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Adair, Thomas W.

Vestner, H.

Texas A&amp;M Repository

PHYSICA j REVIK W A VOLUME 11, NUMBER I JANUARY 1975Conservation of angular momentum in thermomagnetic torque experiments*H. Vestner~ and T. W. Adair IIIDepartment of Physics, Texas 3 4 M University, College Station, Texas 77843(Received 3 June 1974)The conservation of orbital angular momentum has the following consequence for Scott-effectexperiments: The torque exerted on the outer cylinder is equal in size but opposite in directionto the torque on the inner cylinder. This is true for each of the two contributions to the torque,namely, the bulk and slip contribution.I. INTRODUCTIONRecently Wood et al. ' reported an experimentalstudy of the conservation of angular momentum inthermomagnetic torque experiments (Scott ef.—fect'"). Actually, they found no net torque on aninverted double-walled tube filled with nitrogenand placed in an axial magnetic field; a tempera-ture difference between the inner and the outerwall was maintained. This observation, as wellas previous experiments by Scott et al. ,' indicatethat the torque v; exerted on the outer cylinder(in a Scott-effect apparatus) is equal in size butopposite in direction to the torque 7,. on the innercylinder, a result which is not surprising at all,since it is based on the conservation of angularmomentum, as is shown in Sec. II of this note. InSec. III of this note the relation between the flowfield and the torque is discussed.II. CONSERVATION OF ANGULAR MOMENTUMWithin the range of validity of the generalizedBoltzmann equation for a dilute gas of polyatomicmolecules, due to Waldmann and Snider, ' the pres-sure tensor p is symmetric, ~P pp Ppp'Then the conservation equation for linear momen-tumDn mv+V P =6Dtleads to a conservation equation for orbital angularmomentumzz— (Xxmv)+V ~ L =0DtwhereLpp —P~ t6ppl~t g p1is the flux of orbital angular momentum, andD 8-+v VDt Btwhere n~ is the local outer unit normal of the gasat 0~. The force the gas exerts on the unit area ofo~ is f~ =n~ p, and n„~ L =X~ f~ is the torqueper unit area of 0~, i.e.,dg XX f~Eis the total torque exerted on v~. In a steady state,Eq. (3) reduces togT, =O, (4)i.e., there is no net torque on any closed container(volume V) filled with gas. For the application to aScott-effect experiment we take as volume V thespace between the two coaxial cylinders of lengthL and radii R,. and R, (R,. &A,). Since the bottomand top of this volume give no contribution to thetorque, we are left with7, +v,. =0, (5)i.e., the torque v; on the outer cylinder is equal insize and opposite in direction. to the torque ~,. onthe inner cylinder. This fact is a direct conse-quence of the symmetry of the pressure tensor, orequivalently the conservation of orbital angularmomentum.III. SLIP FLOW IN THERMOMAGNETIC TORQUEEXPERIMENTSThe possibility of the existence of an azimuthalflow in the thermomagnetic torque experiments hasbeen discussed since Scott and co-workers dis-is the total (or substantial) time derivative. Hencenot only is the total angular momentum conservedbut also the orbital and intrinsic angular momen-tum are conserved separately. The integration ofEq. (2) over any gas volume V bounded by the sur-faces UE givesrdVn(Xxmv)+ dvnr ~ L =0,dtCONSERVATION OF ANG ULAR MOMENTUM. . .covered this effect. ' The existence of this azi-muthal flow is apparent from measurements ofScott et al.' and Weinstein and Keeney. ' Two dif-ferent mechanisms have been considered. Wald-mann' introduced a "thermomagnetic slip" due toan asymmetry in the distribution of polarized mole-cules reflected from the surface. On the otherhand, Levi and Beenakker' pointed out that thetransverse component of the heat flux (in a magne-tic field) causes a thermal creep along the cylin-ders. Park and Dahler'o presented a kinetic-theory-based approach to these slip problems.The ensuing flow field gives rise to a torque whichmay be called the "slip torque. "Even without any flow of the gas there is a torquedue to thermal stresses arising from second-orderderivatives of temperatures (V V T) This ."bulktorque" has been calculated by Levi et al." anda quantitative evaluation is possible with the knowl-edge obtained from the Senftleben Beenakker ef-fects' of viscosity and thermal conductivity. Thebulk torque has the same direction as the observedtorque but is greater in magnitude, ' " indicatingthat the slip flow gives rise to a torque in theopposite direction. This conclusion is confirmedby the direct observation of the thermomagneticslip effects in experiments on the thermomagneticpressure difference by Hulsman et a/. ,"which givea method to estimate the slip torque, ' since thethermal stresses do not exist in a, gas between flatplates (V VT =0).If the magnetic field direction lj is parallel to thecylinder axis the flow velocity has only Rn azimuth-al component, which depends on the distance rfrom the axis. The Navier-Stokes equation (ne-glecting all inhomogeneous terms arising from thepolarizations") V ~ Vv =0 determines the spatialvariation:v =hxu(C/r+Dr),where u is the unit vector in the radial direction.From the viscous pressure tensor(p„)z =-2q(Vv)~[where ( )~ denotes the symmetric traceless part],the resulting slip torque is calculated:The bulk contribution v,' to the total torquewas calculated by Levi et a/. " from the thermalstresses which are determined by V V T.In Fig. 1 the direction of the flow field and of theresulting slip torque are shown for nitrogen gasin a Scott torque apparatus with a heated cylinder.According to thermomagnetic pressure differencemeasurements by Hulsman et al. ,"the gas turnsin the Kx VT direction; hence the slip torque P, 'has the direction 6 of the magnetic field. Thetotal torque' as well as the bulk torque" have theopposite direction -K.In experiments by Weinstein and Keeney' and byScott et al.' radial baffles were introduced betweenthe two cylinders to disturb the flow field. A de-crease of the torque was observed in cases wherethe distance between the baffle edge and the innercylinder was smaller than about 10 mean freepaths. 4 Such a small clearance interrupts the freecirculation of the gas and causes it to reverse itsdirection at the baffle. As a result of the interplaybetween this counterflow in the central part of thecircular gap and the slip flow along the cylinderwalls, a flow field is established such that thereis no net gas flow. The velocity gradient near theinner cylinder is thus increased by the presenceof a barrier and hence the slip torque (caused bythe flow) is also increased. Consequently, the in-troduction of a baffle causes a reduction in thetotal torque. This explanation of the baffle orbarrier" experiments, together with a quantitativeestimate, was given by Waldmann" as early a.s1970.IV. CONCLUSIONThe symmetry of the pressure tensor, valid fora dilute gas, results in the conservation of orbital~~)' —-Ts' = -4 peal Ch.This result shows that the slip torque on the outercylinder is equal in magnitude but opposite to theslip torque on the inner cylinder. Notice that onlythe x ' term in v contributes to T',.' and that the gasand the inner cylinder rotate in different direc-tions.The constants C and D are determined from theboundary conditions at both surfaces. In the high-pressure limit C is of the order I/P and D van-ishes" (i.e. , is of the order I/P'). Consequently,the velocity varies with x ' and is unidirectional.iysl7 QPIG. 1. Directions of the flow field and the torques forN2 in a Scott-torque apparatus with a heated inner cy-linder; the magnetic field points into the drawing plane.392 H. VESTNER AND T. W. ADAIR IIIangular momentum. It has been shown that, as aconsequence, the torque on the outer cylinder isopposite in direction but equal in magnitude to thetorque exerted on the inner cylinder. Since this istrue for both the bulk and the slip contributionsseparately, the assumption of any additional sourceof angular momentum is not needed in order toexplain any of the torque experiments.*Research supported by The Robert A. Welch Foundation,Houston, Tex.~Permanent address: Institut fur Theoretische Physikder Universitat Erlangen-Nurnberg, Erlangen, Ger-many.~L. T. Wood, C. W. Wells, F. A. Handler, and A. F.Hildebrandt, Phys. Rev. A 9, 2245 (1974).2G. G. Scott, H. W. Sturner, and R. M. Williamson,Phys. Rev. 158, 117 (1967); Phys. Lett. A 25, 573(1967). T. W. Adair III, D. A. Avery, C. F. Squire,and S. Wolfson, Phys. Rev. Lett. 20, 142 (1968).3For a review of the Scott effect and related phenomenasee J. J. M. Beenakker and F. R. McCourt, Annu. Rev.Phys. Chem. 21, 47 (1970).4G. G. Scott, H. W. Sturner, and M. E. Larchez, Phys.Rev. A 2, 792 (1970).5L. Waldmann, Z. Naturforsch. A 12, 660 (1957); A 13609 (1958). R. F. Snider, J. Chem. Phys. 32, 1051(1960).6S. Hess and L. Waldmann, Z. Naturforsch. A 21, 1529(1966).VD. H. Weinstein and J. Keeney, Phys. Rev. Lett. 23,218 (1969).L. Waldmann, Z. Naturforsch. A 22, 1678 (1967).~A. C. Levi and J. J. M. Beenakker, Phys. Lett. A 25,350 (1967).Won-Hoon Park and J. S. Dahler, J. Chem. Phys. 56,1995 (1972). J. S. Dahler and Won-Hoon Park, J. Chem.Phys. 56, 4190 (1972).~~A. C. Levi, F. R. McCourt, and J. J. M. Beenakker,Physica (Utr. ) 42, 363 (1969).A. L. J. Burgmans and T. W. Adair III, J. Chem.Phys. 59, 324 (1973).~3H. Hulsman, F. G. van Kuik, H. F. P. Knaap, andJ. J. M. Beenakker, Physica (Utr. ) 57, 522 (1972).H. Hulsman, G. F. Bulsing, G. E. J. Eggermont,L. J. F. Hermans, and J. J. M. Beenakker, Physica(Utr. ) 72, 287 (1974).H. Vestner, Z. Naturforsch. A 28, 869 (1973); 28, 1554(1973).~~L. Waldmann, Bull. Am. Phys. Soc. 15, 262 (1970).

Conservation of Angular-Momentum in Thermomagnetic Torque Experiments

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Conservation of Angular-Momentum in Thermomagnetic Torque Experiments

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Conservation of Angular-Momentum in Thermomagnetic Torque Experiments

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Texas A&amp;M Repository

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