

The
Observed Properties of Liquid Helium
at the Saturated Vapor
Pressure
Chapter
11. Mutual Friction
This
chapter is somewhat different than the others because there is a review
article (Ref. 13) which already summarizes the data and presents a cubic
spline fit.
Representative values of the dimensionless mutual friction coefficients
B and B are contained in Table 11.3 (as compiled in Ref. 13) which are
adequate for many purposes. We also provide values of
and Z, which are defined
as
However,
experiments that have used B in their analyses have been conducted at
a wide range of counterflow velocities and frequencies. For example, in
counterflow turbulence experiments one uses B to convert from measurements
of the attenuation of second sound (at some frequency) as a function of
heat flux to the vortex line density as a function of heat flux. Experimenters
have used resonances varying from 4 Hz (Ref. 14) to greater than 23 kHz
(Ref. 16) a range of three and one half decades. Table 11.4 shows the
corresponding range of B: at 1.9 K, B varies by more than 50% from 1 Hz
to 10 kHz. The parameter B is also used to convert measurements of temperature
gradient to line density. Here B is a function of the steady counterflow
velocity. Experiments of interest to cryogenic engineers have been performed
with heat fluxes as high as 20 W/cm^{2}, producing a considerable temperature
difference over a length of order 1 cm (Ref. 15). The resulting counterflow
velocity varies with position but is everywhere greater than 160 cm/s.
In the other extreme, quantum turbulence experiments have been carried
out with counterflow velocities as low as 0.1 cm/s. Table 11.5 shows that
the change in B over the relevant velocity range can be of more than a
factor of 2 at 1.9 K.
The theory of the frequency and velocity dependencies of mutual friction
is contained in Ref. 13. Using this theory one can develop a method for
obtaining the mutual friction parameters at arbitrary frequency or counterflow
velocity. One implementation of this idea is contained in the paper by
Ref 16. Some results of this rather technical procedure are contained
in Tables 11.2 and 11.3. New theoretical and experimental work on mutual
friction may soon make this situation clearer and easier to use.
Table 11.3 has been converted to T^{90}, but we have not attempted to convert
the mutual friction data of Tables 11.4 and 11.5 to T^{90} because the current
limited accuracy of these coefficients leads to more uncertainty than
does the temperature scale.
Table
11.1. Knots and coefficients for mutual friction parameter B. The spline
returns log_{10 }B vs. log10 ,
with = 1 –
T / T. For T >
2.167 K use the asymptotic
Knots 
Coefficients 
k(1)=5 

K(2)=5 

K(3)=5

C(1)=1.31928144433 
K(4)=5

C(2)=1.12452707801 
K(5)=2.5

C(3)=0.639314792565 
K(6)=2.0

C(4)=0.313383532495 
K(7)=0.8

C(5)=0.162687403543 
K(8)=0.387958059947

C(6)=0.092047691284 
K(9)=0.387958059947

C(7)=0.188452616588 
K(10)=0.387958059947 

K(11)=0.387958059947 

Table
11.2. Knots and coefficients for mutual friction parameter B.
The spline returns log10 (BZ+15)
vs. log10 ,
with
= 1 – T / T.
For T > 2.134 K use the asymptotic expression .
Knots 
Coefficients 
K(1)=5 

K(2)=5 

K(3)=5

C(1)=
8.47218032526E2 
K(4)=5

C(2)=0.931621715174 
K(5)=3.55

C(3)=0.
973263359433 
K(6)=3.2

C(4)=
1.10543591819 
K(7)=2.5

C(5)=
1.15904485127 
K(8)=1.0

C(6)=
1.18311634566 
K(9)=0.384067377871

C(7)=
1.17480594214 
K(10)=0.384067377871

C(8)=1.19458392766 
K(11)=0.384067377871 

K(12)=0.384067377871 

Table
11.3. Recommended values of the mutual friction coefficients in helium
II.
T90 (K) 
B 
B 


1.30 
1.526 
0.616 
0.034 
1.383E02 
1.35 
1.466 
0.535 
0.042 
1.543E02 
1.40 
1.408 
0.458 
0.051 
1.668E02 
1.45 
1.351 
0.385 
0.061 
1.746E02 
1.50

1.296 
0.317

0.072

1.766E02 
1.55 
1.243 
0.255 
0.084 
1.721E02 
1.60 
1.193 
0.198 
0.097

1.608E02 
1.65 
1.144 
0.149 
0.111 
1.437E02 
1.70 
1.100 
0.107

0.126

1.225E02 
1.75 
1.059 
0.075

0.142 
1.003E02 
1.80

1.024

0.052

0.160

8.211E03 
1.85 
0.996 
0.041

0.181

7.438E03 
1.90 
0.980 
0.040

0.206 
8.340E03 
2.00 
1.008 
0.043 
0.279 
1.198E02 
2.02 
1.031 
0.037 
0.302

1.097E02 
2.04 
1.065 
0.027 
0.330 
8.318E03 
2.06 
1.115 
0.009 
0.366 
3.018E03 
2.08 
1.188 
0.019 
0.414 
6.690E03 
2.10 
1.298 
0.065 
0.481

2.412E02 
2.12

1.476 
0.143

0.581

5.632E02 
2.14
 1.790 
0.297

0.753

1.249E01 
2.16 
2.420 
0.683 
1.097 
3.096E01 
2.162

2.515 
0.755 
1.150

3.453E01 
2.164 
2.622 
0.842 
1.210 
3.883E01 
2.166

2.747

0.949

1.279

4.416E01 
2.168 
2.897 
1.085 
1.362 
5.103E01 
2.170

3.154

1.272 
1.577 
6.358E01 
2.172

3.538 
1.549 
1.769 
7.747E01 
2.174 
4.227 
2.048 
2.113 
1.024E+00 
2.176 
6.391 
3.613 
3.195 
1.807E+00 
2.1761 
6.679 
3.821 
3.339 
1.911E+00 
2.1762

7.027

4.074

3.514

2.037E+00 
2.1763

7.463

4.389 
3.732 
2.194E+00 
2.1764

8.033 
4.801 
4.017

2.401E+00 
2.1765 
8.833 
5.380 
4.417

2.690E+00 
2.1766

10.098

6.295 
5.049 
3.147E+00 
2.1767

12.693

8.172

6.347

4.086E+00 
Table 11.4. Mutual friction
parameter B versus temperature for various second sound frequencies in
the low amplitude limit (after Ref. 16).
T_{58} (K) 
1 Hz 
10 Hz 
100 Hz 
1000 Hz 
10000 Hz 
1 
1.507

1.509 
1.511 
1.513 
1.515 
1.1

1.549 
1.557 
1.564 
1.572 
1.579 
1.2

1.520 
1.538 
1.556

1.574

1.592 
1.3 
1.431 
1.464 
1.499 
1.534 
1.571 
1.4

1.268 
1.318 
1.370 
1.427 
1.487 
1.5

1.098 
1.159 
1.227 
1.302 
1.387 
1.6 
0.958

1.026

1.104 
1.194 
1.300 
1.7

0.855

0.926

1.011 
1.112 
1.236 
1.8 
0.788

0.863

0.953 
1.063 
1.203 
1.9 
0.760

0.836

0.929

1.045

1.194 
2.0

0.788 
0.863 
0.954 
1.067 
1.210 
2.1

1.106

1.197

1.304

1.432

1.588 
2.11

1.192 
1.287 
1.398

1.531 
1.691 
2.12 
1.299

1.398

1.514 
1.651 
1.815 
2.13 
1.436 
1.541

1.661 
1.802

1.967 
2.14

1.623

1.732 
1.856 
1.998 
2.164 
2.15

1.902

2.014

2.140

2.282

2.442 
2.17 
4.600 
4.698

4.798

4.899

5.002 
Table
11.5. Mutual friction parameter B versus temperature for various vortex
linenormal fluid relative velocities in the steady state limit (after
Ref.16).
T_{58} (K) 
0.1 (cm/s) 
1 (cm/s) 
10 (cm/s) 
100 (cm/s) 
1 
1.503 
1.507 
1.511

1.515 
1.1

1.539 
1.553

1.568

1.583 
1.2

1.501

1.536

1.572

1.609 
1.3

1.406

1.472

1.543 
1.619 
1.4

1.244

1.343 
1.456 
1.585 
1.5 
1.080 
1.205 
1.360 
1.556 
1.6

0.949 
1.092 
1.285 
1.555 
1.7 
0.856 
1.012 
1.238

1.590 
1.8

0.798

0.967

1.227 
1.672 
1.9 
0.778 
0.956

1.239

1.756 
2.0 
0.812

0.990

1.268 
1.763 
2.1 
1.139 
1.350 
1.656 
2.140 
2.11 
1.226 
1.445 
1.760 
2.247 
2.12 
1.334 
1.563

1.885

2.369 
2.13

1.473

1.711

2.037

2.511 
2.14

1.662

1.906

2.231 
2.681 
2.15 
1.942 
2.190 
2.506 
2.914 
2.16

2.455

2.700 
2.990

3.332 
2.17 
4.635

4.835

5.039

5.243 
Chronological Bibliography for Mutual Friction
1 
H. E. Hall and W. F. Vinen, "The Rotation of Liquid Helium II. I.
Experiments on the Propagation of Second Sound in Uniformly Rotating
Helium II," Proc. Roy. Soc. A 238, 204214 (1956). 
2 
H.
E. Hall and W. F. Vinen, “The Rotation of Liquid Helium II.
II. The Theory of Mutual Friction in Uniformly Rotating Helium II,”
Proc. Roy. Soc. A 238, 215234 (1956). 
3 
H.
A. Snyder, “Axial Component of Mutual Friction in Uniformly
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4 
H.
A. Snyder and D. M. Linekin, “Measurements of the MutualFriction
Parameter B' in Rotating Helium II,” Phys. Rev. 147, 131139
(1966) 
5 
H.
A. Snyder and Z. Putney, “Angular Dependence of Mutual Friction
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6 
P.
J. Bendt, “Attenuation of Second Sound in Helium II Between
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7 
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8 
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9 
P.
Matheiu, A. Serra, and Y. Simon, “CriticalRegion Measurements
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10 
R.
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11 
E.
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12 
P.
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13 
C.
F. Barenghi, R. J. Donnelly, and W. F. Vinen, “Friction on Quantized
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14 
K.
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15 
J.
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16 
C.
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