Vertex circulation and regularity of compressible Stokes flows: Numerical simulations

Abstract

We investigate the vertex circulation in a cone-like form of compressible Stokes flows and show existence and regularity by constructing the solutions in the infinite sector attached to a fixed vertex, say the origin (0,0). Let ω be the opening angle of the sector. We construct the velocity vector u and the density function ρ of the following formsu(x,y)=rλ(ϕ(θ)e+ψ(θ)e′),θ∈(0,ω),ρ(x,y)=rλ−1σ(θ)where ϕ, ψ and σ are the solutions for a nonlinear boundary value problem with any positive number λ≠nπ/ω for integer n; r=x2+y2,θ are the polar coordinates at the origin, e=(cosθ,sinθ) and e′=(−sinθ,cosθ).The vertex can be a junction point that inflow and outflow meet and may result in a vertex circulation that means a cone-like rotation at vertex. The vertex circulation by compressible Stokes flows may blow up due to the singular behavior of suddenly change at corner. The angular velocity component is found to be positive while for λ ∈ (0, 1) the velocity vector itself vanishes at the vertex and the pressure blows up there. We also demonstrate this phenomena by numerical simulations.