Generalized diffusion of vortex: Self-similarity and the Stefan problem

Journal of Mathematical Sciences

Volumn 161, Issue 5, 2009, Pages 628-647

  Georgievskii, D V ^a  

^a MOSCOW STATE UNIVERSITY   (Russian Federation)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 70350673912     PISSN: 10723374     EISSN: 15738795     Source Type: Journal    
DOI: 10.1007/s10958-009-9593-x     Document Type: Article

References (80)

1. A. V. Aksenov, Nonlinear Periodic Waves in Gas-Similar Media. [in Russian], Doctoral thesis, Moscow (2004), 242 p.
2. M. M. Alvarez, F. J. Muzzio, S. Cerbelli, A. Adrover, and M. Giona, Self-similar spatiotemporal structure of intermaterial boundaries in chaotic flows, Phys. Rev. Lett., 81, No. 16, 3395-3398 (1998).
3. B. D. Annin, V. O. Bytev, and S. I. Senashov, Group Properties of Elastisity and Plasticity Equations [in Russian], Nauka, Novosibirsk (1985).
4. V. G. Baidulov, P. V. Matyushin, and Yu. D. Chashechkin, Evolution of flow induced by the diffusion on the sphere submerged in a continuously stratified fluid, Izv. Ross. Akad. Nauk. MZhG, No. 2 (2007).
5. A. A. Barmin and A. P. Chugainova, Interaction of slow magnetic hydrodynamical shock wave with tangential break, Izv. Ross. Akad. Nauk. MZhG, No. 6. 122-133 (2002).
6. S. P. Bautin and A. A. Eliseev, Many-dimensional analytical heat wave defined by the boundary conditions, Differential Equations,42, No. 5, 1052-1062 (2006).
7. A. Chanda, P. K. Das, A. K. De, and J. Das, Self-similarity in the decay of homogeneous and isotropic turbulence, J. Techn. Phys., 41, No. 1, 43-47 (2000).
8. E. A. Cheblakova, Modelling of convection in domains with free boundaries, Kalkul. Tekhnol., 5, No. 6, 87-98 (2000).
9. X. Chen, M. A. Abramowicz, and J.-P. Lasota, Advection-dominated accretion: global transonic solutions, Astrophys. J., 476, No. 1, Pt. 1, 61-69 (1997).
10. E. M. Dobryshman, Hydrodynamic model of non-stationary processes in the typhoon eye, Meteorology and Hydrology, No. 12, 5-18 (1996).
11. Z. Dursunkaya and S. Nair, Solidification of a finite medium subject to a periodic variation of boundary temperature, Trans. ASME. J. Appl. Mech., 70, No. 5, 633-637 (2003).
12. K. D. Ellenrieder and B. J. Cantwell, Self-similar, slightly compressible, free vortices, J. Fluid Mech., 423, 293-315 (2000).
13. J. D. Evans and J. R. King, Asymptotic results for the Stefan problem with kinetic undercooling, Quart. J. Mech. Appl. Math., 53, No. 3, 449-473 (2000).
14. J. D. Evans and J. R. King, The Stefan problem with nonlinear kinetic undercooling, Quart. J. Mech. Appl. Math., 56, No. 1, 139-161 (2003).
15. D. V. Evdokimov and A. A. Kochubei, The application of boundary element methods for solving the Stefan problem in case of slow phase transitions, Visn. Khar'kov. Nats. Univ., No. 590, 26-31 (2003).
16. D. V. Georgievskii, Break diffusion of the tangential strain on the boundary of a viscoplastic half-plane, Prikl. Mat. Mekh., 70, No. 5, 884-892 (2006).
17. D. V. Georgievskii, Self-similar solutions in the problem on generalized diffusion of vortex, Izv. Ross. Akad. Nauk. MZhG, No. 2 (2007).
18. D. V. Georgievskii, Stability of Deformation Processes of Viscous-Plastic Media [in Russian], URSS, Moscow (1998).
19. D. V. Georgievskii, Some non-one-dimensional problems of viscous-plasticity: rigid zones and stability (a survey), Izv. Ross. Akad. Nauk. MTT, No. 1, 61-78 (2001).
20. D. V. Georgievskii, The stability of a non-stationary shift of a viscous-plastic half-plane with tangential break along the boundary, Vestn. MGU, Ser. 1. Mat., Mekh., No. 3, 65-72 (1996).
21. R. S. R. Gorla, I. Pop, and V. Dakappagari, Three-dimensional flow of a power-law fluid due to a stretching flat surface, ZAMM Z. Angew. Math. Mech., 75, No. 5, 389-394 (1995).
22. H. R. W. Gottlieb, Exact solution of a Stefan problem in a nonhomogeneous cylinder, Appl. Math. Lett., 15, No. 2, 176-172 (2002).
23. T. B. Grankina, Numerical methods of solving the one-phase Stefan problem, Dinamika Sploshn. Sredy, No. 118, 16-20 (2001).
24. Yu. V. Gurkov and A. G. Petrova, Numerical solution of the Stefan problem with non-constant temperature of the phase transition, PMTF, 37, No. 4, 105-112 (1996).
25. C. K. Hsieh, Exact solution of a Stefan problems related to a moving line heat source in a quasi-stationary state, Trans. ASME. J. Heat Transfer, 117, No. 4, 1076-1078 (1995).
26. R. R. Huilgol, On kinematic conditions affecting the existence and nonexistence of a moving yield surface in unsteady uni-directional flows of Bingham fluids, J. Non-Newton. Fluid Mech., 123, Nos. 2-3, 215-221 (2004).
27. A. A. Il'yushin, Some problems of the theory of plastic flow, Izv. AN SSSR., No. 2, 64-86 (1958).
28. A. Yu. Ishlinskii and G. I. Barenblatt, On the blow of a viscous-plastic bar on a rigid barrier, Dokl. Akad. Nauk SSSR, 144, No. 4, 734-737 (1962).
29. O. E. Ivashev and N. N. Smirnov, Formation of a split of hydro-break in a porous medium, Vestn. MGU. Ser. Mat. Mekh., No. 6, 28-36 (2003).
30. C. R. Kaiser and P. Alexander, Heating of the intergalactic medium by FR II radio sources, Mon. Notic. Roy. Astron. Soc., 305, No. 3, 707-723 (1999).
31. I. A. Kaliev, Global solvability of a problem modelling the phase transition gas-solid body, Dinamika Sploshn. Sredy, No. 116, 36-41 (2000).
32. V. K. Kalpakidis, K. G. Balassas, Symmetry groups and similarity solutions for a free convective boundary-layer problem, Internat. J. Non-Linear Mech., 39, No. 10, 1659-1670 (2004).
33. E. Kamke, A Reference Book on Ordinary Differential Equations [Russian translation], Nauka, Moscow (1976).
34. L. P. Kholpanov, S. E. Zakiev, and V. A. Fedotov, The Neumann-Lame-Clapeyron-Stefan problem and its solving by the partial differentiation method, Teor. Osn. Khim. Tekhnol.,37, No. 2, 128-137 (2003).
35. Inwon C. Kin, Uniqueness and existence results on the Hele-Shaw and the Stefan problems, Arch. Ration. Mech. Anal., 168, No. 4, 299-328 (2003).
36. J. R. King and S. M. Cox, Self-similar stagnation point boundary layer flows with suction or injection, Stud. Appl. Math., 115, No. 1, 73-107 (2005).
37. D. M. Klimov, A. G. Petrov, and D. V. Georgievskii, Viscous-Plastic Flows: Dynamical Chaos, Stability, Mixing [in Russian], Nauka, Moscow (2005).
38. N. E. Kochin, I. A. Kibel', and N. V. Rose, Theoretical Hydromechanics. Vol. 2 [in Russian], Fizmatlit, Moscow (1963).
39. N. V. Krasnoshchek, The Stefan problem for degenerating system of equations, Dop. Nats. Akad. Ukrainy, No. 4, 29-32 (1999).
40. S. N. Kruzhkov, On some problems with unknown boundary for the heat equation, Prikl. Mat. Mekh., 31, No. 6, 1009-1020 (1967).
41. S. N. Kruzhkov, On a certain class of problems with unknown boundary for the heat equation, Dokl. Akad. Nauk SSSR, 178, No. 5, 1036-1038 (1968).
42. A. G. Kulikovskii and F. A. Slobodkina, On the choice of an similarity solution in the boundary layer theory, In: Fluid and Gas Mechanics. To the 80th Anniversary of Academician G. G. Chernyi [in Russian], Fizmatlit, Moscow (2003), pp. 621-626. Reprint: Izv. AN SSSR. Mekh. Zhidk. Gas., No. 4, 42-46 (1974).
43. A. A. Lacey and L. A. Herraiz, Macroscopic models for melting derived from averaging microscopic Stefan problems. I. Simple geometries with kinetic undercooling of surface tension, Europ. J. Appl. Math., 11, No. 2, 153-169 (2000).
44. L. G. Loizyanskii, Fluid and Gas Mechanics [in Russian], Drofa, Moscow (2003).
45. Yu. N. Makov, On the universal similarity solution of the Khokhlov-Zabolotskii equation for waves with shock fronts, Akust. Zh.,43, No. 6, 828-833 (1997).
46. S. A. Maslowe and R. J. Spiteri, The continuous spectrum for a boundary layer in a streamwise pressure gradient, Phys. Fluids, 13, No. 5, 1294-1299 (2001).
47. V. N. Monakhov, Exact solutions of heat boundary layer equations for nonlinearly-viscous fluids, Dokl. Ross. Akad. Nauk,367, No. 3, 299-302 (1999).
48. V. N. Monakhov, Similarity problems of two-phase filtration nonlinearly-viscous fluids, Dokl. Ross. Akad. Nauk,412, No. 2, 193-195 (2007).
49. R. Narayan, S. Kato, and F. Honma, Global structure and dynamics of advection-dominated accretion flows around black holes, Astrophys. J., 476, No. 1, Pt. 1, 49-60 (1997).
50. G. A. Nesenenko, Mathematical Modelling of Nonlinear Singularly Perturbed Nonstationary Processes of Heat and Mass Conduction [in Russian], Doctoral thesis, Moscow (2003).
51. V. E. Neuvazhaev and A. F. Podkorytova, Turbulence mixing of a thin layer according to the k-ε model. Comparison with the experiments, In: Vopr. Atom. Nauki i Tekhn. Ser. Mat. Model. Fiz. Proc. [in Russian], No. 1 (2001), pp. 14-19.
52. M. Oberlack, A unified approach for symmetries in plane parallel turbulent shear flows, J. Fluid Mech., 427, 299-308 (2001).
53. P. M. Ogibalov and A. Kh. Mirzadzhanzade, Non-Stationary Motions of Viscous-Plastic Media [in Russian], Nauka, Moscow (1977).
54. N. N. Okulova, On a certain method of solution of the problem on the vortex layer diffusion in a viscoplastic half-plane, Vestn. MGU. Ser. Mat. Mekh., No. 4 (2007).
55. L. V. Ovsyannikov, Group Analysis of Differential Equations [in Russian], Nauka, Moscow (1978).
56. H. Pascal and J. P. Pascal, Similarity solutions to rotating Couette flow with power law fluids, Acta mech., 107, Nos. 1-4, 93-100 (1994).
57. J. P. Pascal and H. Pascal, On some similarity solutions to shear flows of non-Newtonian powerlaw fluids, Acta Mech., 112, Nos. 1-4, 229-236 (1995).
58. B. E. Pobedrya, Mechanics of Composite Materials [in Russian], MGU, Moscow (1984).
59. B. E. Pobedrya, On fractals in mechanics, Vestn. MGU. Ser. Mat. Mekh., No. 1, 40-44 (2000).
60. V. V. Pukhnachev, Symmetries in Navier-Stokes equations, Usp. Mekh., 4, No. 1, 6-76 (2006).
61. L. I. Rubinshtein, Stefan Problem [in Russian], Zvaigzne, Riga (1967).
62. A. I. Safronchik, Non-stationary flow of a viscoplastic material between parallel walls, Prikl. Mat. Mekh., 23, No. 5, 925-935 (1959).
63. A. I. Safronchik, Rotation of a cylinder with variable angular velocity in a viscous-plastic medium, Prikl. Mat. Mekh., 23, No. 6, 1051-1056 (1959).
64. A. Ya. Sagomonyan, Motion of landslides appearing on slopes of heights under the action of the rain, Vestn. MGU, Ser. 1. Mat., Mekh., No. 4, 30-36 (1999).
65. B. A. Saveliev, Glaciology [in Russian], MGU, Moscow (1991).
66. L. I. Sedov, Similarity and Dimension Methods in Mechanics [in Russian], Nauka, Moscow (1987).
67. Z. P. Shulman and B. M. Berkovskii, Boundary Layer of Non-Newtonian Fluids [in Russian], Nauka i Tekhn., Minsk (1966).
68. N. N. Smirnov and V. R. Tagirova, Self-similar solutions of the problem on the formation of a hydro-break split in a porous medium, Izv. Ross. Akad. Nauk. MZhG, No. 1, 70-82 (2007).
69. R. Tamsalu, P. Mälkki, and K. Myrberg, Self-similarity concept in marine system modelling, Geophysica, 33, No. 2, 51-68 (1997).
70. D. A. Tarzia and C. V. Turner, The asymptotic behavior for the two-phase Stefan problem with a convective boundary condition, Commun. Appl. Anal., 7, Nos. 2-3, 313-334 (2003).
71. A. N. Tikhonov and A. A. Samarskii, Equations of Mathematical Physics [in Russian], MGU, Moscow (1999).
72. P. P. Volosevich and E. I. Levanov, Self-similar Solutions of Problems of Gas Dynamics and Heat Conduction [in Russian], MFTI, Moscow (1997).
73. A. N. Vulfson, Self-similarity and propagation of the upper boundary of convective thermics in a neutrally stratified atmosphere, caused by point, linear, and plane heat sources, Izv. Ross. Akad. Nauk. Fiz. Atm. and Okean.,34, No. 4, 557-564 (1998).
74. 2) = 0 with λ ∈ (-1/2; 0) arising in boundary layer theory, Appl. Math. Lett., 17, No. 11, 1261-1265 (2004).
75. Xu Yonglin, Self-similar crack expansion method for two-dimensional cracks under mixed mode loading conditions, Eng. Fract. Mech., 59, No. 2, 165-182 (1998).
76. M. Yürüsoy, Similarity solutions of boundary layer equations for a special non-Newtonian fluid in a special coordinate system, Appl. Math. Mech., 25, No. 5, 587-594 (2004).
77. V. S. Zarubin and M. M. Lukashin, The solution of the Stefan problem by the method of through counting, In: Problems of Gas Dynamics and Heat-mass-exchange in Energy Plants. Vol. 2, MEI, Moscow (2003), pp. 372-375.
78. L. C. Zheng, X. H. Su, and X. Zhang, Similarity solutions for boundary layer flow on a moving surface in a otherwise quiescent fluid medium, Internat. J. Pure Appl. Math., 19, No. 4, 541-552 (2005).
79. Yu. V. Zhernovyi and M. T. Sajchuk, On using the Green function method for the numerical solution of multidimensional Stefan problems, Inzh. Fiz. Zh., 71, No. 5, 910-916 (1998).
80. A. V. Zvyagin, Motion of a viscous fluid in a canal with elastic boundaries, Vestn. MGU. Ser. Mat. Mekh., No. 1, 50-54 (2005).