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MOVING STONE IN THE HELE-SHAW FLOW

Part of: Incompressible viscous fluids Miscellaneous topics - Partial differential equations Elliptic equations and systems

Published online by Cambridge University Press:  08 April 2015

Gennady Mishuris ,
Sergei Rogosin  and
Michal Wrobel
Gennady Mishuris
Affiliation:
Aberystwyth University, Penglais, Aberystwyth SY23 3BZ, U.K. email [email protected]
Sergei Rogosin
Affiliation:
Aberystwyth University, Penglais, Aberystwyth SY23 3BZ, U.K. email [email protected] Belarusian State University, Nezavisimosti Ave., 4, 220030 Minsk, Belarus email [email protected]
Michal Wrobel
Affiliation:
Aberystwyth University, Penglais, Aberystwyth SY23 3BZ, U.K. email [email protected]
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Abstract

Asymptotic analysis of the Hele-Shaw flow with a small moving obstacle is performed. The method of solution utilizes the uniform asymptotic formulas for Green’s and Neumann functions recently obtained by V. Maz’ya and A. Movchan. The theoretical results of the paper are illustrated by numerical simulations.

MSC classification

Primary: 76D27: Other free-boundary flows; Hele-Shaw flows 35J25: Boundary value problems for second-order elliptic equations 35R35: Free boundary problems
Type
Research Article
Information
Mathematika , Volume 61 , Issue 2 , May 2015 , pp. 457 - 474
Copyright
Copyright © University College London 2015 

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References

Antontsev, S. N., Gonçalves, C. R. and Meirmanov, A. M., Local existence of classical solutions to the well-posed Hele-Shaw problem. Port. Math. (N.S.) 59(4) 2002, 435–452.Google Scholar
Antontsev, S. N., Gonçalves, C. R. and Meirmanov, A. M., Exact estimates for the classical solutions to the free-boundary problem in the Hele-Shaw cell. Adv. Differ. Equ. 8(10) 2003, 1259–1280.Google Scholar
Dallaston, M. C. and McCue, S. W., New exact solutions for Hele-Shaw flow in doubly connected regions. Phys. Fluids 24 2012, 052101.CrossRefGoogle Scholar
Driscoll, T. A., Algorithm 756: A MATLAB toolbox for Schwarz–Christoffel mapping. ACM Trans. Math. Software 22 1996, 168–186.CrossRefGoogle Scholar
Driscoll, T. A., Algorithm 843: Improvements to the Schwarz–Christoffel toolbox for MATLAB. ACM Trans. Math. Software 31 2005, 239–251.CrossRefGoogle Scholar
Entov, V. and Etingof, P., On the break of air bubbles in a Hele-Shaw cell. European J. Appl. Math. 22(2) 2011, 125–149.CrossRefGoogle Scholar
Escher, J. and Simonett, G., Classical solutions of multidimensional Hele-Shaw models. SIAM J. Math. Anal. 28(5) 1997, 1028–1047.CrossRefGoogle Scholar
Galin, L. A., Unsteady filtration with a free surface. Dokl. Akad. Nauk USSR 47 1945, 246–249 (in Russian).Google Scholar
Gilbarg, D. and Trudinger, N. S., Elliptic Partial Differential Equations of Second Order, 2nd edn., Springer (Berlin, 2001).CrossRefGoogle Scholar
Gradshteyn, I. S. and Ryzhik, I. M., Table of Integrals, Series, and Products, 7th edn., Elsevier (Amsterdam, 2007).Google Scholar
Gustafsson, B., On a differential equation arising in a Hele-Shaw flow moving boundary problem. Ark. Mat. 22 1984, 251–268.CrossRefGoogle Scholar
Gustafsson, B. and Vasil’ev, A., Conformal and Potential Analysis in Hele-Shaw Cells, Birkhäuser Verlag (Basel, 2006).Google Scholar
Hadamard, J., Sur le problème d’analyse relatif à l’equilibre des plaques élastiques encastrées. In Memoire Couronne en 1907 par l’Academie: Prix Vaillant, Mémoires Présentés par Divers Savants à lÁcademie des Sciences, Vol. 33, No. 4, 1908 (Oeuvres de Jasques Hadamard 2), Centre National de la Recherche Scientifique (Paris, 1968), 515–629.Google Scholar
Hele-Shaw, H. S., The flow of water. Nature 58(1489) 1898, 33–36.Google Scholar
Hohlov, Yu. E. and Reissig, M., On classical solvability for the Hele-Shaw moving boundary problem with kinetic undercooling regularization. European J. Appl. Math. 6 1995, 421–439.CrossRefGoogle Scholar
Maz’ya, V. and Movchan, A., Uniform asymptotics of Green’s kernels for mixed and Neumann problems in domains with small holes and inclusions. In Sobolev Spaces in Mathematics. III: Applications in Mathematical Physics (International Mathematical Series 10) (ed. Isakov, V.), Springer and Tamara Rozhkovskaya Publisher, Novosibirsk (New York, 2009), 277–316.CrossRefGoogle Scholar
Maz’ya, V. and Movchan, A., Uniform asymptotics of Green’s kernels in perforated domains and meso-scale approximation. Complex Var. Elliptic Equ. 57(2) 2012, 137–154.CrossRefGoogle Scholar
Maz’ya, V., Movchan, A. and Nieves, M., Green’s Kernel and Meso-Scale Approximations in Perforated Domains (Lecture Notes in Mathematics 2077), Springer (Heidelberg, 2013).CrossRefGoogle Scholar
Mishuris, G., Rogosin, S. and Wrobel, M., Hele-Shaw flow with a small obstacle. Meccanica 49(9) 2014, 2037–2047, doi:10.1007/S11012-014-9919-8.CrossRefGoogle Scholar
Mishuris, G., Rogosin, S. and Wrobel, M., Moving stone in the Hele-Shaw flow. Preprint, 2014,arXiv:1404.3547v2 [physics.flu-dyn].CrossRefGoogle Scholar
Papamichael, N., Lectures on Numerical Conformal Mapping, University of Cyprus, 2008.Google Scholar
Polubarinova-Kochina, P. Ya., On the motion of the oil contour. Dokl. Akad. Nauk SSSR 47 1945, 254–257 (in Russian).Google Scholar
Reissig, M., The existence and uniqueness of analytic solutions for moving boundary value problem for Hele-Shaw flows in the plane. Nonlinear Anal. 23(5) 1994, 565–576.CrossRefGoogle Scholar
Vasil’ev, A., From the Hele-Shaw experiment to integrable systems: a historical overview. Complex Anal. Oper. Theory 3 2009, 551–585.CrossRefGoogle Scholar