The transient Darcy–Forchheimer flow of a nanofluid over a shrinking permeable inclined rotating disc has several significant industrial applications such as gas turbine cooling systems, performance lubrication, surgical apparatus and high flux heat exchangers. In this investigation, an integrated second-law thermodynamic optimization model is successfully developed and the complex spatial trajectories are visualized with the help of three-dimensional helical streamlines over a slanted spinning surface with partial velocity slip conditions. The nanofluid phase is created by adding aluminium oxide (Al2O3) nanoparticles to a base oil lubricant (polyalphaolefins (PAO)). The algorithm bvp4c from MATLAB is used to numerically solve the transformed ordinary differential system; extensive use is made of a combined adaptive mesh independence study and numerical eigenvalue convergence analysis. The linear stability analysis verifies the existence of two solution sheets (stable upper branch and unstable lower branch) which end at a critical turning locus curve (<InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\varepsilon_{{\text{c}}}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>ε</mi> <mtext>c</mtext> </msub> </math></EquationSource> </InlineEquation>). By using temporal perturbation analysis, it is shown that the upper solution branch is physically stable when minimal eigenvalues are strictly positive (<InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\gamma_{1}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>γ</mi> <mn>1</mn> </msub> </math></EquationSource> </InlineEquation> > 0), while the lower solution branch is untenable (<InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\gamma_{1}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>γ</mi> <mn>1</mn> </msub> </math></EquationSource> </InlineEquation> < 0). Streamline 3D plots show a very tight helical vortex formed and tightly focused towards the disc by the boundary suction forces. In addition, thermodynamic second-law optimization demonstrates that the volumetric entropy production and the Bejan number exhibit distinct variations and reach a sharp maximum around the boundary of the spinning disc, where the viscous shear and Joule dissipation forces are the most dominant, followed by a smooth approach to asymptotic flat profiles in the far-field free-stream region.

Entropy optimization and 3D streamline topology of unsteady Darcy–Forchheimer nanofluid flow over an inclined disc with dual-branch stability analysis

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发表日期：

2026年7月31日

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Afrah Al-Bossly,
Anwar Saeed

摘要

The transient Darcy–Forchheimer flow of a nanofluid over a shrinking permeable inclined rotating disc has several significant industrial applications such as gas turbine cooling systems, performance lubrication, surgical apparatus and high flux heat exchangers. In this investigation, an integrated second-law thermodynamic optimization model is successfully developed and the complex spatial trajectories are visualized with the help of three-dimensional helical streamlines over a slanted spinning surface with partial velocity slip conditions. The nanofluid phase is created by adding aluminium oxide (Al₂O₃) nanoparticles to a base oil lubricant (polyalphaolefins (PAO)). The algorithm bvp4c from MATLAB is used to numerically solve the transformed ordinary differential system; extensive use is made of a combined adaptive mesh independence study and numerical eigenvalue convergence analysis. The linear stability analysis verifies the existence of two solution sheets (stable upper branch and unstable lower branch) which end at a critical turning locus curve ( $\varepsilon_{{\text{c}}}$ $ε_{c}$ ). By using temporal perturbation analysis, it is shown that the upper solution branch is physically stable when minimal eigenvalues are strictly positive ( $\gamma_{1}$ $γ_{1}$ > 0), while the lower solution branch is untenable ( $\gamma_{1}$ $γ_{1}$ < 0). Streamline 3D plots show a very tight helical vortex formed and tightly focused towards the disc by the boundary suction forces. In addition, thermodynamic second-law optimization demonstrates that the volumetric entropy production and the Bejan number exhibit distinct variations and reach a sharp maximum around the boundary of the spinning disc, where the viscous shear and Joule dissipation forces are the most dominant, followed by a smooth approach to asymptotic flat profiles in the far-field free-stream region.