| We derive the energy equation for a perturbation of finite amplitude for Poiseuille flow between two infinite plates, with no-slip boundary conditions on the upper plate and Navier slip boundary conditions on the lower plate. To determine an energy Reynolds number that guarantees the decay of all perturbations, we use the calculus of variations to extremalize a functional associated with the energy of a perturbation. After showing that the Euler-Lagrange equations we obtain for this base flow---and, in fact, any parallel base flow with Navier slip boundary conditions---are the same as the one we would obtain with no-slip boundary conditions, we look for solutions in the form of normal modes and eventually wind up with a coupled system of two ordinary differential equations. The minimum eigenvalue of this system is precisely the energy Reynolds number that we wish to determine. Using Chebyshev interpolation, we employ MATLAB to find this eigenvalue. After briefly examining the energy equations for combined Couette-Poiseuille flow, we adapt the method for the case of Taylor-Couette flow and show once again that the Euler-Lagrange equations we obtain are the same as the one we would obtain with no-slip boundary conditions. Using Lagrange interpolation, we find the energy Reynolds number for Taylor-Couette flow. |
