Fluid Equations in Cartesian Coordinates

Let us adopt the conventional Cartesian coordinate system, ($x$ , $y$ , $z$ ). According to Equation (1.26), the various components of the stress tensor are

$$\sigma_{xx} = -p + 2\,\mu\,\frac{\partial v_x}{\partial x},$$ (1.127)

$$\sigma_{yy} = -p + 2\,\mu\,\frac{\partial v_y}{\partial y},$$ (1.128)

$$\sigma_{zz} = -p + 2\,\mu\,\frac{\partial v_z}{\partial z},$$ (1.129)

$$\sigma_{xy}=\sigma_{yx} = \mu\left(\frac{\partial v_x}{\partial y}+\frac{\partial v_y}{\partial x}\right),$$ (1.130)

$$\sigma_{xz}=\sigma_{zx} = \mu\left(\frac{\partial v_x}{\partial z}+\frac{\partial v_z}{\partial x}\right),$$ (1.131)

$$\sigma_{yz}=\sigma_{zy} = \mu\left(\frac{\partial v_y}{\partial z}+\frac{\partial v_z}{\partial y}\right),$$ (1.132)

where ${\bf v}$ is the velocity, $p$ the pressure, and $\mu$ 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 ${\mit\Delta}=0$ ), become

$$\frac{D\rho}{Dt} =-\rho\,{\mit\Delta},$$ (1.133)

$$\frac{Dv_x}{Dt} = - \frac{1}{\rho}\,\frac{\partial p}{\partial x} - \frac{\partia... ...\nabla^{\,2} v_x + \frac{1}{3}\,\frac{\partial{\mit\Delta}}{\partial x}\right),$$ (1.134)

$$\frac{Dv_y}{Dt} = - \frac{1}{\rho}\,\frac{\partial p}{\partial y} - \frac{\partia... ...\nabla^{\,2} v_y + \frac{1}{3}\,\frac{\partial{\mit\Delta}}{\partial y}\right),$$ (1.135)

$$\frac{Dv_z}{Dt} = - \frac{1}{\rho}\,\frac{\partial p}{\partial z} - \frac{\partia... ...\nabla^{\,2} v_z + \frac{1}{3}\,\frac{\partial{\mit\Delta}}{\partial z}\right),$$ (1.136)

$$\frac{1}{\gamma-1}\left(\frac{D\rho}{Dt} - \frac{\gamma\,p}{\rho}\,\frac{D\rho}{Dt}\right) =\chi + \frac{\kappa\,M}{R}\,\nabla^{\,2}\left(\frac{p}{\rho}\right),$$ (1.137)

where $\rho$ is the mass density, $\gamma$ the ratio of specific heats, $\kappa$ the heat conductivity, $M$ the molar mass, and $R$ the molar ideal gas constant. Furthermore,

$${\mit\Delta} =\frac{\partial v_x}{\partial x} + \frac{\partial v_y}{\partial y} + \frac{\partial v_z}{\partial z},$$ (1.138)

$$\frac{D}{Dt} = \frac{\partial}{\partial t} + v_x\,\frac{\partial }{\partial x} + v_y\,\frac{\partial}{\partial y} + v_z\,\frac{\partial}{\partial z},$$ (1.139)

$$\nabla^{\,2} =\frac{\partial^{\,2}}{\partial x^{\,2}} + \frac{\partial^{\,2}}{\partial y^{\,2}}+\frac{\partial^{\,2}}{\partial z^{\,2}},$$ (1.140)

$$\chi =2\,\mu\left[\left(\frac{\partial v_x}{\partial x}\right)^2+\left... ...\frac{\partial v_x}{\partial y}+\frac{\partial v_y}{\partial x}\right)^2\right.$$

$$\phantom{=}\left.+\frac{1}{2}\left(\frac{\partial v_x}{\partial z... ...frac{\partial v_y}{\partial z}+\frac{\partial v_z}{\partial y}\right)^2\right].$$ (1.141)

Here, $\gamma$ , $\mu$ , $\kappa$ , and $M$ are treated as uniform constants.