<p>This study investigates natural convection in rheological granular flow over an inclined heated surface, with applications in heated powder processing, bulk solid drying, food processing, and thermal polymer rheological systems. The rheological effects are modeled through normal stress differences, while buoyancy forces are incorporated using the Boussinesq approximation. Viscous dissipation is also considered to account for internal frictional heating. The governing equations are derived using a continuum framework with conductive heat transfer, and nonlinear thermal conductivity is modeled via Fourier’s law. The resulting coupled nonlinear differential equations are solved numerically using MATLAB’s bvp4c, based on Newton collocation and adaptive discretization. The numerical results are validated against available studies, and the effects of key material parameters on granular volume fraction, velocity, and temperature distributions are examined graphically. The results show that by increasing the pressure-to-gravity force ratio <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\left({\xi}_{1}\right)$</EquationSource> <EquationSource Format="MATHML"><math> <mfenced close=")" open="("> <msub> <mi>ξ</mi> <mn>1</mn> </msub> </mfenced> </math></EquationSource> </InlineEquation>, thermal buoyancy parameter <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\left({\xi}_{5}\right)$</EquationSource> <EquationSource Format="MATHML"><math> <mfenced close=")" open="("> <msub> <mi>ξ</mi> <mn>5</mn> </msub> </mfenced> </math></EquationSource> </InlineEquation>, and inclination angle <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\left(\gamma \right)$</EquationSource> <EquationSource Format="MATHML"><math> <mfenced close=")" open="("> <mi>γ</mi> </mfenced> </math></EquationSource> </InlineEquation> enhances the granular volume fraction near the inclined heated surface, while reducing it near the free surface. In contrast, higher values of the void distribution force parameter <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\left( {\xi _{2} } \right)$</EquationSource> <EquationSource Format="MATHML"><math> <mfenced close=")" open="("> <msub> <mi>ξ</mi> <mn>2</mn> </msub> </mfenced> </math></EquationSource> </InlineEquation> reduce the volume fraction near the heated surface but enhance it near the free surface. The heat flux ratio parameter <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\left({\xi}_{6}\right)$</EquationSource> <EquationSource Format="MATHML"><math> <mfenced close=")" open="("> <msub> <mi>ξ</mi> <mn>6</mn> </msub> </mfenced> </math></EquationSource> </InlineEquation> suppresses temperature throughout the flow domain, whereas viscous dissipation <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\left({\xi}_{4}\right)$</EquationSource> <EquationSource Format="MATHML"><math> <mfenced close=")" open="("> <msub> <mi>ξ</mi> <mn>4</mn> </msub> </mfenced> </math></EquationSource> </InlineEquation> and inclination angle <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$\left(\gamma \right)$</EquationSource> <EquationSource Format="MATHML"><math> <mfenced close=")" open="("> <mi>γ</mi> </mfenced> </math></EquationSource> </InlineEquation> enhance the temperature distribution.</p>

Analysis of natural convection in dissipative rheological granular flow along an inclined plane with a nonlinear thermal conductivity model

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2026年7月14日

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Muhammad Mubashir Bhatti,
Osman Anwar Bég,
Tasveer Anwar Bég,
Ali Kadir

摘要

This study investigates natural convection in rheological granular flow over an inclined heated surface, with applications in heated powder processing, bulk solid drying, food processing, and thermal polymer rheological systems. The rheological effects are modeled through normal stress differences, while buoyancy forces are incorporated using the Boussinesq approximation. Viscous dissipation is also considered to account for internal frictional heating. The governing equations are derived using a continuum framework with conductive heat transfer, and nonlinear thermal conductivity is modeled via Fourier’s law. The resulting coupled nonlinear differential equations are solved numerically using MATLAB’s bvp4c, based on Newton collocation and adaptive discretization. The numerical results are validated against available studies, and the effects of key material parameters on granular volume fraction, velocity, and temperature distributions are examined graphically. The results show that by increasing the pressure-to-gravity force ratio $\left({\xi}_{1}\right)$ $(ξ_{1})$ , thermal buoyancy parameter $\left({\xi}_{5}\right)$ $(ξ_{5})$ , and inclination angle $\left(\gamma \right)$ $(γ)$ enhances the granular volume fraction near the inclined heated surface, while reducing it near the free surface. In contrast, higher values of the void distribution force parameter $\left( {\xi _{2} } \right)$ $(ξ_{2})$ reduce the volume fraction near the heated surface but enhance it near the free surface. The heat flux ratio parameter $\left({\xi}_{6}\right)$ $(ξ_{6})$ suppresses temperature throughout the flow domain, whereas viscous dissipation $\left({\xi}_{4}\right)$ $(ξ_{4})$ and inclination angle $\left(\gamma \right)$ $(γ)$ enhance the temperature distribution.