Tensor Maxwell pressure

Maxwell pressure tensor, and whose production rate per unit volume is equal to the opposite of the Lorentz force... 

In reality, the Maxwell pressure tensor in relation to the Galilean frame of reference which, at time t and at the point in question, has the velocity v of the fluid, is expressed not by equation [3.60], but rather ... 

The following summary is from Jou and Casas-Vazquez (2001). In the extended nonequilibrium thermodynamics for a binary liquid mixture, the viscous pressure tensor Pv and the diffusion flux J are considered as additional independent variables. The viscous pressure tensor, Pv, by the simplest Maxwell model, is defined by the following constitutive equation ... 

The interaction force P can be calculated by integrating the excess osmotic pressure An and the Maxwell stress tensor T over an arbitrary closed surface 21 enclosing either one of the two interacting particles (Fig. 8.3), which is written as [8]... 

Note that Bell et al. [7] derived Eq. (11.73) by the usual method, that is, by integrating the osmotic pressure and the Maxwell tensor over an arbitrary closed surface enclosing either sphere. Equation (11.73) agrees with the leading term of the exact... 

Note, that if stress is reduced to pressure P,T = —PI, (usual in fluids) this definition gives the classical result (3.203) F = gl, see (3.199). The Eshelby tensor, e.g. gives the condition of phase equilibria (Maxwell relation—equality of chemical potentials (2.116) in fluid phases), namely equality of f[n on both sides of equilibrated solid phases (n is the normal to phase boundary) and may be also used to describe surface phenomena, dislocations, etc. [1, 4, 87]. Eshelby tensors may also be defined in mixtures [2, 3]. 

For ionic surfactant solution the body force tensor, Pb, is not isotropic - it is the Maxwell electric stress tensor, i.e. Pb = f6bEE - i6jE l2, where E = -V is the electric field (Landau and Lifshitz 1960). The density of the electric force plays the role of a spatial body force, f, in the Navier-Stokes equation of motion (3). In the lubrication approximation the pressure in the continuous phase depends on the vertical coordinate, z, only through its osmotic part generated from the electric potential and the pressure in the middle plane (or the pressure, pn, corresponding to the case of zero potential) ... 

In Ihe extended nonequilibrium Ihermodynamics for a binary liquid mixlure, Ihe viscous pressure tensor P and Ihe diffusion flux J are considered as addilional independenl variables. The viscous pressure tensor P, by Ihe simplesl Maxwell model, is defined by Eqn (14.30). In extended nonequilibrium Ihermodynamics of polymer solulions, Ihe generalized extended Gibbs equalion for a fluid characterized by internal energy u and viscous pressure P is... 

These results imply that the extension of equilibrium theories to nonequiUbrium states is not always valid in a straightforward way. Particularly, the diffusion tensor is proportional to the components of the pressure tensor or equivalently to the velocity gradient Vvq, which implies that the amplitude of the noise in the dynamics of the tagged particle is not simply thermal as in equilibrium since the diffusion tensor cannot be characterized entirely by the thermodynamic temperature. In similar manner, Eq. (5) does not depend on the irreversible heat flux. This is an anomaly of the Maxwell potential, for other potentials there will be an additional contribution to the drift vector that would depend on the any temperature gradient in the fluid. 

Equation [19] expresses the condition of equilibrium as a balance between the electrostatic pressure on ions, where the first term is simply the Maxwell electric field stress tensor,and the osmotic pressure with respect to bulk, given by the second term. Only in special cases can the integral in Eq. [17] be evaluated analytically. Several analytical approximations to the planar PB equation for asymmetric electrolytes have been suggested. " We now present three examples possessing exact analytical solutions beginning with the classic Gouy-Chapman solution. 

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