Geodesic
A joint creeping motion of three fluids in a~flat layer: a~priori estimates and convergence to a~stationary regime
Sibirskij žurnal industrialʹnoj matematiki, Tome 19 (2016) no. 1, pp. 3-17.

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We study a partially invariant solution of rank 2 and defect 3e to the equations of a viscous heat-conducting fluid. It is interpreted as a two-dimensional motion of three immiscible fluids in a flat channel bounded by solid walls for which the distribution of temperature is known. From a mathematical point of view, the resulting initial boundary value problem is nonlinear and inverse. Under some assumptions (often fulfilled in practical applications), the problem is replaced by a linear one. We obtain a priori estimates as well as the exact stationary solution and prove that, the solution tends to a stationary regime if the temperatures of the walls stabilize with time.
Keywords: thermocapillarity, a priori estimate, conjugate boundary value problem, asymptotic behavior. 
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V. K. Andreev; E. N. Cheremnykh. A joint creeping motion of three fluids in a~flat layer: a~priori estimates and convergence to a~stationary regime. Sibirskij žurnal industrialʹnoj matematiki, Tome 19 (2016) no. 1, pp. 3-17. http://geodesic.mathdoc.fr/item/SJIM_2016_19_1_a0/

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