Mathematical Analysis of the Poiseuille Flow of a Fluid with Temperature-Dependent Properties

Evgenii S. Baranovskii (), Anastasia A. Domnich and Mikhail A. Artemov
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Evgenii S. Baranovskii: Department of Applied Mathematics, Informatics and Mechanics, Voronezh State University, 394018 Voronezh, Russia
Anastasia A. Domnich: Department of Applied Mathematics, Informatics and Mechanics, Voronezh State University, 394018 Voronezh, Russia
Mikhail A. Artemov: Department of Applied Mathematics, Informatics and Mechanics, Voronezh State University, 394018 Voronezh, Russia

Mathematics, 2024, vol. 12, issue 21, 1-19

Abstract: This article is devoted to the mathematical analysis of a heat and mass transfer model for the pressure-induced flow of a viscous fluid through a plane channel subject to Navier’s slip conditions on the channel walls. The important feature of our work is that the used model takes into account the effects of variable viscosity, thermal conductivity, and slip length, under the assumption that these quantities depend on temperature. Therefore, we arrive at a boundary value problem for strongly nonlinear ordinary differential equations. The existence and uniqueness of a solution to this problem is analyzed. Namely, using the Galerkin scheme, the generalized Borsuk theorem, and the compactness method, we proved the existence theorem for both weak and strong solutions in Sobolev spaces and derive some of their properties. Under extra conditions on the model data, the uniqueness of a solution is established. Moreover, we considered our model subject to some explicit formulae for temperature dependence of viscosity, which are applied in practice, and constructed corresponding exact solutions. Using these solutions, we successfully performed an extra verification of the algorithm for finding solutions that was applied by us to prove the existence theorem.

Keywords: heat and mass transfer model; viscous fluid; Poiseuille flow; Navier slip condition; nonlinear boundary value problem; variable coefficients; weak and strong solutions; exact solutions; Galerkin method; existence and uniqueness theorem (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2024
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