Continuity and Navier Stokes Equations

The following are sets of both continuity and dynamic equations for incompressible Newtonian fluids, in vector notation and Cartesian, Cylindrical and Spherical coordinate systems. Vectorial properties are shown in bold type.

Vector Notation

Continuity

$\frac{D\rho}{Dt} + \rho{\bf\nabla . u} = 0$ in general, or

${\bf\nabla . u} = 0$ for an incompressible fluid

Navier Stokes

$\rho \frac{D{\bf u}}{Dt} = - {\bf\nabla} p + \mu \nabla^{2}{\bf u} + \frac{\mu}{3}{\bf grad}(\nabla . u) + {\bf F}$ in general, or

$\rho \frac{D{\bf u}}{Dt} = - {\bf\nabla} p + \mu \nabla^{2}{\bf u} + {\bf F}$ for an incompressible fluid

For the individual coordinate systems the equations are only quoted for incompressible flow.


Cartesian Coordinates

Continuity

$\frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} +\frac{\partial w}{\partial z} = 0$

Navier Stokes

$\rho[\frac{\partial u}{\partial t} + u \frac{\partial u}{\partial x} + v \frac{... ...partial^{2} u}{\partial y^{2}} + \frac{\partial^{2} u}{\partial z^{2}}] + F_{x}$

$\rho[\frac{\partial v}{\partial t} + u \frac{\partial v}{\partial x} + v \frac{... ...partial^{2} v}{\partial y^{2}} + \frac{\partial^{2} v}{\partial z^{2}}] + F_{y}$

$\rho[\frac{\partial w}{\partial t} + u \frac{\partial w}{\partial x} + v \frac{... ...partial^{2} w}{\partial y^{2}} + \frac{\partial^{2} w}{\partial z^{2}}] + F_{z}$

Cylindrical polar Coordinates

Continuity

$\frac{\partial ( r u_{r})}{r \partial r} + \frac{1}{r} \frac{\partial u_{\phi}}{\partial \phi} + \frac{\partial u_{z}}{\partial z} = 0$

or

$\frac{\partial u_{r}}{\partial r} + \frac{u_{r}}{r} + \frac{1}{r} \frac{\partial u_{\phi}}{\partial \phi} + \frac{\partial u_{z}}{\partial z} = 0$

Navier Stokes

$\rho[\frac{\partial u_{r}}{\partial t} + u_{r} \frac{\partial u_{r}}{\partial r... ...rtial z^{2}} - \frac{2}{r^{2}} \frac{\partial u_{\phi}}{\partial \phi}] + F_{r}$

$\rho[\frac{\partial u_{\phi}}{\partial t} + u_{r} \frac{\partial u_{\phi}}{\par... ...rtial z^{2}} + \frac{2}{r^{2}} \frac{\partial u_{r}}{\partial \phi}] + F_{\phi}$

$\rho[\frac{\partial u_{z}}{\partial t} + u_{r} \frac{\partial u_{z}}{\partial r... ... u_{z}}{\partial \phi^{2}} + \frac{\partial^{2} u_{z}}{\partial z^{2}}] + F_{z}$




Spherical polar Coordinates

Continuity

$\frac{\partial ( r^{2} u_{r})}{r^{2} \partial r} + \frac{1}{r \sin \theta} \fra... ...l \theta} + \frac{1}{r \sin \theta} \frac{\partial u_{\phi}}{\partial \phi} = 0$

or

$\frac{\partial u_{r}}{\partial r} + \frac{2 u_{r}}{r} + \frac{1}{r} \frac{\part... ...theta}{r} + \frac{1}{r \sin \theta} \frac{\partial u_{\phi}}{\partial \phi} = 0$

Navier Stokes

$\rho[\frac{\partial u_{r}}{\partial t} + u_{r} \frac{\partial u_{r}}{\partial r... ... - \frac{2}{r^{2} \sin \theta} \frac{\partial u_{\phi}}{\partial \phi}] + F_{r}$

$\rho[\frac{\partial u_{\theta}}{\partial t} + u_{r} \frac{\partial u_{\theta}}{... ...theta}{r^{2} \sin \theta} \frac{\partial u_{\phi}}{\partial \phi}] + F_{\theta}$

$\rho[\frac{\partial u_{\phi}}{\partial t} + u_{r} \frac{\partial u_{\phi}}{\par... ...theta}{r^{2} \sin \theta} \frac{\partial u_{\theta}}{\partial \phi}] + F_{\phi}$



The terms in $\rho\frac{u_{\theta}^{2}}{r}$ and $\rho\frac{u_{\phi}^{2}}{r}$represent centrifugal forces acting in the r direction due to velocity in the $\theta$ or $\phi$ directions and the terms in $\rho\frac{u_{r} u_{\theta}}{r}$ and $\rho\frac{u_{r} u_{\phi}}{r}$ represent Coriolis forces.

David Balmer
Last modified: Wed Aug 12 12:57:21 BST