Navier-Stokes equations in tensorial form on a general coordinate system

Question

How to write the classical Navier-Stokes equations in tensorial form on a general coordinate system? Any references?

Answer 1

This is the best I have come up with. First, the Euler equation: Let $L_uv = u\cdot \nabla v - v\cdot\nabla u$ denote the Lie derivative. On the space of divergence free vector fields, the formal adjoint of this operator is $$ L_u^* v = -u\cdot\nabla v - v \cdot (\nabla u)^T - \nabla q $$ where $q$ is a scalar field so that $L_u^* v$ is divergence free via the Hodge decomposition. In the particular case $u=v$ you get $$ L_u^* u = -u\cdot\nabla u - \nabla (q + \tfrac12 |u|^2) $$ and so pressure is $q + \frac12 |u|^2$. (Note $q$ is not the head pressure from Bernoulli's formula - the signs are wrong.) Also remember that the formula for $L_u^* u$ is treating $u$ both as a vector field in the subscript, and as a 1-form (i.e. covariant vector). (I Also remember that the formula for $L_u^* u$ is treating $u$ both as a vector field in the subscript, and as a 1-form (i.e. covariant vector). I have seen the notation $u^\flat$ in this context, which I think converts $u$ from a vector to a 1-form.

The viscosity term is rather difficult. This is because the Laplacian is not of a scalar, and so the usual formulas for Laplacian in other coordinate systems don't apply.

Anyway, $\Delta u = - d^* d u$, where $d$ is the exterior derivative of 1-forms, and $d^*$ is the adjoint operator on rank 2 contravarient tensors. Normally the Beltrami-Laplacian operator is $-\Delta = d d^* + d^* d$, but since $u$ is divergence free $d d^* u = 0$. In three dimensions $\Delta u = - \text{curl}(\text{curl} u)$.

I tried really hard to write the Navier-Stokes equation in spherical coordinates using this approach. But in the end I found it easier just to write everything in Mathematica, and have it grind through the formulas.