Program and information

s (in alphabetical order) Michael Dallaston (Imperial College London) ”Interfacial Hele-Shaw flows that develop corner singularities” It is well known that the evolution of a receding free boundary in one-phase Hele-Shaw flow (such as an expanding air bubble surrounded by viscous fluid) is ill-posed and generically forms cusps. The interfacial velocity at the cusp points is unbounded. Imposing a curvature-dependent regularization (such as surface tension) prevents the blow-up of both curvature and velocity, resulting in a well-posed problem. There is also physical and mathematical interest in the effect of a velocity-dependent regularization, often called kinetic undercooling because of its applications to thermodynamical problems of which free boundary Hele-Shaw flow is an ideal approximation. This talk will concern numerical and asymptotic evidence which strongly indicates that the generic behaviour of free boundary Hele-Shaw flow with this regularization, at least in certain regimes, is the formation of corner singularities. These corner singularities represent the blow up of curvature, but not velocity, so do not mean the problem is illposed. Some results and open questions regarding the selection of finger widths by kinetic undercooling will also be presented. Much of the work presented in this talk was published in the following paper: [1] M. C. Dallaston and S. W. McCue, Corner and finger formation in Hele-Shaw flow with kinetic undercooling regularisation, Euro. J. Appl. Math, 25 (2014):707–727 Yasuhide Fukumoto, Ummu Habibah (Kyushu University) ”Motion of a counter-rotating vortex pair in a viscous fluid” We establish a general formula for the traveling speed of a counter-rotating vortex pair, being valid for thick cores, moving in an incompressible fluid with and without viscosity. Two-dimensional motion of vortices with finite cores, interacting with each other, has been extensively studied both analytically and numerically. Mathematical methods and numerical schemes have been highly developed for dealing particularly with vortices of uniform vorticity, called vortex patches [1, 2]. In contrast, this is not the case with vortices with distributed vorticity. The method of matched asymptotic expansions was initiated by Ting and Tung [3] for this purpose. We complete Ting-Tung’s program for a vortex pair, with a general vorticity distribution at the leading order, and derive a surprisingly simple formula for the correction, due to the effect of finite core size, to the traveling speed [4]. The solution of the Navier-Stokes equations is constructed in the form of a power series in a small parameter, the ratio of the core radius to the distance between the core centers. A correction due to the effect of finite thickness of the vortices to the traveling speed makes its appearance at the 5th order. The 5th-order correction is shown to be expressible solely in terms of the 2nd-order quadrupole field. [1] J. Yang J and T. Kubota: SIAM J. Appl. Math. Vol. 54 (1994) 14-25. [2] P. Luzzatto-Fegiz: Fluid Dyn. Res. Vol. 46 (2014) 031408. [3] L. Ting and C. Tung: Phys. Fluids Vol. 8 (1965) 1039-1051. [4] U. Habibah and Y. Fukumoto: submitted to Fluid Dyn. Res.