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:

\[ \underbrace{\frac{\partial \boxed{u}}{\partial x} + \frac{\partial \boxed{v}}{\partial y} + \frac{\partial \boxed{w}}{\partial z} = 0}_{\text{Continuity equation (mass conservation)}} \]

\[ \left. \begin{aligned} \rho \left( \boxed{\frac{\partial u}{\partial t} + u \frac{\partial u}{\partial x} + v \frac{\partial u}{\partial y} + w \frac{\partial u}{\partial z}} \right) &= \rho g_x - \frac{\partial \boxed{p}}{\partial x} + \boxed{\mu \left( \frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} + \frac{\partial^2 u}{\partial z^2} \right)} \\ \rho \left( \frac{\partial v}{\partial t} + u \frac{\partial v}{\partial x} + v \frac{\partial v}{\partial y} + w \frac{\partial v}{\partial z} \right) &= \rho g_y - \frac{\partial p}{\partial y} + \mu \left( \frac{\partial^2 v}{\partial x^2} + \frac{\partial^2 v}{\partial y^2} + \frac{\partial^2 v}{\partial z^2} \right) \\ \rho \left( \frac{\partial w}{\partial t} + u \frac{\partial w}{\partial x} + v \frac{\partial w}{\partial y} + w \frac{\partial w}{\partial z} \right) &= \rho g_z - \frac{\partial p}{\partial z} + \mu \left( \frac{\partial^2 w}{\partial x^2} + \frac{\partial^2 w}{\partial y^2} + \frac{\partial^2 w}{\partial z^2} \right) \end{aligned} \right\} \begin{aligned} &\text{Navier-Stokes equations} \\ &\text{(momentum)} \end{aligned} \]

In principle, these equations can be solved for the velocity field \(\mathbf{u}\) and pressure field \( p \), given sufficient initial and boundary conditions. Note that in general, \( u, v, w, \) and \( p \) all depend on \( x, y, z, \) and \( t \). In practice, there is no general analytic solution to these equations, for the combined effect of a number of reasons (none of which is insurmountable by itself):

They are coupled. The unknowns, \( u, v, w, \) and \( p \), appear in all the equations (\( p \) is not in the continuity equation) and we cannot manipulate the equations to end up with a single equation for any one of the unknowns. Hence we must solve for all unknowns simultaneously.
They are nonlinear. For example, the convective acceleration term \( u \frac{\partial u}{\partial x} + v \frac{\partial u}{\partial y} + w \frac{\partial u}{\partial z}\) has products of \( u \) with itself as well as with \( v \) and \( w \). The consequence of this is that we cannot take one solution to the equations and combine it with a second solution to obtain a third solution. If frictionless flow is assumed, linear equations can be derived to do this combining procedure.
They are second-order partial differential equations. For example, the viscous term \( \mu \left( \frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} + \frac{\partial^2 u}{\partial z^2} \right) \) is second-order in \( u \). These are obviously of a different order of complexity.