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A variational formulation for dissipative fluids with interfaces in an inhomogeneous temperature field

We propose a formalization for dissipative fluids with interfaces in an inhomogeneous temperature field from the viewpoint of a variational principle. Generally, the Lagrangian of a fluid is given by the kinetic energy density minus the internal energy density. The necessary condition for minimizing an action with subject to the constraint of entropy yields the equation of motion. However, it is sometimes to know the proper equation of entropy. Our main purpose is to obtain it by using the three requirements, which are a generalization of Noether's Theorem, the second law of thermodynamics, and well-posedness. To illustrate this approach, we investigate several phenomena in an inhomogeneous temperature field. In the case of vaporization, diffusion and the rotation of a chiral liquid crystals, we clarify the cross effects between the entropy flux and these phenomena via the internal energy.