Viscous pressure tensor

Notation-. T is the temperature, Vi the fluid velocity, II,j the viscous pressure tensor, Jg the heat current density, p its chemical potential, the current density of molecular species a, v J the stoichiometric coefficient (13), and Wp the speed of reaction p. 

Polymer solutions may have the memory effects observed in viscoelastic phenomena. This requires additional relaxation terms in the constitutive equations for the viscous pressure tensor, which may be affected by the changes in the velocity gradient. Besides this, the orientation and stretching of the macromolecules may have an influence on the flow. 

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 respective relaxation times of diffusion flux and viscous pressure tensor are... 

Here, the first equation is the usual Fourier law, the second relates the viscous pressure tensor to the internal variable W, and the last is the evolution of the internal variable. The matrix of the transport coefficients /.(/ is positive definite... 

This is also true for mass flux. Only those terms are important which are associated with the force corresponding to symmetrical non-divergent part of the viscous pressure tensor and space derivative of X . These can become important in capillaries. It seems that linear transport equations have larger domain of validity than expected. 

Complex systems such as solutions of macromolecules, magnetic hysteresis bodies, visco-elastic fluids, polarizable media require some extra variables in the fundamental equation of Gibbs. Dissipative fluxes (heat, diffusion, viscous pressure tensor and viscous pressure) are included in the Gibbs function in new formalism. In the formalism of extended irreversible thermodynamics (EIT), the dissipative fluxes are the independent variables in addition to classical variables of thermostatics [1]. 

Beyond the linear range, it has been shown by Rastogi and Misra [3] that the nonlinear flux equations for mass and heat transport involve non-linear terms containing forces corresponding to the anti-symmetric part of the viscous pressure tensor and the force corresponding to the divergent part of the viscous tensor responsible for bulk viscosity as discussed in Chapter 6. In case of spatially non-uniform systems, a proper expression for Gibbs function should take into account the existence of physical forces as indicated above. 

Based on the extended irreversible thermodynamics, the Gibbs equation for a simple single-component fluid in the presence of a viscous pressure tensor P (up to the second order in P ) is... 

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... 

Table 2.1. Bcdance equations of a non-polarized fluid mixture. B magnetic field, E electrical field, diffusion flux, q heat flux, i diffusion current, Ppressure tensor, II viscous pressure tensor, chemical...

Table 3.1. Balance equations for a polarized reactive fluid medium. H magnetic displacement vector, D electrical displacement vector, B magnetic field, E electrical field, P = D-E E electrical polarization (p per unit mass), M = B-H magnetic polarization (m per unit mass), ( ) following the motion, (2) relative to the dipoles (section 3.1), Jok diffusion flux, q heat flux, i diffusion current, p pressure tensor, pi viscous pressure tensor, chemical production rode, fi distant forces other than those due to the electromagnetic field...

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