Domain of convergence of perturbative solutions for Hele-Shaw flow near interface collapse

Adriana I. Pesci, Raymond E. Goldstein, Michael J. Shelley
1999 Physics of Fluids  
Recent work ͓Phys. Fluids 10, 2701 ͑1998͔͒ has shown that for Hele-Shaw flows sufficiently near a finite-time pinching singularity, there is a breakdown of the leading-order solutions perturbative in a small parameter ⑀ controlling the large-scale dynamics. To elucidate the nature of this breakdown we study the structure of these solutions at higher order. We find a finite radius of convergence that yields a new length scale exponentially small in ⑀. That length scale defines a ball in space
more » ... a ball in space and time, centered around the incipient singularity, inside of which perturbation theory fails. Implications of these results for a possible matching of outer solutions to inner scaling solutions are discussed.
doi:10.1063/1.870139 fatcat:5p6jwpxxzbfodajzetcgimfjjy