M.M. Alimov
Kazan Federal University, Kazan, 420008 Russia
E-mail: Mars.Alimov@kpfu.ru
Received February 15, 2018
Abstract
The problem of the motion of steady fingers in a Hele-Shaw cell in case of nonzero surface tension was posed more than 60 years ago. However, it has not been completely solved. The analysis in the classical formulation by Saffman and Taylor has revealed that the solution has a singularity at the infinity point where the base of the finger lies. This makes it difficult to realize a correct numerical analysis of the problem. A new modified formulation of the problem is proposed. In this formulation, we postulate that the distance at which the finger is completely formed is limited, which removes the potential source of solution singularity. This distance is, in fact, an additional parameter of the modified formulation. For the case of zero surface tension, it has been established that, in the limit, when this distance tends to infinity, the solution of the problem in the modified formulation tends to the Saffman and Taylor finger solution. Specific examples show that it is sufficient to choose a quantity of order 1 as the value of the additional parameter of the modified formulation. This will already ensure the practical coincidence of the finger shape with the form that corresponds to the classical solution for the case of zero surface tension. It has been also established that the principle of minimum energy dissipation rate formulated in the local form can be used when dissipation is estimated only in a small neighborhood of the free boundary. It has been shown that this principle selects a finger of thickness 0.5 from the family of solutions of the problem in the new formulation in full agreement with the results of the experiments in the case of zero surface tension. The revealed qualities of the modified formulation make it promising from the point of view of its generalization to the case of nonzero surface tension and the possibility of its correct numerical analysis.
Keywords: Hele-Shaw flow, free boundary problems, exact solution, minimum energy dissipation rate
Acknowledgments. The work is performed according to the Russian Government Program of Competitive Growth of Kazan Federal University.
References
1. Saffman P.G., Taylor G.I. The penetration of a fluid into a porous medium or Hele-Shaw cell containing a more viscous liquid. Proc. R. Soc. London, Ser. A, 1958, vol. 245, no. 1242, pp. 312–329.
2. McLean J.W., Saffman P.G. The effect of surface tension on the shape of fingers in a Hele-Shaw cell. J. Fluid Mech., 1981, vol. 102, pp. 455–469.
3. Ockendon J.R., Howison S.D. Kochina and Hele-Shaw in modern mathematics, natural science and industry. J. Appl. Math. Mech., 2002, vol. 66, no. 3, pp. 505–512. doi: 10.1016/S0021-8928(02)00060-6.
4. Chapman S.J. On the role of Stokes line in the selection of Saffman–Taylor finger with small surface tension. Eur. J. Appl. Math., 1999, vol. 10, pp. 513–534.
5. Shraiman B. Velocity selection and the Saffman – Taylor problem. Phys. Rev. Lett., 1986, vol. 56, pp. 2028–2031.
6. Tanweer S. Analytic theory for the selection of a symmetric Saffman–Taylor finger in a Hele-Shaw cell. Phys. Fluids, 1987, vol. 30, no. 6, pp. 1589–1605.
7. Combescot R., Hakim V., Dombre T., Pomeau Y., Pumir A. Analytic theory of the Saffman–Taylor fingers. Phys. Rev. A, 1988, vol. 37, no. 4, pp. 1270–1283.
8. Tanvir S. Viscous Displacement in a Hele-Shaw Cell. Segur H. et al. (Eds.). New York, Plenum Press, 1991, pp. 131–153.
9. Tanweer S. The effect of surface tension on the shape of a Hele-Shaw cell bubble. Phys. Fluids, 1986, vol. 29, no. 11, pp. 3537–3548.
10. Alimov M.M.. Numerical analysis of solution of the Hele-Shaw problem for steadily moving bubble. Uchenye Zapiski Kazanskogo Universiteta. Seriya Fiziko-Matematicheskie Nauki, 2012, vol. 154, no. 1, pp. 99–113. (In Russian)
11. Lamb H. Gidrodinamika [Hydrodynamics]. Moscow, Leningrad, Gostekhizdat, 1947. 928 p. (In Russian)
12. Vasconcelos G.L. Exact solutions for N steady fingers in a Hele-Shaw cell. Phys. Rev. E, 1998, vol. 58, no. 5, pp. 6858–6860.
13. Gurevich M.I. Teoriya strui ideal'noi zhidkosti [Theory of Jets in Ideal Fluids]. Moscow, Nauka, 1979. 536 p. (In Russian)
14. Alimov M.M. On steady solutions of Hele-Shaw problem. Uchenye Zapiski Kazanskogo Universiteta. Seriya Fiziko-Matematicheskie Nauki, 2005, vol. 147, no. 3, pp. 33–48. (In Russian)
15. Lavrent'ev M.A., Shabat B.V. Metody teorii funktsii kompleksnogo peremennogo [Methods of Complex Variable Theory]. Moscow, Nauka, 1973. 736 p. (In Russian)
16. Alimov M.M. The principle of minimum energy dissipation rate in steady Hele–Shaw flows. Fluid Dyn., 2013, vol. 48, no. 4, pp. 512–522. doi: 10.1134/S0015462813040108.
For citation: Alimov M.M. A modified formulation of the problem of the steady finger in a Hele-Shaw cell. Uchenye Zapiski Kazanskogo Universiteta. Seriya Fiziko-Matematicheskie Nauki, 2018, vol. 160, no. 3, pp. 462–476. (In Russian)
The content is available under the license Creative Commons Attribution 4.0 License.