The equations are so complex that we can only find analytical solutions for a few relatively simple cases.
Various approximations expand the range of solvable problems somewhat, but detailed solutions for complex flows
were not possible until the development of computers, around the middle of the twentieth century,
and the emergence of computational fluid dynamics (CFD)
For example, even though we may limit ourselves to incompressible flows for which the viscosity is constant,
we still end up with the following equations:
- They are coupled
- They are nonlinear
- They are second-order partial differential equations
\[
\underbrace{\frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} + \frac{\partial w}{\partial z} = 0}_{\text{Continuity equation (mass conservation)}}
\]
\[
\left.
\begin{aligned}
\rho \left( \frac{\partial u}{\partial t} + u \frac{\partial u}{\partial x} + v \frac{\partial u}{\partial y} + w \frac{\partial u}{\partial z} \right) &= \rho g_x - \frac{\partial p}{\partial x} + \mu \left( \frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} + \frac{\partial^2 u}{\partial z^2} \right) \\
\rho \left( \frac{\partial v}{\partial t} + u \frac{\partial v}{\partial x} + v \frac{\partial v}{\partial y} + w \frac{\partial v}{\partial z} \right) &= \rho g_y - \frac{\partial p}{\partial y} + \mu \left( \frac{\partial^2 v}{\partial x^2} + \frac{\partial^2 v}{\partial y^2} + \frac{\partial^2 v}{\partial z^2} \right) \\
\rho \left( \frac{\partial w}{\partial t} + u \frac{\partial w}{\partial x} + v \frac{\partial w}{\partial y} + w \frac{\partial w}{\partial z} \right) &= \rho g_z - \frac{\partial p}{\partial z} + \mu \left( \frac{\partial^2 w}{\partial x^2} + \frac{\partial^2 w}{\partial y^2} + \frac{\partial^2 w}{\partial z^2} \right)
\end{aligned}
\right\}
\begin{aligned}
&\text{Navier-Stokes equations} \\
&\text{(momentum)}
\end{aligned}
\]