Variational formulations for viscous flows which lead to the Navier-Stokes
equation are examined. Since viscosity leads to dissipation and, therefore, to
the irreversible transfer of mechanical energy to heat, thermal degrees of
freedom have been included in the construction of viscous dissipative
Lagrangians, by embedding of thermodynamics aspects of the flow, such as
thermasy and flow exergy. Another approach is based on the presumption that the
pressure gradient force is a constrained force, whose sole role is to maintain
the continuity constraint, with a magnitude that is minimum at every instant.
From these considerations, Lagrangians based on the minimal energy dissipation
principal have been constructed from which the application of the
Euler-Lagrange equation leads to the standard form of the Navier-Stokes
equation directly, or at least they are capable of generating the same
equations of motion for simple steady and unsteady 1 D viscous flows. These
efforts show that there is equivalence between Lagrangian, Hamiltonian, and
Newtonian mechanics as far as the derivation of the Navier-Stokes equation is
concerned. However, one of the conclusions is that the attractiveness of the
variational approach in more complex situations is still an open question for
the applied fluid mechanician.Comment: 13 page