This is for @Chesx
For non-polymeric fluids (liquids and gases), Newton's law of viscosity very accurately describes the state of stress within a fluid as a function of the local fluid deformation and local rate of deformation. It also reduces to the ideal gas law (or other thermodynamics equilibrium equation of state) in the limit of low rates of deformation (i.e., low velocity gradients).
In Cartesian coordinates, the components of the compressive stress tensor are related to the thermodynamic pressure p(v,T) and the fluid velocity gradients by:
$$\sigma_{xx}=p(v,T)-\frac{2}{3}\mu\left[2\frac{\partial u_x}{\partial x}-\frac{\partial u_y}{\partial y}-\frac{\partial u_z}{\partial z}\right]$$
$$\sigma_{yy}=p(v,T)-\frac{2}{3}\mu\left[-\frac{\partial u_x}{\partial x}+2\frac{\partial u_y}{\partial y}-\frac{\partial u_z}{\partial z}\right]$$
$$\sigma_{zz}=p(v,T)-\frac{2}{3}\mu\left[-\frac{\partial u_x}{\partial x}-\frac{\partial u_y}{\partial y}+2\frac{\partial u_z}{\partial z}\right]$$
$$\sigma_{xy}=\mu\left[\frac{\partial u_y}{\partial x}+\frac{\partial u_x}{\partial y}\right]$$
$$\sigma_{xz}=\mu\left[\frac{\partial u_z}{\partial x}+\frac{\partial u_x}{\partial z}\right]$$
$$\sigma_{yz}=\mu\left[\frac{\partial u_z}{\partial y}+\frac{\partial u_y}{\partial z}\right]$$where the u's are the components of gas velocity, v is specific volume, and $\mu$ is the gas viscosity.
In the limit of low gas velocities, these equations reduce to the isotropic thermodynamic equilibrium state values of the stress components:$$\sigma_{xx}=\sigma_{yy}=\sigma_{zz}=p(v,T)$$and$$\sigma_{xy}=\sigma_{xz}=\sigma_{yz}=0$$
For a fluid satisfies the ideal gas thermodynamic equation of state $$p=\frac{RT}{v}$$
We also see from all this that the isotropic thermodynamic pressure p does not capture the state of stress in a gas that is deforming at finite rate. In fact, what we call the pressure is not the force per unit area on an internal surface of arbitrary orientation, but rather it is merely the isotropic part of the overall stress tensor.
