Mixed convection in four-sided lid-driven sinusoidally heated porous cavity using stream function-vorticity formulation

Abstract

This study presents the mixed convection inside a four-sided lid-driven square porous cavity whose right wall is maintained at a sinusoidal temperature condition, the left wall of the cavity is maintained at a cold temperature, while the top and the bottom walls are adiabatic. We have discussed two different cases depending upon the direction of the moving walls. Brinkmann-extended Darcy model is represented in terms of and using the stream function-vorticity formulation to simulate the momentum transfer in the porous medium. This formulation is used to solve the governing equations as a coupled system of equations which consists of the field variables, vorticity () , stream function () , and temperature (T). The velocity components (u, v) are derived from the stream function () whereas the average Nusselt number is derived from temperature. The stability and consistency of the applied numerical scheme to the considered problem has been proven by matrix method. The numerical results are investigated by ranging the various dimensionless numbers such as Grashof number (10 3 ≤ Gr ≤ 10 5) , Darcy number (10 −1 ≤ Da ≤ 10 −5) , Reynolds number (10 ≤ Re ≤ 1000) and keeping the Prandtl number (Pr = 0.7) fixed.

Heat and mass transfer under MHD mixed convection in a four‐sided lid‐driven square cavity

“…, we look at two different scenarios based on heat generation or absorption coefficient. If ϕ is the heat absorption coefficient, that is, ϕ ϕ = − 0 for some ϕ > 0 0 , then this situation can be treated in the same manner as Bagai et al 13,14…”

“…To establish $$\parallel {A}_{2}{A}_{1}\parallel \le 1$$, one can accomplish $$\rho ({A}_{1})\le 1$$, and $$\rho ({A}_{2})\le 1$$ in the same way as given by Ambethkar and Kumar 1 and Patel et al 21 To prove $$\parallel {B}_{4}{B}_{3}\parallel \le 1$$, we look at two different scenarios based on heat generation or absorption coefficient. If $$\phi $$ is the heat absorption coefficient, that is, $$\phi =-{\phi }_{0}$$ for some $${\phi }_{0}\gt 0$$, then this situation can be treated in the same manner as Bagai et al 13,14 case of $$\parallel {G}_{1}{E}_{1}\parallel \le 1$$. The energy equation (17) may be interpreted as follows if $$\phi $$ is the heat‐generating coefficient, that is, $$\phi \gt 0$$.…”

“…Bagai et al 12,13 used the stream function‐vorticity $$(\psi -\xi )$$ formulation to study the mixed or natural convection flow with/without porous materials in square cavities driven by two‐ and four‐sided lids with sinusoidal heat distributions. Bagai et al 14 have studied mixed convection in a four‐sided lid‐driven sinusoidally heated porous cavity under two different cases of the direction of moving lid, namely, antiparallel diagonal and antiparallel antidiagonal. They have found that the change of direction of the moving lid exactly reverses the flow pattern in the cavity.…”

Enhanced thermal performance and entropy generation analysis in a novel cavity design with circular cylinder

“…The dimensionless stream function and vorticity are provided in the following manner, 51 $$$U=\frac{\partial {\rm{\psi }}}{\partial {\rm{Y}}},V=-\frac{\partial \psi }{\partial X} \mathrm{and}\,\frac{\partial V}{\partial X}-\frac{\partial U}{\partial Y}={\rm{\Omega }}.$$$…”

Mixed Convection in a Two-Sided and Four-Sided Lid-Driven Square Porous Cavity

“…Hidayathulla Khan et al [16] investigated the mixed convection under the effect of radiation inside a porous square cavity whose top and bottom walls are kept at a constant temperature, while some portions of the left and right walls of the cavity are partially heated. Vusala and Kumar [17], and Bagai et al [18,19] have used the stream functionvorticity formulation to investigate the mixed convection problem with heat or mass transfer in a two-sided or four-sided lid-driven square porous or without a porous cavity.…”