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Bubble dynamics and velocity selection in a Hele-Shaw cell

Presented by: 
Giovani Vasconcelos
Date: 
Tuesday 17th December 2019 - 14:00 to 15:00
Venue: 
INI Seminar Room 2
Abstract: 
The unsteady motion of a finite assembly of bubbles in a Hele-Shaw channel is studied in the case when surface tension is neglected. A general exact solution is obtained in terms of a conformal map from a multiply connected circular domain to the fluid region exterior to the bubbles. The correspond- ing mapping function is given explicitly in terms of certain special transcen- dental functions, known as the secondary Schottky-Klein prime functions. Exploring the properties of these solutions, we show that steady configura- tions where the bubbles move with a velocity, U, which is twice greater than the velocity, V , of the background flow, i.e., U = 2V , are the only attrac- tor of the dynamics; whereas solutions with U ≠ 2V act as repellors. This demonstrates that the special nature of the solutions with U = 2V is already built-in in the zero-surface-tension dynamics, which is confirmed by the in- clusion of regularization effects. In particular, the case of a single bubble will be discussed in detail and several numerical examples of bubble evolution and bubble selection will be presented.




University of Cambridge Research Councils UK
    Clay Mathematics Institute London Mathematical Society NM Rothschild and Sons