Self-similar asymptotics for a class of Hele–Shaw flows driven solely by surface tension

We investigate the dynamics of relaxation, by surface tension, of a family of curved interfaces between an inviscid and viscous fluids in a Hele–Shaw cell. At t = 0 , the interface is assumed to be of the form | y | = A x m , where A > 0 , m ≥ 0 , and x > 0 . The case of 0 < m < 1 corresponds to a smooth shape, m > 1 corresponds to a cusp, whereas m = 1 corresponds to a wedge. The inviscid fluid tip retreats in the process of relaxation, forming a lobe which size increases with time. Combining analytical and numerical methods we find that, for any m , the relaxation dynamics exhibits self-similar behavior. For m ≠ 1 this behavior arises as an intermediate asymptotics: at late times for 0 ≤ m < 1 , and at early times for m > 1 . In both cases the retreat distance and the lobe size exhibit power-law behaviors in time with different dynamic exponents, uniquely determined by the value of m . In the special case of m = 1 (the wedge) the similarity is exact and holds for the whole interface at all times t > 0 , while the two dynamic exponents merge to become 1/3. Surprisingly, when m ≠ 1 , the interface shape, rescaled to the local maximum elevation of the interface, turns out to be universal (that is, independent of m ) in the similarity region. Even more remarkably, the same rescaled interface shape emerges in the case of m = 1 in the limit of zero wedge angle.