Second-order effects in He-II subcritical flows

Physica 142B (1986) 16-30 North-Holland, Amsterdam

SECOND-ORDER

EFFECTS

IN He-II SUBCRITICAL

FLOWS

R.J. ATKIN and N. FOX Department of Applied and Computational Mathematics,

University of Shefield,

UK

Received 1 April 1986

The dependence of viscosity and entropy on temperature, together with energy dissipation and thermal conductivity, all modify the Poiseuille velocity profile for the normal fluid flow of liquid helium II along a tube. We show that the modifications include a significant temperature gradient in the radial direction, a slightly reduced flow of normal fluid along the tube, and an unexpected diverging radial flow.

1. Introduction

According to the two-fluid theory of liquid helium II, the normal fluid constituent flows as a Newtonian fluid, and carries all the entropy. The superfluid flows independently. Below a certain critical velocity there is no interaction force between the fluids. The flow of normal fluid along a tube at subcritical velocities, under a pressure gradient, is usually assumed to be of the classical Poiseuille type. The velocity is directed along the tube, has parabolic profile, and does not vary along the length. Several factors perturb this simple picture. Viscous dissipation creates entropy thus adding to the normal fluid. The entropy function itself is temperature dependent, and so the temperature gradient associated with the pressure gradient gives rise to an increasing contribution to the entropy along the tube. The viscosity coefficient also varies with the temperature along the tube, and the thermal conductivity contributes to a further heat flux. A one-dimensional discussion of the effects of the temperature dependence of the entropy and viscosity has been given by Brewer and Edwards [l] and Van der Heijden et al. [2]. Our objective here is to give a complete analysis, including the three-dimensional modifications of the flow, arising from these effects. We develop a perturbation scheme based on a small parameter proportional to the temperature difference between the two ends of the tube. The classical Poiseuille flow emerges as a first-order solution of the basic equations. We proceed to a complete second-order solution in which all the above-mentioned effects appear. It we neglect the thermal conductivity, the second-order solution is surprisingly simple. The parabolic profile of the velocity component along the tube is slightly, but increasingly, flattened in the direction of flow, and unexpectedly, a radial component of velocity emerges causing the normal fluid to diverge along the tube. In the modified temperature field there is an increase near the surface of the tube and a decrease near the axis. As far as we are aware, these results are new and as yet unobserved. We also derive the complete second-order solution wifk& neglecting the thermal conductivity, and show that the additional analytical complexity has negligible quantitative effect on most parts of the simpler solution. Only one coefficient is significantly affected.

2. Formulation

of the problem

We first recall the basic equations governing the two-fluid description 0378-4363 I86 / $03.50 0 Elsevier Science Publishers B .V. (North-Holland Physics Publishing Division)

[3,4]. We denote the densities

R. .I. Atkin and N. Fox

I Second-order

effects in He-II subcritical flows

17

and velocities of the normal fluid and superfluid by p,, U” and p,, v”, respectively. We neglect any change in the total mass density p = p, + p, and so the mass balance equation becomes

-& (p,zll

+ p$)

(2.1)

=0.

I

The momentum &I;

P”

n

(

dt+vjx

balance equation for the normal fluid may be written as

au; =$"ij+Psg+ >

1

1

I

1z +$

I

(p,v,s) (VT- US) I

(2.2)

)

where

ati = -s -g + f J$- 2 -& {(VT - u;>>)

(2.3)

-

axi

I

I

I

T, S denote the temperature

and entropy,

and mij are the components

of the stress tensor given by

.

uij =

(2.4)

In (2.4) p denotes the hydrostatic pressure, and A, 71 are the first two coefficients of viscosity. More general theories involve terms arising from the gradients of densities and relative velocity of the normal fluid and superfluid, and a correspondingly larger number of viscosity coefficients. The momentum balance equation for the superfluid is

(2.5) and the energy equation may be written as

where K is the thermal conductivity coefficient. We consider steady-state flow along a tube

r2 = xi +xzS

osx,sz,

a2 ,

G-7)

so that all time derivatives in the basic equations perturbation of an equilibrium state in which nn = *s = 0 )

T=

T,,,

S=S,,

p,=d7

are zero. We suppose that the flow is a small

P,=P:T

P=Po7

(2.8)

where To, So, p”, , pf , PO are constants. We suppose that associated with the steady-state flow there is a temperature difference between the two ends of the tube. We assume that the mean values of T over the end sections are F=

T T;;l

x,=0, + E) ,

x1 = 1.

(2.9)

18

R. J. Atkin and N. Fox

We assume

that the variables

involved

I Second-order

effects in He-II subcritical flows

have asymptotic

expansions

T=T,+EO+E*++..., Un=EVn+2Un+y

us = EVS + &Is

p,=pf+~h+...,

ps = pi - &h +. . . ,

X=X,+&X,

+ ...)

(2.10)

+“‘,

where X denotes any one of the variables S, p, p, 77, A, K. The form of expansion for p, and p, has been adopted to satisfy p = constant identically with p = pi + py. We consider only the temperature dependence of the functions S, p,, p,, 7, A, K. Then for each of these variables X=X(T)=X(T,,+&+...) (2.11) where

Hence,

and,

in particular,

s,=e(FT 1“=g,

(2.12)

where C denotes the specific heat. We suppose that the surface r = a is non-heat-conducting, q*n=O, where

so that (2.13)

to r = a, and q is the heat flux

n is the unit normal qi = p, sT(U;

- U;) -

K

$

(2.14)

. I

We also assume

that the surface

( pnvn + psuS) . n = 0 . Combining

(2.13)-(2.15)

r = a is non-porous

so that (2.15)

gives (2.16)

R. J. Atkin and N. Fox

I Second-order

effects in He-II subcritical flows

19

The ‘no-slip’ condition for the normal fluid on the surface r = a gives v;=o.

(2.17)

3. First-order

solution

The flow of normal fluid in a tube is usually assumed to be approximately of Poiseuille form. Here we show how this type of flow emerges from a first-order solution of the full equations of motion. We seek a first-order solution in which the normal fluid and superfluid velocity have the forms V” = (V;(r),

V” = (V;(r), 0,O) ,

0,O) .

(3.1)

Substituting the expansions (2.10) into eqs. (2.1)-(2.6) and retaining only those terms which are of the first-order in E, we find that (2.1) is satisfied identically, (2.3), (2.5) give (3.2)

and then (2.2) gives d2V; ( x+,nz2

JPI O=-~+% 1

d2VY 3

17

&!!&gL.

(3.3) 2

3

Finally, (2.6) gives d28 -=o. Ko dXidXi

(3.4)

From (3.1)-(3.3)

we deduce de = const , dx1

dP1 = const , dx1

(3.5)

and hence, using (2.9), 0 = Tox,ll . Eq. (3.4) is therefore P=Po

(3.6)

satisfied identically.

atx, =0

If we take (3.7)

so that atx,=O,

Pl =o (3.2),

(3.6)

give

p1 = pSoTox,ll.

(3.8)

R.J.

20

Atkin and N. Fox

/ Second-order

effects in He-II subcritical flows

Eq. (3.3) may now be solved and all the boundary conditions (2.15), (2.16), (2.17) satisfied to first order by taking

v; = g

(r2 - a’) .

(3.9)

This gives the parabolic velocity profile of the usual Poiseuille solution. The superfluid velocity component V’s has not appeared so far in our equations or boundary conditions on r = a. It is determined entirely by the end conditions on x1 = 0, 1. In order to proceed with the second-order solution we need the first-order solution to be complete, and so we assume VT = const .

(3.10)

The magnitude of the constant is determined

4. Second-order

by the externally imposed superfluid flow along the tube.

solution

We now proceed to examine terms of the second order in E in our basic equations (2.1)-(2.6). These give rise to equations for U n, U:, 4, involving the known functions Vl, Vi, 8. Eq. (2.1) gives

$

(ppq + pyJ; + hvy - hv;) = 0.

(4.1)

I

Eq. (2.5) gives

+2

zy--0,

(4.2)

and so (2.3) become (4.3) ()=_S

(4.4)

where cz = 1,2 and v = pSoTollrl, .

(4.5)

Eq. (2.2) gives (4.6) and (2.6) gives pTo&(So~~+S,V:)=~(Ko~+K,~)+rl,(~)2+~~(~)2~ I I

I

(4.7)

R. J. Atkin and N. Fox

Substituting for dp,ldx,

I Second-order

effects in He-II subcritical flows

21

from (4.3), (4.4) into (4.6) gives

where

Differentiating

(4.8) with respect to x1 and (4.9) with respect to x, and adding we find

CT, (A, + 2no)V2D = pSoV2~ - 2~7; + p 12 + $

(4.10)

r2 - r’P”,($az+Iq).

Hence, 2

(A, +2770)D = pS,c#~+

p 7

-2~7;

- ;

&z2 - q”,v;)

;

+ P:v2 & + ‘k >

(4.11)

where (4.12)

V2$=0. Now from (4.7) we find pT,S,D

TO

= ~~~~~ + c

K;

-

(4.13)

p

where K;

=

dK,/dX,

Eliminating

.

D from (4.11) and (4.13) we find

C#J - kV2+ =

(yr4 + @r2 + y -

-

*

PSO

(4.14)

,

where k

=

Ko(ho + 2’70) p2S;To

p=-

P:PS,T;

’

a = -

3212d

7

CT,( A, + 3770) + (A, + 277,JTo

CToa2(A,

+ 412170so

270)

1612~;

412710

4So12770 Y=

+ p:a2pSoT;

+

(ho

+

2’lo)K;

p*s;z

.

+

~o,Tovi + -Tad ___

4170

2170

’

(4.15)

22

R. J. Atkin and N. Fox

I Second-order

effects in He-II subcritical flows

In principle the perturbation solution can now be found by solving (4.12), (4.14) for 4, $ subject to suitable boundary conditions and using (4.13) to solve (4.8), (4.9) for Ul, Uz subject again to appropriate boundary conditions. The components Us are restricted only by (4.1). However, a particularly simple solution emerges if we neglect the terms involving K. The validity of this solution is considered later.

5. The case

K

0

=

If we assume obtain

0, we may substitute for 4 from (4.14) and for D from (4.13) into (4.8), (4.9), to

K =

~ov2uq = -

2 +

(+x1

(5.1)

)

1

qov*u:: = - $ +cwx,

(5.2)

)

a

where

VT,

PCT, (+=2--- I

1

(3 1 “3

dT

G=$

(S,-2c)+

2

( 2T 1 0’

(5.3)

From (4.14) we have

cc,

cp=ar4+pr2+y--

(5.4)

PSO

and we recall

v**=o.

(5.5)

Solutions of (5.1), (5.2) may be written in the form

U;=-+,*+~rzx,+*l, 0

0

(5.6)

_

u:=- &0 x,* + &

r2x, + l(d

,

(5.7)

0

where VQ, =V’& = 0.

(5.8)

We now only need to find harmonic functions $, +1, I,!I~to satisfy the boundary conditions. In view of (2.9), (3.6) and the asymptotic expansion for Tin (2.10), we see that the mean values of C#J over the end sections must vanish. From (5.4), the mean value of I,!Jover the end sections is therefore

(5.9)

R. _I. Atkin and N. Fox

Using (5.6) the ‘no-slip’ condition

I Second-order

effects in He-II subcritical flows

23

(2.17) gives on r = II, (5.10)

The condition

(2.16) with (5.7) gives on T = a, 2

UEx,

=

-

a

t+b+

f

a4 +

2%

qbux, =

0.

(5.11)

0

We satisfy all the conditions

(5.9)-(5.11)

by taking $ to be constant, (5.12)

and (5.13)

J/, =

($ 0

g,xa .

(5.14)

0

Hence,

u; = -g (r2 - U’)Xl

(5.15)

)

0

u:: = -E-(r2 - u2)x, . h-0

In this second-order solution we see that radial components I_Jz of the normal fluid appear in addition to the longitudinal component UT. No solution of the equations is possible without radial components. However the same is not true of the superfluid velocity. The second-order components are restricted only by (4.1), and we consider only those flows for which ZJ” = (US, 0,O) .

(5.17)

We may now substitute for Uy, Ui, Vy, Vi in (4.1) and solve for Us: (J;=-

1*+L@__+

477OPS 477OPS

-Axr2x

4p;7),

+

’

dp, (dTo >I

PS,T;

m

(r2 - a')~,

V"T,dp,

+ ~81 ( dT > 0

where f(r) is an arbitrary function us = (EVS, O,O), then f(r) = 0.

of

f(r) 3

r.

If we take the prescribed

(5.18) value of us at x1 = 0 to be

24

R. J. Atkin and N. Fox

I Second-order

effects in He-II subcritical flows

6. The mean heat flux The rate of flow of heat along the tube per unit of cross-sectional area is pSTut and so the mean value over the length of the tube of the total heat flow along the tube is I

a pSTvy2rr

dr dx, .

(6.1)

x,=0 r=O

Using the expansions

(2.10) and substituting

With a little rearrangement

for 0, Vy, Uy from (3.6), (3.9), (5.15) we find

this may be written in the form

STu:=e(r’-a’)(E+[& (F)o-k(!$)o-i] Substituting

(6.2) into (6.1) and performing

the integrations

~x,s~}+O(E~).

(6.2)

we find

AT -=i!

(6.3)

where AT denotes the difference between x1 = 0 and x1 = I of the mean temperature over the end sections. The relation (6.3) agrees with that obtained by Van der Heijden et al. [2] eq. (4.5) using a purely one-dimensional analysis. These authors base their analysis on the standard relations Ap=pSAT=-T

81710 n-r PST ’

but allow for the variation of approach because each of the provided that Ap is interpreted relations (6.4) become invalid

7. The case

K #

(6.4)

S and 77 with T along the tube. The correct result emerges from this relations (6.4) may be shown to be valid to the second order in E, as the difference in the mean values of p between the end sections. The at some higher order approximations.

0

We return now to the full equations (4.14) for C$may be written as

of section 4 and derive a solution without neglecting

K.

Eq.

x - kV2x = er4 + pr2,

(7.1)

x=f#J+*-y.

(7.2)

where

PSO

R. .I. Atkin and N. Fox

A particular

I Second-order

25

effects in He-II subcritical flows

integral for (7.1) is

x = ar4 + (16kar + /3)(r’ + 4k) . For the complementary

(7.3)

function we require the general solution of

kV*x = x

(7.4)

and we show later that all boundary conditions can be satisfied by taking x as a function of r alone. In this case the general solution of (714) is x = C,Z,(rlG)

+ C*K,(rIfi)

)

(7.5)

where I,,, K,, are modified Bessel functions of zero order. Now 4, the second-order temperature field, must be finite on r = 0, and from (4.11) we see that to avoid singularities in D, the same must be true of I,!L Hence, x must be finite on I =0 and so C, =O. Combining (7.2), (7.3) and (7.5) we find the complete solution for 4 c#l=

C,Z,(rlfi)

+ cxr4 + (16ka + p)(r’ + 4k) + y - t,b/PS, .

Assuming that Cc,depends only on r, and recalling that tj is harmonic, JI -=A,+A,logr, PS,

(7.6) we must have (7.7)

where A,,, A, are constants. Since q%must be finite on r = 0, we must have A, = 0 and the constant A, may be found from the end conditions (2.9). The mean value of T over a cross-section is n

T= 1 2TTrdr. ra* I

(7.8)

0

Substituting for 8 and 4 from (3.6), (7.6) into the expansion for Tin (2.10) and using the property the modified Bessel function

I

rZ,,(rlfi)

dr = aV%Z,(alfi)

of

(7.9)

0

we find that the end conditions A, = q

C,Z,($

>

(2.9) are satisfied if

+$+(16ka+~);+4k(16ka+~)+y.

(7.10)

To find the components of the normal velocity we first use (4.14) to eliminate V24 from (4.13) and then substitute for D in (4.8) and (4.9). We may then reduce the equations for Up and UE to the form

a4 ljov2u; = PSorlo -+PCTo+-~~;)-$, ho + 2770 8-5

(7.11) 0

0

1

26

R. J. Atkin and N. Fox

I Second-order

(Aa+rlo) a* A, + 2770dx,

Tlov2u::= PSorlo --NJ A” + 2770 ax,

770PjtV2

I

8(4, + 2%)

(Aa+ ~0) -@To -

AOdV 2( A, + 2~)

effects in He-II subcritical flows

l2

- 2( A, + 277J

xuy2

2 rlov

0

2

P”U

77OP3+

8( A, + 277J - 2( A, + 277”) ‘a ’

Substituting for C$from (7.6) gives the same equation (5.1) for U 9 as in the for Ui becomes

1

v2u:= : +

=

0 case, but the equation

xa (& 11

PSOCl (Ao

K

1;

+2770)rfi

(7.12)

(7.13)

7

where (y = 1

a

32kwSo770 A, +%a ’

+

(7.14)

For a particular integral of (7.13) we seek a solution of the form

u::= F(r)x,

(7.15)

)

in which case vqJ;

=

(!E39 + $ (cqxa r

r

Writing y = F’lr,

$

(r4y) =

.

(7.16)

r

(7.13) becomes

2

&) ,

r3 + C1r2Zh(

(7.17)

where PSOCl

Cl = (A, + 27$& Two integrations F(r) =

2

(7.18)

’

of (7.17) and use of the properties r2 + y

I,

of the Bessel functions gives the solution (7.19)

0

Hence,

x,+x,, where

(7.20)

R. J. Atkin and N. Fox

I Second-order

effects in He-II subcritical flows

27

The constant C, and the functions x, are to be determined from the boundary conditions. The ‘no-slip’ boundary condition (2.17) is satisfied by taking Uy in the same form (5.15) as in the K = 0 case. Again we seek solutions in which the superfluid velocity has the form (5.17). In this case (2.15) implies U&

=0

(7.21)

)

on r = a, and hence (2.16) implies (7.22) on r = a. These conditions may be satisfied by taking the functions ,Y, to be of the form (7.23)

x, = Kx, > and the constants C, and K to have values (Y+ 2a(16ka K=-a,a2+ f%

Substituting

p&k (4l+

(7.24)

+ p)} ,

{4a3cx + 2a(16ka

+ p)}

(7.25)

.

277cJa

the values of these constants into (7.20), (7.23) we find

z,(r/vx) rZ,(alVT)

The superfluid velocity component

‘a ’

(7.26)

of the second order may be found, as before, from (4.1). - a’)q I, (rlfi) rZ,(alfi)

- g-

S 0

r2x1

Zi(rlfi) - V%Z,(aldE)

(7.27)

where, as before, f(r) is an arbitrary function of r and may be taken to be zero if the prescribed value of us at x1 = 0 is us = (&Vi, 0,O).

8. Discussion

We have shown how the traditional Poiseuille velocity profile for flow of the normal fluid down a tube emerges as a first-order solution in a perturbation scheme based on a small parameter E arising from the temperature difference between the ends of the tube. In the second-order solution, effects emerge which are due to the temperature dependence of the entropy carried by the normal fluid, the temperature dependence of the viscosity, the dissipation, and the thermal conductivity. We now use some typical values of the various parameters to investigate numerically how these effects modify the first-order solution.

28

R. J. Atkin and N. Fox

Integrating to first order, -n VI

=

I Second-order

effects in He-II subcritical POWS

the first-order normal fluid velocity component (3.9) across a cross-section U” = EV”, we find that the mean value of uy across a section is

P&T” -------a&.

and writing,

2

(8.1)

8h

We use the tables of thermodynamic coefficient tabulated by Van der Heijden temperature T, = 1.5 K, -n

u1

=

- 2.57~~~ x 10”’ ms-’

functions given et al. [2]. Then

by Maynard [5], and values of the viscosity for a tube of length 15 cm at an equilibrium

,

(8.2)

where a is measured in metres. For a typical require negative values of E in the range

range

of values

-4x10~4
of 5; up to 10 cm/s.

and a = 0.1 mm.,

we

(8.3)

second-order

effects

are as follows.

(i) ModiJication of the temperature field We see from (2.10) and (3.6) that the first-order temperature field is constant over each crosssection, and decreases linearly with the distance x1 along the length of the tube. At the equilibrium temperature T,, = 1.5 K the temperature drop along the length of the tube, corresponding to E = -4 x 10e4 is 6 X 10m4 K. is constant along the tube but varies over the The second-order perturbation 4, however, cross-section. For the case K =O, and using (5.4) and (5.12), we find

(8.4) and from

(4.15),

(Y = -7.54

taking x

again

1017 mm4 K,

I= 15 cm., p = 1.51 x 10’” rn-* K .

(8.5)

The variation of a24 (again for E = -4 x 10m4) across a section of the tube is shown in fig. 1. There is a temperature increase near the surface and a decrease towards the axis. The largest temperature gradient occurs between r/a = 0.4 and 0.8, and has a value 0.18 K m-‘. This may be compared with the temperature gradient along the tube of 4 x 10 p3 Km-l. The rapid variation of temperature in the cross-section arises from a diverging flow of normal fluid carrying the entropy towards the outer surface. We investigate this in (iii) below. The additional terms in (7.6) arising from the thermal conductivity K may be shown to be negligible for the chosen set of parameters. (ii) Modification of the component of normal fluid velocity along the tube The first-order solution predicts the parabolic Poiseuille velocity profile (3.9) for the normal fluid velocity component along the tube. There is no velocity component perpendicular to the axis of the tube. The second-order perturbation has a component along the tube as well as perpendicular to the axis. The component Uy along the tube has a negative parabolic profile (5.15) of increasing magnitude along the tube. This has the effect of very slightly flattening the parabolic profile of the first-order solution, and these results are independent of the value of K. On r = 0, x1 = 15 cm and for the above values of the parameters

e*U: = -5.45

R. J. Atkin and N. Fox

I Second-order

Fig. 1. Deviation

of temperature

effects in He-II subcritical

from

first-order

flows

solution.

X 10m4m s-l .

This may be compared

29

(8.6)

with the first-order

solution

&VI = 0.21 m s-r .

(8.7)

The second-order effects here amount to only a minor modification of an existing velocity component. However, an entirely new effect emerges from the consideration of the radial velocity component. (iii) Radial component

of the normal fluid velocity

Using (5.16) we see that the second-order normal fluid velocity component perpendicular of the tube may be written lJ:e,, where e, is the unit vector in the radial direction and UT = (~?/877~)(r* - a’)r .

(8.8)

We see from the tables of thermodynamic

functions that over the range of available data

s,-2c
(8.9)

and in the temperature %
to the axis

’

range 1.2K<

T,,<1.7K (8.10)

Hence, from (5.3) we see that (Y< 0, and the radial component (8.8) of the normal fluid velocity is positive. This means that the flow of normal fluid diverges along the tube. As the normal fluid carries the entropy this may account for the increased temperature near the surface of the tube that we commented upon under (i).

R. J. Atkin and N. Fox

30

I Second-order

effects in He-II subcritical flows

In the case K # 0 the radial velocity component is more complicated analytically as we see from (7.26). However, for our set of chosen parameters (we take the value of K from Zinov’eva and Dubrovin [6]) the Bessel function terms are insignificant and the solution closely approximates u: = (‘y1/8rl,)(rZ -

a’)r .

(8.11)

In (8.11) the coefficient (pi replaces Crin (8.8). Now (Y,is significantly affected by increase over &. (Y= -1.09 X 10”’ g mP3 sC2 ,

(Y, = -5.88

X

10” g mm3 s-* .

K,

showing a fivefold

(8.12)

Since the coefficient is negative in each case, both solutions predict a diverging normal fluid velocity. In each case the maximum value occurs when r = a/~‘3. At this value of r, we find the total radial normal fluid velocity is E2UY =

0.60 x lo-’ m s-’ 3.2 x lo-’ m s-i

(K = 0) > #o) .

(8.13)

(K

In this case, the second-order solution predicts an entirely new effect, the diverging flow of normal fluid along the tube. The two values in (8.13) indicate the sources of the effect. The first value (K = 0) arises from the dissipation and the dependence of entropy on temperature. (The temperature dependence of the viscosity has a very small effect.) The second value shows that the thermal conductivity is dominant. We see therefore that the second-order solution predicts some new and unexpected results. Most aspects of the secondary flow are adequately predicted if K is neglected but K dominates the radial flow.

References [I] [2] [3] [4] [5] [6]

D.F. Brewer and D.O. Edwards, Proc. Roy. Sot. A251 (1959) 247. G. van der Heijden, W.J.P. de Voogt and H.C. Kramers, Physica 59 (1972) 473 S.J. Putterman, Superfluid Hydrodynamics (North-Holland, Amsterdam, 1974). I.M. Khalatnikov, Theory of Superfluidity (Benjamin, New York, 1965). J. Maynard, Phys. Rev. B14 (1976) 3686. K.N. Zinov’eva and A.V. Dubrovin, Sov. J. Low Temp. Phys. 9 (1983) 232.