Abstract
A new local form of the principle, in which dissipation is estimated only in a small vicinity of a free interface in steady Hele-Shaw flows, is proposed. It is established that for the problem of bubble propulsion the principle proposed is mathematically equivalent to the variational principle formulated by Saffman and Taylor without physical validation. It is shown that the new local form of the minimum dissipation principle effectively solves the problem of selection of a unique selection in the problems of both bubble and finger propulsion.
Similar content being viewed by others
References
P.G. Saffman and G.I. Taylor, “The Penetration of a Fluid into a Porous Medium or Hele-Shaw Cell Containing a More Viscous Liquid,” Proc. Roy. Soc. London. Ser. A 245(1242), 312 (1958).
G.I. Taylor and P.G. Saffman, “A Note on the Motion of Bubbles in a Hele-Shaw Cell and Porous Medium,” Quart. J. Mech. Appl. Math. 12, 265 (1959).
C.-W. Park, S.R.K. Maruvada, and D.-Y. Yoon, “The Influence of Surfactant on the BubbleMotion in Hele-Shaw Cells,” Phys. Fluids 6, 3267 (1994).
J.R. Okendon and S.D. Howison, “Kochina and Hele-Shaw in Modern Mathematics, Science, and Industry,” Prikl. Mat. Mekh. 66, 515 (2002).
S. Tanweer, “The Effect of Surface Tension on the Shape of a Hele-Shaw Cell Bubble,” Phys. Fluids 29, 3537 (1986).
H. Lamb, Hydrodynamics, Cambridge Univ. Press, Cambridge (1924).
I.A. Charnyi, Underground Hydro- and Gas Dynamics [in Russian], Gostoptekhizdat, Moscow (1963).
K.P. Gurov, Phenomenological Thermodynamics of Irreversible Processes [in Russian], Nauka, Moscow (1978).
I.P. Bazarov, Thermodynamics [in Russian], Vysshaya Shkola, Moscow (1991).
G.I. Barenblatt, V.M. Entov, and V.M. Ryzhik, [in Russian], Liquid and Gas Flows in Natural Formations, Nedra, Moscow (1984).
G.A. Korn and T.M. Korn, Mathematical Handbook for Scientists and Engineers, McGraw-Hill, New York (1968).
M.G. Bernadiner and V.M. Entov, Hydrodynamic Theory of Seepage Flows of Anomalous Fluids [in Russian], Nauka, Moscow (1975).
I.S. Gradshteyn and I.M. Ryzhik, Tables of Integrals, Sums, Series, and Products, Acad. Press, New York (1982).
Additional information
Original Russian Text © M.M. Alimov, 2013, published in Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, 2013, Vol. 48, No. 4, pp. 104–116.
Rights and permissions
About this article
Cite this article
Alimov, M.M. Principle of minimum energy dissipation rate in steady Hele-Shaw flows. Fluid Dyn 48, 512–522 (2013). https://doi.org/10.1134/S0015462813040108
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S0015462813040108