Continuous Media with Microstructure by Bettina Albers

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By Bettina Albers

The contributions to the e-book challenge a variety of features of extension of classical continuum types. those extensions are on the topic of the looks of microstructures either normal in addition to those created by means of tactics. To the 1st category belong numerous thermodynamic types of multicomponent platforms corresponding to porous fabrics, composites, fabrics with microscopic heterogeneities. To the second one category belong essentially microstructures created by way of section ameliorations. Invited authors conceal either fields of thermodynamic modeling and mathematical research of such continua with microstructure. particularly the subsequent matters are lined:

  • thermodynamic modeling of saturated and unsaturated porous and granular media,
  • linear and nonlinear waves in such fabrics,
  • extensions of constitutive legislation by way of inner variables, better gradients and nonequilibrium fields,
  • stochastic methods in porous and fractal fabrics,
  • thermodynamic modeling of composite fabrics,
  • mathematical research of multicomponent structures,
  • phase alterations in solids.

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1 Dimensionless velocities and diffusion temperature fluxes of the constituents versus time. where τv and τT represent relaxation times that for ideal gas assume the expression τv = ρ1 ρ2 T0 ; 0 ρ ψ11 τT = kρ1 ρ2 T02 . 0 (ρ m (γ − 1) + ρ m (γ − 1)) θ11 1 2 2 2 1 1 Starting from these solutions, other field variables can be obtained by means of defining equations: ρ1 v1 + ρ2v2 = ρ v = 0. (1) (2) (1) (2) ρ1 cV T1 + ρ2 cV T2 = (ρ1 cV + ρ2 cV )T (1) (2) = (ρ1 cV + ρ2cV )T0 . It is obvious that, due to dissipative character of the system, all the non-equilibrium variables exponentially decay and converge to their equilibrium values.

Grad θ = 0 and V = 0, we can write r = −φ RV − φ Ggrad θ + o(2), (19) where R and G are material parameters, which are tensor-valued functions of (θ , φ , df , Fs ). Therefore, by the use of (17) through (19), the equations of motion for the fluid constituent (16)1 in the linear theory becomes df v` f = −gradP − k + df g, where k = RV + Ggrad θ − ∂ ψf P grad df . − df df ∂ df (20) (21) The equation of motion for the fluid constituent (20) is a generalization of Darcy’s law. It is shown in [2], that for the case of classical Darcy’s experiment (see [3]), the equation reduces to the original form of the Darcy’s law, where the reciprocal of the resistivity constant R is call the permeability tensor.

There exists a supplementary balance law of entropy with an entropy production non negative:: ∂t ρ S + ∂i(ρ Svi + ϕ i ) = Σ ≥ 0, where ρ S = ∑nα =1 ρα Sα , ϕ i and Σ are the entropy density, the non-convective entropy flux and the entropy production, respectively. For example in the case of a mixture of Eulerian fluids, the entropy production becomes [11]: Σ= n−1 ∑ b=1 μn − 12 u2n μb − 12 u2b − Tn Tb τˆb + un ub − Tn Tb ˆ b+ ·m 1 1 − Tb Tn eˆb ≥ 0. (6) This inequality allows to obtain the following structure of production terms.

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