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Generalized Boussinesq System with Energy Dissipation: Existence of Stationary Solutions

Author

Listed:
  • Evgenii S. Baranovskii

    (Department of Applied Mathematics, Informatics and Mechanics, Voronezh State University, 394018 Voronezh, Russia)

  • Olga Yu. Shishkina

    (Department of Applied Mathematics, Informatics and Mechanics, Voronezh State University, 394018 Voronezh, Russia)

Abstract

In this paper, we investigate the solvability of a boundary value problem for a heat and mass transfer model with the spatially averaged Rayleigh function. The considered model describes the 3D steady-state non-isothermal flow of a generalized Newtonian fluid (with shear-dependent viscosity) in a bounded domain with Lipschitz boundary. The main novelty of our work is that we do not neglect the viscous dissipation effect in contrast to the classical Boussinesq approximation, and hence, deal with a system of strongly nonlinear partial differential equations. Using the properties of the averaging operation and d-monotone operators as well as the Leray–Schauder alternative for completely continuous mappings, we prove the existence of weak solutions without any smallness assumptions for model data. Moreover, it is shown that the set of all weak solutions is compact, and each solution from this set satisfies some energy equalities.

Suggested Citation

  • Evgenii S. Baranovskii & Olga Yu. Shishkina, 2024. "Generalized Boussinesq System with Energy Dissipation: Existence of Stationary Solutions," Mathematics, MDPI, vol. 12(5), pages 1-15, March.
    • Handle: RePEc:gam:jmathe:v:12:y:2024:i:5:p:756-:d:1350479
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    References listed on IDEAS

    as
    1. Evgenii S. Baranovskii & Mikhail A. Artemov, 2016. "Existence of Optimal Control for a Nonlinear-Viscous Fluid Model," International Journal of Differential Equations, Hindawi, vol. 2016, pages 1-6, June.
    2. Exequiel Mallea-Zepeda & Eber Lenes & Elvis Valero, 2018. "Boundary Control Problem for Heat Convection Equations with Slip Boundary Condition," Mathematical Problems in Engineering, Hindawi, vol. 2018, pages 1-14, January.
    3. Mehdi Moslemi & Kourosh Javaherdeh, 2021. "Viscous Dissipation Effect in the Free Convection of Non-Newtonian Fluid with Heat Generation or Absorption Effect on the Vertical Wavy Surface," Journal of Applied Mathematics, Hindawi, vol. 2021, pages 1-14, October.
    4. Evgenii S. Baranovskii, 2017. "On Flows of Bingham-Type Fluids with Threshold Slippage," Advances in Mathematical Physics, Hindawi, vol. 2017, pages 1-6, December.
    Full references (including those not matched with items on IDEAS)

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