Antisymmetric stress

Four novel approaches to contemporary studies of suspensions are briefly reviewed in this final section. Addressed first is Stokesian dynamics, a newly developed simulation technique. Surveyed next is a recent application of generalized Taylor dispersion theory (Brenner, 1980a, 1982) to the study of momentum transport in suspensions. Third, a synopsis is provided of recent studies in the general area of fractal suspensions. Finally, some novel properties (e.g., the existence of antisymmetric stresses) of dipolar suspensions are reviewed in relation to their applications to magnetic and electrorheolog-ical fluid properties. 

D. Antisymmetric Stresses, Internal Spin Fields, and Vortex Viscosity in Magnetic Fluids... 

In their pioneering paper on laminated plates, Reissner and Stavsky investigated an approximate approach (in addition to their exact approach) to calculate deflections and stresses for antisymmetric angie-ply laminated plates [5-27]. Much later, Ashton extended their approach to structural response of more general unsymmetrically laminated plates and called it the reduced stiffness matrix method [5-28]. The attraction of what is now called the Reduced Bending Stiffness (RBS) method is that an unsymmetrically laminated plate can be treated as an orthotropic plate using only a modified D matrix in the solution, i.e.,... 

As stressed at the end of the preceding section, there is no proof that the asymmetric part of the transport matrix vanishes. Casimir [24], no doubt motivated by his observation about the rate of entropy production, on p. 348 asserted that the antisymmetric component of the transport matrix had no observable physical consequence and could be set to zero. However, the present results show that the function makes an important and generally nonnegligible contribution to the dynamics of the steady state even if it does not contribute to the rate of first entropy production. 

Independent evidence for local stress is provided by measurements of the frequency of the antisymmetric stretching vibration of C02 eliminated in crystalline undecanoyl peroxide on photolysis (253). The results indicate that local pressures of tens of kilobars are established. This is very much larger than the lattice strain that has been considered as developing during the polymerization of diacetylenes (254). McBride points out (246) that virtually all reactions cause changes in shape that should create local stress in a solid, so that subsequent... 

In the original design of Binnig and Smith (1986), the deflection voltage is applied to only one of the four quadrants. The stress is no longer antisymmetric with respect to the y = 0 plane, as shown in Fig. 9.11. The general form of the stress should have an additional term ... 

It is important to stress that Eq. (82) does not satisfy Eqs. (48) and (49) because the last boundary term on the rhs of Eq. (82) (vy A//kT) is not antisymmetric against time reversal. Van Zon and Cohen [61-63] have analyzed in much detail work and heat fluctuations in the NESS. They find that work fluctuations satisfy the exact relation... 

It is to be stressed that, although the two-electron submatrix elements in (14.63) and (14.65) are defined relative to non-antisymmetric wave functions, some constraints on the possible values of orbital and spin momenta of the two particles are imposed in an implicit form by second-quantization operators. Really, tensorial products (14.40) and (14.42), when the sum of ranks is odd, are zero. Thus, the appropriate terms in (14.63) and (14.65) then also vanish. 

Owing to the indistinguishability of electrons, the wavefunction of a molecule s electron-cloud must be antisymmetric in the coordinates of the electrons. Hence, in the orbital-approximation, the wavefunction of a molecule (whose state corresponds to a set of complete electronic shells) can be expressed as a Slater-determinant, each column or row of which is written in terms of a single spin-orbital 8>. As pointed out, however, by Fock 9> and Dirac 10>, and later stressed by Lennard-Jones n> and Pople 12 the orbitals of a Slater-determinant are not uniquely determined, mathematically. 

As usual, its isotropic portion, corresponding to the average pressure, is first removed as being without consequence for the incompressible flows of interest to us here. Its antisymmetric part, which is related to any external-body couples exerted on the suspended particles, is also separated out. These operations result in the symmetric and traceless average deviatoric stress,... 

Suspended spherical particles, each containing a permanently embedded dipole (e.g., magnetic), are unable to freely rotate (Brenner, 1984 Sellers and Brenner, 1989) in response to the shear and/or vorticity field that they are subjected to whenever a complementary external (e.g., magnetic) field acts on them. This hindered rotation results from the tendency of the dipole to align itself parallel to the external field because of the creation of a couple arising from any orientational misalignment between the directions of the dipole and external field. In accordance with Cauchy s moment-of-momentum equation for continua, these couples in turn give rise to an antisymmetric state of stress in the dipolar suspension, representable as the pseudovector Tx = — e Ta of the antisymmetric portion Ja — (T — Tf) of the deviatoric stress T = P + Ip. 

With the theoretical underpinning provided by Rosina s theorem, Mazziotti [80] proposes a new reconstruction procedure in which the p-DM is generated from the 2 DM by imposing contraction and p-ensemble representability conditions. For a p-DM to be p-representable, Mazziotti stresses that it must be Hermitian, antisymmetric and positive semidefinite. By his procedure he establishes contact with the earlier studies of Valdemoro [81] (see also the following chapter) and of Nakatsuji and Yasuda [82], His principal conclusion, based on numerical examples, is that... 

Finally, it should be stressed that the position of an index in a sequence is significant, since all operators (and coefficients) will eventually be written in antisymmetrized form. We can shed some light on the sign change for the transformation operator for covariant and contravariant tensors by examining the following equations ... 

This theory has two notable features. The nonlocality of molecular interaction is reflected by the ellipticity of Eq. (19) [cf Eq. (15)]. Thus, the LCP configuration is globally coupled by distortional elasticity. In addition, the elastic stress tensor is asymmetric. The mean-field torque on LCP molecules amounts to a volume torque on the material, which modifies the usual conservation of angular momentum. The antisymmetric part of the stress tensor precisely balances the volume torque computed by averaging the molecular torque. ... 

We have adopted the convention that the position vector be written in component form as r = XiCj. Now we recognize that upon factoring out Xj, three of these terms are a restatement of linear momentum balance, and are thus zero. In addition, the term involving vj also clearly yields zero since it is itself symmetric and is contracted with the antisymmetric Levi-Cevita symbol, The net result is that we are left with = 0 which immediately implies that the stress tensor itself... 

There are two proper explanations, one based on physical intuition and the other based on the principle of material objectivity. The latter is discussed in many books on continuum mechanics.19 Here, we content ourselves with the intuitive physical explanation. The basis of this is that contributions to the deviatoric stress cannot arise from rigid-body motions -whether solid-body translation or rotation. Only if adjacent fluid elements are in relative (nonrigid-body) motion can random molecular motions lead to a net transport of momentum. We shall see in the next paragraph that the rate-of-strain tensor relates to the rate of change of the length of a line element connecting two material points of the fluid (that is, to relative displacements of the material points), whereas the antisymmetric part of Vu, known as the vorticity tensor 12, is related to its rate of (rigid-body) rotation. Thus it follows that t must depend explicitly on E, but not on 12 ... 

Because in Equation 12.17 is symmetric, the antisymmetric component of the geometrical tensor and the antisymmetric component of the electronic stress tensor t >" ( ) should cancel each other out ... 

Antisymmetric electronic stress tensor drives the electron spin through vorticity."... 

This is called the quantum electron spin vorticity principle the time evolution of the electron spin s is driven by the antisymmetric component of the electronic stress tensor x through the vorticity rot . The quantum electron spin vorticity principle is schematically shown in Figure 12.1. 

It should be noted that the vorticity rot appears as a component of the electronic momentum as found in Equation 12.36. This proves the important role of the antisymmetric component of the electronic stress tensor X. It may be further proved that the symmetric component of the electronic stress tensor x plays an important role as tension x = divx compensating the Lorentz force L as... 

Fig. 6.18 Elastic ( ) and Viscoelastic (—) profiles of the stress Tg (in MPa) in an antisymmetric [0/90/04/904/0/90]x AS/3502 composite laminate at various stages of exposure to ambient relative humidity, (a) After a 4h linear cool down from the cure temperature of 452 K (355°F) to the conditioning temperature of 339 K (150°F). (b) After 0.323 days of conditioning at RH = 95%. (c) After 9.76 days at RH = 95%. (d) At saturation, (e) After 0.081 days of drying at RH = 0% past saturation, (f) After 16.96 days of drying at 0% RH past saturation, (g) At total desorption past saturation. (Harper 1983)...

Suppose we wish to know the dipole moment of, say, the HCl molecule, the quantity that tells us important information about the charge distribution. We look up the output and we do not find anything about dipole moment. The reason is that all molecules have the same dipole moment in any of their stationary state y, and this dipole moment equals to zero, see, e.g., Piela (2007) p. 630. Indeed, the dipole moment is calculated as the mean value of the dipole moment operator i.e., ft = (T l/i l ) = ( F (2, q/r,) T), index i runs over all electrons and nuclei. This integral can be calculated very easily the integrand is antisymmetric with respect to inversion and therefore ft = 0. Let us stress that our conclusion pertains to the total wave function, which has to reflect the space isotropy leading to the zero dipole moment, because all orientations in space are equally probable. If one applied the transformation r -r only to some particles in the molecule (e.g., electrons), and not to the other ones (e.g., the nuclei), then the wave function will show no parity (it would be neither symmetric nor antisymmetric). We do this in the adiabatic or Born-Oppenheimer approximation, where the electronic wave function depends on the electronic coordinates only. This explains why the integral ft = ( F F) (the integration is over electronic coordinates only) does not equal zero for some molecules (which we call polar). Thus, to calculate the dipole moment we have to use the adiabatic or the Born-Oppenheimer approximation. 

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