Fluid Equations in Cartesian Coordinates

Let us adopt the conventional Cartesian coordinate system, $x$, $y$, $z$. According to Equation (26), the various components of the stress tensor are

$$\sigma_{xx} = -p + 2\,\mu\,\frac{\partial v_x}{\partial x},$$ (127)

$$\sigma_{yy} = -p + 2\,\mu\,\frac{\partial v_y}{\partial y},$$ (128)

$$\sigma_{zz} = -p + 2\,\mu\,\frac{\partial v_z}{\partial z},$$ (129)

$$\sigma_{xy}=\sigma_{yx} = \mu\left(\frac{\partial v_x}{\partial y}+\frac{\partial v_y}{\partial x}\right),$$ (130)

$$\sigma_{xz}=\sigma_{zx} = \mu\left(\frac{\partial v_x}{\partial z}+\frac{\partial v_z}{\partial x}\right),$$ (131)

$$\sigma_{yz}=\sigma_{zy} = \mu\left(\frac{\partial v_y}{\partial z}+\frac{\partial v_z}{\partial y}\right),$$ (132)

where ${\bf v}$ is the velocity, $p$ the pressure, and $\mu$ the viscosity. The equations of compressible fluid flow, (87)-(89) (from which the equations of incompressible fluid flow can easily be obtained by setting $\Delta=0$), become

$$\frac{D\rho}{Dt} = -\rho\,\Delta,$$ (133)

$$\frac{Dv_x}{Dt} = - \frac{1}{\rho}\,\frac{\partial p}{\partial x} - \frac{\partial{... ...rho}\left(\nabla^2 v_x + \frac{1}{3}\,\frac{\partial\Delta}{\partial x}\right),$$ (134)

$$\frac{Dv_y}{Dt} = - \frac{1}{\rho}\,\frac{\partial p}{\partial y} - \frac{\partial{... ...rho}\left(\nabla^2 v_y + \frac{1}{3}\,\frac{\partial\Delta}{\partial y}\right),$$ (135)

$$\frac{Dv_z}{Dt} = - \frac{1}{\rho}\,\frac{\partial p}{\partial z} - \frac{\partial{... ...rho}\left(\nabla^2 v_z + \frac{1}{3}\,\frac{\partial\Delta}{\partial z}\right),$$ (136)

$$\frac{1}{\gamma-1}\left(\frac{D\rho}{Dt} - \frac{\gamma\,p}{\rho}\,\frac{D\rho}{Dt}\right) = \chi + \frac{\kappa\,{\cal M}}{{\cal R}}\,\nabla^2\left(\frac{p}{\rho}\right),$$ (137)

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

$$\Delta = \frac{\partial v_x}{\partial x} + \frac{\partial v_y}{\partial y} + \frac{\partial v_z}{\partial z},$$ (138)

$$\frac{D}{Dt} = \frac{\partial}{\partial t} + v_x\,\frac{\partial }{\partial x} + v_y\,\frac{\partial}{\partial y} + v_z\,\frac{\partial}{\partial z},$$ (139)

$$\nabla^2 = \frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y^2}+\frac{\partial^2}{\partial z^2},$$ (140)

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

$$\left.+\frac{1}{2}\left(\frac{\partial v_x}{\partial z}+\frac{\pa... ...frac{\partial v_y}{\partial z}+\frac{\partial v_z}{\partial y}\right)^2\right].$$ (141)

In the above, $\gamma$, $\mu$, $\kappa$, and ${\cal M}$ are treated as uniform constants.