Physlib.FluidDynamics.CauchyFlow.NavierStokes

For a $d$-dimensional fluid flow with density field $\rho(t, x)$ and velocity field $\mathbf{u}(t, x)$, given a pressure field, shear viscosity, and second viscosity (which determine the stress tensor $\sigma$), suppose the following differentiability conditions are satisfied: 1. For all points $x$, the density $\rho$ and velocity $\mathbf{u}$ are differentiable with respect to time $t$. 2. For all times $t$, the velocity field $\mathbf{u}$ and the momentum density field $\rho \mathbf{u}$ are differentiable with respect to the spatial coordinates $x$. Then, the Navier-Stokes equations in conservative form: $$\begin{cases} \frac{\partial \rho}{\partial t} + \nabla \cdot (\rho \mathbf{u}) = 0 \\ \frac{\partial (\rho \mathbf{u})}{\partial t} + \nabla \cdot (\rho \mathbf{u} \otimes \mathbf{u}) = \nabla \cdot \sigma + \rho \mathbf{f} \end{cases}$$ hold if and only if the Navier-Stokes equations in convective form: $$\begin{cases} \frac{\partial \rho}{\partial t} + \nabla \cdot (\rho \mathbf{u}) = 0 \\ \rho \left( \frac{\partial \mathbf{u}}{\partial t} + (\mathbf{u} \cdot \nabla) \mathbf{u} \right) = \nabla \cdot \sigma + \rho \mathbf{f} \end{cases}$$ hold, where $\mathbf{f}$ is the external body force and $\otimes$ denotes the tensor product.