On existence of weak solution to a model describing incompressible mixtures with thermal diffusion cross effects

• Published in: Zeitschrift für angewandte Mathematik und Mechanik 2015-06, Vol.95 (6), p.589-619
• Main Authors: Bulíček, M., Havrda, J.
• Format: Article
• Language: English
• Subjects: Chemical potential; Constitutive relationships; cross-effects; Diffusion; Dufour effect; existence of a weak solution; Heat; heat conducting fluid; Heat flux; Heat transfer; incompressible fluid; Mathematical models; Mixture; non-Newtonian fluid; Soret effect; Temperature gradient; Three dimensional
• ISSN: 0044-2267; 1521-4001
• DOI: 10.1002/zamm.201300101

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. The authors present a model describing unsteady flows of a heat conducting mixture composed from L constituents in two and three dimensional bounded domain. They assume that the flow of the mixture is described only by the barycentric velocity, and that the fluid is non‐Newtonian. In addition, they 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.