,...im.

For m = - 1 , {(-~ is also defined as the scalar

t'~-') = ( l / n 0 K ~°)= (l/'q0po(0). (3.5b)

By use of the equation of motion (2.6a) one sees that, for n i> 0, a suitable trace of h ~"+2) is related to i ~"~.

h(,÷2) z~,~ . + ~,(n) (3.6) ~,,i,. . .i . = ~ii, . . . . . ( n + l)(n +2) " i i l ' " i n

i

where the repeated indices are summed over. Because of its symmetry properties the tensor h ~) has only one &her distinct single trace. From the equation of continuity (2.6b) we see that it is

h~") = ~t'} "-2). (3.7) aai2 " • , i n ( 2 " " , J n "

From the pressure equation (2.8) we see that

{~,). . = v 2 / '~ . -2~. ( 3 . 8 )

"'~ ..... (n + 1)(n + 2) -,z ..... •

276 R.B. JONES

The tensors introduced above are not irreducible. Irreducible cartesian tensors of rank m in three dimensions are of two classes: (i) complete ly symmetr ic and traceless tensors, or (ii) tensors ant isymmetr ic in two indices, symmetr ic in the remaining m - 2 indices, and also tracelessS). As an inter-

mediate step in obtaining irreducible tensors define

i i l . . . i n " . . . . iili2"--i-{-"ili2n . . . i , i + ' ' ' + -,,,,,, i,_~], (3.9a)

~ . , ~ = ~r~ ,~ ,~ - h~"~ i,]. (3.9b) U I . I 2 . . . t n t , 2 ) t l ~ i i l i 2 . . . i n , * t l t l 2 . . .

We observe that t ~"), A~"~, and t~") each have suitable symmet ry to be irreducible but they are not yet traceless. Irreducible tensors T ~"~, A ~"~, t ~ arise by removing all t races f rom the tensors t ~"~, .4~"~, {~"~. We indicate this process by writing

i i I . . . i n = - - i i l . . . i n - ] - . . •

A(..,~. .r(,) (3.10) I11.12 . . . i n ~ Z 3 , i i l , i 2 . . . i n - } - • • • ,

t ' , 7 [ ,. = t-~7~ , . + ' " ,

where the omit ted terms are those necessary for removing traces. We remark

that eqs. (3.5a), (3.9a), (3.9b) and (3.10) serve to specify uniquely the normal- ization of the irreducible tensors. By use of eqs. (3.9) and (3.10) we can

express h ~") in the form

h~'~ = T(n) + 2(n + 1)-' u I - • - i n ~ i l l . • . in

X [ A ( . .n) . - - ( n ) A ( n ) . . . . . . 1.12 . . . . • i n - j - z O ' i i 2 . i 3 • i n i l - { - " " " - { - " " U n , q - - • i . - ~ ] + " " " ( 3 . 1 1 )

where the omi t ted terms correspond to those omit ted in eq. (3.10).

We must now briefly consider the nature of the terms not written explicitly in (3.10). As an example consider the reduction of T ~). A typical term involved in removing the single traces might be ~i, i2h!,"~3 i. By use of (3.6) this can be re-expressed in terms of ~c,-2) and h <"-2). There are terms involving multiple t races like ~ i , - ~ h ~ " ~ a ~ ~,. Use of eqs. (3.7) and (3.8) enables such a term to be expressed in terms of it,-4~. In fact , the omit ted terms in (3.10) can all be expressed in terms of lower rank tensors with superscripts n - 2, n - 4, n - 6 . . . . This c i rcumstance means that the calculation of irreducible tensors for n = m requires that we have already carried out the calculation for n = 0, 1,2 . . . . . m - 2 . In the appendix we list the explicit form of the irreducible tensors for n = 0, 1. All other tensors can then be calculated successively. For the case of general n we have also calculated explicitly those terms which would appear in (3.10) to r emove the single traces. These express ions are also given in the appendix.

We close this section by giving expressions for the nth terms in the Taylor ' s

TIME D E P E N D E N T FLOW ABOUT A POLYMER IN SOLUTION 277

series in t roduced above. These express ions include explici t ly the single t race terms given in the appendix as par t of the general t ensor decompos i t ion . For

the ve loc i ty field one has

v(n) T(..n) .Y'. + 2n z(..~). ~ l l l ' ' ' l n ~ 4 1 " " " X i n n "~- 1 " ~ l l l ' 1 2 " ' ' i n X i l " " " x i n

q 2 (n -2) . n ( 2 n + 1) [ x i T sq. . . i,_zxix,~ . . . xi,_2

( n 2 2 ) -2"r(n-2) . - - - - r J i i l . . . i n _ 2 . , , , i l • • • X i n _ 2 ]

(n -- 2 ) q 2 .2 A(n-2) (n -- ' l ) ' )~ Z 1) " l t i l t ' i 2 " " i n - 2 X i l " " xin-2

1 (2n + 1----~ [1 - n + (2n - 1 ) e ] x i t ~ - . z . ! i . _ 2 x : p q , . . . xi._z

( n - 1 ) [ n 2 2 ]_z.(,-2) ,. + ~ - - - • r ~ i i t . . . i . _ : ~ i , . • • x i . 5+" • " (3.12a)

(2n 1)

For the pressure the result is

l ~ t ( . _ l ) , , ( n - 1 )

p(o "+1) = "q~(n + , . i , . . . i . * i t . . • x i . + "q~ 2(2n - 1)

t v r C t ("-3) ,~ (3.12b) X ,,. j i t . . . in_7~it . . . X i n _ 2 "~- " " " .

These express ions reveal that a given i rreducible tensor , say T ("), will occur in infinitely many different terms of the Taylor ' s series, n = m, m + 2, m + 4 . . . . . By consider ing what will happen on summing the Taylor ' s series fo r v0, we see that the t ensor T (") will cont r ibute in the fol lowing manner :

[ (n + 4) (qr) 2 + ] "(") . x. x,. 1 + 2(n + 2)(2n + 5 ) " ' " l~ t . . . . . . t - - .

[ q2 ] .,.(,) .,.x~x,t x,." (3.13a) - (n + 2)(2n + 5) +" " " x i l j i t . . . . .

For the tensor A (") we obtain an express ion

2n ~r l (qr)2 ] A L"," "x" n + 1 / t + +" " " ,,ta . . . . . . , . . . x~.. (3.13b) 2(2n + 3)

The tensor l("~ occurs in the veloci ty field as

278

in the form

rll(n + 2)[1

R.B. J O N E S

(ur)2 .] t (") x x + 2(2n + 5) ~-'" ii,..i, i i~.. .xi , . (3.13d)

4. Symmetric, transverse and pressure parts of the velocity field

Taking the to ~ 0 limit in (3.12a) gives the previously derived static results4). In the static limit the tensor reduction of the nth term, v(0 "~, could be done completely in terms of T t"), A t") and t ("-2). This gave an elegant decomposition of v(0 ") into three parts called the symmetric, transverse, and pressure fields4), each corresponding respectively to one of the three tensors T ("~, A t"), t t"-2). There is an exactly analogous decomposition in the present case but it is complicated by the partial resummation of the Taylor's series indicated at the end of the previous section. Fortunately, this resummation can be carried out to all orders by using the equations of motion together with (3.13) to give suitable boundary conditions.

From the results of the previous section it is clear that there is a decomposition of v0 of the form

v0 = ~ v~")s + ~ v(0")v+ ~ v~ ")', (4.1) n=0 n=l n=l

where the field v~0 "~s (symmetric field) is linear in the tensor T ("), the field v(0 ")T (transverse field) is linear in the tensor A ("~, and the field V(o n)P (pressure field) is linear in the tensor t ("-z>. The superscript n on these fields corresponds to the lowest term in the Taylor's series from which we get a contribution. From eqs. (3.5a), (3.12b) it is clear that the hydrostatic pressure p is expressed entirely in terms of the tensors t ("~ and is independent of the tensors T (n~, A ~'~. Because of this fact it is clear that the symmetric and transverse fields obey equations of motion corresponding to zero hydrostatic pressure,

(V 2 - q2)V(on)S ---- 0 , V • V~ ")s = 0 , (4.2)

(7 2- q2)V(O")T = O, V • V~ )v = O. (4.3)

Because v~0 ")T, v(0 ")s are divergenceless fields linear in the tensors A ("), T ~") respectively, we may write

2n ) (n) v(~, : = ~ A,(r)Ai io2. . . i ,x i , . . .x i , , (4.4)

and


B' = + r B ; ( r ) ~ T (" ) v(o7 )s B . ( r ) n + 2 ] - " ' inXil " " " xin

_13 Xr.___)) . x .T(. . , , . x . ' . ( n + 2 ) r ' I , . . . . . r - , , - . • x l . . ( 4 . 5 )

Here A , ( r ) , B , ( r ) are spherically symmetric functions of r and the prime denotes differentiation with respect to r. From (3.13a, b) we can read off the first two terms of a series expansion of A , ( r ) , B , ( r ) . From eq. (4.4) we note the obvious property of the transverse field,

r - v(0 "~T = 0 . ( 4 . 6 )

The field V(o ")P is not divergenceless and couples to the pressure field p(o ~ = p(o n~P. The equations of motion become

( V 2 - - qZ)v (on)P = (1/rh) Vp(o ")p, (4.7a)

•" V (n)P= (~:/rh)P(o raP. (4.7b)

Since both v(0 ")P and p(0 ")P are linear in t ("-2~, we may write, for n 1> 1,

(.~P _ (.-2) (.-2~ (4.8a) Voi - C . ( r ) t ii, . . . i._2xi, . . . xi._2 + D . ( r ) x i t ii~ . . . i._2xjxi~ • • • x xi._2,

p(n)P (n-2) (4.8b) = r l l n E . ( r ) t i i , . . , i ._2xjxi , . • . x i ,_v

Again from (3.13c) and (3.13d) we can read off the first terms in a series expansion of the spherically symmetric functions C . ( r ) , D . ( r ) , E . ( r ) .

The unknown functions A., B., C., D., E. can now be determined to all orders by using the equations of motion for the various fields. For example, from (2.8) we see that p(0 "~P satisfies

(V 2 - v2)p(0n)P = 0. (4.9a)

If we insert (4.8b) into this equation we get an ordinary differential equation for E , ( r ) ,

E " ( r ) + 2-~n r E ' , ( r ) - v 2 E , ( r ) = 0, (4.9b)

where primes denote differentiation with respect to the radial variable r. The solution of this equation which is non-singular at r = 0 and which agrees with the first two terms of the series in (3.13d) is

E . ( r ) = 2 " - t t 2 F ( n + ~) y.-U2 I.-j/2(y), (4.10a)

with

y = yr . (4.10b)

28O R.B. JONES

I ,- ln is a bessel function and F(n + ½) is the gamma function. The equations of motion (4.2), (4.3) for v~0 "~s, v~0 ")T lead at once to

2n+t/2F(n + ~) A . ( r ) = z n + l / 2 In+uz(z), (4.11)

2"+312F(n + ~) B.(r ) = z.+31z I.+312(z), (4.12)

where

z = qr. (4.13)

The equations of motion (4.7a) and (4.7b) for v~o n)P, p(o nJP lead to a coupled

set of ordinary differential equations,

C"-~ 2(n - 1) C ' - qZCn + 2(n - I)Dn = n(n - I)En, r

D " + 2(n + 1 ) D , - - - q2D, = n (E ' / r ) , (4.14)

r

C'./r + rD" + (n + 2)/9. = EnE..

The non-singular solution of these which agrees with the terms given explicitly in (3.13c) is

1 - ~ [ rB'.2 ] Cn = --~-- n(n - 1 ) Bn-2 + n - E . , (4.15)

q2 (n - 1) rn-2+ n . (4.16)

To conclude this section we remark that we have found a general represen- tation of any unperturbed flow satisfying eqs. (2.6a) and (2.6b) in an infinite unbounded region. The representat ion is covariantly expressed in terms of irreducible tensors constructed from the derivatives of the unperturbed fields evaluated at a point.

5. Solution for the perturbed flow by cartesian ansatz

Once we have the decomposit ion of the unperturbed flow, the solution of the time dependent Debye-Bueche -Br inkman equations (2.11a) and (2.11b) follows at once by ansatz just as in the static case4). Because of the linearity


of the equations we can assume that

n=0 n=l n=l

n=0 n=l

The field ~ is associated with the rigid body rotation of the polymer and satisfies

V2~ _ (q2 + k2(r))~ = _k2(r) / ] x r, (5.2a)

V. ~ = 0. (5.2b)

We make the ansatz

= R ( r ) ~ x r, (5.2c)

to reduce (5.2a) to the ordinary differential equation

R " + ( 4 / r ) R ' - (q2 + k 2 ( r ) ) R ( r ) = _ k2(r). (5.2d)

The transverse field v~ n)T satisfies

V2v~n)T _ (q2 + k2( r ) ) v~)T = k2(r)V(o~)T ' (5.3a)

g . v~ ~)T = 0. (5.3b)

The ansatz

v~'T = fnT(r) 2(~---~-~) a(~) iit,i2.. " i X i l . . . Xir t ( 5 . 3 C )

reduces (5.3a) to the ordinary differential equation

[ r + 2(n + 1)/r /~ '- (q2 + k2(r ) ) [~ = k 2 ( r ) A n ( r ) . (5.3d)

The flow (~/, 3 a) is associated with the rigid body translation of the polymer.

T h i s f l o w s a t i s f i e s

V2ad _ 1 I r ~ - (q2 + k2 (r ) )ad = _ k2(r)U, ( 5 . 4 a ) r/t

V. ad = ~-- ~. (5.4b) rh

The ansatz

qli = [ l ( r ) Ui + [ 2 ( r ) x i ( U • r ) , (5.4c)

= m g ( r ) ( U • r ) ,

282 R.B. JONES

r e d u c e s the e q u a t i o n s o f m o t i o n to the o r d i n a r y c o u p l e d s y s t e m

f '( + (21r)f~ + 2f2 - g - (q2 + k2(r) ) f l = _ k2(r),

f'~ + ( 6 / r ) f ' 2 - ( l l r ) g ' - (qZ + kZ(r))f2 = 0, (5.4d)

( l l r ) f ' l + r /~+ 4/z = eg.

T h e s y m m e t r i c flow (v] ")s, p].)s) sat isf ies

V2v~,)s _ ( l / r h ) Vp ~,~s _ ( q2 + kZ(r))v~,)s = k2(r )v ~o.)S ' (5.5a)

V • v~ ")s = e--~ p(~ °s. (5.5b) */t

T h e a n s a t z

/)~n)S ~__. f ( n ) S t ~ T ( n ) - L f ( n ) S g ~ T ( n ) J l ~ l ] a i i l . . . i n X i l • • " X i n v J 2 ~ t I ~ i a j i l . . . i n X j X i l . • • X i n ,

(5.5c) p~.)s = rhgS(r)T(n~

- - P l , " - i n X I X i l " " " X i n '

r e d u c e s the a b o v e e q u a t i o n s to the s y s t e m

f],)s,,+ (2(n + 1)/r)f]")s' + 2(n + l)f~ ")s - (n + l )g s - (q2 + k2(r))f].)s

= k2 ( r ) (B , + r B ' l ( n + 2)),

f~,)s', + (2(n + 3)lr)f(2 "~s'- ( l / r ) g s' - ( q2 + kZ(r))f(2.~s

= - k 2 ( r ) B ' , / [ ( n + 2)r] ,

( l / r)f]")s ' + r/t2")s' + (n + 4)f~ "'s = Eg s. (5.5d)

F ina l ly , the f low s y s t e m (v] "~P, p]")P) o b e y s the e q u a t i o n s o f m o t i o n

V210] n)P -- L Vp ~n)P -- (q2 + k2(r))v]n)P = kZ(r)vCon)P ' (5.6a) rh

• . v]n)P = ~ p]n)P. (5.6b) r/l

An a n s a t z s imilar to tha t f o r the s y m m e t r i c field,

/.)~n)P f ( n ¿ P t ( n 2) A_ ,f(n)P t ( n 2) ~ v ~ - J I t ii t . . . i , _ 2 X i l . . . X i n 2 r J 2 X i t ji I . . . i n 2 ~ J ' ~ i l • • " X i n _ 2,

(5.6c) p ~.)r P (.-2) = rhnga t iq . . , i . 2 X j X i l . . • X i n _ 2 ,

r e d u c e s (5.6a, b) to

/]")P" + (2(n - 1) /r ) f ]"}v+ 2(n - l)ft2 "~P - n ( n - 1)g,"

_ (qZ + k2(r))f(~)p = kZ(r)C, .

f~")P" + (2(n + 1)/r)ft2 "~P' - (n/r)gP. ' - (q2 + k2(r))f~2.~p = k2 ( r )D .

( l /r)[ t~)P'+ rf~")P'+ (n + 2)/t2 "~p = ~ng P. (5.6d)


When n = 1 there is no function f~l)P, but ft21)P and gl P still obey eq. (5.6d) with f~l~P put equal to zero.

The solution of these systems of ordinary differential equations will evidently depend upon the inverse permeability A (r) assumed for the polymer molecule. The unperturbed flow contributes to the solution through the inhomogeneous terms in the equations and because the unperturbed flow sets the scale of the irreducible tensors. Boundary conditions for the ordinary differential equations follow from the original field equations where we have the two conditions that the flow (Vl, p0 should vanish at infinity and that the pressure tensor ~r should be continuous everywhere (for piecewise continuous A(r)). This means that all of the functions R(r), fT(r), fl(r), f2(r), g(r), f~"~S(r), f~2n~S(r), gS(r), [~n~P(r), f~2"~P(r), g~(r) should vanish at infinity. Because ~r depends upon p and upon the derivatives of velocity one sees that g(r), gS(r), gP(r) should be continuous everywhere while all the other functions should be everywhere continuous together with their first derivatives. For a hard sphere of radius R the same systems of ordinary differential equations hold in the region r > R (with k2(r)=O), but at r = R one applies the appropriate stick or mixed stick-slip boundary conditions. The notation here is such that in the static limit to -->0 the functions f~, gS, g~ introduced above reduce to those functions already defined in the creeping motion case treated earlier3).

6. Conclusions

The general solution presented here is more complicated than in the static case owing chiefly to the compressibility. The divergence condition (2.11b) makes the coupled systems (5.4d), (5.5d) and (5.6d) more difficult to solve than in the static limit. Nevertheless, these ordinary differential equations are quite amenable to analytic solution in special cases and to numerical integration otherwise. The covariant form of the solution enables it to be readily adapted to the calculation of the effect of boundaries at finite distance through the method of reflections9).

We may now utilize this general solution to establish Faxen type theorems 3) for polymers in time dependent flow. In the following article we will derive such theorems for the force, torque, and symmetric force dipole exerted on a polymer immersed in an arbitrary unperturbed flow. Such theorems will enable one to calculate the force and torque exerted on a polymer by a sound wave and to obtain the frequency dependent intrinsic viscosity of a dilute solution of polymers.

284 R.B. JONES

Acknowledgement

I wish to thank Prof . B.U. Fe lde rhof for a helpful conver sa t ion during the

course of this work.

Appendix

We first list explici t ly the low rank irreducible tensors in t roduced in sect ion

3. We begin by not ing that fo r the scalar /'(-1) we can write trivially

t(-1) = f(-l) = (1/*h)p0(0). (A. 1)

For n = 0 we have

T~0)= ~!0,= h!O), (A.2)

2

t! °) t-I m h( 2~ q hm) = " = - - i a a - - 2 - " i , (A.3)

For n = 1 we start with

2 t t ~ i i I , * i l D .

To r emove the t race we write

l i i I - - 3 l ~ t a ° J i l "

Using (3.7) we have

T 2 = h 2 = ~t ~-'),

so that finally we m a y write

T!~ ) = T ! ~ - -~ t(-')~i,,. (A.4)

The an t i symmet r i c tensor A~), ) is a l ready t raceless giving

. " i i I = 2 1 _ " lit - - ° . q , J " (A.5)

For the remaining n = 1 t ensor we start with

q2 = _

• * i a a i I - 6

F r o m (3.8) we calculate that

V 2

/'t22 = ~- t '- ').


H e n c e , r e m o v i n g the t race , we wri te

. 1., 2

= ig t l - " 8 " , " (A.6)

N e x t we cons ider t ensors for genera l n and wri te expl ic i t ly the t e rms which r e m o v e the single t races . The s imples t case is t I"). A s imple ca lcula t ion gives

t !.") . /'~") - ( 2 n + 1 ) -~ i i l . . . l n ~ , i i I . . . i n

i = n - 1

j~< 8. i! ~). X Zj4k a a u l . . . i~_lt/+l . . . ik_lik+l . . . in + " • ".

/ = 0

In this exp res s ion io = i and the t e rms not wr i t ten expl ic i t ly are those

involving mult iple t races . By use of (3.8) [~2,, . can be e x p r e s s e d in t e rms of t[i~72!. Fur the r by (3.10) we k n o w that t ~"-2) begins l inear ly with ~/.-2). The re - fore we finally obta in

,,, . i , = . ; i , . . i , ( n + l ) ( n + 2 ) ( 2 n + l )

i = n - I

X X ~iiikt!~-.2.!i,-,ii+,...ik-,i*+,...i. + ' " ". (A.7) j<k j=0

Note that (A.6) is the n = 1 case of (A.7). For the tensors 7"/") a similar argument goes through, but now both ]-1,-2)

and t t"-2) occur in the single trace terms. The result is

T~..~) "~(") - (2n + 1) -1 ul . . . i n ~- l iil . . . ln

j = n - I ~ q 2 , - r , (n-2)

X X ~iiik t n ( n ~ l ) liil'"i'-Ilj+l'"ik-lik+l'"in" j<k / = 0

+ [1 - 2(1 - ¢) / (n + l)]t~r,?2!~j_,~+, ~,-,,k+, in } + ' " "" (A.8)

Again one sees that (A.4) is the n = 1 case of (A.8). Fo r the t ensors A I"> the expres s ion one gets a f te r r e m o v i n g single t races

f rom At , ) i s more compl i ca t ed to wri te down.

in). . X ( n ) . A i i l , t 2 . . . i n .-~ z a i i l , i 2 .. i n

q 2 j = n - I

(n - 1)n(2n - I) ~ 61ji~ j = 2

1 x A!~,~22!.. #_,ii+ ' .... "k-,ik+,... i. + of.

2n

X ~ [ 3 t tn -2 ) ~ t (n -2 ) t i,~ i l i 2 . . . i i _ l i j+ l . . . i n - ill i i i 2 . . . i i _ l i i+ l . . . i J

i=2

286 R.B. JONES

( n + l ) q 2 ~] [~iijhl~i_2) ii I i j + l "in-- ¢~ilii + 2 n ( 2 n - ' ] ~ - - n - 1)i=2 2 . . . . .

q2 X l . ( n - 2 ) .in] -t "ii2""iJ Iii+l'" 2n(2n - l )n(n - 1)

f j=n--I X ~ ~ [~iiih~:il!!.i i ,ii+l...ik_lik+l...in-'}- ¢~ii k

i :2

X h (n. 2) . . i3 , * l it 1 . , . I / - i I / + I . . . i k I i k + l • . .

- same express ion with i *-* il} + • • .. (A.9)

One may further use (3.11) to replace h I"-z) by a linear combinat ion of T ~"-2)

and A ~"-z). To obtain the rather simple result in (3.12a) one must contract the a b o v e tensor express ion with the posi t ion vector c o m p o n e n t s xqx~2.., xi.. After contract ion (A.9) b e c o m e s much simpler and readily yields the terms quoted in sect ion 3.

References

1) P. Debye and A.M. Bueche, J. Chem. Phys. 16 (1948) 573. 2) B.U. Felderhof and J.M. Deutch, J. Chem. Phys. 62 (1975) 2391. 3) B.U. Felderhof and R.B. Jones, Physica 93A (1978) 457. 4) R. Schmitz and B.U. Felderhof, Physica 92A (1978) 423. 5) R.B. Jones, Physica 92A (1978) 545,557,571. 6) L.D. Landau and E.M. Lifshitz, Fluid Mechanics (Pergamon Press, London, New York, 1959)

ch. 2. 7) A.R. Edmonds, Angular Momentum in Quantum Mechanics (Princeton University Press, 1957)

p. 81. 8) M. Hamermesh, Group Theory and its Application to Physical Problems (Addison-Wesley.

New York, 1962). 9) P. Reuland, B.U. Felderhof and R.B. Jones, Physica 93A (1978) 465.

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Time dependent compressible flow about a spherically symmetric polymer in solution