COEFFICIENTS OF VISCOSITY FOR A FLUID IN A MAGNETIC FIELD OR IN A ROTATING SYSTEM

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1 Hooyman, G.J. Physica XXI Mazur, P. " Groot, S. R. de 1955 COEFFICIENTS OF VISCOSITY FOR A FLUID IN A MAGNETIC FIELD OR IN A ROTATING SYSTEM by G. J. HOOYMAN, P. MAZUR and S. R. DE GROOT Instituut voor theoretische natuurkunde, Universiteit, Utrecht, Nederland lnstituut-lorentz, Universiteit, Leiden, Nederland Synopsis The linear equations between the elements of the viscous pressure tensor and the rates of deformation are investigated for the case of an isotropic fluid in an external magnetic field or for the equivalent case of a rotating fluid. Since these equations can be incorporated within the thermodynamics of irreversible processes, the Onsager reciprocity relations hold for the scheme of phenomenological coefficients. For the present case the viscous behaviour is seen to be described by 8 coefficients between which one Onsager relation exists. The remaining 7 independent coefficients can be combined in a linear way so as to yield 5 coefficients of ordinary viscosity, the other 2 coefficients then describing the volume viscosity and a cross-effect between the ordinary and the volume viscosity, respectively. For the special case of vanishing volume viscosity the equations are compared with those derived from kinetic theory by Chapman and Cowling for an ionized gas in a magnetic field. The macroscopic description of viscosity can be developed from the viewpoint of the thermodynamics of irreversible processes 1)2)8). In this theory an expression for the entropy production a (per unit time and volume) due to the irreversible phenomena occurring within a system is derived by means of the conservation laws and the second law of thermodynamics. For the contribution av of viscous flour one then finds Ta~ = -- II : Grad v, (1) where T is the temperature, II the viscous pressure tensor, v the barycentric velocity and : denotes the interior product of two tensors, contracted twice. We shall restrict ourselves to the case usually met with that II is a symmetric tensor and denote the six independent cartesian components as 975xx ~ ~"~1, "rryz ~ 2T'4J.-ryy = ~r 2, :r,, = ~5, (2)

2 356 G.J. HOOYMAN, P. MAZUR AND S. R. DE GROOT In (1) the tensor Grad v then can be replaced by its symmetric part which we shall denote by ~ with components Exx ~,S1, 6yy ~ 6.2, Oy: ~ ½1~4 ' ~'xz ~- 1E5, (3) such that we have :z ~ 83, Exy ~ ½,$6, N~6 Ta~, = -- H :, = -- ~i= 1 2li ~'i. (4) According to the thermodynalnics of irreversible processes we next assume linear relationships between the elements of II and c which occur in (4) as 'fluxes' and 'forces' in the thermodynamic sense. These 'phenomenological equations' can be written as ve Z,, e k, (i ), (5) --"rri ~--- "'Jk=l ' "" "' (we shall not consider cross-effects between viscosity and other irreversible phenomena although such effects might exist). We now suppose the fluid (e.g., an ionized gas) to be placed in a homogeneous external magnetic field or to rotate with a constant angular velocity. The magnetic field strength or the angular velocity will be denoted by the comprehensive symbol H. Supposing that the fluid itself is isotropic we then want to investigate the symmetry properties of the phenomenological equations (5). a. Spatial symmetry. If we choose the x-axis in the direction of H it follows from the isotropy of the fluid that the relations (5) are invariant with respect to rotations about the x-axis. By straightforward calculation (e.g., introducing an infinitesimal rotation) one then finds for the scheme of phenomenological coefficients ~:1 E2 E 3 E 4 85 E 6 --2"t I -- ~T 2 --2/: 3 Lll LI2 L m L21 L22 L2a L L2t L23 L22 --L L24 L24 ½(L22--L23 ) Ls5 L L56 Ls5 in which only 8 coefficients are left. (6) b. Parity. Since any axis perpendicular to the x-direction is a 2-fold axis of rotation the relations (5) are invariant for a rotation of the coordinate system by an angle zt about the z-axis. This leads to Lik(H ) = (-- l)" Zik (-- I-I), (7)

3 COEFFICIENTS OF VISCOSITY FOR A FLUID IN A MAGNETIC FIELD 357 if the index z figures n times in 3; and e, together (c/. (2) and (3)). Hence we find that LlI, LI2, L2,, [-22, L23 and Lss are even functions of n, [ L2.1 and Ls6,, odd,, n. / (8) c. The Onsager relations. For the phenomenological coefficients the ()nsager reciprocal relations s) 4) L,.,(H) = Lki (- t-i) (9) hold. In view of (6) and the parity relations (8) we are left with only one true Onsager relation, viz., Li2 = L21, (10) by which the number of independent coefficients is further reduced to 7 (5 of them being even functions of I-t and 2 odd). d. Ordinary viscosity, volume viscosity and their cross-effect. Each of the tensors II and c can be split up into a tensor with zero trace and a scalar multiple of the unit tensor 8 where ~ and 0 are the traces n = ft + / = ~+ ~.08, ~ (11) ~1 '~3 = ":"i=1 ei = div v. J (12) The expression (4) for the entropy production then can be rewritten as -- T G-- l:i : ~ +{-st0. (13) We now can write the phenolnenological equations in a form corresponding to (13). From (5) and (I 2) we find, using also (6) and (10), -~ (L,1 +2L12)~1 + (L12+L22+L23)(~2+ e3)+'~(l,, +4L~2+2L22+2L23)O, (14) and therefore --2 i 1 --,--~( L,,-- L 12),-:~( e 2L,2-- L 22-- L 23)( e 2+@+:~-(LII+LI2--L22--L:3)O,, -~2--~(-LII[-LI2) _1 ei-]-~(--li2-~2l22--l23)'~2-}-a( L f2-- L 22 Jl- 2L 23) e3t[ 'J + 1(_ LI ' _ L, 2 + L22 + L23)t9 -}- L24e4, }(15) --st3=:~(-- Ljl + LJ2) t:l + ½(-- L12-- L22 + 2L23) e2 + +s(--l12+2l22--l23)~3+ I(-- LI1 L12+ L22+ L23 ) 19 L24F 4. 1

4 358 G. J. HOOYMAN, P. MAZUR AND S. R. DE GROOT Since ~l + d2 + ~3 = 0 these equations can be given a more symmetrical form. Writing 2Lll -- 4L12 + L22 + L2a =-- 6pl, L;1 -- 2L12 + 2L22 -- L23 =-- 6p2, Lll + 4L12 + 2L22 + 2L23-=-9ffo, Lll + L12 -- L22 -- L23 ~ 3~', L55 ~ P3, ] L24 ~ ~h, we find the following scheme of phenomenological coefficients connecting the two sets of quantities which occur in (13)" ~; d2 ~3 e4 es e6 ~0 / t / (is) --z~ 3 --~4 -- ~5 -- Yf6 -- 2ff I ~, 0 2ff2 2(ffl--if2) 'rh ~" 0 2(lq--if2) 2ff2 --r h ~ 0 --~1 ~'/1 21~2--1t I tt,3 'r/ P3 0 2~ --~ --~ if,, (17) /q, if2, t'3, if, and ~ are even functions of H, /]1 and ~12 are odd. The coefficients/q, P2, P3, */1 and 772 describe ordinary viscosity, p,, is the coefficient of volume (or bulk) viscosity and ~ describes a cross-effect between ordinary and volume viscosity. With regard to symmetry and parity the scheme (17) can be compared with the equations given by C h a p m a n and C o w 1 i n g s) for the stress tensor of a simple gas in a magnetic field, derived from kinetic theory. The scheme is in agreement with these equations apart from an apparent error of sign in the latter (the coefficients of 2 eyz in py:, and p,,, resp., should be the opposite of the coefficients of dyy and d,:, resp., in py:; this follows from the spatial symmetry and is confirmed by the Onsager relations (9)). It may be noted that in Chapman and Cowling's approximation #~ and vanish. e. The case o~ isotropy. If H -~- 0 the above equations reduce to the wellknown linear relationships between the stresses and rates of deformation in an isotropic system. As a matter of fact, for complete isotropy we have in addition to (6) LI2 = L21 = L23, L L , ] gll = L22, L5 s = L44, ~ (18)

5 COEFFICIENTS OF VISCOSITY FOR A FLUID IN A MAGNETIC FIELD 359 so that only two independent coefficients are left (the Onsager "relations become trivial for this case). By (16) this means ~ll = [12 = [t3' ~ ~-- ~1 ~ ~2 = O, (19) and (17) reduces to a diagonal scheme, pertaining to the equations or - fl = 2it1 I -- ~ = 3/%0, ] (20) -- H : 2tq~ + tq, O 8-2/q ~ , (21) where ;t is the 'second coefficient of viscosity' defined by p,, ~ 2 + ~ffl. (22) The authors wish to thank Professor I. Prigogine for a remark which led to this note. Received REFERENCES 1) G r o o t, S. R. d e, Thermodynamics o/irreversible processes, North Holland Publishing Company, Amsterdam, and Interscience Pulbishers, Inc., New York, ) G r o o t, S. R. d e, Hydrodynamics and thermodynamics, Proc. Fourth Syrup. oll Appl. Math., McGraw Hill, New York (1953) 87. 3) Groot, S. R. de, and Mazur, P., Phys. Rex,. 94 (1954) 218; Mazur, P. andgroot, S. R. de, Phys. Rev. 94 (1954) ) Groot, S. R., de and Kampen, N. G. van, Physica2l (1955) 39. 5) Chapman, S. and Cowling, T. G., The mathematical theory o! non-uni[orm gases. University Press, Cambridge (1939) 338.