Simulation of the droplet-to-bubble transition in a two-fluid system

Abstract

Recent experiments by Burton and Taborek have demonstrated a droplet-to-bubble transition in the pinchoff behavior of one inviscid fluid inside another. With D the relative densities ρ E /ρ I , they find transition from (D = 0) droplet-to-bubble behavior at D ≈ 4. Numerical simulations of this two-fluid system, up to and beyond the initial breakup of the inner fluid, have been carried out utilizing level set and boundary integral methods. A droplet-to-bubble transition is predicted: For D sufficiently large, the volume of the satellite droplet shrinks to zero and there is no overturning of the fluid at separation. The calculated self-similar scaling exponents and the pinchoff region shapes match the known behavior at the droplet and bubble extremes (D = 0, D = 100). For intermediate D values, the simulations presented here indicate that the transition range between droplet and bubble behavior depends upon initial drop geometry. When the neck separates two nonequal inner fluid masses the transition is mild and occurs in the range 4 < D < 10, whereas in the case of equal masses an abrupt transition occurs at D ≈ 4 in perfect agreement with the above mentioned experiments.

Boundary integral analysis for non-homogeneous, incompressible Stokes flows

“…It would be of interest to repeat the successful potential flow studies of coalescence [6] and Rayleigh-Taylor break-up [4,5] with the more complex fluids mentioned above. The Laplace calculations employ cylindrical coordinates {ρ, θ, z}, and with θ integrated out they become two-dimensional {ρ, z} analyses.…”

Moving Boundary Problems for Quasi-Steady Conduction Limited Melting

“…where r 0 can be interpreted as a measure of the depth of the pinch in the middle of the peanut. Following Garzon et al [19], the third initial condition considered is a dumbbell shaped interface of the form s(θ, 0) = (ρ * (θ) 2 + z * (θ) 2 ) 1/2 , where z * (θ) = 1 + r 0 sin 2 (θ/2),…”

“…We note that since the governing equation for temperature satisfies Laplace's equation, an alternative approach for computing the temperature u is the boundary integral method, which can be coupled with the level set method to solve problems where changes in topology occur [19]. However, an advantage of using a finite difference stencil is that it can easily be adapted to problems where the boundary integral method is not applicable.…”

Interfacial dynamics and pinch-off singularities for axially symmetric Darcy flow

“…Furthermore, on its own the BIM cannot easily handle changes in topology. Success has been achieved using 'front capturing methods', which are generally not as accurate but have greater flexibility [51][52][53]. The numerical scheme we use in this article implements the level set method [54], a numerical framework that describes the evolution of moving interfaces by representing them as the zero level set of a higher dimensional hyper-surface.…”