Mutual friction and second sound

Physica 69 (1973) 245-250 © North-HollandPublishing Co.

MUTUAL FRICTION AND SECOND SOUND H. C. KRAMERS, T. M. WIARDA and G. VAN DER HEIJDEN Kamerlingh Onnes Laboratorium der Rijksuniversiteit Leiden, Leiden, Nederland

(Commun. Suppl. No. 129) Received 25 June 1973

Synopsis The attenuation of second sound is investigated in flow conditions in a capillary. In particular results for "pure" superfluid flow and "pure" normal flow are given and compared.

1. Introduction. The idea of Gorter and Mellink to ascribe the peculiar flow resistance, occurring in liquid helium II, to a mutual force or "friction" between the counterparts of the two-fluid system has been a very fruitful one. Further, the fundamental work of Onsager and Feynman and the subsequent study of Hall and Vinen have allowed for a deeper physical insight into its nature by explaining it as an interaction between the thermal excitations of the normal fluid and the quantized vortex lines or rings essentially moving with the superfluid. In particular the picture of vortex lines parallel to the axis of rotation in a rotating liquid-filled bucket could satisfactorily be analysed by observing the attenuation of second sound, i.e., of a periodic relative motion of the two fluids1). On the other hand, notwithstanding a wealth of experiments and of theoretical studies, no complete quantitative understanding of the resistance phenomena occurring in the flow through tubes or channels can be claimed at the present time. Neither the exact form of the vortex-excitation interaction, nor the conditions of creation and annihilation of vortices are satisfactorily understood. From an experimental point of view a big problem still is the acquisition of sufficiently welldefined physical conditions; therefore, contradictory results are a rather common feature of this field. In our laboratory an extensive study has been made of adiabatic flow through metal tubes with diameters around 1 mm. Some of the results have been published2), other results can only be found in the theses of two of the present authors3'4). In the plesent paper some peculiar features of second-sound attenuation results will be discussed.

245

246

H.C. KRAMERS, T. M. WIARDA AND G. VAN DER HEIJDEN

2. The model. In order to describe the experimental results, it is necessary to summarize the general picture which, of course, is based on the two-fluid model. Mutual friction may occur above certain critical values of either the superfluid velocity vs or the relative velocity v, - v~; its magnitude may further depend on the normal-fluid velocity vn as well. The superfluid is supposed to "carry" a certain amount of vorticity (vortex lines) or "turbulence", the density of which may in general depend on vn, v~ and T. The bars indicate values averaged over the region of the tube or channel and over an appropriate time, connected with the build-up of the vortices. In general the normal fluid might also be turbulent or turbulence might be a condition of the liquid as a whole. However, in the present study, which is limited to velocities of the order of 10 cm/s, there has been found rather strong evidence for simultaneous laminar properties of the normal fluid and "turbulent" properties of the superfluid. This has been discussed elsewhere4). In the present paper discussion will be limited to measurements of the attenuation of second sound as a function of 0n, f~ and T. Observations of the gradients of temperature and the chemical potential have to be referred to other papers 5' 6). The attenuation of second sound was always found to be linear, i.e., independent of the amplitude. Therefore, it seems to be reasonable to use linearized quasiequations of motion for the analysis of the results. It should be emphasized that, though these equations show a close resemblance to the Landau equations of motion, the mutual-friction terms are insufficiently founded from a theoretical point of view. The equations can be written as ~v~ _ St

1 grad p + S grad T + G(vn - v~), p ps

~ V n __

1

at

P

grad p - P~ S g r a d T _ a~ (vn - vs) + ~-- Avn. Pn Pn Pn

(1)

(2)

Subtraction gives the "equation of motion" for the relative velocity: 8(Vn-- Vs)__ t

P gradT-Pn

G

p (vnPn P~

v~) + tl AVn Pn

(3)

from which, by combination with the equations of conservation of mass and entropy (valid in this linearized approximation), the attenuation and velocity of second sound can be deduced. The velocities vn and vs are the time-dependent velocities in the second-sound wave; the mutual-friction parameter G may depend on the average steady-flow velocities gn and Os and on T. In the original G o r t e r Mellink form of the friction force, G was set equal to Ap,,ps(v ,1- os) 2 which form is, however, not compatible with the present experimental results. The value of G is, clearly, a measure for the vortex density of the superfluid, if the suggested model

MUTUAL FRICTION AND SECOND SOUND

247

is accepted. The last term ofeqs. (2) and (3) is the ordinary viscosity term, mainly responsible for the attenuation in zero steady flow.

3. The experiment. The method used to produce at will any steady-flow condition in a capillary, i.e., any combination of ~, and ~s, has been described in ref. 3. Second-sound attenuation was determined by measuring the damping of a secondsound resonator of the "double" Helmholtz type connected in series with the capillary (see fig. 1). Its resonant frequency corresponded with v,2 = (v2/4nZ)(O/l)[(1/V1) + (1/Vz)],

(4)

with O, l, V 1 and V z indicating the effective cross section and length of the connecting tube and the two "capacitive" volumes; vn is the second-sound velocity, given by

vz = (pJp.)SZT/C.

(5)

The main contributions to the attenuation are: a) the viscous damping of the normal fluid motion which can be written as an appropriate width of the resonance curve

Av,,

v,(plps)alr,

=

(6)

with r, the radius of the tube and 6, the viscous penetration depth. The latter can be written as 6 = (tl./~vp.) ½. b) the damping due to the mutal friction if steady flow is present, its resonance width corresponds to

Avm (p/p.ps)c.

(7)

=

A discussion of other (smaller) contributions to the damping will be omitted here. Further, as will be suggested in the following, the two main contributions may not always be additive. H2

Vl

0

V2

Fig. 1. Double Helmholtz resonator for second sound (v ~,, 100 Hz). Hx H2-"heaters; T: thermometer detector; connecting tube: length 8 mm, diameter 0.51 mm.

248

H . C . K R A M E R S , T. M. W I A R D A A N D G. V A N D E R H E I J D E N

4. Results. Fig. 2 gives a review of the measurements at T = 1.5 K. Plotted are values of the observed resonance width Av divided by its magnitude without steady flow (A v) 0, as a function of the average relative steady velocity. Fully drawn lines indicate loci of constant ~n, dotted lines loci of' constant ~s. It is clear that the original Gorter-Mellink formula containing a quadratic dependence on the relative velocity is not sufficient for a description of the results. Most promising for a better understanding are the curves for ~, = 0 and ~s = 0. Due to the rather large cross section of the tube, no critical velocity can be observed for pure superfluid flow. Pure normal-fluid flow shows, however, an extended region of absence of mutual friction, i.e., an apparent vortex-free region. Careful 6~

I

'

L

'

I

'

I

'

I /

I

/

r

(~v) ° rl I Q/ i -4

-'" ,

i

.

~,=o I , I 0 ~n-gs 2

,

-2

,

I , 4-cm/s

I 6

F i g . 2. D a m p i n g o f second s o u n d a t T =

1.5 K against the relative velocity ~7. - - 77~. Fully-drawn lines: loci o f constant ~n; dotted lines: loci o f constant ~Ts. 50

m

Hz

10

1

-k

0"C~1

"~ I

i

r

i

i

itl[

i

1

i

i

cm/s

i

~

i*,

I0

Fig. 3. Resonance width for some cases against ~Tn and *Ts. T = 1.5 K. o: 77n = 0 plotted is ~iv - - Avq; × : *7, = 0 plotted is ~iv. T = 2 . 0 K. A: ~7, = 0 plotted is ~iv - - d v n ; + : 77s = 0 plotted is d r .

MUTUAL FRICTION AND SECOND SOUND

249

measurements show that this situation is limited to within a very small critical value of ~sFor two temperatures the double-logarithmic drawing of fig. 3 gives in one graph Av for pure normal flow together with Av - Av,~ for pure superflow. The peculiar outcome is that, by plotting in this way, the turbulent parts of both curves apparently coincide, i.e., the straight mutual-friction curve for ~n -- 0 is joined by the ~s = 0 line at exactly the point where it cuts the horizontal laminar stretch of the latter. As is clear, in particular for the lower temperature ( T = 1.5 K), this laminar part can be extended to the right if sufficient care is taken. Actually there is a small region of instability or rather bistability. The noted correlation between the ~s = 0 curve and the ~n = 0 curve is not yet understood. It is perhaps not surprising that the two contributions to the attenuation are additive if the average value of v, is zero; the amount of vorticity (strength of turbulence) in the superftuid may then be expected to depend only on vs and the profiles involved are practically flat. The situation for a pure steady normal flow may be more complicated. For instance the profile for vn has a tendency to be parabolic, though the shortness of the resonator tube makes a fully developed Poiseuille profile improbable. In what way this can influence the energy dissipation of the second-sound wave is not clear. The conclusion must be that, for pure normal flow, energy loss of the secondsound wave initially takes place by the viscous process only; above the crossing point, however, the number of vortices is adjusted in such a way that dissipation in pure normal flow corresponds to the one effected by mutual friction only, as is found in pure superfluid motion of the same magnitude. It appears rather as if the viscous dissipation completely drops out for the pure normal flow as soon as the muttml friction becomes effective, but this seems rather improbable. This result was obtained at all temperatures, provided sufficient care was taken to allow for establishment of equilibrium. The slope of the turbulent curve does not agree with a quadratic dependence as was suggested by Gorter and Mellink. The power dependence on the temperature is given in table I. TABLE I Power dependence on temperature T (K)

1.3

1.4

1.5

1.6

1.7

1.8

1.9

2.0

Power

1.66

1.69

1.84

1.90

1.97

2.20

2.30

2.64

The conclusion of this m a y be that it might be profitable to try to understand the two types of flow with their peculiar interrelation as discussed here, before attempts are made to explain more complicated flow conditions. Perhaps one may suggest that the flow condition most frequently studied, i.e., that of zero-mass flow or "pure heat conduction", may belong to the more complicated class.

250

H . C . KRAMERS, T. M. W I A R D A A N D G. VAN DER HEIJDEN REFERENCES

1) Progress in Low Temperature Physics, C. J. Gorter, ed., vol. I, North-Holland Publ. Comp. (Amsterdam, 1955), chap. I and II; vol. III (1961), chap. I. See for further references other volumes of this series and Wilks, J., The Properties of Liquid and Solid Helium, Clarendon Press (Oxford, 1967). 2) Kramers, H. C., Superfluid Helium, J. F. Allen, ed., Academic Press (London and New York 1966) p. 199. 3) Wiarda, T. M., Thesis (Leiden, 1967). 4) Van der Heijden, G., Thesis (Leiden, 1972). 5) Van der Heijden, G., De Voogt, W. J. P. and Kramers, H. C., Physica 59 (1972) 473 (Commun. Kamerlingh Onnes Lab., Leiden No. 392a). 6) Van der Heijden, G., Giezen, J. J. and Kramers, H. C., Physica 61 (1972) 566 (Commun. Kamerlingh Onnes Lab., Leiden No. 394c). 7) Van der Heijden, G., Van der Boog, A. G. M. and Kramers, H. C., Physica, to be published.