On three-dimensional stable long-wavelength convection in the presence of Dirichlet thermal boundary conditions
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On three-dimensional stable long-wavelength convection in the presence of Dirichlet thermal boundary conditions Alaric Rohl · Layachi Hadji
Received: 20 February 2020 / Accepted: 26 August 2020 © Springer Nature B.V. 2020
Abstract It is a well-known fact that the onset of Rayleigh–Bénard convection occurs via a long-wavelength instability when the horizontal boundaries are thermally insulated. The aim of this paper is to quantify the exact dimensions of a cylinder of rectangular cross-section wherein stable three-dimensional Rayleigh–Bénard convection sets in via a long-wavelength instability from the motionless state at the same value of the critical Rayleigh number as the corresponding horizontally unbounded problem when the bounding horizontal walls have infinite thermal conductance. Hence, we consider three-dimensional Rayleigh–Bénard convection in a cell of infinite extent in the x-direction, confined between two vertical walls located at y = ±H and horizontal boundaries located at z = 0 and z = d. Our analysis predicts the existence of the sought stable state for experimental velocity boundary conditions at the vertical walls provided the aspect ratio δ = H/d takes a certain value. In the limit H → ∞, we retrieve the stability characteristics of the horizontally unbounded problem. As expected, the analysis predicts two counterrotating rolls aligned along the y-direction of period 2π/δ equal to the period of the roll in the y-direction of the corresponding unbounded problem. A long-scale asymptotic analysis leads to the derivation of an evolution partial differential equation (PDE) that is fourth order in space and contains a single bifurcation parameter. The PDE, valid for a specific value of δ, is analyzed analytically and numerically as function of the bifurcation parameter and for a variety of velocity boundary conditions at the vertical walls to seek the stable steady-state solutions. The same analysis is also extended to the case of convection in a fluid-saturated porous medium. Keywords Dirichlet thermal boundary condition · Long-wavelength convection · Rayleigh–Bénard convection 1 Introduction Rayleigh–Bénard convection refers to an experiment wherein a fluid is confined between two horizontal plates, the lower of which is warmer than the top. The fluid motion that results is due to an interplay between the stabilizing effects of heat and viscous dissipation and the destabilizing effects due to the fluid’s thermal expansion. The resulting competition between these effects leads to the emergence of a dimensionless parameter, namely the Rayleigh number, A. Rohl · L. Hadji (B) Department of Mathematics, The University of Alabama, Tuscaloosa, AL 35487, USA e-mail: [email protected] A. Rohl e-mail: [email protected]
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A. Rohl, L. Hadji
the minimum value of which is an indicator of the convection onset and its intensity. Moreover, its dependence on the Nusselt number is a subject of intense studies aimed at understanding the mechanism of heat transfer in convection. From the analysis standpoint, the
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