Friction term in Navier-Stokes equation

Question

The friction term in Navier-Stokes equation assumes that the viscosity coefficients are the same for the longitudinal and transverse directions. This doesn't seem intuitive, because the former is essentially a bulk modulus while the latter doesn't involve any compression of the fluid. How is the assumption justified?

Answer 1

Actually, there are two different viscosity coefficients. You can see this from the stress tensor $$ \sigma_{ij} = -p_0 \delta_{ij} + \eta \left( \frac{\partial v_i}{\partial x_j} + \frac{\partial v_j}{\partial x_i} - \frac{2}{3} \delta_{ij} \frac{\partial v_k}{\partial x_k} \right) + \zeta \delta_{ij} \frac{\partial v_k}{\partial x_k} $$ which has the two coefficients of viscosity $\eta$ and $\zeta$ (see Landau & Lifshitz, Fluid Mechanics, for example). The pressure $p_0$ is given by the thermodynamic equation of state, but this is not the whole pressure $p$. The latter is given by the mean normal stress $$ p = - \frac{1}{3} \sigma_{ii} = p_0 - \zeta \frac{\partial v_k}{\partial x_k} $$ so that the stress tensor is $$ \sigma_{ij} = -p \delta_{ij} + \eta \left( \frac{\partial v_i}{\partial x_j} + \frac{\partial v_j}{\partial x_i} - \frac{2}{3} \delta_{ij} \frac{\partial v_k}{\partial x_k} \right) .$$

That's why sometimes you don't see the coefficient $\zeta$ (often called second viscosity) in the Navier-Stokes equation. It is hidden in the pressure, but it's there.