Rapidly dissolving dense bodies in an inviscid fluid

Abstract

We study the rapid dissolution of a dense rigid body (a cylinder or sphere; density ρ p ), fixed in a uniform flow of speed U (density ρ f ), within an inviscid framework. The body consists of N shells, and dissolves through a series of discrete steps, where after each shell dissolves, it is pushed out by a radial flow. The flow through the surface of the body pushes bound vorticity into the ambient flow, creating free vorticity and satisfying, at the same time, Kelvin's circulation theorem. The circulation of each shell is described by a Lagrangian equation describing the position of each shell and an integral relationship between the circulation of the shells dissolved and the bound vorticity. The thermal component is neglected from this analysis. The impulse of the exterior fluid increases to compensate for the decrease in the momentum of the body. The bodies are fixed as they dissolve so that the sum of the flow impulse and body momentum is only approximately conserved. The impulse of the vorticity field created after the body has disappeared, I T , is approximately equal to the initial momentum of the body, ρ p UV (where V is the initial volume of the body). The total circulation in the upper half-plane is Γ ≈ Γ 0 + C Γ ( ρ p / ρ f ) 1/ N a 0 U , where N =2, 3 for a cylinder, sphere, respectively, and C Γ ∼ O (1) a constant. The total kinetic energy associated with the vorticity field after the body has disappeared is T ∼(1/2) ρ p C T VU 2 , where C T <1. The sum of the kinetic energy of the flow and body is not conserved. After the cylinder/sphere has disappeared, the vorticity distribution rolls up to produce a dipolar vortex/vortex ring. The properties of these dipolar structures are estimated assuming a final form and the conservation of T , Γ and I T . A dense cylinder dissolves to create a vortex that has a radius 1.3 a 0 ( ρ p / ρ f ) 1/2 and moves with a speed 0.28 U , while a dense sphere dissolves to create a vortex of radius 0.65 a 0 ( ρ p / ρ f ) 1/3 , which moves with a speed 0.50 U .