Bubble break-off in Hele-Shaw flow: Integrability and matrix model.

摘要

Consider a situation in Hele-Shaw flow in which the domain of inviscid fluid breaks off and changes its connectivity. Asymptotically close to the breaking-off the evolution is universally characterized by (i) self-similarity and (ii) two real parameters. The solutions outside the above category exist but require a fine tuning of the initial shape and, therefore, are unlikely to occur. A generic solutions for break-off turns out to satisfy the dispersionless AKNS hierarchy of equations, i.e. an infinite set of partial differential equations that are mutually compatible. The Hele-Shaw equation (Darcy's Law) and the shape of the boundary become the string equation and the spectral curve appearing in the hierarchy, respectively.; The dispersionful AKNS hierarchy introduces a small parameters h to regularize the Hele-Shaw system. This is the set of an infinite number of equations which becomes the dispersionless AKNS hierarchy of equations at the limit of zero h. With non-zero h, the break-off is described by an ordinary differential equation called the Painleve II equation. Microscopically, the normal matrix model (NMM) describes the above regularized Hele-Shaw flow. The same Painleve equation appears in the scaling limit of NMM.