<p>This study investigates the unsteady mixed convective flow of an electrically conducting Williamson tetra hybrid nanofluid at the stagnation point of a rotating sphere, taking into account viscous dissipation, quadratic convection, quadratic thermal radiation, Joule heating, and Darcy–Forchheimer effects. The motivation stems from the growing demand for efficient thermal management in engineering and industrial applications, where enhanced heat transfer, reduced entropy generation, and improved cooling efficiency are critical. The tetra hybrid nanofluid is composed of four nanoparticles (<InlineEquation ID="IEq1"> <EquationSource Format="TEX">${\text{Ag}}$</EquationSource> <EquationSource Format="MATHML"><math> <mtext>Ag</mtext> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq2"> <EquationSource Format="TEX">${\text{TiO}}_{2}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mtext>TiO</mtext> <mn>2</mn> </msub> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq3"> <EquationSource Format="TEX">${\text{Al}}_{2} {\text{O}}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mtext>Al</mtext> <mn>2</mn> </msub> <mtext>O</mtext> </mrow> </math></EquationSource> </InlineEquation>, and <InlineEquation ID="IEq4"> <EquationSource Format="TEX">${\text{CuO}}$</EquationSource> <EquationSource Format="MATHML"><math> <mtext>CuO</mtext> </math></EquationSource> </InlineEquation>) dispersed in water, exhibiting superior thermal conductivity and performance compared to conventional and hybrid nanofluids. The governing PDEs are transformed into ODEs using similarity transformations and solved numerically via the fourth-order Runge–Kutta method with the shooting technique. The influence of key parameters on velocity, temperature, entropy generation, Bejan number, and streamline patterns is systematically examined through graphs, tables, and three-dimensional plots. Results indicate that increasing the parameter <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$M\left( { = 1.0,3.0,5.0} \right)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>M</mi> <mfenced close=")" open="("> <mrow> <mo>=</mo> <mn>1.0</mn> <mo>,</mo> <mn>3.0</mn> <mo>,</mo> <mn>5.0</mn> </mrow> </mfenced> </mrow> </math></EquationSource> </InlineEquation> enhances primary velocity while reducing secondary velocity, whereas a higher Weissenberg number <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$We\left( { = 0.5,0.6,0.7} \right)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>W</mi> <mi>e</mi> <mfenced close=")" open="("> <mrow> <mo>=</mo> <mn>0.5</mn> <mo>,</mo> <mn>0.6</mn> <mo>,</mo> <mn>0.7</mn> </mrow> </mfenced> </mrow> </math></EquationSource> </InlineEquation> decreases primary velocity and increases secondary velocity by 5–9%. Thermal radiation <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$Rd\left( { = 0.01,0.02,0.03} \right)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>R</mi> <mi>d</mi> <mfenced close=")" open="("> <mrow> <mo>=</mo> <mn>0.01</mn> <mo>,</mo> <mn>0.02</mn> <mo>,</mo> <mn>0.03</mn> </mrow> </mfenced> </mrow> </math></EquationSource> </InlineEquation> intensifies temperature by 5–12%, and higher Brinkman numbers <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$Br\left( { = 0.1,0.2,0.3} \right)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>B</mi> <mi>r</mi> <mfenced close=")" open="("> <mrow> <mo>=</mo> <mn>0.1</mn> <mo>,</mo> <mn>0.2</mn> <mo>,</mo> <mn>0.3</mn> </mrow> </mfenced> </mrow> </math></EquationSource> </InlineEquation> lead to an 8–20% increase in entropy generation. Moreover, statistical analyses including multiple linear regression, correlation, and probable error assessment, validate the numerical results and confirm the strong dependence of flow and thermal behaviour on physical quantities. Comparative evaluation demonstrates that tetra hybrid nanofluids achieve superior heat transfer efficiency, providing practical insights for power generation, electronics cooling, and advanced thermal management systems.</p>

Multiple linear regression-based numerical analysis of unsteady MHD tetra hybrid nanofluid flow over a rotating sphere for advanced thermal management

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2025年12月23日

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M. Keerthiga,
P. Bala Anki Reddy,
K. Sakkaravarthi

摘要

This study investigates the unsteady mixed convective flow of an electrically conducting Williamson tetra hybrid nanofluid at the stagnation point of a rotating sphere, taking into account viscous dissipation, quadratic convection, quadratic thermal radiation, Joule heating, and Darcy–Forchheimer effects. The motivation stems from the growing demand for efficient thermal management in engineering and industrial applications, where enhanced heat transfer, reduced entropy generation, and improved cooling efficiency are critical. The tetra hybrid nanofluid is composed of four nanoparticles ( ${\text{Ag}}$ $Ag$ , ${\text{TiO}}_{2}$ ${TiO}_{2}$ , ${\text{Al}}_{2} {\text{O}}$ ${Al}_{2} O$ , and ${\text{CuO}}$ $CuO$ ) dispersed in water, exhibiting superior thermal conductivity and performance compared to conventional and hybrid nanofluids. The governing PDEs are transformed into ODEs using similarity transformations and solved numerically via the fourth-order Runge–Kutta method with the shooting technique. The influence of key parameters on velocity, temperature, entropy generation, Bejan number, and streamline patterns is systematically examined through graphs, tables, and three-dimensional plots. Results indicate that increasing the parameter $M\left( { = 1.0,3.0,5.0} \right)$ $M (= 1.0, 3.0, 5.0)$ enhances primary velocity while reducing secondary velocity, whereas a higher Weissenberg number $We\left( { = 0.5,0.6,0.7} \right)$ $W e (= 0.5, 0.6, 0.7)$ decreases primary velocity and increases secondary velocity by 5–9%. Thermal radiation $Rd\left( { = 0.01,0.02,0.03} \right)$ $R d (= 0.01, 0.02, 0.03)$ intensifies temperature by 5–12%, and higher Brinkman numbers $Br\left( { = 0.1,0.2,0.3} \right)$ $B r (= 0.1, 0.2, 0.3)$ lead to an 8–20% increase in entropy generation. Moreover, statistical analyses including multiple linear regression, correlation, and probable error assessment, validate the numerical results and confirm the strong dependence of flow and thermal behaviour on physical quantities. Comparative evaluation demonstrates that tetra hybrid nanofluids achieve superior heat transfer efficiency, providing practical insights for power generation, electronics cooling, and advanced thermal management systems.