On existence of weak solution to a model describing incompressible mixtures with thermal diffusion cross effects
Abstract
We present a model describing unsteady flows of a heat conducting mixture composed from L constituents in two and three dimensional bounded domain. We assume that the flow of the mixture is described only by the barycentric velocity, and that the fluid is non-Newtonian.
In addition, we assume that the diffusion flux depends also on the temperature gradient, describing the Soret effect, and that the heat flux depends also on the chemical potentials gradient, describing the Dufour effect. We briefly show under which assumptions on the constitutive equations the model obeys the first and the second laws of thermodynamics and for a large class of physically well-motivated constitutive relations we establish the existence of a weak solution.
For simplicity we restrict ourselves only onto the linear models, i.e., the diffusion and the heat flux depend linearly on the temperature and chemical potentials gradients. (C) 2014 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim