Abstract
We prove non-existence of nontrivial uniformly subsonic inviscid irrotational flows around several classes of solid bodies with two protruding corners, in particular vertical and angled flat plates; horizontal plates are the only case where solutions exists. This fills the gap between classical results on bodies with a single protruding corner on one hand and recent work on bodies with three or more protruding corners. Thus even with zero viscosity and slip boundary conditions solids can generate vorticity, in the sense of having at least one rotational but no irrotational solutions. Our observation complements the commonly accepted explanation of vorticity generation based on Prandtl's theory of viscous boundary layers.
Highlights
There are infinitely many solutions, parametrized by circulation Γ, but only one of them satisfies the Kutta-Joukowsky condition, namely that the velocity is bounded at the corner
Calculating the resulting pressure forces yields a well-known formula for lift, component of the force perpendicular to the velocity at infinity
The formula is in reasonable agreement1 with experimental data at least in some physical regimes
Summary
The isentropic Euler equations are 0 = ∂t + ∇ · ( v), 0 = ∂t( v) + ∇ · ( v ⊗ v) + ∇P, where v is velocity while pressure P = P( ) is a strictly increasing function of density. We only consider the polytropic pressure law. P( ) = γ with isentropic coefficient γ greater than 1. Assuming sufficient regularity the equations can be expanded into 0 = Dt + ∇ · v, Dt = ∂t + v · ∇,.
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