I'm trying to get to the classical compressible Navier-Stokes equation from the relativistic tensors in the Landau/Eckart frame. I understand these have their own problems about entropy.
In several sources, including:
- Landau and Lifshitz Volume 6
- Weinberg Gravitation Part 1 chapter 11
- this answer here
and various others, they all derive the shear part of the stress-energy tensor to have preceding minus signs.
$$ \tau_{\mu\nu} = -\eta \left[\partial_\mu u_\nu+\partial_\nu u_\mu - u_\mu u^\sigma \partial_\sigma u_\nu - u_\nu u^\sigma\partial_\sigma u_\mu\right] - \lambda\partial_\sigma u^\sigma(g_{\mu\nu}-u_\mu u_\nu) $$
Where I have replaced the bulk viscosity coefficient with $\lambda = \zeta - \frac{2}{3}\eta$. Weinberg says this is to have a positive "rate of entropy product per unit volume at P" (From page 55 where the formula is for $\frac{\partial S^\alpha}{\partial x^\alpha}$). I'm happy so far.
My question is how does this match back onto the classical Navier-Stokes equation? I seem to get a different sign and I can't understand why. I'll use downstairs indices on the stress-energy tensor to be consistent with the sources and remove the pressure term for simplicity.
\begin{equation} T_{\mu\nu} = w v_\mu v_\nu - \lambda \left(g_{\mu\nu} - v_\mu v_\nu \right)\partial_\sigma v^\sigma - \eta\left(\partial_\mu v_\nu + \partial_\nu v_\mu - v_\mu v^\sigma \partial_\sigma v_\nu - v_\nu v^\sigma\partial_\sigma v_\mu \right). \end{equation}
Using $\partial^\mu T_{\mu\nu} = 0$ and dropping terms cubic in $v$ (suppressed by $\frac{1}{c^2}$) we get
$$ -w v_\mu \partial^\mu v_\nu = w v_\nu\partial^\mu v_\mu - \lambda\partial_\nu \partial_\sigma v^\sigma - \eta \partial^\mu \partial_\mu v_\nu - \eta\partial_\nu\partial^\mu v_\mu.$$
Now setting $\nu=j$ to get the spatial components ($v_\nu \to -\mathbf v$), and using a non-relativistic limit so that
$$v^\mu \partial_\mu \approx \frac{\partial}{\partial t} + \mathbf v \cdot \nabla \quad \text{and} \quad \partial_\mu v^\mu \approx \nabla \cdot v \quad \text{and} \quad \partial_\mu \partial^\mu = \square \approx -\nabla^2 .$$
We get to
$$ w \frac{\partial \mathbf v}{\partial t} + w (\mathbf v \cdot \nabla) \mathbf v = -w \mathbf v (\nabla \cdot \mathbf v) - \eta\nabla^2 \mathbf v - (\eta + \lambda)\nabla (\nabla \cdot \mathbf v).$$
This so nearly looks like the compressible Navier-Stokes except the $\eta$ and $\lambda$ have a negative sign. But Weinberg insists that the negative is required. What's gone wrong in my working?
