3568
Ind. Eng. Chem. Res. 1996,34, 3568-3572
The Symmetry of the Stress Tensor Gerard D. C.Kuikent Laboratory for Aero- and Hydrodynamics, Rotterdamseweg 145, NL 2628 AL Delfc, The Netherlands
It is argued that from the continuum point of view the stress tensor is symmetric, except for a class of suspensions in which torques could be produced on each particle in the carrier fluid by a n external field. Statistical theories are shortly reviewed. A new set of macroscopic angular momentum equations is obtained for polar media by considering the order of magnitude of each contribution to the integral angular momentum balance. It is found that couple stresses and microinertia can always be neglected with respect to the contribution of the stress tensor. The angular momentum balance is not discussed in the now 35 years old and famous book Transport Phenomena. On pp 114 and 115 angular momentum arguments are used t o show, by letting an elementary volume go to zero, that the couples produced by the shear stresses are not counterbalanced, unless the stress tensor is symmetric. Bird (1993) remarked that there is no experimental evidence that a nonsymmetric stress tensor is needed and that in the kinetic theory for dilute monatomic gases and for flexible and rodlike polymers the stress tensor is symmetric. To quote also Lodge (1974, p 119): “For polymeric materials, however, the assumption [that the stress tensor is symmetric] appears to be valid: We know of no evidence, experimental or theoretical, which casts doubt on its validity, although, of course, the possibility of requiring a nonsymmetric stress tensor must always be borne in mind.” On the other hand, it is not directly evident that the Kramers form for the stress tensor yields a symmetric stress tensor (Bird et al., 1987, p 69). The remarks of Bird and Lodge are in agreement with the Cauchy second law of motion: When there are no external torques and no couple stresses, a necessary and sufficient condition for the balance of moment of momentum in a body where the linear momentum is balanced is that the stress tensor is symmetric (Truesdell, 1960, p 546). This condition is sufficient, but the symmetry of the macroscopic stress tensor is satisfied for less stringent conditions if one requires that the coordinates that characterize the observed microstructure of the whole body have the same value for all observers. Capriz (1989, p 24) showed that then the stress tensor is necessarily symmetric even if couple stresses are present. Green and Naghdi (1995) argue that the symmetry of the stress tensor can be regarded as a consequence of the invariance under superposed rigid-body motions. Likewise, Benach and Miiller (1974, eq 2.35) use the invariance of the internal energy under Euclidean transformations to show that the stress tensor in a magnetizable dielectric mixture is necessarily symmetric. However, this paper does not consider the possibility of an external supply of angular momentum per unit of mass due to electromagnetic fields. From the statistical theory developed by Irving and Kirkwood (1950)for monatomic liquids, a nonsymmetric stress tensor (eq 5.11) arises if the potential between pairs of particles is not central or depends on the internal angular momenta and the positions of the particles. Dahler and Scriven (1963) have generalized the results of Irving and Kirkwood, and these authors state on p 509 of their paper that in the statistical +
e-mail:
[email protected],NL. 0888-5885/95/2634-3568$09.00l0
theory of real materials asymmetric states of stress appear quite naturally as the rule rather than the exception. From the statistical calculations the asymmetric states of stress arise from the Lorentz magnetic force due to the motion of the charged molecules and from noncentral forces between pairs of molecules. Since noncentral forces and polarization by interaction with magnetic fields are the alleged sources of an asymmetric stress tensor, an asymmetry should have been noticed in careful measurements of the flow of polyatomic gases in electromagnetic fields, in which angular momentum polarizations also occur due to the noncentral forces from nonspherical interactions. It is known that the transport properties of polyatomic gases are influenced by a magnetic field. Nonspherical intermolecular interactions occur because of the precession of the molecular angular momentum, which is coupled to the rotational magnetic moment of the molecule. Changes in the viscosity tensor due to these effects are known as the viscomagnetic effects. These effects have been measured (Beenakker, 1974; McCourt et al., 1991). The changes in the viscosity due to electric fields have also been measured. However, the definition of the stress tensor in a gas of rotating molecules that follows from the classical and quantum mechanical generalization of the Boltzmann equation (WaldmannSnider equation; see McCourt et al., 1991, section 4.5) leads to a symmetric although anisotropic stress tensor, and the number of independent viscosity coefficients in the fourth-order viscosity tensor increases from two to seven (De Groot and Mazur, 1962, p 311; McCourt et al., 1991, Chapter 5). We mention parenthetically that it has been found experimentally that for most gases the viscosity decreases in a magnetic field, and when an increase is found, this is due to a more complicated structure of the molecules and not due to the presence of a strong electric dipole moment (Van Ditzhuyzen et al., 1977). Anomalous field effects can be measured in the transport coefficients under the influence of either a magnetic or an electric field. These anomalous effects are caused by the coupling of inner spin and rotational angular momenta, but there has been no indication that an asymmetry in the stress tensor is needed to describe the experimental results. Coupling between the stress tensor and the heat flux has never been observed either in polyatomic gases (McCourt et al., 1991, p 229, Chapter 8) in the presence of an electric field. In media without electric and magnetic polarization the Maxwell electromagnetic pressure tensor is symmetric (De Groot and Mazur, 1962, p 378). De Groot and Mazur show also that the definition of the Maxwell stress tensor is to some extent arbitrary, but without polarization the stress tensor is modified neither from
0 1995 American Chemical Society
Ind. Eng. Chem. Res., Vol. 34,No. 10,1995 3589 the macroscopic nor from the statistical point of view. One might suspect that electromagnetic fields could lead to a nonsymmetric stress tensor in materials with inner (atomic, molecular) angular momenta, electric and magnetic multipoles. Anonsymmetric part of the stress tensor can then be obtained (De Groot and Suttorp 1972, p 641,but it has also been proven that in these systems the total internal angular momentuminner angular momentum plus orbital angular momentumis conserved. The total angular momentum is conserved not only in the classical approximation but also in the semirelativistic quantum statistical description (De Groot and Suttorp 1972,pp 65 and 508). Since the total angular momentum is conserved for the electromagnetic behavior of composite particles, no asymmetric part of the macroscopic stress tensor arises in electric and magnetic polarized media due to the applied electromagnetic fields. The electromagnetic atomic and molecular angular moments are balanced locally and do not show up on a macroscopic scale. Likewise, Condiff and Dahler (1964)have neglected in their discussion of the fluid mechanical effects of asymmetric stresses the atomic and submolecular forms of angular momentum. With reference to the theory of Irving and Kirkwood (19501,which predicts that for noncentral forces between monatomic molecules a nonsymmetric stress tensor follows, it has been suggested by McLennan (1966)that a symmetric stress tensor can always be constructed by adding divergenceless couple stresses. Irving and Kirkwood have already contemplated the possibility of adding divergenceless tensors to the stress tensor, but they remark that their definition of the stress tensor is the only choice in accord with the physical d e f ~ t i o nof the stress tensor as the force transmitted per unit area. Although the added divergenceless couple stress tensor does not affect the boundary conditions for the stress tensor, this added tensor has to be consistent with the boundary conditions for the couple stresses. The boundary conditions are not discussed in McLennan's paper. Snider and Lewchuk (1967)remark that the advantages of such formulations are not yet clear, since terms have to be added to the mass flux as well as to the external forces to obtain an equivalent set of balance equations. Capriz (1989) showed that the stress tensor will be symmetric by requiring only the null divergence for some kind of "structure tensor", provided that microinertia and body actions on the microstructure of the continuum are discarded. Condiff and Dahler (1964)obtained a symmetric stress tensor under equivalent conditions but remark, in addition, that the spin boundary conditions have to be satisfied. They argue that the asymmetric part of the stress tensor in fluid systems, having the velocity E, is coupled with the difference between the fluid vorticity '/z curl ii and the mean angular velocity G of the constituent particles at each point in the fluid. De Groot and Mazur (1962,p 309) argued that this antisymmetric contribution to the stress tensor usually vanishes quickly. The asymmetric contribution is not observable since the relaxation time is very short, and the quantities 2G and curl ii are equalized aRer a time that is much shorter than the macroscopic times related to the flow. However, the situation changes for interpenetrating continua in which couples can be exerted on the internal structure by a n external field (Brenner, 1970, 1984; Rosensweig, 1985). These continua are called polar continua. For example, in the ferrofluid suspension (Rosensweig, 1985),magnetic particles are
I
I
log(dvX
log AV
Figure 1. Space average of the masa AM in AV as a function of an arbitrary eontrol volume AV around some point in the media on a logarithmic scale. The smallest volume in which a fluctuation of the density cannot be measured mammpically is denoted by (Am*. Mawempic theories: region 111. High-energy physics and quantum mechanics: region I. Molecular theories: region 11.
suspended in a liquid and torques can be applied to these particles by a magnetic field not parallel to the vorticity. Then the suspended particles do not rotate with the vorticity, and a n asymmetric stress can be related to this difference in rotation. In turbulent flow of a ferrofluid no net continuous body torque is generated and no magnetic field effect is found (Kamiyama et al., 1983). The turbulent Reynolds stress remains symmetric. In the macroscopic description the physical quantities are based on tiny volumes (AW* just large enough that the quantities show no macroscopic fluctuation, a s is sketched in Figure 1. The continuum approach applies if the characteristic length scale of the variation of the macroscopic quantity is much larger than the length scale I* of the tiny volume. The macroscopic length scales (region 111)are also much larger than the length scales used in the atomic (region I) and molecular descriptions (region 11) of the quantities. The actions of material on material are given by the tractions and from the suggestion of Voigt in 1887 also by couples (Truesdell, 1966). The tractions and couples are linearly related to the unit normal of the surface by ii*nand iim, respectively, where ii denotes the outwardly directed unit normal, z the stress tensor with the pressure defined as negative, and rn the couple stress tensor. The internal angular momentum balance in the continuum description then reads
(1) where DIDt denotes the substantial derivative, the internal angular momentum per unit mass, t the external torque per unit mass, and 6 the third-order alternatingunit tensor. As the sum of inner and orbital electromagnetic angular momentum is conserved on a microscopic scale (De Groot and Suttorp, 1972),there are no contributions to the asymmetry of the macroscopic stress tensor from electromagnetic sources, and these microscopic contributions do not show up in eq 1. However, from the macroscopic point of view a Maxwell electromagnetic stress tensor can be defined, which might be asymmetric when the medium is polarized. For polarized media the definition of the Maxwell stress tensor is not unique (De Groot and Mazur, 1962),and consequently the possible nonsymmetric states of stress in a polarized medium subjected to an applied electromagnetic field are not uniquely defined.
3570 Ind. Eng. Chem. Res., Vol. 34, No. 10, 1995
With reference t o e e l s h e symmetry of the stress tensor follows only if L, t, and m vanish or can be neglected. Without these quantities the fluid is called nonpolar in the continuum theory. In the polar theories (Eringen, 1964; Eringen and Kafadar, 1976; Ariman et al., 1973) all quantities are considered bounded and independent of the size of the tiny subvolumes implied by the continuum approach. Then by integration over the tiny volume (AW* the magnitude of each integrated quantity is of the order 1*3, but it will be argued in the next paragraphs that L and m are of higher order in 1* than the contributjon of the stress tensor. Except for some suspensions t is also of higher order in 1*, and as a consequence a symmetric stress tensor results even if there are couple stresses. In suspensions in which gravity (Brenner, 1970, 1984) or a magnetic field (Rosensweig, 1985) exerts a couple on the suspended particles, an asymmetric stress is needed to counterbalance this couple. The global momentum balance can be applied locally, that is, to a tiny volume (Am* with the linear dimension I* and velocity li.By assum-ing that the acceleration p DliDt and the body forces pf are bounded, these contributions are of the order 1*3, while the contribution of the bounded surface tractions is of the order 1*2. Since 1* is very small, the surface integral of the tractions has to go to zero to balance the volume contributions. By assuming continuity of the tractions, the Cauchy flux theorem (Kuiken, 1994, p 63; Truesdell and Toupin, 1960, p 543) shows that the tractions are expressed as Z.X. The surface integral is transformed into a volume integral by using the Gauss theorem, and then the familiar momentum equation follows p
- + div n
Dij -= pf Dt
where all contributions stem from terms of the order 1*3 in the continuum limit. The terms in the internal angular momentum balance (1)are not all of the same order in I*. It will be argued that the contribution of the stress tensor comes from terms of the order l*3. The other contributions are of order 1*4 and 1*5 with the exception that in polar continua body couples of the order 1*3 might be found. Let us first consider the surface tractions in the lowest order approximation, which is strictly local from the macroscopic point of view. The tractions are assumed to be short-range forces having an influence sphere of radius h where hll*
