Variational principle for a viscous heat-conducting fluid with relaxation

Abstract

In continuum mechanics a large number [1] of variational principles are known, but only some of them can be used for continua with energy dissipation and heat conduction. A classical example is the Helmholtz minimum energy dissipation principle for creeping motion. It is well known [1] that there is no holonomic variational principle from which the Navier-Stokes equations follow. Nevertheless, the local potential method [2], which represents a certain variational approach to obtaining these equations, has been developed. The disadvantages of this method include the dependence of the functional on the varied variables satisfying the Navier-Stokes equation, which, essentially, must also be obtained by minimizing the functional. Accordingly, the local potential method requires the use of a certain convergent iterative procedure that minimizes the functional. In this article an alternative approach is considered. The variational principle is holonomic; therefore its extremals are not solutions of the Navier-Stokes equations. However, it is possible to construct the unknown solutions by imposing simple constraints on the extremals obtained. This approach requires an extension of the space on which the varied variables are defined.