Natural convection of Casson fluid in a square enclosure
M.S. Aghighi, Christel Métivier, H. Masoumi
Abstract
Purpose The purpose of this paper is to analyze the natural convection of a yield stress fluid in a square enclosure with differentially heated side walls. In particular, the Casson model is considered which is a commonly used model. Design/methodology/approach The coupled conservation equations of mass, momentum and energy related to the two-dimensional steady-state natural convection within square enclosures are solved numerically by using the Galerkin's weighted residual finite element method with quadrilateral, eight nodes elements. Findings Results highlight a small degree of the shear-thinning in the Casson fluids. It is shown that the yield stress has a stabilizing effect since the convection can stop for yield stress fluids while this is not the case for Newtonian fluids. The heat transfer rate, velocity and <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"><m:mrow><m:msub><m:mi>Y</m:mi><m:mi>c</m:mi></m:msub></m:mrow></m:math> obtained with the Casson model have the smallest values compared to other viscoplastic models. Results highlight a weak dependence of <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"><m:mrow><m:msub><m:mi>Y</m:mi><m:mi>c</m:mi></m:msub></m:mrow></m:math> with the Rayleigh number: <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"><m:mrow><m:mo> </m:mo><m:msub><m:mi>Y</m:mi><m:mi>c</m:mi></m:msub><m:mo>∼</m:mo><m:mi mathvariant="italic">R</m:mi><m:msup><m:mi mathvariant="italic">a</m:mi><m:mrow><m:mn>0.07</m:mn></m:mrow></m:msup></m:mrow></m:math> . A supercritical bifurcation at the transition between the convective and the conductive regimes is found. Originality/value The originality of the present study concerns the comprehensive and detailed solutions of the natural convection of Casson fluids in square enclosures with differentially heated side walls. It is shown that there exists a major difference between the cases of Casson and Bingham models, and hence using the Bingham model for analyzing the viscoplastic behavior of the fluids which follow the Casson model (such as blood) may not be accurate. Finally, a correlation is proposed for the mean Nusselt number <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"><m:mrow><m:mover accent="true"><m:mrow><m:mi>N</m:mi><m:mi>u</m:mi></m:mrow><m:mo stretchy="true">¯</m:mo></m:mover></m:mrow></m:math> .