Brinkman–Bénard Convection with Rough Boundaries and Third-Type Thermal Boundary Conditions
The Brinkman–Bénard convection problem is chosen for investigation, along with very general boundary conditions. Using the Maclaurin series, in this paper, we show that it is possible to perform a relatively exact linear stability analysis, as well as a weakly nonlinear stability analysis, as normal...
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MDPI AG
2023-07-01
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| Series: | Symmetry |
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| author | Pradeep G. Siddheshwar Mahesha Narayana David Laroze C. Kanchana |
| author_facet | Pradeep G. Siddheshwar Mahesha Narayana David Laroze C. Kanchana |
| author_sort | Pradeep G. Siddheshwar |
| collection | DOAJ |
| description | The Brinkman–Bénard convection problem is chosen for investigation, along with very general boundary conditions. Using the Maclaurin series, in this paper, we show that it is possible to perform a relatively exact linear stability analysis, as well as a weakly nonlinear stability analysis, as normally performed in the case of a classical free isothermal/free isothermal boundary combination. Starting from a classical linear stability analysis, we ultimately study the chaos in such systems, all conducted with great accuracy. The principle of exchange of stabilities is proven, and the critical Rayleigh number, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>R</mi><msub><mi>a</mi><mi>c</mi></msub></mrow></semantics></math></inline-formula>, and the wave number, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>a</mi><mi>c</mi></msub></semantics></math></inline-formula>, are obtained in closed form. An asymptotic analysis is performed, to obtain <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>R</mi><msub><mi>a</mi><mi>c</mi></msub></mrow></semantics></math></inline-formula> for the case of adiabatic boundaries, for which <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>a</mi><mi>c</mi></msub><mo>≃</mo><mn>0</mn></mrow></semantics></math></inline-formula>. A seemingly minimal representation yields a generalized Lorenz model for the general boundary condition used. The symmetry in the three Lorenz equations, their dissipative nature, energy-conserving nature, and bounded solution are observed for the considered general boundary condition. Thus, one may infer that, to obtain the results of various related problems, they can be handled in an integrated manner, and results can be obtained with great accuracy. The effect of increasing the values of the Biot numbers and/or slip Darcy numbers is to increase, not only the value of the critical Rayleigh number, but also the critical wave number. Extreme values of zero and infinity, when assigned to the Biot number, yield the results of an adiabatic and an isothermal boundary, respectively. Likewise, these extreme values assigned to the slip Darcy number yield the effects of free and rigid boundary conditions, respectively. Intermediate values of the Biot and slip Darcy numbers bridge the gap between the extreme cases. The effects of the Biot and slip Darcy numbers on the Hopf–Rayleigh number are, however, opposite to each other. In view of the known analogy between Bénard convection and Taylor–Couette flow in the linear regime, it is imperative that the results of the latter problem, viz., Brinkman–Taylor–Couette flow, become as well known. |
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| spelling | doaj.art-3a0490eb986d4d63aac3ad6e68cf8d7d2023-11-19T03:10:43ZengMDPI AGSymmetry2073-89942023-07-01158150610.3390/sym15081506Brinkman–Bénard Convection with Rough Boundaries and Third-Type Thermal Boundary ConditionsPradeep G. Siddheshwar0Mahesha Narayana1David Laroze2C. Kanchana3Centre for Mathematical Needs, Department of Mathematics, CHRIST (Deemed to be University), Bengaluru 560029, IndiaDepartment of Mathematics, The University of the West Indies, Mona Campus, Kingston 7, JamaicaInstituto de Alta Investigación, Universidad de Tarapacá, Casilla 7 D, Arica 1000000, ChileInstituto de Alta Investigación, Universidad de Tarapacá, Casilla 7 D, Arica 1000000, ChileThe Brinkman–Bénard convection problem is chosen for investigation, along with very general boundary conditions. Using the Maclaurin series, in this paper, we show that it is possible to perform a relatively exact linear stability analysis, as well as a weakly nonlinear stability analysis, as normally performed in the case of a classical free isothermal/free isothermal boundary combination. Starting from a classical linear stability analysis, we ultimately study the chaos in such systems, all conducted with great accuracy. The principle of exchange of stabilities is proven, and the critical Rayleigh number, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>R</mi><msub><mi>a</mi><mi>c</mi></msub></mrow></semantics></math></inline-formula>, and the wave number, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>a</mi><mi>c</mi></msub></semantics></math></inline-formula>, are obtained in closed form. An asymptotic analysis is performed, to obtain <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>R</mi><msub><mi>a</mi><mi>c</mi></msub></mrow></semantics></math></inline-formula> for the case of adiabatic boundaries, for which <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>a</mi><mi>c</mi></msub><mo>≃</mo><mn>0</mn></mrow></semantics></math></inline-formula>. A seemingly minimal representation yields a generalized Lorenz model for the general boundary condition used. The symmetry in the three Lorenz equations, their dissipative nature, energy-conserving nature, and bounded solution are observed for the considered general boundary condition. Thus, one may infer that, to obtain the results of various related problems, they can be handled in an integrated manner, and results can be obtained with great accuracy. The effect of increasing the values of the Biot numbers and/or slip Darcy numbers is to increase, not only the value of the critical Rayleigh number, but also the critical wave number. Extreme values of zero and infinity, when assigned to the Biot number, yield the results of an adiabatic and an isothermal boundary, respectively. Likewise, these extreme values assigned to the slip Darcy number yield the effects of free and rigid boundary conditions, respectively. Intermediate values of the Biot and slip Darcy numbers bridge the gap between the extreme cases. The effects of the Biot and slip Darcy numbers on the Hopf–Rayleigh number are, however, opposite to each other. In view of the known analogy between Bénard convection and Taylor–Couette flow in the linear regime, it is imperative that the results of the latter problem, viz., Brinkman–Taylor–Couette flow, become as well known.https://www.mdpi.com/2073-8994/15/8/1506asymptotic analysisBrinkman–Bénard convectionBiot numberDarcy–Rayleigh numbergeneralized Lorenz modelMaclaurin series |
| spellingShingle | Pradeep G. Siddheshwar Mahesha Narayana David Laroze C. Kanchana Brinkman–Bénard Convection with Rough Boundaries and Third-Type Thermal Boundary Conditions Symmetry asymptotic analysis Brinkman–Bénard convection Biot number Darcy–Rayleigh number generalized Lorenz model Maclaurin series |
| title | Brinkman–Bénard Convection with Rough Boundaries and Third-Type Thermal Boundary Conditions |
| title_full | Brinkman–Bénard Convection with Rough Boundaries and Third-Type Thermal Boundary Conditions |
| title_fullStr | Brinkman–Bénard Convection with Rough Boundaries and Third-Type Thermal Boundary Conditions |
| title_full_unstemmed | Brinkman–Bénard Convection with Rough Boundaries and Third-Type Thermal Boundary Conditions |
| title_short | Brinkman–Bénard Convection with Rough Boundaries and Third-Type Thermal Boundary Conditions |
| title_sort | brinkman benard convection with rough boundaries and third type thermal boundary conditions |
| topic | asymptotic analysis Brinkman–Bénard convection Biot number Darcy–Rayleigh number generalized Lorenz model Maclaurin series |
| url | https://www.mdpi.com/2073-8994/15/8/1506 |
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