In the variational principle leading to the Euler equation for a perfect fluid, we can use the method of undetermined multiplier for holonomic constraints representing mass conservation and adiabatic condition. For a dissipative fluid, the latter condition is replaced by the constraint specifying how to dissipate. Noting that this constraint is nonholonomic, we can derive the balance equation of momentum for viscous and viscoelastic fluids by using a single variational principle. We can also derive the associated Hamiltonian formulation by regarding the velocity field as the input in the framework of control theory.Comment: 15 page

Equations for a perfect fluid can be obtained by means of the variational principle both in the Lagrangian description and in the Eulerian one. It is known that we need additional fields somehow to describe a rotational isentropic flow in the latter description. We give a simple explanation for these fields; they are introduced to fix both ends of a pathline in the variational calculus. This restriction is imposed in the former description, and should be imposed in the latter description. It is also shown that we can derive a canonical Hamiltonian formulation for a perfect fluid by regarding the velocity field as the input in the framework of control theory.Comment: 15 page

We derive the equations governing the motion of fluids with interfaces in an inhomogeneous temperature filed, by employing a variational principle. Generally, the Lagrangian of a fluid is given by the kinetic energy density minus the internal energy density. The dynamics also obeys the equation of entropy. Then the necessary condition for minimizing an action with subject to the constraint of entropy yields the equation of motion. In this way, this method provides the equations of a fluid when the kinetic and internal energies and the equation of entropy are given. However, it is sometimes to know the proper equation of entropy. Our main purpose of this article is to determine 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.