Numerical Computation of MHD Thermal Flow of Cross Model over an Elliptic Cylinder: Reduction of Forces via Thickness Ratio
Afraz Hussain Majeed, Rashid Mahmood, Waqas Sarwar Abbasi, Kamran Usman
Abstract
The present work is concerned with a comprehensive analysis of hydrodynamic forces, under MHD and forced convection thermal flow over a heated cylinder in presence of insulated plates installed at walls. The magnetic field is imposed in the transverse direction of flow. The Galerkin finite element (GFE) scheme has been used to discretize the two-dimensional system of nonlinear partial different equations. The study is executed for the varying range of flow behavior index <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" id="M1"> <a:mfenced open="(" close=")" separators="|"> <a:mrow> <a:mi>n</a:mi> </a:mrow> </a:mfenced> </a:math> from 0.4 to 1.6, Hartmann number <f:math xmlns:f="http://www.w3.org/1998/Math/MathML" id="M2"> <f:mfenced open="(" close=")" separators="|"> <f:mrow> <f:mtext>Ha</f:mtext> </f:mrow> </f:mfenced> </f:math> from 0 to 100, Reynolds number <k:math xmlns:k="http://www.w3.org/1998/Math/MathML" id="M3"> <k:mfenced open="(" close=")" separators="|"> <k:mrow> <k:mi mathvariant="normal">Re</k:mi> </k:mrow> </k:mfenced> </k:math> from 10 to 50, Grashof number <q:math xmlns:q="http://www.w3.org/1998/Math/MathML" id="M4"> <q:mfenced open="(" close=")" separators="|"> <q:mrow> <q:mtext>Gr</q:mtext> </q:mrow> </q:mfenced> </q:math> from 1 to 10, thickness ratio <v:math xmlns:v="http://www.w3.org/1998/Math/MathML" id="M5"> <v:mfenced open="(" close=")" separators="|"> <v:mrow> <v:mi>e</v:mi> </v:mrow> </v:mfenced> </v:math> from 0.5 to 1.0, and Prandtl number <ab:math xmlns:ab="http://www.w3.org/1998/Math/MathML" id="M6"> <ab:mfenced open="(" close=")" separators="|"> <ab:mrow> <ab:mi mathvariant="normal">Pr</ab:mi> </ab:mrow> </ab:mfenced> </ab:math> from 1 to 10, respectively. A coarse hybrid computational grid is developed, and further refinement is carried out for obtaining the highly accurate solution. The optimum case selection is based on flow patterns, drag and lift coefficients, and pressure drop reduction against cylinder thickness ratios and average Nusselt numbers. Drag coefficient increases with an increase in thickness ratio <gb:math xmlns:gb="http://www.w3.org/1998/Math/MathML" id="M7"> <gb:mfenced open="(" close=")" separators="|"> <gb:mrow> <gb:mi>e</gb:mi> </gb:mrow> </gb:mfenced> <gb:mo>.</gb:mo> </gb:math> The drag force reduction for <lb:math xmlns:lb="http://www.w3.org/1998/Math/MathML" id="M8"> <lb:mi>e</lb:mi> <lb:mo>=</lb:mo> <lb:mn>0.5</lb:mn> </lb:math> and <nb:math xmlns:nb="http://www.w3.org/1998/Math/MathML" id="M9"> <nb:mi>e</nb:mi> <nb:mo>=</nb:mo> <nb:mn>0.75</nb:mn> </nb:math> is also observed for a range of the power law index as compared with <pb:math xmlns:pb="http://www.w3.org/1998/Math/MathML" id="M10"> <pb:mi>e</pb:mi> <pb:mo>=</pb:mo> <pb:mn>1.0</pb:mn> </pb:math> cylinder. Maximum pressure drop over the back and front points of cylinder is reported at <rb:math xmlns:rb="http://www.w3.org/1998/Math/MathML" id="M11"> <rb:mtext>Ha</rb:mtext> <rb:mo>=</rb:mo> <rb:mn>100</rb:mn> </rb:math> .