There is no general solution for the equations summarized in section 6.8, because they are inherently nonlinear. The main source of nonlinearity is in the advective part of the material derivative, e.g., $$[\vec{u}\cdot\vec{\nabla}]\vec{u}$$, which stymies standard solution methods as well as accounting for many of the most fascinating aspects of fluid motion. To make analytical progress, we must restrict our attention to very simple flow geometries. In recent decades numerical methods of solution have become increasingly important. While allowing progress on complex flows, numerical solutions have an important limitation. Each numerical solution pertains only to a single set of assumed parameter values. If we want to know how a flow varies with some parameter, we must create many such solutions, and we can never be sure that we’ve captured all of the variability.
An analytical solution, even if it requires an extreme simplification of the physics, provides us with a mathematical description that we can examine in as much detail as we wish. For example, we can find the mountain height that maximizes wind speed simply by differentiating the solution. In the mountain example, the most useful solution follows from assumptions of this sort: "The flow varies mainly in the streamwise $$(x)$$ direction and in height $$z$$, so derivatives with respect to $$t$$ and $$y$$ can be discarded."