Studying the Model of Air and Water Filtration in a Melting or Freezing Snowpack

The article is devoted to a theoretical study of a non-stationary problem on thermomechanical processes in snow taking into account the effects of melting and freezing. Snow is modeled as a continuous medium consisting of water, air and porous ice skeleton. The governing equations of snow are based on the fundamental conservation laws of continuum mechanics. For the one-dimensional setting, the Rothe scheme is constructed as an approximation of the considered problem and the Rothe method is formally justified, i.e., convergence of approximate solutions to the solution of the considered problem is established under some additional regularity requirements.

Keywords

filtration; phase transition; snow; conservation laws; Rothe method.

References

1. Daanen R.P., Nieber J.L. Model for Coupled Liquid Water Flow and Heat Transport with Phase Change in a Snowpack. Journal of Cold Regions Engineering, 2009, vol. 23, no. 2, pp. 43-68. DOI: 10.1061/(ASCE)0887-381X(2009)23:2(43)
2. Sibin A.N., Papin A.A. Heat and Mass Transfer in Melting Snow. Journal of Applied Mechanics and Technical Physics, 2021, vol. 62, no. 1, pp. 96-104. DOI: 10.1134/S0021894421010120
3. Papin A.A., Tokareva M.A. [Dynamics of Melting Deformable Snow-Ice Cover]. Vestnik of Novosibirsk State University. Mathematics, Mechanics, Informatics, 2012, vol. 12, no. 4, pp. 107-113. (in Russian)
4. Tokareva M.A., Papin A.A. Mathematical Model of Fluids Motion in Poroelastic Snow-Ice Cover. Journal of Siberian Federal University. Mathematics and Physics, 2021, vol. 14, no. 1, pp. 47-56. DOI: 10.17516/1997-1397-2021-14-1-47-56
5. Antontsev S.N., Kazhikhov A.V., Monakhov V.N. Boundary Value Problems in Mechanics of Nonhomogeneous Fluids. Amsterdam, North Holland, 1990.
6. Kuchment L.S., Demidov V.N., Motovilov Yu.G. Formirovaniye Rechnogo Stoka. Fiziko-Matematicheskiye Modeli [River Flow Formation. Physical and Mathematical Models]. Moscow, Nauka, 1983. (in Russian)
7. Papin А.А. Solvability of a Model Problem of Heat and Mass Transfer in Thawing Snow. Journal of Applied Mechanics and Technical Physics, 2008, vol. 49, no. 4, pp. 527-536. DOI: 10.1007/s10808-008-0070-y
8. Hartman P. Ordinary Differential Equations: Second Edition. Classics in Applied Mathematics. Philadelphia, SIAM, 1982.
9. Ladyzhenskaya O.A. The Boundary Value Problems of Mathematical Physics. Applied Mathematical Sciences. New York, Springer, 1985.