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A comparison of normal fluid and superfluid vorticity in Taylor-Couette flow of helium II
PHYSICA
Physica B 194-196 (1994) 495-496 North-Holland
A Comparison o f N o r m a l Fluid and Superfluid Vorticity in T a y l o r - C o u e t t e F l o w o f Helium II Chris J. Swanson a, Yuanpeng B. You and Russell J. Donnellyb aDepartment of Physics, Westmont College, 955 La Paz Rd., Santa Barbara, CA 93108 bDepartment of Physics, University of Oregon, Eugene, OR 97401
We have compared the normal fluid vorticity and the superfluid vorticity in Taylor-Couette flow of He II. We have found that when the normal fluid vorticity is large the method of free energy minimization correctly predicts the superfluid vorticity to be nearly equal to the normal fluid vorticity. When the normal fluid vorticity is small, however, free energy minimization is no longer valid. The lowest free energy state becomes a limit which the system approaches but never quite achieves. The evidence suggests that the dynamics of vortex growth and decay and the presence of residual vorticity determine the superfluid vortex state.
1. INTRODUCTION It is well known that in solid body rotation of helium II the superfluid vorticity is nearly equal to the normal fluid vorticity.l,2,3 What has been less appreciated is that even in the presence of shear the superfluid vorticity, cos, follows the normal fluid vorticity, con. The first evidence of the equivalence of o~s and co, in shear appeared in a paper by Bendt4 which studied attenuation of second sound in Taylor-Couette flow, flow between concentric rotating cylinders. We have recently made measurements of second sound attenuation in Taylor-Couette flow which extend that initial study to a variety of velocity ratios and rotational configurations not examined before. Consider first laminar Taylor-Couette flow in which the velocity of the normal fluid is v, =(Cr+DIr)~ where C and D depend on the angular velocities ~ and ~2 and the radii R n and R2 of the cylinders. The normal fluid vorticity is given by 2C. It has been believed that in solid body rotation of the annulus (D = 0) the superfluid vorticity can be calculated by minimizing the free energy of the system.2,5 The analysis shows that the superfluid vorticity is equal to the normal fluid
vorticity everywhere except along thin strips at the boundaries of the container which are free of vorticity. If we denote the width of these strips as AR, we obtain the superfluid vorticity,
where d is the gap between the walls. 2 For our measurements d = .298 cm and R1 = 1.950 cm. Free energy minimization expects that the width of the vortex free strips is given by AR = (~I¢ 12C)1/2 where PCis the quantum of circulation and ~ is a parameter depending on the vortex spacing and core parameter. Thus the difference between co, and ~o, becomes small as C increases. Although studies of the above theory have been done and qualitative agreement has been found, the predictions have not been rigorously verified. Using the same formalism, we have generalized the study to include shear flow and in so doing found support for the free energy theory and an explanation for the existing discrepancies in flow without shear. 2. M E A S U R E M E N T S
We have made measurements of the attenuation of resonant second sound modes in the
0921-4526/94/$07.00 © 1994 - Elsevier Science B.V. All rights reserved SSDI 0921-4526(93)E0827-4
496 annular cavity. The details of the apparatus and technique are discussed elsewhere. 6 The attenuation of second sound is related to the average superfluid vortex density aligned perpendicular to the direction of propagation by means of the mutual friction coefficient B.1 I will denote this average superfluid vorticity as ~ . The various rotations we explored were inner cylinder rotation only, outer cylinder rotation, counter rotation with ~ / f 2 2 fixed and a variety of rotations in which f22 was held fixed while ~ l was varied over a range of values. The fixed values of f22 were 0.1, -0.3, and -0.7 s-1 and the values of ~"/~ ranged from 0 to 0.2 s-1. All of these measurements were made at a temperature of 2.12 K. Our results corroborated and extended the findings of Bendt. We found that as long as the normal fluid remained laminar and as long as [CI was greater than about 0.1, the superfluid vorticity matched the calculated vorticity given by Eq. 1 with adjusted to 5.47. Thus we determined experimentally that the normal fluid vorticity and superfluid vorticity are closely related in shear flow for relatively large values of the normal fluid vorticity. When ICI is very small, certain discrepancies do occur between predictions using free energy minimization and our experimental results. To investigate the differences, measurements of ~ , were made for solid body rotation and potential flow. We found that in solid body rotation the free energy was not minimized and that the vorticity was determined instead by the dynamics of superfluid vortex decay.6 The impact of vortex dynamics is seen first in a quiescent bath of helium. In such a bath a certain amount of residual vorticity exists which will not decay in an experimentally achievable time frame. 7 Thus the fluid does not even begin in a state of lowest free energy. As the cylinders are accelerated from rest the residual vorticity grows until a constant angular velocity is achieved. As the vorticity subsequently decays, the density of vortex
lines approaches the lowest free energy state but never seems to achieve it. In potential flow where there is no normal fluid vorticity we expect that there would also be no superfluid vorticity. Indeed for values of f22 < 0.2 s-~ there was no measurable vorticity. Yet, some supeffluid vorticity was generated as the angular velocities of the cylinders increased. Thus we conclude that although the free energy minimization seems to work at higher values of C, the dynamics of vortex decay determines the vorticity at lower values of C. 3. CONCLUSIONS We find that as we move away from the region where C is very small, He II begins to act more and more like a single classical fluid. The normal fluid which behaves classically drives the flow. The superfluid, whose vorticity is easily measured, follows the normal flow. The two fluids become coupled together acting as a Navier-Stokes fluid but still exhibiting superfluidity. REFERENCES
1. R. J. Donnelly, High Reynolds Number Flows Using Liquid and Gaseous Helium, SpringerVerlag, New York, 1991. 2. Stauffer, D., and Fetter, A. L., Phys. Rev. 168, 156 (1968). 3. Shenk, D. S., and Mehl, J. B., Phys. Rev. Lett. 27, 1703 (1971). 4. Bendt, P. J., Phys. Rev. 153, 280 (1967). 5. Donnelly, R. J., and Fetter, A. L., Phys. Rev. Lett. 17, 747 (1966) and Fetter, A. L., Phys. Rev. 153,285 (1967). 6. Swanson, C. J., and Donnelly, R. J., 1992. Accepted for publication in J. Low Temp. Phys. May 1993. 7. Awschalom, D. D. and Schwarz, K. W. Phys. Rev. Lett. 52, 49 (1984)
Physica B 194-196 (1994) 495-496 North-Holland
A Comparison o f N o r m a l Fluid and Superfluid Vorticity in T a y l o r - C o u e t t e F l o w o f Helium II Chris J. Swanson a, Yuanpeng B. You and Russell J. Donnellyb aDepartment of Physics, Westmont College, 955 La Paz Rd., Santa Barbara, CA 93108 bDepartment of Physics, University of Oregon, Eugene, OR 97401
We have compared the normal fluid vorticity and the superfluid vorticity in Taylor-Couette flow of He II. We have found that when the normal fluid vorticity is large the method of free energy minimization correctly predicts the superfluid vorticity to be nearly equal to the normal fluid vorticity. When the normal fluid vorticity is small, however, free energy minimization is no longer valid. The lowest free energy state becomes a limit which the system approaches but never quite achieves. The evidence suggests that the dynamics of vortex growth and decay and the presence of residual vorticity determine the superfluid vortex state.
1. INTRODUCTION It is well known that in solid body rotation of helium II the superfluid vorticity is nearly equal to the normal fluid vorticity.l,2,3 What has been less appreciated is that even in the presence of shear the superfluid vorticity, cos, follows the normal fluid vorticity, con. The first evidence of the equivalence of o~s and co, in shear appeared in a paper by Bendt4 which studied attenuation of second sound in Taylor-Couette flow, flow between concentric rotating cylinders. We have recently made measurements of second sound attenuation in Taylor-Couette flow which extend that initial study to a variety of velocity ratios and rotational configurations not examined before. Consider first laminar Taylor-Couette flow in which the velocity of the normal fluid is v, =(Cr+DIr)~ where C and D depend on the angular velocities ~ and ~2 and the radii R n and R2 of the cylinders. The normal fluid vorticity is given by 2C. It has been believed that in solid body rotation of the annulus (D = 0) the superfluid vorticity can be calculated by minimizing the free energy of the system.2,5 The analysis shows that the superfluid vorticity is equal to the normal fluid
vorticity everywhere except along thin strips at the boundaries of the container which are free of vorticity. If we denote the width of these strips as AR, we obtain the superfluid vorticity,
where d is the gap between the walls. 2 For our measurements d = .298 cm and R1 = 1.950 cm. Free energy minimization expects that the width of the vortex free strips is given by AR = (~I¢ 12C)1/2 where PCis the quantum of circulation and ~ is a parameter depending on the vortex spacing and core parameter. Thus the difference between co, and ~o, becomes small as C increases. Although studies of the above theory have been done and qualitative agreement has been found, the predictions have not been rigorously verified. Using the same formalism, we have generalized the study to include shear flow and in so doing found support for the free energy theory and an explanation for the existing discrepancies in flow without shear. 2. M E A S U R E M E N T S
We have made measurements of the attenuation of resonant second sound modes in the
0921-4526/94/$07.00 © 1994 - Elsevier Science B.V. All rights reserved SSDI 0921-4526(93)E0827-4
496 annular cavity. The details of the apparatus and technique are discussed elsewhere. 6 The attenuation of second sound is related to the average superfluid vortex density aligned perpendicular to the direction of propagation by means of the mutual friction coefficient B.1 I will denote this average superfluid vorticity as ~ . The various rotations we explored were inner cylinder rotation only, outer cylinder rotation, counter rotation with ~ / f 2 2 fixed and a variety of rotations in which f22 was held fixed while ~ l was varied over a range of values. The fixed values of f22 were 0.1, -0.3, and -0.7 s-1 and the values of ~"/~ ranged from 0 to 0.2 s-1. All of these measurements were made at a temperature of 2.12 K. Our results corroborated and extended the findings of Bendt. We found that as long as the normal fluid remained laminar and as long as [CI was greater than about 0.1, the superfluid vorticity matched the calculated vorticity given by Eq. 1 with adjusted to 5.47. Thus we determined experimentally that the normal fluid vorticity and superfluid vorticity are closely related in shear flow for relatively large values of the normal fluid vorticity. When ICI is very small, certain discrepancies do occur between predictions using free energy minimization and our experimental results. To investigate the differences, measurements of ~ , were made for solid body rotation and potential flow. We found that in solid body rotation the free energy was not minimized and that the vorticity was determined instead by the dynamics of superfluid vortex decay.6 The impact of vortex dynamics is seen first in a quiescent bath of helium. In such a bath a certain amount of residual vorticity exists which will not decay in an experimentally achievable time frame. 7 Thus the fluid does not even begin in a state of lowest free energy. As the cylinders are accelerated from rest the residual vorticity grows until a constant angular velocity is achieved. As the vorticity subsequently decays, the density of vortex
lines approaches the lowest free energy state but never seems to achieve it. In potential flow where there is no normal fluid vorticity we expect that there would also be no superfluid vorticity. Indeed for values of f22 < 0.2 s-~ there was no measurable vorticity. Yet, some supeffluid vorticity was generated as the angular velocities of the cylinders increased. Thus we conclude that although the free energy minimization seems to work at higher values of C, the dynamics of vortex decay determines the vorticity at lower values of C. 3. CONCLUSIONS We find that as we move away from the region where C is very small, He II begins to act more and more like a single classical fluid. The normal fluid which behaves classically drives the flow. The superfluid, whose vorticity is easily measured, follows the normal flow. The two fluids become coupled together acting as a Navier-Stokes fluid but still exhibiting superfluidity. REFERENCES
1. R. J. Donnelly, High Reynolds Number Flows Using Liquid and Gaseous Helium, SpringerVerlag, New York, 1991. 2. Stauffer, D., and Fetter, A. L., Phys. Rev. 168, 156 (1968). 3. Shenk, D. S., and Mehl, J. B., Phys. Rev. Lett. 27, 1703 (1971). 4. Bendt, P. J., Phys. Rev. 153, 280 (1967). 5. Donnelly, R. J., and Fetter, A. L., Phys. Rev. Lett. 17, 747 (1966) and Fetter, A. L., Phys. Rev. 153,285 (1967). 6. Swanson, C. J., and Donnelly, R. J., 1992. Accepted for publication in J. Low Temp. Phys. May 1993. 7. Awschalom, D. D. and Schwarz, K. W. Phys. Rev. Lett. 52, 49 (1984)