Continuation of a single layer flow over an obstacle

In this text I will investigate steady single layer flow over an obstacle of which the height will be increased slowly to see what this does to the solutions and eigenvalues of the model. If the obstacle is not too high we will have completely subcritical flow. However, at a certain height the flow may become supercritical. First I will introduce the basic equations for this problem. These will be derived from the Navier-Stokes equations. Using some assumptions and simplifications we arrive at the equations for our specific problem. After that I will derive the analytical solution to the linearized equations and look at what this solution can learn us about the transition from sub- to supercritical flow. This means we will look at the eigenvalue problem for a perturbation of the linear equations. Furthermore we take a look at the model that we need for the numerical analysis and what methods are used. We can come up with an eigenvalue problem related to the discretized equations and this is important for the numerical research. After that I will investigate numerical solutions of the equations and look what happens with the eigenvalues when we go from sub- to supercritical flow