A general class of exact solutions is presented for a time-evolving bubble in a two-dimensional slow viscous flow in the presence of surface tension. These solutions can describe a bubble in a linear shear flow as well as an expanding or contracting bubble in an otherwise quiescent flow. In the case of expanding bubbles, the solutions have a simple behaviour in the sense that for essentially arbitrary initial shapes the bubble its asymptote is expanding circle. Contracting bubbles, on the other hand, can develop narrow structures (‘near-cusps’) on the interface and may undergo ‘breakup’ before all the bubble fluid is completely removed. The mathematical structure underlying the existence of these exact solutions is also investigated.
In this paper, we review some aspects of viscous fingering in a Hele-Shaw cell that at first sight appear to defy intuition. These include singular effects of surface tension relative to the corresponding zero-surface-tension problem both for the steady and unsteady problem. They also include a disproportionately large influence of small effects like local inhomogeneity of the flow field near the finger tip, or of the leakage term in boundary conditions that incorporate realistic thin-film effects. Through simple explicit model problems, we demonstrate how such properties are not unexpected for a system approaching structural instability or ill-posedness.
Moore (1979) demonstrated that the cumulative influence of small nonlinear effects on the evolution of a slightly perturbed vortex sheet is such that a curvature singularity can develop at a large, but finite, time. By means of an analytical continuation of the problem into the complex spatial plane, we find a consistent asymptotic solution to the problem posed by Moore. Our solution includes the shape of the vortex sheet as the curvature singularity forms. Analytic results are confirmed by comparison with numerical solutions. Further, for a wide class of initial conditions (including perturbations of finite amplitude), we demonstrate that 3/2-power singularities can spontaneously form at t=0+ in the complex plane. We show that these singularities propagate around the complex plane. If two singularities collide on the real axis, then a point of infinite curvature develops on the vortex sheet. For such an occurrence we give an asymptotic description of the vortex-sheet shape at times close to singularity formation.
Numerical and asymptotic solutions are found for the steady motion of a symmetrical bubble through a parallel-sided channel in a Hele–Shaw cell containing a viscous liquid. The degeneracy of the Taylor–Saffman zero surface-tension solution is shown to be removed by the effect of surface tension. An apparent contradiction between numerics and perturbation arises here as it does for the finger. This contradiction is resolved analytically for small bubbles and is shown to be the result of exponentially small terms. Numerical results suggest that this is true for bubbles of arbitrary size. The limit of infinite surface tension is also analyzed.
We show that the tritronquée solution yt of the Painlevé equation P I that behaves algebraically for large z with arg z = π/5, is analytic in a region containing the sector z = 0, arg z ∈ − 3π 5 , π and the disk z : |z| < 37 20 . This implies the Dubrovin conjecture, an important open problem in the theory of Painlevé transcendents. The method, building on a technique developed in [4], is general and constructive. As a byproduct, we obtain the value of the tritronquée and its derivative at zero, also important in applications, within less than 1/100 rigorous error bounds. arXiv:1209.1009v2 [math.CA] 23 Oct 2014 24 O. COSTIN 1 , M. HUANG 2 , S. TANVEER 1 8.1. Values of intermediate constants for ρ = 3. The numerical values of these constants might be helpful to the reader who would like to double-check the estimates. These are: J M = 0.282580···, jm = 0.64374···, Y 1,M = 1.16314···, Y 1,R,M = 0.132618···, E M = 0.0490292··· z 2,R,M = 0.54226···, z 2,M = 0.91863···, Mq = 0.066702···, M L,q = 0.075708···, V M = 0.2239···, T M = 0.0385··· M 1 = 1.13838···, M 2 = 0.04303···, M 3 = 0.28346···, M 4 = 0.45227···, M 5 = 0.05430···, M 6 = 0.00231···, M 7 = 0.02018··· 9. Acknowledgments
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