Reduced Second Order Constitutive Theory of Fluids

A fully second order continuum theory of fluids has been previously developed (Continuum Mech. Thermodyn. (2022) 34:185-215). The constitutive equations depend on density, temperature and velocity, and their derivatives up to second order and satisfy the second law. These equations have been shown to be consistent but much more general than other known results. Since they contain a number of material derivatives terms of a number of quantities (e.g., strain rate and temperature gradient), we call this form of the equations as original. Here, using order of magnitude analysis in viscosity, we obtain reduced equations by replacing the material derivatives terms with the spatial gradients. This substitution maintains second order accuracy in viscosity. We also obtain corresponding simpler equations valid for simple fluids. The reduced constitutive equations equations are used to obtain results for the shock structure and thermal stress problems and compare the results with those from kinetic theory.