A Variational Clebsch Formulation for Viscous Fluid Dynamics

This work introduces a consistent phenomenological variational formulation for viscous and dissipative fluid dynamics based on an extended Clebsch decomposition of the velocity field. Starting from the classical action principle for ideal fluids, a non-conservative contribution governing entropy production is incorporated to account for irreversible processes associated with shear/bulk viscosities and thermal conduction. The resulting framework naturally integrates fundamental thermodynamic variables—such as specific entropy, temperature, and internal energy—into the Eulerian Lagrangian description. The corresponding Euler-Lagrange equations yield a modified momentum balance equation that recovers the Navier-Stokes structure alongside additional thermodynamic driving forces.