BrinkmanBard Convection with Rough Boundaries and Third-Type Thermal Boundary Conditions

Title

BrinkmanBard Convection with Rough Boundaries and Third-Type Thermal Boundary Conditions

Creator

Siddheshwar P.G.; Narayana M.; Laroze D.; Kanchana C.

Description

The BrinkmanBard 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, (Formula presented.), and the wave number, (Formula presented.), are obtained in closed form. An asymptotic analysis is performed, to obtain (Formula presented.) for the case of adiabatic boundaries, for which (Formula presented.). 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 HopfRayleigh number are, however, opposite to each other. In view of the known analogy between Bard convection and TaylorCouette flow in the linear regime, it is imperative that the results of the latter problem, viz., BrinkmanTaylorCouette flow, become as well known. 2023 by the authors.

Source

Symmetry, Vol-15, No. 8

Date

2023-01-01

Publisher

Multidisciplinary Digital Publishing Institute (MDPI)

Subject

asymptotic analysis; Biot number; BrinkmanBard convection; DarcyRayleigh number; generalized Lorenz model; Maclaurin series; Robin boundary condition; rough boundaries; slip Darcy number

Coverage

Siddheshwar P.G., Centre for Mathematical Needs, Department of Mathematics, CHRIST (Deemed to be University), Bengaluru, 560029, India; Narayana M., Department of Mathematics, The University of the West Indies, Mona Campus, 7, Kingston, Jamaica; Laroze D., Instituto de Alta Investigaci, Universidad de Tarapac Casilla 7 D, Arica, 1000000, Chile; Kanchana C., Instituto de Alta Investigaci, Universidad de Tarapac Casilla 7 D, Arica, 1000000, Chile

Rights

All Open Access; Gold Open Access; Green Open Access

Relation

ISSN: 20738994

Format

Online

Language

English

Type

Article

Identifier

https://doi.org/10.3390/sym15081506

https://www.scopus.com/inward/record.uri?eid=2-s2.0-85168920179&doi=10.3390%2fsym15081506&partnerID=40&md5=eb92adebdcb72fd9c088f09536177206