Eventos Anais de eventos
EPTT 2024
14th Spring School on Transition and Turbulence
Boundary integral simulations based on the vortex sheet formalism applied to track droplet interfaces in Hele-Shaw cells
Submission Author:
Rafael Menezes de Oliveira , RJ , Brazil
Co-Authors:
Rafael Menezes de Oliveira, Pedro Anjos
Presenter: Rafael Menezes de Oliveira
doi://10.26678/ABCM.EPTT2024.EPT24-0056
Abstract
We track nonlinear deformations of liquid droplets confined between the two parallel plates of a Hele-Shaw cell. The viscous droplet is surrounded by a fluid of different viscosity, and its interface is governed by surface tension. Both fluids satisfy Darcy’s law, which makes flow in their bulk potential and irrotational. As a result, vorticity is concentrated at the sharp interface between them. This allows for the discretization of the interface based on an accurate boundary integral method written in terms of the vortex sheet. This quantity measures the jump of the tangential velocity component as one crosses the interface. It is related to and better suited than vorticity to track sharp interface deformations. Application of Darcy'law and the Young-Laplace pressure jump boundary condition allows to write a Birkhoff-Rott integral equation for the distribution of vortex sheet in every time step, which is discretized by the modified point vortex approximation. Time evolution of the interfacial shape is conducted by integrating in time the interface velocity written in terms of the vortex sheet. This is accomplished by writing the evolution equations in terms of the tangent angle and arclength. This change of variables is motivated by the Frenet-Serret identity that relates derivatives of the tangent angle with planar curvatures. This reduces the order of the differential equations and, alongside the small-scale decomposition, also removes stiffness. The final evolution equations have spectral accuracy, are linear, and are solved by the Adams-Bashforth algorithm. We discuss this methodology and present results for different Hele-Shaw flow configurations, including (i) Growth of viscous fingers by radial injection; (ii) Droplet deformations related to adhesion phenomena and driven by air invasion when the top plate is lifted; and (iii) Deformations of magnetic droplets subjected to crossed magnetic fields.
Keywords
Boundary integral method, Hele-Shaw cell, fingering instability
