Thermal and flow dynamics of unsteady MHD nanofluid convection in a partially heated porous cavity

Abstract This study numerical investigates unsteady freeconvection in a square permeable cavity, driven by magnetohydrodynamic flow of TiO 2 -H 2 O nanofluid. The cavity features partially heated vertical walls, with adiabatic bottom and inclined upper walls, a scenario with widespread applications in fields such as heating, ventilation, and air conditioning (HVAC), electronic cooling, solar collectors, building insulation, automotive engineering, geothermal energy, and aerospace technology. The governing equations are discretized using the Marker and Cell (MAC) method, with spatial discretization achieved through the finite difference technique. A staggered grid system is employed to enhance stability and accuracy, and the resulting equations are solved iteratively to capture the unsteady behavior of the system.The study examines the range of Darcy number $$\left( {10^{ - 1} \le {\text{Da}} \le 10^{ - 3} } \right),$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mfenced> <mml:mrow> <mml:msup> <mml:mn>10</mml:mn> <mml:mrow> <mml:mo>-</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msup> <mml:mo>≤</mml:mo> <mml:mtext>Da</mml:mtext> <mml:mo>≤</mml:mo> <mml:msup> <mml:mn>10</mml:mn> <mml:mrow> <mml:mo>-</mml:mo> <mml:mn>3</mml:mn> </mml:mrow> </mml:msup> </mml:mrow> </mml:mfenced> <mml:mo>,</mml:mo> </mml:mrow> </mml:math> Rayleigh number $$\left( {10^{3} \le {\text{Ra}} \le 10^{6} } \right),$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mfenced> <mml:mrow> <mml:msup> <mml:mn>10</mml:mn> <mml:mn>3</mml:mn> </mml:msup> <mml:mo>≤</mml:mo> <mml:mtext>Ra</mml:mtext> <mml:mo>≤</mml:mo> <mml:msup> <mml:mn>10</mml:mn> <mml:mn>6</mml:mn> </mml:msup> </mml:mrow> </mml:mfenced> <mml:mo>,</mml:mo> </mml:mrow> </mml:math> Hartmann number $$\left( {10 \le Ha \le 30} \right),$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mfenced> <mml:mrow> <mml:mn>10</mml:mn> <mml:mo>≤</mml:mo> <mml:mi>H</mml:mi> <mml:mi>a</mml:mi> <mml:mo>≤</mml:mo> <mml:mn>30</mml:mn> </mml:mrow> </mml:mfenced> <mml:mo>,</mml:mo> </mml:mrow> </mml:math> Prandtl number $$\left( {\Pr = 6.2} \right),$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mfenced> <mml:mrow> <mml:mo>Pr</mml:mo> <mml:mo>=</mml:mo> <mml:mn>6.2</mml:mn> </mml:mrow> </mml:mfenced> <mml:mo>,</mml:mo> </mml:mrow> </mml:math> inclined magnetic field inclination angle $$\left( {\gamma = \frac{\pi }{6},\frac{\pi }{4},\frac{\pi }{3}} \right),$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mfenced> <mml:mrow> <mml:mi>γ</mml:mi> <mml:mo>=</mml:mo> <mml:mfrac> <mml:mi>π</mml:mi> <mml:mn>6</mml:mn> </mml:mfrac> <mml:mo>,</mml:mo> <mml:mfrac> <mml:mi>π</mml:mi> <mml:mn>4</mml:mn> </mml:mfrac> <mml:mo>,</mml:mo> <mml:mfrac> <mml:mi>π</mml:mi> <mml:mn>3</mml:mn> </mml:mfrac> </mml:mrow> </mml:mfenced> <mml:mo>,</mml:mo> </mml:mrow> </mml:math> heat source/sink parameter $$\left( { - 6 \le Q \le 6} \right)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mfenced> <mml:mrow> <mml:mo>-</mml:mo> <mml:mn>6</mml:mn> <mml:mo>≤</mml:mo> <mml:mi>Q</mml:mi> <mml:mo>≤</mml:mo> <mml:mn>6</mml:mn> </mml:mrow> </mml:mfenced> </mml:math> . Parametric modeling explores, streamline and isotherm plots reveal rising $${\text{Ra}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mtext>Ra</mml:mtext> </mml:math> enhance heat transfer, while increased Darcy numbers intensify fluid flow and broaden temperature gradients. Larger heat source/sink parameters expand isotherm contours, amplifying convection. Validation against prior studies confirms the model's reliability and provides insights into the intricate thermal dynamics of the system.