Journal o[ Low Temperature Physics, Vol. 38, Nos. 3/4, 1980

Normal Fluid Heat-Exchange Drag in Liquid Helium II

Robert Lynch

Physics Department, University o f Petroleum and Minerals, Dhahran, Saudi Arabia

(Received June 18, 1979)

An examination is made of the modification to Stokes'law for a heated sphere in liquid helium II, as calculated by Springett. His calculation yields an extra term in the drag, over and above the ordinary viscous drag and the Penney- Hunt heat-exchange drag. It is shown that this term is an artifact of the handling of the inertial term in Oseen's approximation to the full Navier- Stokes equation, and thus Springett's result cannot be regarded as valid. The problem is also considered by means of the Stokes theory, and it is shown that in this approximation the extra term is zero. Another approach, boundary layer theory, is mentioned, but not considered; it is cautioned that in the use of this theory careful attention should be paid to the treatment of the inertial terms.

1. I N T R O D U C T I O N

Springett 1 has calculated the modification of Stokes' law for a heated sphere in liquid helium II, due to heat-exchange forces. 2 The calculation was carried out by means of Oseen's approximation to the full Navier-Stokes equation, valid at low Reynolds number.

Assume that at infinity in addition to the radial flow due to the heat input, the normal fluid has a uniform flow velocity U along the x axis, and further that the superfluid flow is taken to be Vs. At r = a normal to the surface of the sphere, p.vn + psv~ = 0, where 0 = pSTv. is the rate of heating, assumed uniform over the sphere. Letting *i and v. = *liP. be the viscosity and kinematic viscosity, respectively. Springett's result for the drag force on the sphere is

FD = F v + FpH + FN (1)

where Fv=67ra*iU( l+3aU/8u , ) i is the ordinary viscous drag; FpH---- 4~psa2vs(U]+Vs) is the Penney-Hunt 2 result arising from the tangential

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312 Robert Lynch

forces of the heat-exchange theory; and

FN = (4vr/3)a2p.v.U'i (2)

is a new term, supposedly resulting from the modification of the flow about the sphere due to viscosity.

It is quite easy to show, however, that the FN term is in reality an artifact of the Oseen approximation. This can perhaps be most clearly seen by considering the approximation in the limit of zero viscosity.

2. OSEEN APPROXIMATION FOR ZERO VISCOSITY

The full Navier-Stokes equation is

aV. /a t + (V. �9 V)Vn = - ( 1 / p . ) VP + v. V2V., V . V . = 0 (3) O

For steady flow, 8V~/Ot=O, Oseen's approximation results from the linearization ( V . . V)V.--> ( U i " V)V.. The FN term is then found from Springett's detailed solution of (3) in this approximation.

Now consider the approximation for v. = 0. We have

( U i . V)V. = - ( 1 / p . ) V P (4)

Setting V. = -V~b, we find

P = P o + p . ( U i " V)$ (5)

Consider the effect of the normal fluid pressure term only. It is not difficult to show that the proper form for ~b is

q5 = - -Vna2 / r -- (r + a 3 / 2 r 2 ) O c o s 0 (6)

Inserting (6) into (5) and calculating

Fo = - f s P d a (7)

where da is the element of surface area, we find FD = FN. However, in the limit v. = 0, considering only the pressure term, we

should find Fo = 0, i.e., d 'Alembert 's paradox. The difficulty is caused by the approximate treatment of the inertial term (V. �9 V)V.. Consider instead of (4) the exact equation

(37.. V)V. = - ( 1 / p . ) VP (8)

from which follows the Bernoulli law 1 2 P = const -~p. V . (9)

Normal Fluid Heat-Exchange Drag in Liquid Hefium II 313

Substituting (6) into (9) and calculating FD from (7), we find Fo = 0, as expected.

Thus FN as calculated by Springett is simply an artifact of the handling of the inertial term in the Oseen approximation, and hence (2) cannot be regarded as a valid result.

3. O T H E R A P P R O X I M A T I O N S

It is unfortunate that Oseen 's approach does not work in this case, since there are few approximation schemes for this kind of problem, and exact solutions are impossible to obtain.

One can consider, instead of Oseen 's approximation, Stokes' approach, 3 in which the inertial term is dropped altogether:

77 V2Vn = VP (10)

However , letting Vn = VnH + V,s, where the first term is a pure radial outflow of normal fluid, and the second term the Stokes solution for viscous flow about a sphere, we have P = P ~ + Ps. Since XrPH = r/Xr2Vnn = 0, PH is a constant over the sphere; if one then calculates the drag force, including viscous stresses, one finds FN = 0. This result is to be expected of the Stokes theory, since it is strictly linear, allowing no "mixing" of the counterflow and ordinary flow.

Another approach which might prove more fruitful but will not be considered in this paper would be to attack the problem via boundary layer theory. In view of the difficulties found above, it would perhaps be wise when tackling the problem from this angle to keep careful track of the inertial terms in such a theory.

A C K N O W L E D G M E N T

The author would like to thank M. A. Farooqi for his help in the preparat ion of the manuscript.

R E F E R E N C E S

1. B. E. Springett, J. Low Temp. Phys. 5, 45 (1971). 2. R. Penney, Phys. Fluids 10, 2147 (1967); R. Penney and A. W. Overhauser, Phys. Rev. 164,

268 (1967); R. Penney and T. K. Hunt, Phys. Rev. 169, 228 (1968). 3. L. D. Landau and E. M. Lifshitz, Fluid.;Vlechanics (Pergamon Press, Oxford, 1959), p, 63.