Axisymmetric boundary integral formulation for a two‐fluid system

Abstract

SUMMARY A 3D axisymmetric Galerkin boundary integral formulation for potential flow is employed to model two fluids of different densities, one fluid enclosed inside the other. The interface variables are the velocity potential and the normal velocity, and they can be solved for separately, the second linear system being symmetric. The algorithm is validated by comparing with the analytic solutions for a static interior spherical drop over a range of values for the relative densities DMathClass-rel=ρscriptEMathClass-bin/ρscriptI of exterior and interior fluids and various boundary conditions. For time‐dependent simulations utilizing a level set method for the interface tracking, the accuracy has been checked by comparing against the known oscillation frequency of the sphere. Pinch‐off profiles corresponding to an initial two‐lobe geometry drop and D = 6 are also presented. Published in 2011 by John Wiley & Sons, Ltd.

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

“…For the free surface representation equally spaced nodes were chosen with fixed s = 0.03, which led to N p ≈ 250 for all cases. For this set of numerical experiments, the conservation of the total system energy, calculated as in [11], was also monitored; the relative error at pinchoff time e E is shown in Table IV Table IV and Fig. 25, where the computed and experimental scaling exponents are plotted for this drop geometry, we can conclude that the transition region is sharp and it occurs at D ≈ 4, in perfect agreement with Burton and Taborek findings.…”

“…In particular, a complete numerical analysis comparing the numerical results with the analytical solution of an oscillating sphere was presented in [11]. In what follows we present the numerical experiments and results for the breakup and post breakup dynamics of a two lobe geometry drop.…”

“…As the total arc length of the free boundary changes with time, the boundary mesh size also changes, being s ≈ 0.033 and s ≈ 0.025 for cases (a) and (b), respectively. The time step is 0.001 initially and changed adaptively according to the scheme described in [11].…”

“…The presentation below is kept as brief as possible; further details can be found in [8,10,11]. Consider a fluid of density ρ I immersed in an (infinite) exterior fluid of density ρ E .…”

“…The direct boundary integral formulation for the two-fluid problem that is employed in this work has been discussed in detail in [11], and thus this section provides only a brief summary of this approach. The method is direct in that, in contrast to previous work [6,7], the potentials and normal derivatives for the two fluids are the basic variables.…”