Hydrodynamic Anisotropy Effects on Radiation-Mixed Convection Interaction in a Vertical Porous Channel

Abstract

The effects of hydrodynamic anisotropy on the mixed-convection in a vertical porous channel heated on its plates with a thermal radiation are investigated analytically for fully developed flow regime. The porous medium is anisotropic in permeability whose principal axes are oriented in a direction that is oblique to the gravity. The generalized Brinkman-extended Darcy model which allows the no-slip boundary-condition on solid wall is used in the formulation of the problem. The flow reversal, the thermal radiation influence for natural, and forced convection are considered in the limiting cases for low and high porosity media. It was found that the anisotropic permeability ratio, the orientation angle of the principal axes of permeability and the radiation parameter affected significantly the flow regime and the heat transfer.

The role of stably stratified fluid and anisotropy porous medium in modeling fluid flow through an asymmetrically heated/cooled vertical annulus

“…´. At t > 0 ʹ , the temperatures 11 where a vertical channel is saturated with an anisotropic porous matrix and Deka and Paul 26 where stably stratified fluid through a moving vertical cylinder is taken into account. Thus, symbolically, the mathematical model governing the present investigation is given by the following dimensional coupled momentum and energy equations:…”

“…The formulation of the mathematical model for this present investigation is based on the works of Degan et al, 11 where a vertical channel is saturated with an anisotropic porous matrix and Deka and Paul 26 where stably stratified fluid through a moving vertical cylinder is taken into account. Thus, symbolically, the mathematical model governing the present investigation is given by the following dimensional coupled momentum and energy equations: $$$\frac{\partial {U}^{^{\prime} }}{\partial {t}^{^{\prime} }}=\phantom{\rule{}{0ex}}{\upsilon }_{{eff}}\left(\frac{{\partial }^{2}{U}^{^{\prime} }}{\partial {r}^{{\prime} 2}}+\frac{1}{{r}^{^{\prime} }}\frac{\partial {U}^{^{\prime} }}{\partial {r}^{^{\prime} }}\right)\phantom{\rule{}{0ex}}+\phantom{\rule{}{0ex}}g\beta ({T}^{^{\prime} }-{T}_{m})-\phantom{\rule{}{0ex}}\frac{{U}^{^{\prime} }\nu }{\mathop{K}\limits^{̿}},$$$ $$$\frac{\partial {T}^{^{\prime} }}{\partial {t}^{^{\prime} }}=\phantom{\rule{}{0ex}}\alpha \left(\frac{{\partial }^{2}{T}^{^{\prime} }}{\partial {r}^{{\prime} 2}}+\frac{1}{{r}^{^{\prime} }}\frac{\partial {T}^{^{\prime} }}{\partial {r}^{^{\prime} }}\right)\phantom{\rule{}{0ex}}-\phantom{\rule{}{0ex}}{S}^{^{\prime} }{U}^{^{\prime} }.$$$ Subject to the flowing dimensional initial and boundary physical constraints: …”

“…The research additionally proved that for a large value of anisotropic permeability ratio $$({k}^{* }\gt 1)$$, a weak convective heat transfer and a high penetrative fluid flow occurred. In another research, Degan et al 11 showed that in such type of directionally dependent porous matrix, the permeability ratio, the orientation angle of the principal axes of the permeability and the radiation parameter significantly affect the flow regime and the heat transfer. Other fascinating results on anisotropy are documented by Lu et al, 12 Yadav, 13 and Yadav et al 14 …”

Time‐dependent natural convection fluid flow of stably stratified fluid in an asymmetrically heated vertical channel filled with an anisotropic porous material

“…Motivated by the works of Deka and Paul, 1 Karmakar and Sekhar, 17 and Degan et al, 18 the fully developed one‐dimensional time dependent governing mathematical model representing the present problem assumes the following form: $$$\frac{\partial U^{\prime} }{\partial t^{\prime} }={\upsilon }_{\text{eff}}\frac{{\partial }^{2}U^{\prime} }{\partial y{{\prime} }^{2}}-\frac{U^{\prime} \upsilon }{\mathop{K}\limits^{̿}}+g\beta (T^{\prime} -{T}_{1}),$$$ $$$\frac{\partial T^{\prime} }{\partial t^{\prime} }=\alpha \frac{{\partial }^{2}T^{\prime} }{\partial y{{\prime} }^{2}}-\xi ^{\prime} U^{\prime} ,$$$where $$\alpha =\frac{k}{{\rho C}_{p}}$$ and $$\mathop{K}\limits^{̿}$$ is the symmetrical second‐order permeability tensor defined as: …”

“…Motivated by the works of Deka and Paul, 1 Karmakar and Sekhar, 17 and Degan et al, 18 the fully developed one-dimensional time dependent governing mathematical model representing the present problem assumes the following form:…”