Dear Mohammad, Navier-Stokes equations are derived on particular hypothesis about the expression of internal fluid Cauchy stress; if you change these hypothesis, you change the equations because you are studying other typed of fluids (e.g. thermoelastic fluids). Gianluca

The Navier-Stokes equations for ideal gases can be derived from the Boltzmann equation under the assumption of small Knudsen numbers, by what is known as Chapman-Enskog expansion to first order. If you include higher orders, you arrive at what is known as Burnett equations (2nd order), or super-Burnett equations (3rd order). Higher orders describes effects that appear in rarefied gases. The Burnett-type equations have the form you are looking for, but are unstable. There are alternative approaches to this based on moment methods, but these requires more equations and variables.

See:

Article Macroscopic Transport Models for Rarefied Gas Flows: A Brief Review

Article Modeling Nonequilibrium Gas Flow Based on Moment Equations

I agree with Gianluca, the original system of the NS equations requires closure model such as Newton and Fourier constitutive laws, as well as a model for the gas properties. In the common formulation of the NS, no third order derivatives appear in the mass, momentum and total energy. However, we disregard some effects that can be conversely relevant in specific flow conditions of fluid model.

Then, some specific models for flow motion can have third order derivatives as in the Korteweg-de Vries equations