Stationary flow past a semi-infinite flat plate: analytical and numerical evidence for a symmetry-breaking solution

We consider the incompressible stationary flow of a fluid past a semi-infinite flat plate. This is a very old and well studied problem and is discussed in most introductory texts on fluid mechanics. Indeed, an easy scaling argument shows that far downstream the flow should be to leading order described by the so called Blasius solution, and this has been confirmed to good precision by experiments. However, there still exists no mathematical proof of the existence of a solution of the Navier-Stokes equations for this situation. Here, we do not prove existence of a solution either, but rather show that the problem might be even more complicated than hitherto thought, by providing solid arguments that a solution with broken symmetry should exist. Namely, by using techniques from dynamical system theory we analyze in detail the vorticity equation for this problem, and show that a symmetry-breaking term fits naturally into a downstream asymptotic expansion of a solution. This new term replaces the symmetric second order logarithmic term found in the literature. In contrast to all earlier work our expansion produces order by order smooth divergence free vector fields satisfying all the boundary conditions. To check that our asymptotic expressions can be completed to a solution of the Navier-Stokes equations we also solve the problem numerically, by using our results to prescribe artificial boundary conditions for a sequence of truncated domains. The results of these numerical computations are clearly compatible with the existence of a symmetry-breaking solution. Mathematics Subject Classification (2000). 76D05, 76D25, 76M10, 41A60, 35Q35.