Navier-Stokes Equations

The Navier-Stokes equations describe the motion of viscous fluid substances. They are a set of nonlinear partial differential equations that predict how the velocity field of a fluid evolves over time: $$\rho \left( \frac{\partial \mathbf{u}}{\partial t} + \mathbf{u} \cdot \nabla \mathbf{u} \right) = -\nabla p + \nabla \cdot \mathbf{T} + \mathbf{f}$$ where $\mathbf{u}$ is the velocity field, $p$ is the pressure, $\mathbf{T}$ is the stress tensor, and $\mathbf{f}$ represents body forces.