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 J. Sib. Fed. Univ. Math. Phys., 2020, Volume 13, Issue 5, Pages 574–582 (Mi jsfu864)

Layered motion of two immiscible liquids with a free boundary

Elena N. Lemeshkova

Institute of computational modelling SB RAS, Krasnoyarsk, Russian Federation

Abstract: The unidirectional motion of two viscous immiscible incompressible liquids in a flat channel is studied. An unsteady temperature gradient is set on the bottom solid wall, and the upper wall is a free boundary. Liquids contact on a flat interface. The motion is caused by the combined action of thermogravitational and thermocapillary forces and a given total unsteady flow rate in the layers. The corresponding initial boundary value problem is conjugate and inverse, since the pressure gradient along the channel is determined together with the velocity and temperature field. An exact stationary solution was found for it. In Laplace images, the solution of the non-stationary problem is found in the quadrature forms. It was established that if the temperature on the bottom wall and the flow rate stabilize with time, then the motion goes to a stationary state with time. This fact indicates the stability of the stationary solution with respect to unidirectional unsteady perturbations. The calculation results showing various methods of controlling motion by setting the temperature on the wall are given.

Keywords: thermocapillary, interface, Oberbek–Boussinesq equations.

 Funding Agency Grant Number Ministry of Science and Higher Education of the Russian Federation 075-02-2020-1631 This work is supported by the Krasnoyarsk Mathematical Center and financed by the Ministry of Science and Higher Education of the Russian Federation in the framework of the establishment and development of regional Centers for Mathematics Research and Education (Agreement no. 075-02-2020-1631).

DOI: https://doi.org/10.17516/1997-1397-2020-13-5-574-582

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UDC: 517.9
Received in revised form: 06.06.2020
Accepted: 17.07.2020
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Citation: Elena N. Lemeshkova, “Layered motion of two immiscible liquids with a free boundary”, J. Sib. Fed. Univ. Math. Phys., 13:5 (2020), 574–582

Citation in format AMSBIB
\Bibitem{Lem20} \by Elena~N.~Lemeshkova \paper Layered motion of two immiscible liquids with a free boundary \jour J. Sib. Fed. Univ. Math. Phys. \yr 2020 \vol 13 \issue 5 \pages 574--582 \mathnet{http://mi.mathnet.ru/jsfu864} \crossref{https://doi.org/10.17516/1997-1397-2020-13-5-574-582} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000580315300006}