Geodesic
Unsteady flow of two binary mixtures in a cylindrical capillary with changes in the internal energy of the interface
Žurnal Sibirskogo federalʹnogo universiteta. Matematika i fizika, Tome 15 (2022) no. 5, pp. 623-634.

Voir la notice de l'article provenant de la source Math-Net.Ru

The problem of two-dimensional unsteady flow of two immiscible incompressible binary mixtures in a cylindrical capillary in the absence of mass forces is studied. The mixtures are contacted through a common interface on which the energy condition is taken into account. The temperature and concentration of mixtures are distributed according to the quadratic law. It is in good agreement with the velocity field of the Hiemenz type. The resulting conjugate boundary value problem is a non-linear problem. It is also an inverse problem with respect to the pressure gradient along the axis of the cylindrical tube. To solve the problem the tau-method is used. It was shown that with increasing time the solution of the non-stationary problem tends to a steady state. It was established that the effect of increments of the internal energy of the inter-facial surface significantly affects the dynamics of the flow of mixtures in the layers.
Keywords: non-stationary solution, binary mixture, energy condition, internal energy, inverse problem, tau-method
Mots-clés : interface, pressure gradient, thermal diffusion. 
@article{JSFU_2022_15_5_a8,
     author = {Natalya L. Sobachkina},
     title = {Unsteady flow of two binary mixtures in a cylindrical capillary with changes in the internal energy of the interface},
     journal = {\v{Z}urnal Sibirskogo federalʹnogo universiteta. Matematika i fizika},
     pages = {623--634},
     publisher = {mathdoc},
     volume = {15},
     number = {5},
     year = {2022},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/JSFU_2022_15_5_a8/}
}
TY  - JOUR
AU  - Natalya L. Sobachkina
TI  - Unsteady flow of two binary mixtures in a cylindrical capillary with changes in the internal energy of the interface
JO  - Žurnal Sibirskogo federalʹnogo universiteta. Matematika i fizika
PY  - 2022
SP  - 623
EP  - 634
VL  - 15
IS  - 5
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/JSFU_2022_15_5_a8/
LA  - en
ID  - JSFU_2022_15_5_a8
ER  - 
%0 Journal Article
%A Natalya L. Sobachkina
%T Unsteady flow of two binary mixtures in a cylindrical capillary with changes in the internal energy of the interface
%J Žurnal Sibirskogo federalʹnogo universiteta. Matematika i fizika
%D 2022
%P 623-634
%V 15
%N 5
%I mathdoc
%U http://geodesic.mathdoc.fr/item/JSFU_2022_15_5_a8/
%G en
%F JSFU_2022_15_5_a8
Natalya L. Sobachkina. Unsteady flow of two binary mixtures in a cylindrical capillary with changes in the internal energy of the interface. Žurnal Sibirskogo federalʹnogo universiteta. Matematika i fizika, Tome 15 (2022) no. 5, pp. 623-634. http://geodesic.mathdoc.fr/item/JSFU_2022_15_5_a8/

[1] J.F. Harper, M.D. Woor, J.R.A. Pearson, “The effect of the variation of surface tension with temperature on the motion of drops and bubbles”, J. Fluid Mech., 27 (1967), 361 | DOI

[2] Ch. Sore, “Sur l'etat d'ecuilibre que prend au point de vue de sa concentration une dissolution saline primitivement homohene dont deux parties sont portees a des temperatures differentes”, Arch Sci. Phis. Nat., 2 (1879), 48–61

[3] L.G. Napolitano, “Plane Marangoni-Poiseulle flow two immiscible fluids”, Acta Astronautica, 7:4-5 (1980), 461–478 | DOI | Zbl

[4] V.K. Andreev, V.E. Zakhvataev, E.A. Ryabitsky, Thermocapillary instability, Nauka, Novosibirsk, 2000 (in Russian) | MR

[5] F.E. Torres, H. Elborzheimer, “Temperature gradients and drag effects produced by convections of interfacial internal energy around bubbles”, Phys. Fluids A, 5:3 (1993), 537–549 | DOI | Zbl

[6] V.K. Andreev, M.E. Pagdenko, “Two-dimensional stationary flow of immiscible fluids in a cylinder taking into account the internal energy of the interface”, Journal of physics: Conference series, 1268 (2019), 012045 | DOI | MR

[7] E.P. Magdenko, “The influence of changes in the internal energy of the interface on a two-layer flow in a cylinder”, Journal of Siberian Federal University. Mathematics and Physics, 12:2 (2019), 213–221 | MR | Zbl

[8] E.N. Lemeshkova, “Two-dimensional plane thermocapillary flow of two immiscible liquids”, Journal of Siberian Federal University. Mathematics and Physics, 12:3 (2019), 310–316 | DOI | MR | Zbl

[9] M.V. Efimova, “The effect of interfacial heat transfer energy on a two-layer creeping flow in a flat channel”, Journal of physics: Conference series, 1268 (2019), 012022 | DOI

[10] V.K. Andreev, N.L. Sobachkina, “Two-layer stationary flow in a cylindrical capillary taking into account changes in the internal energy of the interface”, Journal of Siberian Federal University. Mathematics and Physics, 14:4 (2021), 507–518 | DOI | MR | Zbl

[11] S. Wiegend, “Thermal diffusion in liquid mixtures and polymer solutions”, J. Phys.: Condens. Matter., 16 (2004), 357–379 | MR

[12] K. Hiemenz, “Die Grenzschicht an einem in den gleichformigen Flussigkeitsstrom eingetauchten geraden Kreiszylinder”, Dinglers Politech. J., 326 (1911), 314–321

[13] V.K. Andreev, B.V. Bekezhanova, Stability of non-isothermal liquids, monograph, Siberian Federal University, Krasnoyarsk, 2010 (in Russian)

[14] K. Fletcher, Numerical methods based on the Galerkin method, Mir, M., 1988 (in Russian) | MR

[15] Yu. Luke, Special mathematical functions and their approximations, Mir, M., 1980 (in Russian) | MR

[16] G. Sege, Orthogonal polynomials, Fizmatlit, M., 1962 (in Russian)

[17] N.L. Sobachkina, Solution of initial boundary value problems on the motion of binary mixtures in cylindrical regions, Diss. cand. phys.-math. sciences, Siberian Federal University, Krasnoyarsk, 2009 (in Russian)