Viscous stresses and heat conduction

It has been assumed to this point that there are no viscous stresses, nor any heat conduction. Thus, the dynamics of the ideal fluid of the preceding chapters are compatible with an isotropic distribution of molecular velocities. In fact, anisotropy is always present in a real assembly of molecules, owing to the walls of the fluid container, fields of force or sources of heat. The Navier-Stokes equations for a real fluid may be derived from Boltzmann's equation for a dilute gas using the Chapman-Enksog expansion (Chapman and Cowling, 1970), which assumes a molecular velocity distribution close to an isotropic equilibirum. A simpler derivation, requiring less physical insight, follows from the general principles of continuum mechanics by adopting Newton's and Fourier's laws as the constitutive relations. The essential aspect of these constitutive relations is that they are local in the Eulerian framework: the viscous stress tensor is proportional to the Eulerian rate of strain tensor, while the heat flux is proportional to the Eulerian temperature gradient. The Navier-Stokes equations are accordingly expressed naturally in Eulerian form, while the Lagrangian form can only be derived by “cheating.” That is, it cannot be derived from Boltzmann's equation. Cheating can be minimized (see Aside in Section 3.2), but in the interest of moving forward, let us cheat in full.