# Navier-Stokes equation

From Soft-Matter

The Navier-Stokes equation describes the motion of fluids. It is derived from applying Newton's second law to fluid systems, and assumes that the fluid stress is the sum of a viscous term and a pressure term. The general form of the equations of fluid motion is

- <math>\rho \left(\frac{\partial \mathbf{v}}{\partial t} + \mathbf{v} \cdot \nabla \mathbf{v}\right) = -\nabla p + \nabla \cdot\mathbb{T} + \mathbf{f},</math>

where <math>\mathbf{v}</math> is the flow velocity, <math>\rho</math> is the fluid density, *p* is the pressure, <math>\mathbb{T}</math> is the stress tensor, and <math>\mathbf{f} </math> represents body forces per unit volume acting on the fluid.