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Energy and stability of vortex rings in liquid helium II; critical velocities
Physica
44 (1969) 415-436
ENERGY
AND
IN LIQUID A. G. VAN Philips
0 North-Holland
Publishing
STABILITY
HELIUM
VIJFEIJKEN,
II;
Co., Amsterdam
OF VORTEX CRITICAL
VELOCITIES
A. WALRAVEN
Research Laboratories, N. V. Philips’ Eindhoven, Nederland
RINGS
and F. A. STAAS Gloeilampenfabrieken,
Received 7 January 1969
Synopsis In this paper quantized vortex rings in tubes of circular cross section are studied. The energy and momentum associated with confined vortex rings are calculated. It is shown that a confined ring carries no momentum. This fact makes the application of Landau’s criterion for the critical velocity at which vortex rings are created, doubtful. A new criterion for vC, based on mechanical equilibrium for a ring, is proposed and applied. It leads to an expression for vCwhich contains both the diameter of the tube and the distance from the wall at which a ring is created. Fom a comparison with experiments this distance is estimated. In addition, the time needed for the creation of a ring is calculated.
1, Introduction. It is generally accepted that the destruction of superfluid flow of liquid helium II arises from the creation and subsequent motion of vortex lines or vortex rings. At flow velocities z, less than a certain critical velocity ZJ~a truly persistent flow is possible, i.e. friction and dissipation are absent, while for v 2 v, the flow is dissipative. The idea of the existence of a critical velocity has been confirmed experimentally in various configurations : film flow, flow through slits and capillaries i-4). In this connection the use of ions as a tool for studying critical velocities should also be mentioned5-7). Hitherto all attempts to derive theoretical expressions for vc are based on suggestions by Landaus) and Feynmang) that vc is given by the minimum value of the ratio E/P, where E is the energy of a vortex ring of a given radius and P its momentum. Feynman in his pioneering papera) considered a vortex ring of radius ra and circulation K:
(1) where ti is the mass of a 4He atom. His calculation of the energy of a ring was only qualitative: using the known energy (T) of a straight vortex line 415
A. G. VAN VIJFEIJKEN,
416
A. WALRAVEN
AND F. A. STAAS
of unit length the energy for a ring was set equal to E = 27troT = &&e
10
In -. a
(2)
Here a is the radius of the vortex core and
:
vC=-ff- In-f!xd
2a
’
As the core radius a is a weak function of the temperature and as it appears in eq. (5) under the logarithm, eq. (5) predicts an essentially temperatureindependent critical velocity, which is practically inversely proportional to the tube diameter d. Various attempts to refine Feynman’s calculation have been made. Geilikmanli) put forward an expression for the energy E of a vortex ring which takes into account the effect of the bending. The problem is, in fact, completely analogous to calculating the magnetic energy associated with a circular current loop in vacuum. Geilikman’s result is E =
ips~%‘o
8~0 In-a
-
7 ‘I ‘.
4J
(6)
By using eqs. (3) to (5) and assuming that the rings are created inside the *) By “the radius of the tube” we always mean the hydraulic radius which is defined as: 2 x (cross-sectional area)/(perimeter). For tubes of circular cross section this is equal to the actual radius.
VORTEX
RINGS
IN He II
417
tube with radii equal to the radius of the tube, he arrived at an equation for vc of the form
As long as we have d > u, the results of eqs. (5) and (7) are rather similar. Peshkovis) made another calculation for vc, using a model which is essentially different from those used by Feynman and Geilikman. We return to his calculation in section 4 of this paper. Both Geilikman and Peshkov assumed that the surrounding walls have only a minor effect on the energy of a vortex ring and this effect was, therefore, neglected in their calculations. But, as was first remarked by Fineman and Chasers), this assumption is not justified. To illustrate this, let us consider a ring in a cylindrical tube with its centre lying on the axis of the tube. The kinetic energy-density associated with the ring is highest near the core of the ring. If now the radius of the ring approaches the radius of the tube, the distortion of the velocity-field due to the walls is large in regions close to the core, i.e. in regions of high energy-density. Therefore we should expect the energy of the enclosed ring to be profoundly different from its value if the fluid were unbounded, in all cases where the ring is at a distance from the wall which is small compared to its radius. Fineman and Chase were able to make this idea quantitative and they carried out a numerical calculation of the energy E of an enclosed vortex ring as a function of the ratio ra/rb, i.e. the ratio of the ring radius to the tube radius. This was done for several values of the parameter a/rb. The authors mentioned plotted the dimensionless quantity E/(p&,$) ve?%%s?‘a/Yb and indeed found a clear maximum, located at ra/rb M 0.9. This shows that E is reduced when 70 approaches 7b. Their calculation also showed that E goes to zero when (7b a). This result, together with Landau’s condition of eq. (4), would then imply that vc is zero. The authors therefore conclude that Landau’s condition alone is not sufficient to explain the destruction of superfluid flow by means of vortex rings.
70 +
A calculation very similar to that of Fineman and Chase was made by Raja Gopall”). Although Raja Gopal’s results are very close to those of Fineman and Chase, there is one important difference: Raja Gopal finds a finite nonvanishing value for E in the limit of 70 --f (7b - a). As a consequence, Gopal deduces from eq. (4) a non-vanishing vc. Explicitly, according to Raja Gopal, vc is given by vc = & Again
(2 In 2 + 4).
a temperature-independent
(8) critical
velocity
is predicted,
while the
418
A. G. VAN
VI JFEI JKEN,
A. WALRAVEN
AND
F. A. STAAS
dependence on the tube diameter d is almost the same as that found from the calculations by Feynman and Geilikman mentioned earlier. We shall now first briefly consider some experimental results. Critical velocity experiments can conveniently be classified in two categories: a. Experiments involving only heat flow and no mass flow. b. Experiments involving mass flow which may or may not be accompanied by heat flow. As these two classes of experiments need not necessarily yield the same critical velocities, one should consider them separately. For the time being we shall restrict our attention to experiments of class b. Numerous investigators have observed critical velocities which appeared to be almost totally temperature-independent and which varied with d approximately as 1Id for d-values larger than about 10-s cm. For detailed information concerning this point the reader is referred to refs. 1, 3 and 4. The ideas which are at the basis of the above equations for zlc seem to be qualitatively correct, at least for large enough tube diameters. Quantitatively, there is a discrepancy : the values of ZIPobtained from eqs. (7) and (8) are about 5 times too small. There is, however, a difficulty in the interpretation of observed critical velocities. It may be that what is observed as a critical velocity is not a critical velocity for the onset of vorticity in the superfluid component but rather a critical velocity for the onset of turbulence in the normal component. This point has been stressed particularly by Staasls). Now, the latter phenomenon is determined by the Reynolds number, which for tubes of circular cross section reads
where p is the total mass density, 21, the flow velocity of the normal component and q,, its coefficient of viscosity, As soon as Re exceeds a critical value, Re,, the normal component becomes turbulent. For cylindrical tubes Re, equals about 1200. From the critical Reynolds number we obtain for the critical velocity for turbulence in the normal fluid, Dnc, the expression V nc
=
$
Rec.
(10)
So Vnc also varies as l/d, just as V, does, see eqs. (7) and (8). The question, therefore, arises which critical velocity is observed in a given experiment; the study of the dependence on channel diameter does not answer this question. There is, however, one difference between vnC as given by eq. (10) and vc as given by eq. (7) or (8), i.e. v nc is temperature dependent through qn while vc is predicted to be temperature independent. But unfortunately the temperature dependence of 7 n is extremely weak for temperatures in between 1.4 K and 1.9 K, which is the temperature range in which critical
VORTEX
RINGS
IN He II
419
velocities are mostly measured. So in order to settle the problem raised above by means of a study of the T-dependence of observed critical velocities, one should go to temperatures which are either at about 2 K and greater or are considerably lower than 1.4 K. As regards the order of magnitude of the two critical velocities: from eq. (7) or (8) we find the product v,d to be approximately equal to 4 x 1O-3 cm3/s, while from eq. (10) we obtain, using q’n w 15 micropoise and Re, = 1200 vncd m 1.3 x 10-l cm3/s. Therefore, for a given d-value vnc is about 2 orders of magnitude greater than vC predicted from the above mentioned theories. This fact enables one, in principle, to distinguish between the two critical velocities. It is illuminating to briefly consider counterflow experiments at this point In experiments of this kind of which ordinary heat conduction is the most common example, we have psvs = pnvn. Let us suppose that in a given counterflow experiment turbulence in the normal component occurs before vorticity in the superfluid component has set in. The value for vs at the point where the normal component has become turbulent, is then Pn
Vs = _
Vnc.
PS
Therefore, vs
.d=pnv Ps
nc .a .
Since vncd is of the order of 10-i cm3/s and pn/ps is about 10-3 at T M 1 K, we obtain for v,d a value of about 10-3 cm3/s. In other words, we arrive at the same value as we would calculate from the assumption that vorticity in the superfluid component had been observed. This is another illustration of the difficulty in interpreting an observed critical velocity. The results of counterflow flow experiments are usually given in a plot showing v,d velsZts d. This plot exhibits a plateau for d > 10-3 cm, which plateau is frequently erroneously interpreted as a proof of the presence of Feynman vortex rings. The fact that the plateau lies at about 10-3 cm3/s proves nothing, as outlined above. Moreover, it has been observed that the height of the plateau is temperature-dependent, which shows that we are not dealing here with Feynman vortices. In the search for an experimental detection of vC Van Alphen et al. 16-19) used a channel with one or two superleaks placed in the flow path, together with a vacuum jacket, designed to keep the normal fluid component at rest. It is true that in such a configuration an observed critical velocity can’ be unambiguously identified as a critical velocity for the occurrence of
420
A. G. VAN
VI JFEIJKEN,
A. WALRAVEN
AND
F. A. STAAS
vorticity, i.e. as vc, provided the superleaks used are ideal entropy-filters. In practice, however, it is exceedingly difficult to make ideal superleaks. Let us suppose that we have a heat flux Q through the superleak of 10 erg/s at a temperature T = 0.6 K. Then it follows from the relation
(11) and eq. (9) that the Reynolds number for a tube connected to the superleak with Yb = 10-s cm is already about 4000, that is to say much larger than Re,. And a heat influx of 10 erg/s is in practice a very low value. Staas and Severijns have carried out some preliminary measurements regarding the heat flux through superleaks. These measurements made it clear that Q may be as high as a few milliwatts, i.e. 0 = 104 erg/s. It is our belief, that the measurements of critical velocities in superfluid windtunnels without using a vacuum jacket are open to considerable doubt. In other words, a vacuum jacket should always be used. In addition, if experiments are done at T < 0.7 K, the superleaks should be nearly ideal, because otherwise, since ST is small (see eq. (1 l)), the normal fluid velocity readily exceeds vnc. As mentioned already the dependence of vc on the tube diameter d as predicted by eqs. (7) and (8) is not born out by experiment for values of d less than 1O-s cm. Van Alphen and co-workersla), gathering data concerning measured v,-values from several experiments in which the normalcomponent is said to be sufficiently clamped, found that the quantity v,d is a continuous function of d, where the d-values used varied over a wide range: 10-T < d< < 1(cm). This empirical result suggests that the mechanism governing vc is one and the same for all practical values of d. According to the authors of ref. 19 the experimental points can very well be represented by the relation vcdf = c,
(12)
where C is a constant: C = 1 cm’/’ s-1. Craigsa) has produced an argument which supports the result given by eq. (12). Craig’s reasoning is essentially a refinement of the ideas originally introduced by Peshkovis). The only difference between Craig’s calculation and the one by Peshkov is that Craig uses for the energy of a vortex ring the result obtained by Fineman and Chaseis) (which includes the effect of the boundary), while Peshkov used for this energy the expression given by eq. (2) (which, as said already, does not account for the walls). Although two papers have already been published dealing with the problem of the energy of a vortex ring in an enclosure, i.e. the papers by Fineman-Chaseia) and by Raja GopallJ), there is room for an additional treatment. Firstly, there is the discrepancy mentioned previously between the results of the two papers. Secondly, the present analysis uses a method which, as the .authors see it, reveals much more explicitly the effects of
VORTEX
RINGS
IN He II
421
the boundaries than do the methods used in the papers of refs. 13 and 14. The main difference is that we describe the velocity field associated with a vortex ring by a vector potential, while Fineman-Chase and Raja Gopal used a scalar potential. This may seem to be merely a matter of taste; we hope to show it is not. Thirdly, once the energy of an enclosed vortex is calculated, we derive vc not by using Landau’s criterion, i.e. eq. (4), but instead by requiring mechanical stability for the ring. In order to show that, in the framework in which a vector potential is used, the calculation of the energy of an enclosed ring is a straightforward extension of the calculation for a ring in an unbounded fluid, we start in section 2 with the unenclosed ring. Section 3 then deals with the energy for the enclosed ring. In section 4 a condition for mechanical equilibrium is introduced, from which vc is derived, using the result of section 3. The critical velocity obtained is, of course, a function of the tube radius Yb. Moreover, our expression for v, contains a parameter A, which describes the distance from the wall at which the ring is created. This parameter is determined from a comparison of the calculated and measured v,-values. A possible explanation of the origin of the distance A is then suggested. In section 4 we finally make an estimate for the time needed to create a vortex ring. 2. A vortex ring ilz an unbowded fluid. We start by considering a vortex ring of radius 70 embedded in an unbounded-isothermal fluid. The origin of of our coordinate frame coincides with the centre of the ring and the z axis is along the symmetry-axis of the ring. Cylindrical coordinates (Y, q, z) are used. Since the fluid as a whole is incompressible and the normal component is supposed to be at rest (vn = 0), it follows that the superfluid flow is divergence-free, i.e. div up,= 0. Therefore, us can be derived from a vector potential Ao : us = curl Ao.
(‘3)
For A0 the usual gauge is chosen: div Ao = 0. The velocity field us has only r and z components. This means that for A0 we can take: A0 = e,Ao(r, z), where e, is a unit vector in the v direction. Taking the curl of eq. (13),and using the fact that div A0 = 0, we obtain for the vector potential associated with an unenclosed vortex ring the differential equation -K+
-
9’0)
d(Z).
(14
The source term on the right-hand side of this equation is self-explanatory The components of (usare given by vsr = -->
aA
aZ
A. G. VAN
422
VIJFEIJKEN,
A. WALRAVEN
It is convenient to doubly transform transform is used, i.e. A&,
z) = [A&
z) rJr(sr)
The inverse transformation
AND
F. A. STAAS
Ao(r, z). For the variable
r the Hankel
dr.
(16a)
reads
Ao(r, 2) = OrAo(s, 2) sJ&)
ds.
(16b)
In these expressions Jr is the ordinary Bessel function of the first kind, of order 1. Next the z variable is Fourier-transformed, i.e. Aa(s, +) = jza(s, --oo
z) eipz dz.
The inverse of this is +oO 1 AO(s,z)
=
~
AO(s, p) e-ipz
2x s
(17a)
d$.
(17W
--m
We now transform
the whole eq. (14). This gives
KroJl (syo)
Ao(saP) =
s2 + p2
08)
-
For the derivation of this result we have used the fact that rAa(r, z) and (raAo(r, z)Py) g o t o zero both as Y + 0 and Y --f co. It follows from eq. (15a) that the condition at r -+ 0 is satisfied. This is so because vsr(O, z) = 0, so Ao(r = 0, z) is independent of z. Since Aa(r = 0, z + fm) = 0 we see that Aa is zero everywhere on the z axis. Consequently, rAo(r, z) --f 0 as r + 0. Concerning the other limit, r -+ 00, it is known from the theory of electromagnetism that the vector potential for a current distribution of finite extension behaves at infinity as l/G. Therefore, rA a(~, z) and raAa(r, z)/& go to zero when Y +ca as l/r and l/r2 respectively. Using eq. (18) we now transform back. Taking first the variable from eq. (17b) AO(s, z) =
Regarding
r0 K-Jl(sr0) e-slzl.
p we get
(‘9a)
2s
the variable s, we find from eq. (16.b), Q)
Ao(r, 2) = 9
s
0
Jr(sro)
Jr(sr)
e-slzl ds.
(19b)
VORTEX
The integral can be evaluated,
Ao(r,.z)
+{(I
RINGS
IN He II
423
yielding
- ;)K(k)-E(k)).
(20)
Here K(k) and E(K) are the complete elliptic integrals of the first and second kind, respectively, n/2
742
s -“,“z
K(k) =. and
Jl
sin20
E(K) =
’
s
Jl
-
Ks sin20 de,
0
k2 = 4rro/{(r + ro)2 + z”>. These expressions enable us to check the behaviour of Ao(r, Z) for I --f 00. In this limit k becomes very small, i.e. K -+ 2(re/r)f. Because of this we may expand the factor appearing in curly brackets and retain only the first term. We get for this factor: (x/32) k4. Therefore, Aa(r, z) is proportional to (Q,/Y)~ as r + co, so A@, z) indeed behaves as stated previously. The expression (20) is, of course, fully analogous to the expression for the electromagnetic vector potential of a current loop in vacuum21). Combining eqs. (15a) and (15b) with eq. (20) we find the velocity components to be given by KZ/2Y % -
J(Yo + Y)2 + 22
(K(k)
+ ( z
T :yzi2)
E(k)}
(2la)
and
K/2 vsz -
J(yo
+
r)2 +
22
1 ( K(k)
+
Y; - 12 - 22
(Yo -
r)2 +
22
>
E(k)
1
*
(21b)
Inspection of eq. (21a) shows that vS,. vanishes along the z axis. This result is obvious from symmetry arguments. In addition,
V&Y = 0,z) =
4 zf 2 (Yi + 22y
’
so Vusz(Y= 0, 2) + 0 as z-3 if z --f foe.
One notes also that v&, z) = = v&r, - z), which feature is also clear from symmetry considerations. We next want to briefly consider the total momentum of the fluid that is associated with a vortex ring. This is given by PO =
JP~Q
&X
=
ezps J vsz &z.
Here e, is a unit vector along the z axis. It is easily recognized component vsr does not contribute to PO.
(234
that the
A. G. VAN
424
The integral most
VIJFEIJKEN,
of eq. (23a)
easily
seen in the PO = ezPo, where
Pa = 2xp.9
rdr
s
s
A. WALRAVEN
is not, however,
following
dz ;
&
0
F. A. STAAS
absolutely
Using
eq.
convergent.
(15b)
This is
we can
write :
(rAo(r, z)) =
= 2xp, lim dr dz i Zcl’rns s R-m
way.
AND
(rAe(r, z)).
(23b)
--I0
The factor ATEcomes from the integration
variable q.We also have the relation
CC-3
j Ao(r, z) eQz dz = Aa(r, #), where Ao(r, $) is the Fourier-transform fore,
of Aa(r, z) for the variable
z. There-
r
lim Aa(r, z) dz = lim Aa(r, fi). BO-+ca--Bo P+o
(23c)
On the other hand, Aa(r, p) can be obtained from eq. (18) by carrying the inverse transformation with respect to the variable s. We then get w
s
s
00
~o(r,
K"OJl(sf'O) sJ1
Ao(s, 9) sJl(=) ds =
fi) =
s2 + $2
b’) ds -
0
0 The last integral
out
can be evaluated
to give
AO(r, 9) =
Kf'O~l(@') Kl(@O),
if
0 .s 7 If,O>
(234
AO(r, 9) =
KT0~1@'0)
if
ro <
(234
Kl(@),
1.
In this expression Ii is the first order modified Bessel function of the first kind and Ki is the first order Hankel function. If we combine eq. (23c), (23d) and (23e) we can write for PO occurring in eq. (23b) : TO PO =
27CKp#O
lim P-+0iT
dr $
(r~i(@)
Ki(@o))
+
or, (23f)
VORTEX
RINGS
The two stock terms corresponding
IN He II
to r = ~0 cancel each other. The stock
term corresponding to r M 0 vanishes, since Ii(x) leaves only the term with r = R. Therefore, PO
=
~~KpslOp~~~
{m@yO)
425
Kl(+R)}.
M x/2 for small x. This
Pd
R-m,
Let us first take the limit fi -+ 0. Since Ki(x) lim Rll($ro)
Kl(#R)
P-t0
pro 1
= R -*2
=
l/x at small x we find that
10
j5R =-?
With this we obtain for PO PO =
XK&.
(24)
This is a result, already obtained by Lord Kelvin; see ref. 10. If, however, we invert the order of the limits in eq. (23g), we derive a different result. Using the fact that 63
a
() It
Kl(x)
-
2x
e-z
for
X-CO,
we see that lim Rl1(p5ro) Kl($R)
= 0.
R-
The result is then PO = 0.
(25)
Recalling that taking the limit of $J + 0 corresponds to letting z go to infinity, we can summarize the foregoing computation in the following way: If we first do the z-integration and then the r-integration we arrive at the familiar result given in most books on hydrodynamics. If we invert the order of integration we find that a vortex ring in an unbounded fluid carries no momentum. In fact, hydrodynamicists have been aware of this ambiguity for a long timess). Lord Kelvin himself carefully avoided speaking about the momentum of a vortex ring. Instead, he introduced the concept of impulse by which is meant the space-time integral of the force-density needed to set up the motion of the fluid associated with a vortex ring, when the fluid is originally at rest. The impulse has the dimension of a momentum and is indeed given by the right-hand side of eq. (24). But the fact that the impulse is uniquely defined and has a finite value does not mean that the momentum created by the impulse is retained in the fluid. The momentum created can be transmitted partly or wholly to infinity, depending on the boundary conditions.
426
A. G. VAN VIJFEIJKEN,
A. WALRAVEN
AND F. A. STAAS
The close similarity between impulse and momentum has led people to replace the momentum P in Landau’s criterion, eq. (4), by the impulse and then to manipulate with this quantity as if it were a momentum. The present authors feel this is a rather unsatisfactory procedure because the derivation of Landau’s criterium rests basically on the conservation of momentum. The momenta involved in this conservation law ought to be true physical momenta and not quasimomenta of some sort. It is precisely the ambiguity of the momentum of a vortex ring which led us to search for a different criterion for oC. This criterion will be introduced in section 4. Next we evaluate the energy of an unenclosed vortex ring. For this we have : I& = &,p,j v:dax = Bps j (curl Ao)2 da%.
(26a)
The volume integration extends over all space except the space occupied by the vortex core. In the integration volume no vorticity is supposed to be present, so there we have; curl A0 = 0. Due to this fact we can write: (curl Ao)2 = div (A 0 A curl Ao), which relation enables us to transform the three-dimensional integral of eq. (26a) into a surface integral. So, _??a= $ps S (A0 A curl Ao) *da.
s.
(26b)
In the last integral SC is the surface of the vortex core and do is a vector perpendicular to the core surface and pointing into the core. Substituting Aa from eq. (20) and using the fact that the core radius a is a very small quantity, we obtain for Ea:
(27) Usually one finds in the literature a constant -$ instead of -2. The reason for this is probably attributable to the use of the analogy with the energy stored in a current-loop of inductance L, which most textbooks on electromagnetism give as: L = 4xra{ln 8ro/a - z}. If one looks carefully, however, one finds: L = 4xra{ln 8ro/a - 2 + ,uJ4}, where pr is the relative permeability of the current-carrying wire. The term with ,uLraccounts for the magnetic energy stored in the wire itself. In the case of a vortex line or vortex ring in He II one always assumes a hollow core, which means that no kinetic energy is stored in the core, corresponding to ,u,. = 0. This is why we should have -2 in eq. (27). Using z is equivalent to the assumption of solid body rotation inside the core. In fact, the term (-2) is borne out by eq. (26) if one makes an expansion in powers of (a/ra). 3. A vortex ring in an elzclosure. In this section we consider a vortex ring of radius y. which is enclosed in a cylinder of radius ?‘b. The cylinder is
VORTEX
RINGS IN He II
427
Fig. 1. A vortex ring in a tube. Ring 1 is the true ring while 2 indicates the sense of rotation of the vorticity covering the wall of the tube. The function f(z) describes the distribution of wall vorticity. supposed to be infinitely long. Fig. 1 gives the geometry. We again introduce a vector potential A, which, as before, can be chosen as : A = e,A (I, z). The velocity components v6r and vaZare derived from A as given by eqs. (15a) and (15b). The difference as compared with the previous section is that we now have to satisfy a boundary condition at the surface of the tube, i.e. at r = lb. This condition is simply “w(“b,
2) =
0,
Pa)
for all z. From eq. (15a) it then follows that A must be constant at the surface of the tube. Since we consider the case where no net current is present, A (?‘b, .z + &oo) = 0, so we can also put the boundary condition in the form A(?‘b, 2) = 0.
(28b)
This boundary condition for A can be satisfied by introducing into the differential equation for A an extra source term. We then write this equation in the form
a2A
1
aA
1+rx-$+== = -Kd(V
a2A
?‘I))d(z) + Kd(Y -
rb) f(2).
(29)
The first source term is the same as that for the unenclosed ring. The second source term describes vorticity 012 the wall. This wall vorticity, as we shall call it, has a certain spread in the z direction. In fig. I a possible distribution function f(z) is drawn. Our task now is to determine f(z) from the boundary condition (28b). But before proceeding with eq. (29) we want to make some brief remarks to substantiate the statement in section 1 about the advantages of the vector potential over the scalar potential. Firstly, the differential equations for A (c.q. Ao) holds in all space, whereas the differential equation for the scalar velocity-potential holds only outside the core region. Secondly and more important, the boundary conditions can more easily be accounted for if one works with A. Each boundary simply introduces a new source term in
428
A. G. VAN
VI JFEI JKEN,
A. WALKAVEN
AND
F. A. STAAS
eq. (29). If, for instance, we considered a vortex ring in the space between two concentric tubes, we would get a third source term in eq. [29), of the same shape as the one we already have. So it is easy to generalize eq. (29) to more complicated geometries. We now proceed with the solution of eq. (29). As in section 2 it is convenient to doubly transform A (Y, z), yielding A (s, $). From eq. (29) it follows that A(s, fi) is given by the following expression: A(% P) =
K~OJl(sQ) 9
+
_
KYbJl(Srb) f(p) s2+p2
p2
(30)
.
The first term has already been encountered, c.f. eq. (18), while the second term is due to the wall vorticity. In eq. (30) f(p) is, of course, the Fourier transform of f(z) . In order to obtain f(p) it is suitable to transform A(s, p) back as far as concerns the variable s. One gets A@‘,$) = 'dl(p~)
Kl[prO)
-
Krbll(pr) Kl(plb)
= KYOIl(pYO) Kl(pr) - Krbll(pY) Kl(prb) =K?'OIl(prO)
Kl(pr)
-
f(p),
if
f(p), if
Kf'bIl(prb) Kl(pr)f(p),
if
Olr
(3la)
TO<
(31b)
lb
7 < yb,
(314
Here 11 is the modified Bessel function of the first kind of order 1 and Kr is the first-order Hankel function. At I = ~0 the first two expressions become identical while at r = Yb the last two become the same. This behaviour is as it should be. Since A (Yb, z) = 0 for all z and A (I, p) is the Fourier transform of A (Y, z), it follows that A (Yb, p) is zero for all 6. Using this, it follows from eq. (3 1b) (or, for that matter, eq. (31~)) that f(p) is given by:
(32) It is seen that
f(p) --f 1 if
10 --f Yb. In that limiting
case we have for f(z) :
+m
f(z) =
&- 1f(p)
emipz dp = 6(z).
-co
This is exactly what one would expect. The distribution function f(z) gets broader when re/rb decreases. Another interesting property of f(z) must be mentioned. From eq. (32) we find :
VORTEX
This holds for arbitrary f(# = 0) =17(z) --00 Combining
values of
RINGS IN He II
On the other hand, we have
rO/rb.
dz.
these two expressions,
KY$(Z) -on
429
we can derive the following
relation:
dz = ICY;.
(34)
Now, the right-hand side of this equation is the analogue of the magnetic moment of a current-loop, while the left-hand side is the analogue of a magnetic moment induced on the wall. In this sense eq. (34) states that the inducing vortex ring and the induced vorticity are of equal “strength”. If we were to consider a vortex ring enclosed in between two concentric cylinders, we would find that the total vorticity induced on both walls would have “strength” K. Using eq. (32), we rewrite eqs. (31) : 11(fi70) Kl@7b)
A(7> 9) = K7Oll(j'7)
Il@7) A(79 f') =
K7OI1(#70)
A(7, $) =
0
if
I1 (@lb)
(35a)
if 70<7<7b,
(35b)
Kl(fi7b)
I1 (fi7b)
if
0<7<70,
y,, 5 7.
PC)
From eq. (35~) it follows that A(r, z) is identically zero for r > Yb. Due to this we can now work in free space which is convenient for the evaluation of certain integrals to appear later in the calculation. Let us now consider the momentum P associated with an enclosed ring. We now have instead of eq. (23b), P = 2xps
s
r dr
dz +
s
(r/l (I, z)) =
--03
0
9% =
$
27rpglim
a0
dr
20+00 s 0
dz G
(7A (7, s)).
(364
s -2%
Eq. (23~) is again valid but now we are dealing with A(r, p), which is given by the equations (35a) to (35~). We, therefore, get
P =
hKps70
p-0 k-II
+9.1
711(@7) Il(ib70) Kl(fi7b) 7~1($7)K1(fi70)
I1
0
r~l(@o) K&7)
-
7~1($7O)ll(?7)&(f'7b) Il(p7b)
@‘7d
II
*
1+ Wb)
A. G. VAN
430
VIJFEIJKEN,
A. WALRAVEN
AND
F. A. STAAS
The two contributions coming from I = 10 cancel each other, whereas the contribution coming from I = 0 vanishes, again due to the fact that II(X) M x/2 for small x. Therefore, we are left with:
P = 2XKps?$ lim
rbll(jh’0)
Kl(plb)
-
Ibll(~lO)~l(PYb)Kl(Plb)
-
P-4
which is identically
, (37a) ll(f'rb)
zero. Thus
P = 0.
(W
So in the case of an enclosed vortex ring there exists no ambiguity value of its momentum, contrary vortex ring. But the fact that P Landau criterion, c.J. eq. (4), does Next the energy of an enclosed
in the to what was found for an unenclosed = 0 for any value of 10 means that the not work at all for an enclosed ring. ring is calculated. We again have:
E=~~psSu~dg~=~psS(A~curlA)*da. s.
(38a)
The symbol SC stands, as before, for the surface of the core. The surface of the cylinder does not contribute to E, as at this surface A = 0. To evaluate E it is convenient to write A as: A@, z) = A@, z) + Al@, z). Here A&, z) is the vector potential of the unenclosed vortex ring, given in section 2, and Al(r, z) is the vector potential due to the wall vorticity. The Fourier transforms of Aa(r, z) and Ar(r, z) are given by the first and second terms of eqs. (35a) and (35b). For curl A we write: curl A = us, and vs is decomposed similarly into: ZJ~= ~,a + ~~1. With this decomposition we get (AAcurlA).da=(Aa~Zf~a)*do+(Aa~2r~r).du+ + (AI A v,o).du
+ (Al A Q-da.
(38b)
Of the four terms appearing in eq. (38b) only the first and third are important, which is easily seen as follows. The integration over da brings in a factor a, being the core radius. At the core surface A1 and TJ~~ are finite, therefore A1 A us1 - da is proportional to a and we can thus neglect this term. Similarly, at the core surface, Aa is proportional to In a, so Ao A %l*du vanishes as a In a. The first term gives simply the energy Eo of an unenclosed vortex ring, which diverges as In a. The term Al A ~,o.du is the true interaction term and is independent of a since vsa brings in a factor u-l and du a factor a. Finally, we make use of the fact that at SC the vectors Ao, t&o and da form a right-hand side rectangular coordinate frame. All these considerations then lead us to write:
[A
A
curl A).du
Substituting
w A ovso do -
A lvso da.
(38~)
this expression into eq. (38a) we obtain
E = E. - $jAlvsodu. s.
(384
VORTEX
For the interaction J Alus s.
RINGS IN He II
431
integral we have:
da = AI(ro, z = 0) 27~~~Y$ZI,~dl = 2x/croAr(ro, z = 0).
But
With this the expression E becomes O”
E = Eo -
2 II@‘(O)
Kl(@d
psK%‘o
s
dP
II
(39) '
0
The integral can only be numerically evaluated. The results of our computer calculation are given in fig. 2. Here we have plotted the energy E, normalized to psK2f’b, versus the quantity 6 = (Yb - ra)/rb. This quantity gives the distance of the ring from the cylinder-wall, normalized to ?‘b. of creation presumably have radii ~0 As vortex rings at the “moment” very close to Yb, the energetics of the creation process are determined by the behaviour of E’ z E/(p,K2rb) for small 6. This is why it is advantageous to
7.0 I 6.01
.“““,
10-7
\
/
.,,I
10-G
,,,,,
,,,,,(,,,
lo-
5
,,,,,,,,,
10-b
Fig. 2. Plot of the reduced energy E/(pst&)
,I
n-3
),,,,,,
,,,,,,,,,
10-2
*
,,,
lo-'
as a function of 6 = (Y,, - q,)/q, several values of the parameter U/Q,.
1
for
432
A. G. VAN
VIJFEIJKEN,
A. WALRAVEN
AND
F. A. STAAS
plot E’ as a function of S and to use a logarithmic scale for 6, rather than to plot E’ verszls ~0 and take for ~0 a linear scale. The different curves in fig. 2 have different values of the parameter U/Q,. The important features of fig. 2 are the following. Firstly for 10
if
6 --f ajrb.
(40)
This is in agreement with the results of Raja Gopal14) and it disagrees with the calculation by Fineman and Chase 1s). Moreover, it is seen from fig. 2 that, for small 6, E’ is strictly proportional to In 6 with a slope independent of the parameter U/lb. The above result for E’ as a function of 6 will now be used in the following section to construct a condition, for the critical velocity. 4. Stability condition for vortex rings. As argued in the preceding sections, the Landau criterion for vc, i.e. eq. (4), is not applicable to a vortex ring. It is not applicable either to a ring in an unbounded fluid or to an enclosed ring. In this section an alternative condition for vc is proposed. It is obtained essentially by requiring mechanical stability for the ring once the ring is created. We thus look at all forces that act on each element of the ring and require the net force to be zero. Now, there are three forces, all of them of the Magnus type, acting on a given element of the ring. Firstly a given element is subjected to a force due to all other elements of the ring. The result of this force, which we can view also as an elastic force, is to push the given element away from the cylindrical wall. Secondly the wall vorticity also exerts a force on our given element, and this force tries to pull the element towards the wall. The sum of these two forces is, per unit length of the ring, simply given by F1=---
1
aE
__.
27F~oI aro I
(41)
Physically it is clear that the radial force F1 tries to let the ring shrink if r-0is small compared to rb, whereas F1 tends to let the ring grow in the case where YOM lb. This is what is also borne out by fig. 2. Finally, a drift velocity is present, which we shall call ve in order to distinguish it from the previously considered superflow-velocity vosdue to the vortex ring and its
VORTEX
RINGS
IN He II
433
image. The velocity field ve exerts also a force on the ring, which, again per unit length of the ring, is given by Fs =
@KVO.
(42)
This force tends to push the element away from the wall. It should be noted that by taking the expression of eq. (42) for the force on the ring-elements due to the transport current, we have assumed that the ring, at the creation, does not move along the axis of the tube. It might well be that the ring attains an axial velocity at a later time; we only make an assumption regarding its axial velocity at the creation. Each element of the ring is now in mechanical equilibrium if the two forces F1 and Fs balance each other, i.e. if
(43) The quantity on the left-hand side is a function not only of the tube radius 7b but also of the ring radius ~0. This means that, if we identify vo from eq. (43) with the critical velocity vc, we obtain vc = %(rb, YO) = %(rb, A),
(44)
where we have introduced: A = lb - ~0. So A is the distance from the wall at which the ring is created. If we evaluate the left-hand side of eq. (43) from eqs (39) and (27) we obtain for vC after doing some elementary algebra : K ”
2x(d -
=
.
I 1
_
In
24) ’
4(d- 24 a
+
.
4
(45)
1
Here we have introduced the diameter d of the tube: d = 2Yb. The dependence of vC on the two parameters d and A is clearly exhibited. A direct comparison of eq. (45) with experimental results is not possible as we do not know beforehand what A is. In order to get some information concerning this length we did the following. We took the experimental data collected by Van Alphen et a1.19) which give vC as a function of d. For each pair of values (vC, d), we then calculated A from eq. (45) and in this way we obtained A as a function of d. This was done for d-values ranging from d = 10-s cm to d = 1 cm. The result is given in fig. 3; the drawn curve containing the crosses gives A. In the figure the Van Alphen line for vCd is also given for comparison. As mentioned already in section 1, this line yields the
A. G. VAN VI JFEI JKEN, A. WALRAVEN
434
AND F. A. STAAS
Fig. 3. The length A and the creation time 7 as a function of a tube diameter d.
empirical expression vcd* = constant. As is seen from fig. 3, d M IO-6 cm for d = 10-S cm, while LI M lo-4 cm for d = 1 cm. Thus, d increases by 2 decades if d increases by 5 decades. The actual values for A, together with the fact that A increases slowly with d suggest that A might be the average surface roughness of the walls. It is well known that for the onset of turbulence in a normal fluid the surface conditions are important. The suggestion that this will also be the case for the onset of vorticity in a superfluid is therefore quite plausible. In order to test our suggestion it seems worthwhile to carry out v,-measurements on capillaries for which the surface roughnesses are reasonably well known. Next we consider the time needed to create a vortex ring. For this we start from an argument presented some years ago in a paper by Peshkov 12). The essence of Peshkov’s paper is that the energy E stored in a vortex ring is acquired at the expense of the kinetic energy stored in the transported current. Associated with the latter is an energy &~~z$nr~per unit length of the capillary. Peshkov then assumes that a segment of the capillary of length (Yb + ~07) contributes to the vortex ring to be formed. Here T is the creation time for a vortex ring. The length mentioned contains an amount of energy equal to -&@‘U~Wb(~b+ ~0,). If 01is the fraction of energy extracted to create the ring, we have, at ~10= Q,
Peshkov YO =
Yb,
took E as given by the simple Feynman-expression, eq. (2) with and established in this way the dependence of vC on lb with OL and r
VORTEX
as adaptable
parameters.
RINGS IN He II
By choosing
able to obtain a quantitative
435
01M 0.1 and r w 1O-4 s he was also
fit for vC versus lb.
We have used a modified version of eq. (46.a) to calculate T as a function of d, z.e. we work with the relation +&J&
= E.
(46b)
By taking the values for vC and d from Van Alphen’s straight line, we can read from fig. 3 the corresponding A-values and then from fig. 2 the corresponding E-values. The result of this part of the calculation is drawn in fig. 3, z.e. the curve with the open squares. It is seen that T decreases with increasing d, with an overall variation of about a factor of 10 in the range of d-values used. It should be possible to measure 7, for instance by measuring the attenuation of ultrasonic sound. One final comment can be made. In view of the fact that a finite time is involved in the creation of a ring, and therefore a finite segment of the capillary, the length A is really to be understood as the average roughness of the walls, i.e. the average of a segment of length vc7. So in the picture drawn in this paper a single sharp spike on the surface is never responsible for the creation of a ring.
REFERENCES 1) Wilks, J., The properties of liquid and solid helium (Clarendon Press, Oxford, 1967). (The most recent survey containing numerous references to critical velocitymeasurements.) See in particular p. 383-399. 2) Peshkov, V. P., Prog. Low Temp. Phys., Vol. IV, ed. C. J. Gorter (North-Holland Publ. Co., Amsterdam 1964) p. 1. 3) Hammel, E. F. and Keller, W. E., Superfluid Helium, Proc. Symp. St. Andrews, ed. J. F. Allen (Academic Press, London, New York, 1966) p. 121. 4) Hammel, E. F. and Keller, W. E., Proc. Xth Int. Conf. Low Temp. Phys. Vol. I, ed. in chief M. P. Malkov (Moscow, 1967) p. 30. 5) Careri, G., Prog. Low Temp. Phys., Vol. III, ed. C. J. Gorter (North-Holland Publ. Co., Amsterdam, 1961). p. 58 (a review of the earlier experiments involving ions in He II). 6) Careri, G., Dupre, F. and Mazzoldi, P., Quantum Fluids, Proc. Sussex Univ. Symp., ed. D. F. Brewer (North-Holland Publ. Co., Amsterdam, 1966), p. 305 (a survey of more recent work). 7) Reif, F., Proc. Xth Int. Conf. Low Temp. Phys., Vol. I, ed. in chief M. P. Malkov (Moscow, 1967) p. 86. 8) Landau, L. D., J. Phys. (U.S.S.R.) 5 (1941) 71. 9) Feynman, R. P., Progr. Low Temp. Phys., Vol. I, ed. C. J. Gorter (North-Holland Publ. Co., Amsterdam, 1955) p. 19. 10) Lamb, H., Hydrodynamics (Dover Publication, New York, 1945) chapter 7. 11) Geilikman, B. T., Zh. eksper. theor. Fiz. 37 (1959) 891 (English translation: Soviet Physics- JETP 10 (1960) 635). 12) Peshkov, V. P., Programme VIIth Intern. Conf. Low Temp. Phys. (Univ. of
436
13) 14) 15) 16)
17) 18) 19) 20) 21) 22)
VORTEX
RINGS IN He II
Toronto Press, Toronto, 1960) p. 83; Zh. eksper. theor. Fiz. (U.S.S.R.) 40 (1961) 379 (English translation: Soviet Physics - JETP 13 (1961) 259. Fineman, J. C. and Chase, C. E., Preprints VIIIth Int. Conf. Low Temp. Phys. (Butterworth & Co., Ltd., London, 1962) p. 222; Phys. Rev. 129 (1963) 1. Raja Gopal, E. S., Ann. Physics 25 (1963) 196. Staas, F. A., Taconis, K. W. and Van Alphen, W. M., Physica 27 (196 1) 893. Van Alphen, W. M., Vermeer, W., Taconis, K. W. andde BruynOuboter, R., Proc. IXth Int. Conf. Low Temp. Phys. ed. J. G. Daunt et al. (Plenum Press, New York, 1965) p. 323. Van Alphen, W. M., Olijhoek, J. F., De Bruyn Ouboter, R. and Taconis, K. W., Physica 32 (1966) 1901; 35 (1967) 483. Vermeer, W., Van Alphen, W. M., Olijhoek, J. F., Taconis, K. W. and De Bruijn Ouboter, R., Phys. Letters 18 (1965) 65. Van Alphen, W. M., van Haasteren, G. J., De BruynOuboter, R. and Taconis, K. W., Phys. Letters 20 (1966) 474. Graig, P. P., Phys. Letters 21 (1966) 385. See for instance: Landau-Lifshitz, Course of Theor. Physics, Vol. 8, p. 125. See C. C. Lin in Proc. Int. School of Physics “Enrico Fermi”, Course XXI, 1961, ed. G. Careri (Acad. Press, New York, 1963) p. 98.
44 (1969) 415-436
ENERGY
AND
IN LIQUID A. G. VAN Philips
0 North-Holland
Publishing
STABILITY
HELIUM
VIJFEIJKEN,
II;
Co., Amsterdam
OF VORTEX CRITICAL
VELOCITIES
A. WALRAVEN
Research Laboratories, N. V. Philips’ Eindhoven, Nederland
RINGS
and F. A. STAAS Gloeilampenfabrieken,
Received 7 January 1969
Synopsis In this paper quantized vortex rings in tubes of circular cross section are studied. The energy and momentum associated with confined vortex rings are calculated. It is shown that a confined ring carries no momentum. This fact makes the application of Landau’s criterion for the critical velocity at which vortex rings are created, doubtful. A new criterion for vC, based on mechanical equilibrium for a ring, is proposed and applied. It leads to an expression for vCwhich contains both the diameter of the tube and the distance from the wall at which a ring is created. Fom a comparison with experiments this distance is estimated. In addition, the time needed for the creation of a ring is calculated.
1, Introduction. It is generally accepted that the destruction of superfluid flow of liquid helium II arises from the creation and subsequent motion of vortex lines or vortex rings. At flow velocities z, less than a certain critical velocity ZJ~a truly persistent flow is possible, i.e. friction and dissipation are absent, while for v 2 v, the flow is dissipative. The idea of the existence of a critical velocity has been confirmed experimentally in various configurations : film flow, flow through slits and capillaries i-4). In this connection the use of ions as a tool for studying critical velocities should also be mentioned5-7). Hitherto all attempts to derive theoretical expressions for vc are based on suggestions by Landaus) and Feynmang) that vc is given by the minimum value of the ratio E/P, where E is the energy of a vortex ring of a given radius and P its momentum. Feynman in his pioneering papera) considered a vortex ring of radius ra and circulation K:
(1) where ti is the mass of a 4He atom. His calculation of the energy of a ring was only qualitative: using the known energy (T) of a straight vortex line 415
A. G. VAN VIJFEIJKEN,
416
A. WALRAVEN
AND F. A. STAAS
of unit length the energy for a ring was set equal to E = 27troT = &&e
10
In -. a
(2)
Here a is the radius of the vortex core and
:
vC=-ff- In-f!xd
2a
’
As the core radius a is a weak function of the temperature and as it appears in eq. (5) under the logarithm, eq. (5) predicts an essentially temperatureindependent critical velocity, which is practically inversely proportional to the tube diameter d. Various attempts to refine Feynman’s calculation have been made. Geilikmanli) put forward an expression for the energy E of a vortex ring which takes into account the effect of the bending. The problem is, in fact, completely analogous to calculating the magnetic energy associated with a circular current loop in vacuum. Geilikman’s result is E =
ips~%‘o
8~0 In-a
-
7 ‘I ‘.
4J
(6)
By using eqs. (3) to (5) and assuming that the rings are created inside the *) By “the radius of the tube” we always mean the hydraulic radius which is defined as: 2 x (cross-sectional area)/(perimeter). For tubes of circular cross section this is equal to the actual radius.
VORTEX
RINGS
IN He II
417
tube with radii equal to the radius of the tube, he arrived at an equation for vc of the form
As long as we have d > u, the results of eqs. (5) and (7) are rather similar. Peshkovis) made another calculation for vc, using a model which is essentially different from those used by Feynman and Geilikman. We return to his calculation in section 4 of this paper. Both Geilikman and Peshkov assumed that the surrounding walls have only a minor effect on the energy of a vortex ring and this effect was, therefore, neglected in their calculations. But, as was first remarked by Fineman and Chasers), this assumption is not justified. To illustrate this, let us consider a ring in a cylindrical tube with its centre lying on the axis of the tube. The kinetic energy-density associated with the ring is highest near the core of the ring. If now the radius of the ring approaches the radius of the tube, the distortion of the velocity-field due to the walls is large in regions close to the core, i.e. in regions of high energy-density. Therefore we should expect the energy of the enclosed ring to be profoundly different from its value if the fluid were unbounded, in all cases where the ring is at a distance from the wall which is small compared to its radius. Fineman and Chase were able to make this idea quantitative and they carried out a numerical calculation of the energy E of an enclosed vortex ring as a function of the ratio ra/rb, i.e. the ratio of the ring radius to the tube radius. This was done for several values of the parameter a/rb. The authors mentioned plotted the dimensionless quantity E/(p&,$) ve?%%s?‘a/Yb and indeed found a clear maximum, located at ra/rb M 0.9. This shows that E is reduced when 70 approaches 7b. Their calculation also showed that E goes to zero when (7b a). This result, together with Landau’s condition of eq. (4), would then imply that vc is zero. The authors therefore conclude that Landau’s condition alone is not sufficient to explain the destruction of superfluid flow by means of vortex rings.
70 +
A calculation very similar to that of Fineman and Chase was made by Raja Gopall”). Although Raja Gopal’s results are very close to those of Fineman and Chase, there is one important difference: Raja Gopal finds a finite nonvanishing value for E in the limit of 70 --f (7b - a). As a consequence, Gopal deduces from eq. (4) a non-vanishing vc. Explicitly, according to Raja Gopal, vc is given by vc = & Again
(2 In 2 + 4).
a temperature-independent
(8) critical
velocity
is predicted,
while the
418
A. G. VAN
VI JFEI JKEN,
A. WALRAVEN
AND
F. A. STAAS
dependence on the tube diameter d is almost the same as that found from the calculations by Feynman and Geilikman mentioned earlier. We shall now first briefly consider some experimental results. Critical velocity experiments can conveniently be classified in two categories: a. Experiments involving only heat flow and no mass flow. b. Experiments involving mass flow which may or may not be accompanied by heat flow. As these two classes of experiments need not necessarily yield the same critical velocities, one should consider them separately. For the time being we shall restrict our attention to experiments of class b. Numerous investigators have observed critical velocities which appeared to be almost totally temperature-independent and which varied with d approximately as 1Id for d-values larger than about 10-s cm. For detailed information concerning this point the reader is referred to refs. 1, 3 and 4. The ideas which are at the basis of the above equations for zlc seem to be qualitatively correct, at least for large enough tube diameters. Quantitatively, there is a discrepancy : the values of ZIPobtained from eqs. (7) and (8) are about 5 times too small. There is, however, a difficulty in the interpretation of observed critical velocities. It may be that what is observed as a critical velocity is not a critical velocity for the onset of vorticity in the superfluid component but rather a critical velocity for the onset of turbulence in the normal component. This point has been stressed particularly by Staasls). Now, the latter phenomenon is determined by the Reynolds number, which for tubes of circular cross section reads
where p is the total mass density, 21, the flow velocity of the normal component and q,, its coefficient of viscosity, As soon as Re exceeds a critical value, Re,, the normal component becomes turbulent. For cylindrical tubes Re, equals about 1200. From the critical Reynolds number we obtain for the critical velocity for turbulence in the normal fluid, Dnc, the expression V nc
=
$
Rec.
(10)
So Vnc also varies as l/d, just as V, does, see eqs. (7) and (8). The question, therefore, arises which critical velocity is observed in a given experiment; the study of the dependence on channel diameter does not answer this question. There is, however, one difference between vnC as given by eq. (10) and vc as given by eq. (7) or (8), i.e. v nc is temperature dependent through qn while vc is predicted to be temperature independent. But unfortunately the temperature dependence of 7 n is extremely weak for temperatures in between 1.4 K and 1.9 K, which is the temperature range in which critical
VORTEX
RINGS
IN He II
419
velocities are mostly measured. So in order to settle the problem raised above by means of a study of the T-dependence of observed critical velocities, one should go to temperatures which are either at about 2 K and greater or are considerably lower than 1.4 K. As regards the order of magnitude of the two critical velocities: from eq. (7) or (8) we find the product v,d to be approximately equal to 4 x 1O-3 cm3/s, while from eq. (10) we obtain, using q’n w 15 micropoise and Re, = 1200 vncd m 1.3 x 10-l cm3/s. Therefore, for a given d-value vnc is about 2 orders of magnitude greater than vC predicted from the above mentioned theories. This fact enables one, in principle, to distinguish between the two critical velocities. It is illuminating to briefly consider counterflow experiments at this point In experiments of this kind of which ordinary heat conduction is the most common example, we have psvs = pnvn. Let us suppose that in a given counterflow experiment turbulence in the normal component occurs before vorticity in the superfluid component has set in. The value for vs at the point where the normal component has become turbulent, is then Pn
Vs = _
Vnc.
PS
Therefore, vs
.d=pnv Ps
nc .a .
Since vncd is of the order of 10-i cm3/s and pn/ps is about 10-3 at T M 1 K, we obtain for v,d a value of about 10-3 cm3/s. In other words, we arrive at the same value as we would calculate from the assumption that vorticity in the superfluid component had been observed. This is another illustration of the difficulty in interpreting an observed critical velocity. The results of counterflow flow experiments are usually given in a plot showing v,d velsZts d. This plot exhibits a plateau for d > 10-3 cm, which plateau is frequently erroneously interpreted as a proof of the presence of Feynman vortex rings. The fact that the plateau lies at about 10-3 cm3/s proves nothing, as outlined above. Moreover, it has been observed that the height of the plateau is temperature-dependent, which shows that we are not dealing here with Feynman vortices. In the search for an experimental detection of vC Van Alphen et al. 16-19) used a channel with one or two superleaks placed in the flow path, together with a vacuum jacket, designed to keep the normal fluid component at rest. It is true that in such a configuration an observed critical velocity can’ be unambiguously identified as a critical velocity for the occurrence of
420
A. G. VAN
VI JFEIJKEN,
A. WALRAVEN
AND
F. A. STAAS
vorticity, i.e. as vc, provided the superleaks used are ideal entropy-filters. In practice, however, it is exceedingly difficult to make ideal superleaks. Let us suppose that we have a heat flux Q through the superleak of 10 erg/s at a temperature T = 0.6 K. Then it follows from the relation
(11) and eq. (9) that the Reynolds number for a tube connected to the superleak with Yb = 10-s cm is already about 4000, that is to say much larger than Re,. And a heat influx of 10 erg/s is in practice a very low value. Staas and Severijns have carried out some preliminary measurements regarding the heat flux through superleaks. These measurements made it clear that Q may be as high as a few milliwatts, i.e. 0 = 104 erg/s. It is our belief, that the measurements of critical velocities in superfluid windtunnels without using a vacuum jacket are open to considerable doubt. In other words, a vacuum jacket should always be used. In addition, if experiments are done at T < 0.7 K, the superleaks should be nearly ideal, because otherwise, since ST is small (see eq. (1 l)), the normal fluid velocity readily exceeds vnc. As mentioned already the dependence of vc on the tube diameter d as predicted by eqs. (7) and (8) is not born out by experiment for values of d less than 1O-s cm. Van Alphen and co-workersla), gathering data concerning measured v,-values from several experiments in which the normalcomponent is said to be sufficiently clamped, found that the quantity v,d is a continuous function of d, where the d-values used varied over a wide range: 10-T < d< < 1(cm). This empirical result suggests that the mechanism governing vc is one and the same for all practical values of d. According to the authors of ref. 19 the experimental points can very well be represented by the relation vcdf = c,
(12)
where C is a constant: C = 1 cm’/’ s-1. Craigsa) has produced an argument which supports the result given by eq. (12). Craig’s reasoning is essentially a refinement of the ideas originally introduced by Peshkovis). The only difference between Craig’s calculation and the one by Peshkov is that Craig uses for the energy of a vortex ring the result obtained by Fineman and Chaseis) (which includes the effect of the boundary), while Peshkov used for this energy the expression given by eq. (2) (which, as said already, does not account for the walls). Although two papers have already been published dealing with the problem of the energy of a vortex ring in an enclosure, i.e. the papers by Fineman-Chaseia) and by Raja GopallJ), there is room for an additional treatment. Firstly, there is the discrepancy mentioned previously between the results of the two papers. Secondly, the present analysis uses a method which, as the .authors see it, reveals much more explicitly the effects of
VORTEX
RINGS
IN He II
421
the boundaries than do the methods used in the papers of refs. 13 and 14. The main difference is that we describe the velocity field associated with a vortex ring by a vector potential, while Fineman-Chase and Raja Gopal used a scalar potential. This may seem to be merely a matter of taste; we hope to show it is not. Thirdly, once the energy of an enclosed vortex is calculated, we derive vc not by using Landau’s criterion, i.e. eq. (4), but instead by requiring mechanical stability for the ring. In order to show that, in the framework in which a vector potential is used, the calculation of the energy of an enclosed ring is a straightforward extension of the calculation for a ring in an unbounded fluid, we start in section 2 with the unenclosed ring. Section 3 then deals with the energy for the enclosed ring. In section 4 a condition for mechanical equilibrium is introduced, from which vc is derived, using the result of section 3. The critical velocity obtained is, of course, a function of the tube radius Yb. Moreover, our expression for v, contains a parameter A, which describes the distance from the wall at which the ring is created. This parameter is determined from a comparison of the calculated and measured v,-values. A possible explanation of the origin of the distance A is then suggested. In section 4 we finally make an estimate for the time needed to create a vortex ring. 2. A vortex ring ilz an unbowded fluid. We start by considering a vortex ring of radius 70 embedded in an unbounded-isothermal fluid. The origin of of our coordinate frame coincides with the centre of the ring and the z axis is along the symmetry-axis of the ring. Cylindrical coordinates (Y, q, z) are used. Since the fluid as a whole is incompressible and the normal component is supposed to be at rest (vn = 0), it follows that the superfluid flow is divergence-free, i.e. div up,= 0. Therefore, us can be derived from a vector potential Ao : us = curl Ao.
(‘3)
For A0 the usual gauge is chosen: div Ao = 0. The velocity field us has only r and z components. This means that for A0 we can take: A0 = e,Ao(r, z), where e, is a unit vector in the v direction. Taking the curl of eq. (13),and using the fact that div A0 = 0, we obtain for the vector potential associated with an unenclosed vortex ring the differential equation -K+
-
9’0)
d(Z).
(14
The source term on the right-hand side of this equation is self-explanatory The components of (usare given by vsr = -->
aA
aZ
A. G. VAN
422
VIJFEIJKEN,
A. WALRAVEN
It is convenient to doubly transform transform is used, i.e. A&,
z) = [A&
z) rJr(sr)
The inverse transformation
AND
F. A. STAAS
Ao(r, z). For the variable
r the Hankel
dr.
(16a)
reads
Ao(r, 2) = OrAo(s, 2) sJ&)
ds.
(16b)
In these expressions Jr is the ordinary Bessel function of the first kind, of order 1. Next the z variable is Fourier-transformed, i.e. Aa(s, +) = jza(s, --oo
z) eipz dz.
The inverse of this is +oO 1 AO(s,z)
=
~
AO(s, p) e-ipz
2x s
(17a)
d$.
(17W
--m
We now transform
the whole eq. (14). This gives
KroJl (syo)
Ao(saP) =
s2 + p2
08)
-
For the derivation of this result we have used the fact that rAa(r, z) and (raAo(r, z)Py) g o t o zero both as Y + 0 and Y --f co. It follows from eq. (15a) that the condition at r -+ 0 is satisfied. This is so because vsr(O, z) = 0, so Ao(r = 0, z) is independent of z. Since Aa(r = 0, z + fm) = 0 we see that Aa is zero everywhere on the z axis. Consequently, rAo(r, z) --f 0 as r + 0. Concerning the other limit, r -+ 00, it is known from the theory of electromagnetism that the vector potential for a current distribution of finite extension behaves at infinity as l/G. Therefore, rA a(~, z) and raAa(r, z)/& go to zero when Y +ca as l/r and l/r2 respectively. Using eq. (18) we now transform back. Taking first the variable from eq. (17b) AO(s, z) =
Regarding
r0 K-Jl(sr0) e-slzl.
p we get
(‘9a)
2s
the variable s, we find from eq. (16.b), Q)
Ao(r, 2) = 9
s
0
Jr(sro)
Jr(sr)
e-slzl ds.
(19b)
VORTEX
The integral can be evaluated,
Ao(r,.z)
+{(I
RINGS
IN He II
423
yielding
- ;)K(k)-E(k)).
(20)
Here K(k) and E(K) are the complete elliptic integrals of the first and second kind, respectively, n/2
742
s -“,“z
K(k) =. and
Jl
sin20
E(K) =
’
s
Jl
-
Ks sin20 de,
0
k2 = 4rro/{(r + ro)2 + z”>. These expressions enable us to check the behaviour of Ao(r, Z) for I --f 00. In this limit k becomes very small, i.e. K -+ 2(re/r)f. Because of this we may expand the factor appearing in curly brackets and retain only the first term. We get for this factor: (x/32) k4. Therefore, Aa(r, z) is proportional to (Q,/Y)~ as r + co, so A@, z) indeed behaves as stated previously. The expression (20) is, of course, fully analogous to the expression for the electromagnetic vector potential of a current loop in vacuum21). Combining eqs. (15a) and (15b) with eq. (20) we find the velocity components to be given by KZ/2Y % -
J(Yo + Y)2 + 22
(K(k)
+ ( z
T :yzi2)
E(k)}
(2la)
and
K/2 vsz -
J(yo
+
r)2 +
22
1 ( K(k)
+
Y; - 12 - 22
(Yo -
r)2 +
22
>
E(k)
1
*
(21b)
Inspection of eq. (21a) shows that vS,. vanishes along the z axis. This result is obvious from symmetry arguments. In addition,
V&Y = 0,z) =
4 zf 2 (Yi + 22y
’
so Vusz(Y= 0, 2) + 0 as z-3 if z --f foe.
One notes also that v&, z) = = v&r, - z), which feature is also clear from symmetry considerations. We next want to briefly consider the total momentum of the fluid that is associated with a vortex ring. This is given by PO =
JP~Q
&X
=
ezps J vsz &z.
Here e, is a unit vector along the z axis. It is easily recognized component vsr does not contribute to PO.
(234
that the
A. G. VAN
424
The integral most
VIJFEIJKEN,
of eq. (23a)
easily
seen in the PO = ezPo, where
Pa = 2xp.9
rdr
s
s
A. WALRAVEN
is not, however,
following
dz ;
&
0
F. A. STAAS
absolutely
Using
eq.
convergent.
(15b)
This is
we can
write :
(rAo(r, z)) =
= 2xp, lim dr dz i Zcl’rns s R-m
way.
AND
(rAe(r, z)).
(23b)
--I0
The factor ATEcomes from the integration
variable q.We also have the relation
CC-3
j Ao(r, z) eQz dz = Aa(r, #), where Ao(r, $) is the Fourier-transform fore,
of Aa(r, z) for the variable
z. There-
r
lim Aa(r, z) dz = lim Aa(r, fi). BO-+ca--Bo P+o
(23c)
On the other hand, Aa(r, p) can be obtained from eq. (18) by carrying the inverse transformation with respect to the variable s. We then get w
s
s
00
~o(r,
K"OJl(sf'O) sJ1
Ao(s, 9) sJl(=) ds =
fi) =
s2 + $2
b’) ds -
0
0 The last integral
out
can be evaluated
to give
AO(r, 9) =
Kf'O~l(@') Kl(@O),
if
0 .s 7 If,O>
(234
AO(r, 9) =
KT0~1@'0)
if
ro <
(234
Kl(@),
1.
In this expression Ii is the first order modified Bessel function of the first kind and Ki is the first order Hankel function. If we combine eq. (23c), (23d) and (23e) we can write for PO occurring in eq. (23b) : TO PO =
27CKp#O
lim P-+0iT
dr $
(r~i(@)
Ki(@o))
+
or, (23f)
VORTEX
RINGS
The two stock terms corresponding
IN He II
to r = ~0 cancel each other. The stock
term corresponding to r M 0 vanishes, since Ii(x) leaves only the term with r = R. Therefore, PO
=
~~KpslOp~~~
{m@yO)
425
Kl(+R)}.
M x/2 for small x. This
Pd
R-m,
Let us first take the limit fi -+ 0. Since Ki(x) lim Rll($ro)
Kl(#R)
P-t0
pro 1
= R -*2
=
l/x at small x we find that
10
j5R =-?
With this we obtain for PO PO =
XK&.
(24)
This is a result, already obtained by Lord Kelvin; see ref. 10. If, however, we invert the order of the limits in eq. (23g), we derive a different result. Using the fact that 63
a
() It
Kl(x)
-
2x
e-z
for
X-CO,
we see that lim Rl1(p5ro) Kl($R)
= 0.
R-
The result is then PO = 0.
(25)
Recalling that taking the limit of $J + 0 corresponds to letting z go to infinity, we can summarize the foregoing computation in the following way: If we first do the z-integration and then the r-integration we arrive at the familiar result given in most books on hydrodynamics. If we invert the order of integration we find that a vortex ring in an unbounded fluid carries no momentum. In fact, hydrodynamicists have been aware of this ambiguity for a long timess). Lord Kelvin himself carefully avoided speaking about the momentum of a vortex ring. Instead, he introduced the concept of impulse by which is meant the space-time integral of the force-density needed to set up the motion of the fluid associated with a vortex ring, when the fluid is originally at rest. The impulse has the dimension of a momentum and is indeed given by the right-hand side of eq. (24). But the fact that the impulse is uniquely defined and has a finite value does not mean that the momentum created by the impulse is retained in the fluid. The momentum created can be transmitted partly or wholly to infinity, depending on the boundary conditions.
426
A. G. VAN VIJFEIJKEN,
A. WALRAVEN
AND F. A. STAAS
The close similarity between impulse and momentum has led people to replace the momentum P in Landau’s criterion, eq. (4), by the impulse and then to manipulate with this quantity as if it were a momentum. The present authors feel this is a rather unsatisfactory procedure because the derivation of Landau’s criterium rests basically on the conservation of momentum. The momenta involved in this conservation law ought to be true physical momenta and not quasimomenta of some sort. It is precisely the ambiguity of the momentum of a vortex ring which led us to search for a different criterion for oC. This criterion will be introduced in section 4. Next we evaluate the energy of an unenclosed vortex ring. For this we have : I& = &,p,j v:dax = Bps j (curl Ao)2 da%.
(26a)
The volume integration extends over all space except the space occupied by the vortex core. In the integration volume no vorticity is supposed to be present, so there we have; curl A0 = 0. Due to this fact we can write: (curl Ao)2 = div (A 0 A curl Ao), which relation enables us to transform the three-dimensional integral of eq. (26a) into a surface integral. So, _??a= $ps S (A0 A curl Ao) *da.
s.
(26b)
In the last integral SC is the surface of the vortex core and do is a vector perpendicular to the core surface and pointing into the core. Substituting Aa from eq. (20) and using the fact that the core radius a is a very small quantity, we obtain for Ea:
(27) Usually one finds in the literature a constant -$ instead of -2. The reason for this is probably attributable to the use of the analogy with the energy stored in a current-loop of inductance L, which most textbooks on electromagnetism give as: L = 4xra{ln 8ro/a - z}. If one looks carefully, however, one finds: L = 4xra{ln 8ro/a - 2 + ,uJ4}, where pr is the relative permeability of the current-carrying wire. The term with ,uLraccounts for the magnetic energy stored in the wire itself. In the case of a vortex line or vortex ring in He II one always assumes a hollow core, which means that no kinetic energy is stored in the core, corresponding to ,u,. = 0. This is why we should have -2 in eq. (27). Using z is equivalent to the assumption of solid body rotation inside the core. In fact, the term (-2) is borne out by eq. (26) if one makes an expansion in powers of (a/ra). 3. A vortex ring in an elzclosure. In this section we consider a vortex ring of radius y. which is enclosed in a cylinder of radius ?‘b. The cylinder is
VORTEX
RINGS IN He II
427
Fig. 1. A vortex ring in a tube. Ring 1 is the true ring while 2 indicates the sense of rotation of the vorticity covering the wall of the tube. The function f(z) describes the distribution of wall vorticity. supposed to be infinitely long. Fig. 1 gives the geometry. We again introduce a vector potential A, which, as before, can be chosen as : A = e,A (I, z). The velocity components v6r and vaZare derived from A as given by eqs. (15a) and (15b). The difference as compared with the previous section is that we now have to satisfy a boundary condition at the surface of the tube, i.e. at r = lb. This condition is simply “w(“b,
2) =
0,
Pa)
for all z. From eq. (15a) it then follows that A must be constant at the surface of the tube. Since we consider the case where no net current is present, A (?‘b, .z + &oo) = 0, so we can also put the boundary condition in the form A(?‘b, 2) = 0.
(28b)
This boundary condition for A can be satisfied by introducing into the differential equation for A an extra source term. We then write this equation in the form
a2A
1
aA
1+rx-$+== = -Kd(V
a2A
?‘I))d(z) + Kd(Y -
rb) f(2).
(29)
The first source term is the same as that for the unenclosed ring. The second source term describes vorticity 012 the wall. This wall vorticity, as we shall call it, has a certain spread in the z direction. In fig. I a possible distribution function f(z) is drawn. Our task now is to determine f(z) from the boundary condition (28b). But before proceeding with eq. (29) we want to make some brief remarks to substantiate the statement in section 1 about the advantages of the vector potential over the scalar potential. Firstly, the differential equations for A (c.q. Ao) holds in all space, whereas the differential equation for the scalar velocity-potential holds only outside the core region. Secondly and more important, the boundary conditions can more easily be accounted for if one works with A. Each boundary simply introduces a new source term in
428
A. G. VAN
VI JFEI JKEN,
A. WALKAVEN
AND
F. A. STAAS
eq. (29). If, for instance, we considered a vortex ring in the space between two concentric tubes, we would get a third source term in eq. [29), of the same shape as the one we already have. So it is easy to generalize eq. (29) to more complicated geometries. We now proceed with the solution of eq. (29). As in section 2 it is convenient to doubly transform A (Y, z), yielding A (s, $). From eq. (29) it follows that A(s, fi) is given by the following expression: A(% P) =
K~OJl(sQ) 9
+
_
KYbJl(Srb) f(p) s2+p2
p2
(30)
.
The first term has already been encountered, c.f. eq. (18), while the second term is due to the wall vorticity. In eq. (30) f(p) is, of course, the Fourier transform of f(z) . In order to obtain f(p) it is suitable to transform A(s, p) back as far as concerns the variable s. One gets A@‘,$) = 'dl(p~)
Kl[prO)
-
Krbll(pr) Kl(plb)
= KYOIl(pYO) Kl(pr) - Krbll(pY) Kl(prb) =K?'OIl(prO)
Kl(pr)
-
f(p),
if
f(p), if
Kf'bIl(prb) Kl(pr)f(p),
if
Olr
(3la)
TO<
(31b)
lb
7 < yb,
(314
Here 11 is the modified Bessel function of the first kind of order 1 and Kr is the first-order Hankel function. At I = ~0 the first two expressions become identical while at r = Yb the last two become the same. This behaviour is as it should be. Since A (Yb, z) = 0 for all z and A (I, p) is the Fourier transform of A (Y, z), it follows that A (Yb, p) is zero for all 6. Using this, it follows from eq. (3 1b) (or, for that matter, eq. (31~)) that f(p) is given by:
(32) It is seen that
f(p) --f 1 if
10 --f Yb. In that limiting
case we have for f(z) :
+m
f(z) =
&- 1f(p)
emipz dp = 6(z).
-co
This is exactly what one would expect. The distribution function f(z) gets broader when re/rb decreases. Another interesting property of f(z) must be mentioned. From eq. (32) we find :
VORTEX
This holds for arbitrary f(# = 0) =17(z) --00 Combining
values of
RINGS IN He II
On the other hand, we have
rO/rb.
dz.
these two expressions,
KY$(Z) -on
429
we can derive the following
relation:
dz = ICY;.
(34)
Now, the right-hand side of this equation is the analogue of the magnetic moment of a current-loop, while the left-hand side is the analogue of a magnetic moment induced on the wall. In this sense eq. (34) states that the inducing vortex ring and the induced vorticity are of equal “strength”. If we were to consider a vortex ring enclosed in between two concentric cylinders, we would find that the total vorticity induced on both walls would have “strength” K. Using eq. (32), we rewrite eqs. (31) : 11(fi70) Kl@7b)
A(7> 9) = K7Oll(j'7)
Il@7) A(79 f') =
K7OI1(#70)
A(7, $) =
0
if
I1 (@lb)
(35a)
if 70<7<7b,
(35b)
Kl(fi7b)
I1 (fi7b)
if
0<7<70,
y,, 5 7.
PC)
From eq. (35~) it follows that A(r, z) is identically zero for r > Yb. Due to this we can now work in free space which is convenient for the evaluation of certain integrals to appear later in the calculation. Let us now consider the momentum P associated with an enclosed ring. We now have instead of eq. (23b), P = 2xps
s
r dr
dz +
s
(r/l (I, z)) =
--03
0
9% =
$
27rpglim
a0
dr
20+00 s 0
dz G
(7A (7, s)).
(364
s -2%
Eq. (23~) is again valid but now we are dealing with A(r, p), which is given by the equations (35a) to (35~). We, therefore, get
P =
hKps70
p-0 k-II
+9.1
711(@7) Il(ib70) Kl(fi7b) 7~1($7)K1(fi70)
I1
0
r~l(@o) K&7)
-
7~1($7O)ll(?7)&(f'7b) Il(p7b)
@‘7d
II
*
1+ Wb)
A. G. VAN
430
VIJFEIJKEN,
A. WALRAVEN
AND
F. A. STAAS
The two contributions coming from I = 10 cancel each other, whereas the contribution coming from I = 0 vanishes, again due to the fact that II(X) M x/2 for small x. Therefore, we are left with:
P = 2XKps?$ lim
rbll(jh’0)
Kl(plb)
-
Ibll(~lO)~l(PYb)Kl(Plb)
-
P-4
which is identically
, (37a) ll(f'rb)
zero. Thus
P = 0.
(W
So in the case of an enclosed vortex ring there exists no ambiguity value of its momentum, contrary vortex ring. But the fact that P Landau criterion, c.J. eq. (4), does Next the energy of an enclosed
in the to what was found for an unenclosed = 0 for any value of 10 means that the not work at all for an enclosed ring. ring is calculated. We again have:
E=~~psSu~dg~=~psS(A~curlA)*da. s.
(38a)
The symbol SC stands, as before, for the surface of the core. The surface of the cylinder does not contribute to E, as at this surface A = 0. To evaluate E it is convenient to write A as: A@, z) = A@, z) + Al@, z). Here A&, z) is the vector potential of the unenclosed vortex ring, given in section 2, and Al(r, z) is the vector potential due to the wall vorticity. The Fourier transforms of Aa(r, z) and Ar(r, z) are given by the first and second terms of eqs. (35a) and (35b). For curl A we write: curl A = us, and vs is decomposed similarly into: ZJ~= ~,a + ~~1. With this decomposition we get (AAcurlA).da=(Aa~Zf~a)*do+(Aa~2r~r).du+ + (AI A v,o).du
+ (Al A Q-da.
(38b)
Of the four terms appearing in eq. (38b) only the first and third are important, which is easily seen as follows. The integration over da brings in a factor a, being the core radius. At the core surface A1 and TJ~~ are finite, therefore A1 A us1 - da is proportional to a and we can thus neglect this term. Similarly, at the core surface, Aa is proportional to In a, so Ao A %l*du vanishes as a In a. The first term gives simply the energy Eo of an unenclosed vortex ring, which diverges as In a. The term Al A ~,o.du is the true interaction term and is independent of a since vsa brings in a factor u-l and du a factor a. Finally, we make use of the fact that at SC the vectors Ao, t&o and da form a right-hand side rectangular coordinate frame. All these considerations then lead us to write:
[A
A
curl A).du
Substituting
w A ovso do -
A lvso da.
(38~)
this expression into eq. (38a) we obtain
E = E. - $jAlvsodu. s.
(384
VORTEX
For the interaction J Alus s.
RINGS IN He II
431
integral we have:
da = AI(ro, z = 0) 27~~~Y$ZI,~dl = 2x/croAr(ro, z = 0).
But
With this the expression E becomes O”
E = Eo -
2 II@‘(O)
Kl(@d
psK%‘o
s
dP
II
(39) '
0
The integral can only be numerically evaluated. The results of our computer calculation are given in fig. 2. Here we have plotted the energy E, normalized to psK2f’b, versus the quantity 6 = (Yb - ra)/rb. This quantity gives the distance of the ring from the cylinder-wall, normalized to ?‘b. of creation presumably have radii ~0 As vortex rings at the “moment” very close to Yb, the energetics of the creation process are determined by the behaviour of E’ z E/(p,K2rb) for small 6. This is why it is advantageous to
7.0 I 6.01
.“““,
10-7
\
/
.,,I
10-G
,,,,,
,,,,,(,,,
lo-
5
,,,,,,,,,
10-b
Fig. 2. Plot of the reduced energy E/(pst&)
,I
n-3
),,,,,,
,,,,,,,,,
10-2
*
,,,
lo-'
as a function of 6 = (Y,, - q,)/q, several values of the parameter U/Q,.
1
for
432
A. G. VAN
VIJFEIJKEN,
A. WALRAVEN
AND
F. A. STAAS
plot E’ as a function of S and to use a logarithmic scale for 6, rather than to plot E’ verszls ~0 and take for ~0 a linear scale. The different curves in fig. 2 have different values of the parameter U/Q,. The important features of fig. 2 are the following. Firstly for 10
if
6 --f ajrb.
(40)
This is in agreement with the results of Raja Gopal14) and it disagrees with the calculation by Fineman and Chase 1s). Moreover, it is seen from fig. 2 that, for small 6, E’ is strictly proportional to In 6 with a slope independent of the parameter U/lb. The above result for E’ as a function of 6 will now be used in the following section to construct a condition, for the critical velocity. 4. Stability condition for vortex rings. As argued in the preceding sections, the Landau criterion for vc, i.e. eq. (4), is not applicable to a vortex ring. It is not applicable either to a ring in an unbounded fluid or to an enclosed ring. In this section an alternative condition for vc is proposed. It is obtained essentially by requiring mechanical stability for the ring once the ring is created. We thus look at all forces that act on each element of the ring and require the net force to be zero. Now, there are three forces, all of them of the Magnus type, acting on a given element of the ring. Firstly a given element is subjected to a force due to all other elements of the ring. The result of this force, which we can view also as an elastic force, is to push the given element away from the cylindrical wall. Secondly the wall vorticity also exerts a force on our given element, and this force tries to pull the element towards the wall. The sum of these two forces is, per unit length of the ring, simply given by F1=---
1
aE
__.
27F~oI aro I
(41)
Physically it is clear that the radial force F1 tries to let the ring shrink if r-0is small compared to rb, whereas F1 tends to let the ring grow in the case where YOM lb. This is what is also borne out by fig. 2. Finally, a drift velocity is present, which we shall call ve in order to distinguish it from the previously considered superflow-velocity vosdue to the vortex ring and its
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RINGS
IN He II
433
image. The velocity field ve exerts also a force on the ring, which, again per unit length of the ring, is given by Fs =
@KVO.
(42)
This force tends to push the element away from the wall. It should be noted that by taking the expression of eq. (42) for the force on the ring-elements due to the transport current, we have assumed that the ring, at the creation, does not move along the axis of the tube. It might well be that the ring attains an axial velocity at a later time; we only make an assumption regarding its axial velocity at the creation. Each element of the ring is now in mechanical equilibrium if the two forces F1 and Fs balance each other, i.e. if
(43) The quantity on the left-hand side is a function not only of the tube radius 7b but also of the ring radius ~0. This means that, if we identify vo from eq. (43) with the critical velocity vc, we obtain vc = %(rb, YO) = %(rb, A),
(44)
where we have introduced: A = lb - ~0. So A is the distance from the wall at which the ring is created. If we evaluate the left-hand side of eq. (43) from eqs (39) and (27) we obtain for vC after doing some elementary algebra : K ”
2x(d -
=
.
I 1
_
In
24) ’
4(d- 24 a
+
.
4
(45)
1
Here we have introduced the diameter d of the tube: d = 2Yb. The dependence of vC on the two parameters d and A is clearly exhibited. A direct comparison of eq. (45) with experimental results is not possible as we do not know beforehand what A is. In order to get some information concerning this length we did the following. We took the experimental data collected by Van Alphen et a1.19) which give vC as a function of d. For each pair of values (vC, d), we then calculated A from eq. (45) and in this way we obtained A as a function of d. This was done for d-values ranging from d = 10-s cm to d = 1 cm. The result is given in fig. 3; the drawn curve containing the crosses gives A. In the figure the Van Alphen line for vCd is also given for comparison. As mentioned already in section 1, this line yields the
A. G. VAN VI JFEI JKEN, A. WALRAVEN
434
AND F. A. STAAS
Fig. 3. The length A and the creation time 7 as a function of a tube diameter d.
empirical expression vcd* = constant. As is seen from fig. 3, d M IO-6 cm for d = 10-S cm, while LI M lo-4 cm for d = 1 cm. Thus, d increases by 2 decades if d increases by 5 decades. The actual values for A, together with the fact that A increases slowly with d suggest that A might be the average surface roughness of the walls. It is well known that for the onset of turbulence in a normal fluid the surface conditions are important. The suggestion that this will also be the case for the onset of vorticity in a superfluid is therefore quite plausible. In order to test our suggestion it seems worthwhile to carry out v,-measurements on capillaries for which the surface roughnesses are reasonably well known. Next we consider the time needed to create a vortex ring. For this we start from an argument presented some years ago in a paper by Peshkov 12). The essence of Peshkov’s paper is that the energy E stored in a vortex ring is acquired at the expense of the kinetic energy stored in the transported current. Associated with the latter is an energy &~~z$nr~per unit length of the capillary. Peshkov then assumes that a segment of the capillary of length (Yb + ~07) contributes to the vortex ring to be formed. Here T is the creation time for a vortex ring. The length mentioned contains an amount of energy equal to -&@‘U~Wb(~b+ ~0,). If 01is the fraction of energy extracted to create the ring, we have, at ~10= Q,
Peshkov YO =
Yb,
took E as given by the simple Feynman-expression, eq. (2) with and established in this way the dependence of vC on lb with OL and r
VORTEX
as adaptable
parameters.
RINGS IN He II
By choosing
able to obtain a quantitative
435
01M 0.1 and r w 1O-4 s he was also
fit for vC versus lb.
We have used a modified version of eq. (46.a) to calculate T as a function of d, z.e. we work with the relation +&J&
= E.
(46b)
By taking the values for vC and d from Van Alphen’s straight line, we can read from fig. 3 the corresponding A-values and then from fig. 2 the corresponding E-values. The result of this part of the calculation is drawn in fig. 3, z.e. the curve with the open squares. It is seen that T decreases with increasing d, with an overall variation of about a factor of 10 in the range of d-values used. It should be possible to measure 7, for instance by measuring the attenuation of ultrasonic sound. One final comment can be made. In view of the fact that a finite time is involved in the creation of a ring, and therefore a finite segment of the capillary, the length A is really to be understood as the average roughness of the walls, i.e. the average of a segment of length vc7. So in the picture drawn in this paper a single sharp spike on the surface is never responsible for the creation of a ring.
REFERENCES 1) Wilks, J., The properties of liquid and solid helium (Clarendon Press, Oxford, 1967). (The most recent survey containing numerous references to critical velocitymeasurements.) See in particular p. 383-399. 2) Peshkov, V. P., Prog. Low Temp. Phys., Vol. IV, ed. C. J. Gorter (North-Holland Publ. Co., Amsterdam 1964) p. 1. 3) Hammel, E. F. and Keller, W. E., Superfluid Helium, Proc. Symp. St. Andrews, ed. J. F. Allen (Academic Press, London, New York, 1966) p. 121. 4) Hammel, E. F. and Keller, W. E., Proc. Xth Int. Conf. Low Temp. Phys. Vol. I, ed. in chief M. P. Malkov (Moscow, 1967) p. 30. 5) Careri, G., Prog. Low Temp. Phys., Vol. III, ed. C. J. Gorter (North-Holland Publ. Co., Amsterdam, 1961). p. 58 (a review of the earlier experiments involving ions in He II). 6) Careri, G., Dupre, F. and Mazzoldi, P., Quantum Fluids, Proc. Sussex Univ. Symp., ed. D. F. Brewer (North-Holland Publ. Co., Amsterdam, 1966), p. 305 (a survey of more recent work). 7) Reif, F., Proc. Xth Int. Conf. Low Temp. Phys., Vol. I, ed. in chief M. P. Malkov (Moscow, 1967) p. 86. 8) Landau, L. D., J. Phys. (U.S.S.R.) 5 (1941) 71. 9) Feynman, R. P., Progr. Low Temp. Phys., Vol. I, ed. C. J. Gorter (North-Holland Publ. Co., Amsterdam, 1955) p. 19. 10) Lamb, H., Hydrodynamics (Dover Publication, New York, 1945) chapter 7. 11) Geilikman, B. T., Zh. eksper. theor. Fiz. 37 (1959) 891 (English translation: Soviet Physics- JETP 10 (1960) 635). 12) Peshkov, V. P., Programme VIIth Intern. Conf. Low Temp. Phys. (Univ. of
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13) 14) 15) 16)
17) 18) 19) 20) 21) 22)
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RINGS IN He II
Toronto Press, Toronto, 1960) p. 83; Zh. eksper. theor. Fiz. (U.S.S.R.) 40 (1961) 379 (English translation: Soviet Physics - JETP 13 (1961) 259. Fineman, J. C. and Chase, C. E., Preprints VIIIth Int. Conf. Low Temp. Phys. (Butterworth & Co., Ltd., London, 1962) p. 222; Phys. Rev. 129 (1963) 1. Raja Gopal, E. S., Ann. Physics 25 (1963) 196. Staas, F. A., Taconis, K. W. and Van Alphen, W. M., Physica 27 (196 1) 893. Van Alphen, W. M., Vermeer, W., Taconis, K. W. andde BruynOuboter, R., Proc. IXth Int. Conf. Low Temp. Phys. ed. J. G. Daunt et al. (Plenum Press, New York, 1965) p. 323. Van Alphen, W. M., Olijhoek, J. F., De Bruyn Ouboter, R. and Taconis, K. W., Physica 32 (1966) 1901; 35 (1967) 483. Vermeer, W., Van Alphen, W. M., Olijhoek, J. F., Taconis, K. W. and De Bruijn Ouboter, R., Phys. Letters 18 (1965) 65. Van Alphen, W. M., van Haasteren, G. J., De BruynOuboter, R. and Taconis, K. W., Phys. Letters 20 (1966) 474. Graig, P. P., Phys. Letters 21 (1966) 385. See for instance: Landau-Lifshitz, Course of Theor. Physics, Vol. 8, p. 125. See C. C. Lin in Proc. Int. School of Physics “Enrico Fermi”, Course XXI, 1961, ed. G. Careri (Acad. Press, New York, 1963) p. 98.