# Navier-Stokes Equation

## $\LARGE\rho\left(\frac{\partial\mathbf{u}}{{\partial}t}+(\mathbf{u}\cdot\boldsymbol{\nabla})\mathbf{u}\right)=-\boldsymbol{\nabla}p+\eta\nabla^2\mathbf{u}+\mathbf{f}$

### The Terms

$\rho$ - Fluid density [Units: kg m-3]

$\mathbf{u}$ - Fluid flow velocity [Units: m s-1]

$\boldsymbol{\nabla}$ - Gradient operator, $\boldsymbol{\nabla}=\partial_x\mathbf{i}+\partial_y\mathbf{j}+\partial_z\mathbf{k}$

$p$ - Fluid pressure [Units: Pa]

$\eta$ - Dynamic viscosity [Units: Pa s]

$\nabla^2$ - Laplacian operator, $\nabla^2\mathbf{u}=\partial_{xx}^2\mathbf{u}+\partial_{yy}^2\mathbf{u}+\partial_{zz}^2\mathbf{u}$

$\mathbf{f}$ - Body forces per unit volume (e.g. gravity per unit volume) [Units: N m-3]

### What Does It Mean?

The Navier-Stokes Equation describes the flow of fluid substances. The equation given here is particular to incompressible flows of Newtonian fluids. Incompressible flows are flows where the divergence of the velocity field $\mathbf{u}$ is zero, i.e. $\boldsymbol{\nabla}\cdot\mathbf{u}=0$. Newtonian fluids are those for which the stress $\tau$ is directly proportional to the velocity perpendicular to the direction of shear (called the strain rate), $du/dt$, with proportionality constant $\eta$, the dynamic viscosity.

The solution of this equation is a velocity field or flow field; a description of the velocity of the fluid at a particular point in space and time.

### Did You know?

That the Navier-Stokes equation can be combined with the low-frequency version of Maxwell's equations for electromagnetic fields by adding the magnetic Lorentz force j x B as a force per volume. This equation describes macroscopically the momentum balance of plasmas and is a central part of the theory of magnetohydrodynamics (MHD) and is used to study the behaviour of plasmas ranging from the corona of the Sun, galactic jets to fusion experiments

### Further information at Warwick

This equation is derived and applied to solve fluid dynamical problems in the second year module "PX264 Physics of Fluids" and the third year module "MA3D1 Fluid Dynamics". Combined with electromagentism it is a core component of PX420 Solar Magnetohydrodynamics