A Variational Principle for Dissipative Fluid Dynamics

TL;DR

This paper develops a variational principle framework for dissipative fluid dynamics, extending classical methods to include nonholonomic constraints for viscous and viscoelastic fluids, and derives related Hamiltonian formulations.

Contribution

It introduces a novel variational principle for dissipative fluids using nonholonomic constraints, enabling derivation of momentum balance and Hamiltonian formulations.

Findings

Derived the momentum balance equations for viscous and viscoelastic fluids.

Established a Hamiltonian formulation incorporating dissipation.

Extended variational methods to nonholonomic constraints in fluid dynamics.

Abstract

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.