Invariance of flows in doubly-connected domains with the same modulus

Abstract

We consider systems of elliptic partial differential equations in divergence form with Dirichlet's boundary conditions in doubly-connected domain of the plane with modulus μ. We prove an invariance property of the corresponding global flows in the class of domains with the same modulus. Applications are given to the problem of electrical heating of a conductor whose thermal and electrical conductivities depend on the temperature and to the flow of a viscous fluid in a porous medium, taking into account the Soret and Dufour's effects.Keywords Systems of PDE in divergence form · Doubly-connected plane domain · Thermistor problem · Porous media · Darcy law · Soret effect · Dufour effect Mathematics Subject Classification 30E30 · 35J66 · 35J57

Weak solutions for a thermoelectric problem with power-type boundary effects

“…Existence of solutions for parabolic-elliptic systems with nonlinear no-flux boundary conditions is not a new idea if taking constant coefficients into account [4]. Application of elliptic PDE system in divergence form with Dirichlet boundary conditions in doubly-connected domain of the plane are given in [7] to the problem of electrical heating of a conductor whose thermal and electrical conductivities depend on the temperature and to the flow of a viscous fluid in a porous medium, taking into account the Soret and Dufour effects. In [8], the authors deal with a traditional RLC circuit in which a thermistor has been inserted, representing the microwave heating process with temperature-induced modulations on the electric field.…”