Soft Matter and Complex Systems Seminar
sala 1.40, ul. Pasteura 5
2017-04-21 (09:30) 
Mark Mineev-Weinstein (Federal University of Rio Grande do Norte, Brazil)
Laplacian growth and Selection in Non-Equilibrium Physics
Selecting a single member from continuum of stationary solutions of the nonlinear Laplacian growth equation (LGE), so that the selected member corresponds to the observable asymptotic pattern, is highly non-trivial and has attracted a lot of attention. This problem was solved in 1986 by adding surface tension and using the WKB-like theory, called “Asymptotics beyond all orders”, developed by Kruskal and others.
Finite-parametric exact solutions of LGE, obtained due to its integrability, made possible to challenge this traditional approach by selecting the correct member from a continuous family without surface tension. This was done in 1998 for a relatively simple geometric pattern, namely the finger propagating in a long rectangular Hele-Shaw channel.
After surveying this background, I will demonstrate very recent (and strange) selection results, obtained in 2014-2016 in a multi-connected moving domain. Using exact solutions for this geometry, we obtained that an arbitrary number of moving bubbles reach (after nonlinear interaction) the same asymptotic velocity, which is precisely twice the velocity of a background flow. (In the singular limit this result is reduced to the finger problem mentioned above.)
Finite-parametric exact solutions of LGE, obtained due to its integrability, made possible to challenge this traditional approach by selecting the correct member from a continuous family without surface tension. This was done in 1998 for a relatively simple geometric pattern, namely the finger propagating in a long rectangular Hele-Shaw channel.
After surveying this background, I will demonstrate very recent (and strange) selection results, obtained in 2014-2016 in a multi-connected moving domain. Using exact solutions for this geometry, we obtained that an arbitrary number of moving bubbles reach (after nonlinear interaction) the same asymptotic velocity, which is precisely twice the velocity of a background flow. (In the singular limit this result is reduced to the finger problem mentioned above.)