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} $$