The unsteady, incompressible, three-dimensional magnetohydrodynamic (MHD) flow of Powell–Eyring nanofluid over a paraboloid surface with variable porosity, nonlinear thermal radiation, and velocity slip is investigated. The study explores the effects of key physical parameters on flow dynamics, heat transfer, and nanofluid behavior in porous media. The mathematical model is formulated based on the Buongiorno model, incorporating magnetic field effects, slip conditions, and non-linear thermal radiation. Using the boundary layer approach, the governing equations are transformed into nonlinear ordinary differential equations via similarity transformations. Numerical solutions are obtained through the finite element method, specifically the Galerkin method, to compute velocity, temperature, and concentration profiles. The results indicate that the velocity profile decreases with increasing values of the magnetic parameter M and velocity slip parameter <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="41939_2025_955_Article_IEq1.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="23" /> </InlineMediaObject> <EquationSource Format="TEX">$ A_n $</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>A</mi> <mi>n</mi> </msub> </math></EquationSource> </InlineEquation> but increases with higher values of the Darcy number <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="41939_2025_955_Article_IEq2.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="23" /> </InlineMediaObject> <EquationSource Format="TEX">$ D_a $</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>D</mi> <mi>a</mi> </msub> </math></EquationSource> </InlineEquation> and fluid parameter <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="41939_2025_955_Article_IEq3.gif" Format="GIF" Height="10" Rendition="HTML" Resolution="72" Type="Linedraw" Width="14" /> </InlineMediaObject> <EquationSource Format="TEX">$ \alpha $</EquationSource> <EquationSource Format="MATHML"><math> <mi>α</mi> </math></EquationSource> </InlineEquation>. The temperature profile intensifies with an increase in the thermal radiation parameter <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="41939_2025_955_Article_IEq4.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="26" /> </InlineMediaObject> <EquationSource Format="TEX">$ M_d $</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>M</mi> <mi>d</mi> </msub> </math></EquationSource> </InlineEquation>, temperature ratio parameter <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="41939_2025_955_Article_IEq5.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="19" /> </InlineMediaObject> <EquationSource Format="TEX">$ \theta _w $</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>θ</mi> <mi>w</mi> </msub> </math></EquationSource> </InlineEquation>, unsteady parameter <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="41939_2025_955_Article_IEq6.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="20" /> </InlineMediaObject> <EquationSource Format="TEX">$ A_i $</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>A</mi> <mi>i</mi> </msub> </math></EquationSource> </InlineEquation>, and magnetic parameter, while it decreases with a higher Prandtl number Pr and temperature slip parameter <InlineEquation ID="IEq7"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="41939_2025_955_Article_IEq7.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="26" /> </InlineMediaObject> <EquationSource Format="TEX">$ A_m $</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>A</mi> <mi>m</mi> </msub> </math></EquationSource> </InlineEquation>. The concentration profile accelerates with the thermophoresis parameter Nt but decreases with the chemical reaction parameter <InlineEquation ID="IEq8"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="41939_2025_955_Article_IEq8.gif" Format="GIF" Height="12" Rendition="HTML" Resolution="72" Type="Linedraw" Width="23" /> </InlineMediaObject> <EquationSource Format="TEX">$ \kappa _m $</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>κ</mi> <mi>m</mi> </msub> </math></EquationSource> </InlineEquation> and Lewis number Le. Among the analyzed flow parameters, 56.67% lead to an increase in fluid flow characteristics, while 43.33% cause a decrease. The accuracy of the present method is verified against previously published results, demonstrating excellent agreement. Additionally, grid independence tests confirm the reliability and consistency of the numerical solutions. The study also explores the effects of viscous dissipation and external forces on the flow, with particular emphasis on variations in temperature, concentration, and velocity profiles near the paraboloidal surface. It provides new insights into the combined effects of MHD, non-Newtonian fluid behavior, and porous media interactions. These findings provide valuable insights into the effects of key parameters on MHD nanofluid flow over non-planar geometries, aiding in the optimization of industrial processes involving porous media and thermal management applications.

Magnetic Powell–Eyring nanofluid flow past a paraboloid surface with variable porosity

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发表日期：

2025年7月2日

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Tadesse Lamesse,
Wubshet Ibrahim

摘要

The unsteady, incompressible, three-dimensional magnetohydrodynamic (MHD) flow of Powell–Eyring nanofluid over a paraboloid surface with variable porosity, nonlinear thermal radiation, and velocity slip is investigated. The study explores the effects of key physical parameters on flow dynamics, heat transfer, and nanofluid behavior in porous media. The mathematical model is formulated based on the Buongiorno model, incorporating magnetic field effects, slip conditions, and non-linear thermal radiation. Using the boundary layer approach, the governing equations are transformed into nonlinear ordinary differential equations via similarity transformations. Numerical solutions are obtained through the finite element method, specifically the Galerkin method, to compute velocity, temperature, and concentration profiles. The results indicate that the velocity profile decreases with increasing values of the magnetic parameter M and velocity slip parameter $ A_n $ $A_{n}$ but increases with higher values of the Darcy number $ D_a $ $D_{a}$ and fluid parameter $ \alpha $ $α$ . The temperature profile intensifies with an increase in the thermal radiation parameter $ M_d $ $M_{d}$ , temperature ratio parameter $ \theta _w $ $θ_{w}$ , unsteady parameter $ A_i $ $A_{i}$ , and magnetic parameter, while it decreases with a higher Prandtl number Pr and temperature slip parameter $ A_m $ $A_{m}$ . The concentration profile accelerates with the thermophoresis parameter Nt but decreases with the chemical reaction parameter $ \kappa _m $ $κ_{m}$ and Lewis number Le. Among the analyzed flow parameters, 56.67% lead to an increase in fluid flow characteristics, while 43.33% cause a decrease. The accuracy of the present method is verified against previously published results, demonstrating excellent agreement. Additionally, grid independence tests confirm the reliability and consistency of the numerical solutions. The study also explores the effects of viscous dissipation and external forces on the flow, with particular emphasis on variations in temperature, concentration, and velocity profiles near the paraboloidal surface. It provides new insights into the combined effects of MHD, non-Newtonian fluid behavior, and porous media interactions. These findings provide valuable insights into the effects of key parameters on MHD nanofluid flow over non-planar geometries, aiding in the optimization of industrial processes involving porous media and thermal management applications.