Node320 - CFD SUPPORT

Previous: Equation of state Up: Appendix: Fluid Mechanics Next: System of Navier-Stokes equations

This is an automatically generated documentation by LaTeX2HTML utility. In case of any issue, please, contact us at info@cfdsupport.com.

System of Navier-Stokes equations

From the above equations we can obtain system of Navier-Stokes equations for incompressible, viscous fluid:

$\displaystyle \frac{\partial u}{\partial x} +\frac{\partial v}{\partial y}+\frac{\partial w}{\partial z}$	$\displaystyle =$	0	(21.15)
$\displaystyle \rho \Big( \frac{\partial u}{\partial t}+ u \frac{\partial u}{\partial x}+ v \frac{\partial u}{\partial y}+w \frac{\partial u}{\partial z}\Big)$	$\displaystyle =$	$\displaystyle - \frac{\partial p}{\partial x} +\frac{\partial}{\partial x} \Big( \eta \frac{\partial u}{\partial x} \Big) +\frac{\partial}{\partial y} \Big( \eta \frac{\partial u}{\partial y} \Big) +\frac{\partial}{\partial z} \Big( \eta \frac{\partial u}{\partial z} \Big)+f_1$
$\displaystyle \rho \Big( \frac{\partial v}{\partial t}+ u \frac{\partial v}{\partial x}+ v \frac{\partial v}{\partial y}+w \frac{\partial v}{\partial z}\Big)$	$\displaystyle =$	$\displaystyle - \frac{\partial p}{\partial y} +\frac{\partial}{\partial x} \Big( \eta \frac{\partial v}{\partial x} \Big) +\frac{\partial}{\partial y} \Big( \eta \frac{\partial v}{\partial y} \Big) +\frac{\partial}{\partial z} \Big( \eta \frac{\partial v}{\partial z} \Big)+f_2$
$\displaystyle \rho \Big( \frac{\partial w}{\partial t}+ u \frac{\partial w}{\partial x}+ v \frac{\partial w}{\partial y}+w \frac{\partial w}{\partial z}\Big)$	$\displaystyle =$	$\displaystyle - \frac{\partial p}{\partial z} +\frac{\partial}{\partial x} \Big( \eta \frac{\partial w}{\partial x} \Big) +\frac{\partial}{\partial y} \Big( \eta \frac{\partial w}{\partial y} \Big) +\frac{\partial}{\partial z} \Big( \eta \frac{\partial w}{\partial z} \Big)+f_3$

These equations create the complete system, we will solve in this work. For more simplicity we will consider the volume forces to be equal .