# 6.10: Solution methods

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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.

For example, suppose we want to know how the wind speed over a mountain depends on the mountain’s height. We could construct numerical solutions for mountains of height 1000 m, 2000 m, 3000 m, etc., plot the results on a graph and draw a smooth curve connecting them. But what if something completely different happens for a mountain of height 1500 m? No matter how closely we space our heights, we can never be certain that we are seeing the real picture. At what height is the speed a maximum? We can simulate forever and never be sure. The task is further complicated because wind speed over a mountain depends on many other parameters such as the width of the mountain and the upstream velocity. We can easily find ourselves doing thousands of simulations to describe one fairly simple flow geometry. Laboratory experiments, incidentally, suffer exactly the same limitation.

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."

In practice, progress in understanding fluids results from a combination of numerical solutions, analytical solutions and laboratory experiments, all of which must be compared with real-world observations to assess the validity of the underlying assumptions.

In what follows we will construct analytical solutions for a few very simple flow geometries that model phenomena we witness in everyday life. We do this to gain insight into the workings of these phenomena, but more importantly to test the validity of our model of Newtonian fluid mechanics by comparing its predictions with the behavior we observe.