User:Dream mind/Sandbox/Navier-stokes equation

Differential momentum equation
$$\rho\ g_x+ \left ( \frac{\partial \sigma _x\ _x}{\partial \ _x} \right )+\left ( \frac{\partial \tau _y\ _x}{\partial \ _y} \right )+\left ( \frac{\partial \tau _z\ _x}{\partial \ _z} \right )=\rho\left ( \frac{\partial u}{\partial t}+ u \frac{\partial u}{\partial x}+v \frac{\partial u}{\partial y}+w \frac{\partial u}{\partial z}\right ) $$

$$\rho\ g_y+ \left ( \frac{\partial \tau _x\ _y}{\partial \ _x} \right )+\left ( \frac{\partial \sigma _y\ _y}{\partial \ _y} \right )+\left ( \frac{\partial \tau _z\ _y}{\partial \ _z} \right )=\rho\left ( \frac{\partial v}{\partial t}+ u \frac{\partial v}{\partial x}+v \frac{\partial v}{\partial y}+w \frac{\partial v}{\partial z}\right ) $$

$$\rho\ g_y+ \left ( \frac{\partial \tau _x\ _z}{\partial \ _x} \right )+\left ( \frac{\partial \tau _y\ _z}{\partial \ _z} \right )+\left ( \frac{\partial \sigma _z\ _z}{\partial \ _z} \right )=\rho\left ( \frac{\partial w}{\partial t}+ u \frac{\partial w}{\partial x}+v \frac{\partial w}{\partial y}+w \frac{\partial w}{\partial z}\right ) $$