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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/cm², 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 T⁹⁰, but we have not attempted to convert the mutual friction data of Tables 11.4 and 11.5 to T⁹⁰ 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 log₁₀ 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)= -8.47218032526E-2
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	1.383E-02
1.35	1.466	0.535	0.042	1.543E-02
1.40	1.408	0.458	0.051	1.668E-02
1.45	1.351	0.385	0.061	1.746E-02
1.50	1.296	0.317	0.072	1.766E-02
1.55	1.243	0.255	0.084	1.721E-02
1.60	1.193	0.198	0.097	1.608E-02
1.65	1.144	0.149	0.111	1.437E-02
1.70	1.100	0.107	0.126	1.225E-02
1.75	1.059	0.075	0.142	1.003E-02
1.80	1.024	0.052	0.160	8.211E-03
1.85	0.996	0.041	0.181	7.438E-03
1.90	0.980	0.040	0.206	8.340E-03
2.00	1.008	0.043	0.279	1.198E-02
2.02	1.031	0.037	0.302	1.097E-02
2.04	1.065	0.027	0.330	8.318E-03
2.06	1.115	0.009	0.366	3.018E-03
2.08	1.188	-0.019	0.414	-6.690E-03
2.10	1.298	-0.065	0.481	-2.412E-02
2.12	1.476	-0.143	0.581	-5.632E-02
2.14	1.790	-0.297	0.753	-1.249E-01
2.16	2.420	-0.683	1.097	-3.096E-01
2.162	2.515	-0.755	1.150	-3.453E-01
2.164	2.622	-0.842	1.210	-3.883E-01
2.166	2.747	-0.949	1.279	-4.416E-01
2.168	2.897	-1.085	1.362	-5.103E-01
2.170	3.154	-1.272	1.577	-6.358E-01
2.172	3.538	-1.549	1.769	-7.747E-01
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).

T58 (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 line-normal fluid relative velocities in the steady state limit (after Ref.16).

T58 (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, 204-214 (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, 215-234 (1956).
3	H. A. Snyder, “Axial Component of Mutual Friction in Uniformly Rotating Liquid Helium,” Phys. Fluids 6, 755-764 (1963).
4	H. A. Snyder and D. M. Linekin, “Measurements of the Mutual-Friction Parameter B' in Rotating Helium II,” Phys. Rev. 147, 131-139 (1966)
5	H. A. Snyder and Z. Putney, “Angular Dependence of Mutual Friction in Rotating Helium II,” Phys. Rev. 150, 110-117 (1966).
6	P. J. Bendt, “Attenuation of Second Sound in Helium II Between Rotating Cylinders,” Phys. Rev. 153, 280-284 (1967).
7	J. A. Lipa, C. J. Pearce, and P. D. Jarman, “Second-Sound Attenuation in Rotating Helium II Close to the Lambda Point,” Phys. Rev. 155, 75-77 (1967).
8	P. Lucas, “Mutual Friction Measurements in Uniformly Rotating Liquid Helium,” J. Phys. C. 3, 1180-1192 (1970).
9	P. Matheiu, A. Serra, and Y. Simon, “Critical-Region Measurements of the Mutual Friction Parameters in Rotating Helium II,” Phys. Rev. B 14, 3755-3761 (1976).
10	R. J. Miller, I. H. Lynall, and J. B. Mehl, “Velocity of Second Sound and Mutual Friction in Rotating Helium II,” Phys. Rev. B 17, 1035-1045 (1978).
11	E. J. Yarmchuck and W. I. Glaberson, “Counterflow in Rotating Superfluid Helium,” J. Low Temp. Phys. 36, 381-430 (1979).
12	P. Matheiu and Y. Simon, “Frequency Dependence of Mutual Friction in Rotating Helium II,” Phys. Rev. B 26, 1233-1243 (1982).
13	C. F. Barenghi, R. J. Donnelly, and W. F. Vinen, “Friction on Quantized Vortices in Helium II. A Review,” J. Low Temp. Phys. 52, 189-247 (1983).
14	K. P. Martin and J. T. Tough, “Evolution of Superfluid Turbulence in Thermal Counterflow,” Phys. Rev. B 27, 2788-2799 (1983).
15	J. M. Pfotenhaurer and R. J. Donnelly, “Heat Transfer in Liquid Helium,” Advances in Heat Transfer 17, 65-158 (1985).
16	C. E. Swanson, W. T. Wagner, R. J. Donnelly, and C. F. Barenghi, “Calculation of Frequency- and Velocity-Dependent Mutual Friction Parameters in Helium II,” J. Low Temp. Phys. 5 & 6, 263-267 (1987).