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Fluid Equations in Cartesian Coordinates
Let us adopt the conventional Cartesian coordinate system, (
,
,
). According to Equation (1.26), the various components
of the stress tensor are
![$\displaystyle \sigma_{xx}$](img440.png) |
![$\displaystyle = -p + 2\,\mu\,\frac{\partial v_x}{\partial x},$](img441.png) |
(1.127) |
![$\displaystyle \sigma_{yy}$](img442.png) |
![$\displaystyle = -p + 2\,\mu\,\frac{\partial v_y}{\partial y},$](img443.png) |
(1.128) |
![$\displaystyle \sigma_{zz}$](img444.png) |
![$\displaystyle = -p + 2\,\mu\,\frac{\partial v_z}{\partial z},$](img445.png) |
(1.129) |
![$\displaystyle \sigma_{xy}=\sigma_{yx}$](img446.png) |
![$\displaystyle = \mu\left(\frac{\partial v_x}{\partial y}+\frac{\partial v_y}{\partial x}\right),$](img447.png) |
(1.130) |
![$\displaystyle \sigma_{xz}=\sigma_{zx}$](img448.png) |
![$\displaystyle = \mu\left(\frac{\partial v_x}{\partial z}+\frac{\partial v_z}{\partial x}\right),$](img449.png) |
(1.131) |
![$\displaystyle \sigma_{yz}=\sigma_{zy}$](img450.png) |
![$\displaystyle = \mu\left(\frac{\partial v_y}{\partial z}+\frac{\partial v_z}{\partial y}\right),$](img451.png) |
(1.132) |
where
is the velocity,
the pressure, and
the viscosity. The equations of compressible
fluid flow, (1.87)-(1.89) (from which the equations of incompressible fluid flow
can easily be obtained by setting
), become
![$\displaystyle \frac{D\rho}{Dt}$](img346.png) |
![$\displaystyle =-\rho\,{\mit\Delta},$](img454.png) |
(1.133) |
![$\displaystyle \frac{Dv_x}{Dt}$](img455.png) |
![$\displaystyle = - \frac{1}{\rho}\,\frac{\partial p}{\partial x} - \frac{\partia...
...\nabla^{\,2} v_x + \frac{1}{3}\,\frac{\partial{\mit\Delta}}{\partial x}\right),$](img456.png) |
(1.134) |
![$\displaystyle \frac{Dv_y}{Dt}$](img457.png) |
![$\displaystyle = - \frac{1}{\rho}\,\frac{\partial p}{\partial y} - \frac{\partia...
...\nabla^{\,2} v_y + \frac{1}{3}\,\frac{\partial{\mit\Delta}}{\partial y}\right),$](img458.png) |
(1.135) |
![$\displaystyle \frac{Dv_z}{Dt}$](img459.png) |
![$\displaystyle = - \frac{1}{\rho}\,\frac{\partial p}{\partial z} - \frac{\partia...
...\nabla^{\,2} v_z + \frac{1}{3}\,\frac{\partial{\mit\Delta}}{\partial z}\right),$](img460.png) |
(1.136) |
![$\displaystyle \frac{1}{\gamma-1}\left(\frac{D\rho}{Dt} - \frac{\gamma\,p}{\rho}\,\frac{D\rho}{Dt}\right)$](img461.png) |
![$\displaystyle =\chi + \frac{\kappa\,M}{R}\,\nabla^{\,2}\left(\frac{p}{\rho}\right),$](img462.png) |
(1.137) |
where
is the mass density,
the ratio of specific heats,
the heat conductivity,
the molar mass, and
the molar ideal gas constant. Furthermore,
![$\displaystyle {\mit\Delta}$](img463.png) |
![$\displaystyle =\frac{\partial v_x}{\partial x} + \frac{\partial v_y}{\partial y} + \frac{\partial v_z}{\partial z},$](img464.png) |
(1.138) |
![$\displaystyle \frac{D}{Dt}$](img465.png) |
![$\displaystyle = \frac{\partial}{\partial t} + v_x\,\frac{\partial }{\partial x} + v_y\,\frac{\partial}{\partial y} + v_z\,\frac{\partial}{\partial z},$](img466.png) |
(1.139) |
![$\displaystyle \nabla^{\,2}$](img467.png) |
![$\displaystyle =\frac{\partial^{\,2}}{\partial x^{\,2}} + \frac{\partial^{\,2}}{\partial y^{\,2}}+\frac{\partial^{\,2}}{\partial z^{\,2}},$](img468.png) |
(1.140) |
![$\displaystyle \chi$](img469.png) |
![$\displaystyle =2\,\mu\left[\left(\frac{\partial v_x}{\partial x}\right)^2+\left...
...\frac{\partial v_x}{\partial y}+\frac{\partial v_y}{\partial x}\right)^2\right.$](img470.png) |
|
|
![$\displaystyle \phantom{=}\left.+\frac{1}{2}\left(\frac{\partial v_x}{\partial z...
...frac{\partial v_y}{\partial z}+\frac{\partial v_z}{\partial y}\right)^2\right].$](img471.png) |
(1.141) |
Here,
,
,
, and
are treated as uniform constants.
Next: Fluid Equations in Cylindrical
Up: Mathematical Models of Fluid
Previous: Dimensionless Numbers in Compressible
Richard Fitzpatrick
2016-01-22