Writing the vector equation explicitly,
\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) = -\frac{\partial p}{\partial x} + \mu \left(\frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} + \frac{\partial^2 u}{\partial z^2}\right) + \rho g_x
\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) = -\frac{\partial p}{\partial y} + \mu \left(\frac{\partial^2 v}{\partial x^2} + \frac{\partial^2 v}{\partial y^2} + \frac{\partial^2 v}{\partial z^2}\right) + \rho g_y
\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) = -\frac{\partial p}{\partial z} + \mu \left(\frac{\partial^2 w}{\partial x^2} + \frac{\partial^2 w}{\partial y^2} + \frac{\partial^2 w}{\partial z^2}\right) + \rho g_z.
Note that gravity has been accounted for as a body force, and the values of gx,gy,gz will depend on the orientation of gravity with respect to the chosen set of coordinates.
The continuity equation reads:
{\partial \rho \over \part t} + {\partial (\rho u ) \over \partial x} + {\partial (\rho v) \over \partial y} + {\partial (\rho w) \over \partial z} = 0.
When the flow is at steady-state, ρ does not change with respect to time. The continuity equation is reduced to:
{\partial (\rho u) \over \partial x} + {\partial (\rho v) \over \partial y} + {\partial (\rho w) \over \partial z} = 0.
When the flow is incompressible, ρ is constant and does not change with respect to space. The continuity equation is reduced to:
{\partial u \over \partial x} + {\partial v \over \partial y} + {\partial w \over \partial z} = 0.
The velocity components (the dependent variables to be solved for) are typically named u, v, w. This system of four equations comprises the most commonly used and studied form. Though comparatively more compact than other representations, this is still a nonlinear system of partial differential equations for which solutions are difficult to obtain.
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