On the Use and Misuse of the Oberbeck–Boussinesq Approximation

Barletta, Antonio; Celli, Michele; Rees, D. Andrew S.

doi:10.3390/physics5010022

Cited by 9 publications

(2 citation statements)

References 22 publications

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“…Note that there are situations where the Boussinesq approximation may lead to incorrect results (for example, in astrophysical MHD simulations). The limitations of the use of the Boussinesq approximation have recently been described in detail (see [31][32][33]). The detailed derivation of the dimensionless form of Equations ( 2)-( 4) and the analysis of the Boussinesq approximation are found elsewhere (see, for example, [29]), and are not shown here for brevity.…”

Section: Mathematical Formulation Of the Problemmentioning

confidence: 99%

On the Stability of a Convective Flow with Nonlinear Heat Sources

Gritsans,

Kolyshkin,

Sadyrbaev

et al. 2023

Mathematics

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The linear stability of a convective flow in a vertical fluid layer caused by nonlinear heat sources in the presence of cross-flow through the walls of the channel is investigated in this paper. This study is relevant to the analysis of factors that affect the effectiveness of biomass thermal conversion. The nonlinear problem for the base flow temperature is investigated in detail using the Krasnosel’skiĭ–Guo cone expansion/contraction theorem. It is shown that a different number of solutions can exist depending on the values of the parameters. Estimates for the norm of the solutions are obtained. The linear stability problem is solved numerically by a collocation method based on Chebyshev polynomials. It is shown that the increase in the cross-flow intensity stabilizes the flow, but there is also a small region of the radial Reynolds numbers where the flow is destabilized.

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Section: Mathematical Formulation Of the Problemmentioning

confidence: 99%

On the Stability of a Convective Flow with Nonlinear Heat Sources

Gritsans,

Kolyshkin,

Sadyrbaev

et al. 2023

Mathematics

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“…The first equations to describe the convective motion of a viscous fluid were obtained experimentally and theoretically from the Oberbeck-Boussinesq equations. In their derivation, a linear dependence of density on temperature was used [7][8][9][10][11][12][13][14]. It was further established that an impurity (solute) in a fluid similarly causes convection [15][16][17].…”

Section: Introductionmentioning

confidence: 99%

Exact Solutions of the Oberbeck–Boussinesq Equations for the Description of Shear Thermal Diffusion of Newtonian Fluid Flows

Ershkov,

Burmasheva,

Leshchenko

et al. 2023

Symmetry

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We present a new exact solution of the thermal diffusion equations for steady-state shear flows of a binary fluid. Shear fluid flows are used in modeling and simulating large-scale currents of the world ocean, motions in thin layers of fluid, fluid flows in processes, and apparatuses of chemical technology. To describe the steady shear flows of an incompressible fluid, the system of Navier–Stokes equations in the Boussinesq approximation is redefined, so the construction of exact and numerical solutions to the equations of hydrodynamics is a very difficult and urgent task. A non-trivial exact solution is constructed in the Lin-Sidorov-Aristov class. For this class of exact solutions, the hydrodynamic fields (velocity field, pressure field, temperature field, and solute concentration field) were considered as linear forms in the x and y coordinates. The coefficients of linear forms depend on the third coordinate z. Thus, when considering a shear flow, the two-dimensional velocity field depends on three coordinates. It is worth noting that the solvability condition given in the article imposes a condition (relation) only between the velocity gradients. A theorem on the uniqueness of the exact solution in the Lin–Sidorov–Aristov class is formulated. The remaining coefficients of linear forms for hydrodynamic fields have functional arbitrariness. To illustrate the exact solution of the overdetermined system of Oberbeck–Boussinesq equations, a boundary value problem was solved to describe the complex convection of a vertical swirling fluid without its preliminary rotation. It was shown that the velocity field is highly stratified. Complex countercurrents are recorded in the fluid.

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Rigorous Derivation of the Oberbeck–Boussinesq Approximation Revealing Unexpected Term

Bella

Feireisl

Oschmann

2023

Commun. Math. Phys.

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We consider a general compressible viscous and heat conducting fluid confined between two parallel plates and heated from the bottom. The time evolution of the fluid is described by the Navier–Stokes–Fourier system considered in the regime of low Mach and Froude numbers suitably interrelated. Surprisingly and differently to the case of Neumann boundary conditions for the temperature, the asymptotic limit is identified as the Oberbeck–Boussinesq system supplemented with non-local boundary conditions for the temperature deviation.

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On the Use and Misuse of the Oberbeck–Boussinesq Approximation

Cited by 9 publications

References 22 publications

On the Stability of a Convective Flow with Nonlinear Heat Sources

On the Stability of a Convective Flow with Nonlinear Heat Sources

Exact Solutions of the Oberbeck–Boussinesq Equations for the Description of Shear Thermal Diffusion of Newtonian Fluid Flows

Rigorous Derivation of the Oberbeck–Boussinesq Approximation Revealing Unexpected Term

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