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

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 mu. 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.