User:GreyTrafalgar/Sandbox/Derivation of the Navier-Stokes equations

Beginning with conservation of momentum in differential form:

$\rho {\frac {D\mathbf {v} }{Dt}}=\rho \mathbf {b} +\nabla \cdot {\boldsymbol {\sigma }}$

or in Einstein summation convention:

$\rho {\frac {Dv_{i}}{Dt}}=\rho b_{i}+\nabla _{j}\sigma _{ij}$

where $\sigma _{ij}$ is the surface stress tensor. In the case of a Newtonian fluid, the stress tensor is defined as:

$\sigma _{ij}=\mu \left({\frac {\partial v_{i}}{\partial x_{j}}}+{\frac {\partial v_{j}}{\partial x_{i}}}\right)+\lambda {\frac {\partial v_{k}}{\partial x_{k}}}\delta _{ij}$