Wire coating, fiber drawing, and the extrusion of filled thermoplastics all draw a slender continuous filament through a hot, particle-laden viscoelastic melt, often in a packed die and under an imposed magnetic field. This paper formulates that configuration as a steady two-phase boundary layer flow of a radiative magnetohydrodynamic (MHD) dusty Maxwell nanofluid over a stretching or shrinking cylinder in a Darcy–Forchheimer porous medium, retaining a micrometer-scale dust phase with velocity and thermal slip alongside a Buongiorno nanoparticle description, nonlinear Rosseland radiation, and an Arrhenius reaction. Similarity reduction yields a tenth-order system in which transverse curvature enters through the parameter $ \gamma $, multiplying every diffusion operator by $ (1+2\gamma\eta) $ and adding a viscoelastic term with no plane-sheet analog. A collocation solver with fold-following continuation, verified against published stretching cylinder data to within $ 0.004\% $ at matched truncation, resolves both solution branches for the decelerating suction-assisted regime. Curvature is found to govern the problem principally through feasibility: Because the coefficient of the highest derivative is $ (1+2\gamma\eta)-\beta f^2 $, curvature relaxes the condition for a physically admissible solution, enlarging the admissible region of the suction–Deborah plane by about two-thirds at $ \gamma = 0.5 $ and doubling or tripling the largest attainable Deborah number at fixed suction. It also contracts the range of shrinking rates supporting dual solutions, moving the critical value from $ \lambda_c = -1.8336 $ on a plane sheet to $ -1.6840 $ at $ \gamma = 0.5 $, with the contraction strengthening as suction increases. A linear temporal stability analysis of the complete disturbance system confirms the upper branch to be stable and the lower to be unstable; the linearized operator is shown to be block lower-triangular, so the customary restriction to the momentum–dust subsystem is exact rather than assumed. Finally, a machine learning surrogate is assessed against classical alternatives on equal data. A Gaussian process built on 10 reference solutions attains $ 0.18\% $ accuracy for under three seconds of construction and repays itself after 11 evaluations, against roughly 875 for a network trained on the differential residual. The same Gaussian process degrades by nearly three orders of magnitude approaching the turning point, erring by $ 12.3\% $ at a distance of $ 0.036 $ from the fold, where the branch has a vertical tangent and smooth interpolation is the wrong hypothesis class. The case for embedding the governing equations in a surrogate rests on that neighborhood, and not on the cost.
Citation: Bilal Ahmad, Muhammad Abubakar, Amber Sultan, Syed Baqar Hussain, Yazeed Yasin Ghadi. Physics-informed machine learning approach for radiative MHD dusty Maxwell nanofluid flow over a stretching/shrinking surface in a non-Darcy porous medium: A case of dual solutions[J]. AIMS Mathematics, 2026, 11(11): 36026-36064. doi: 10.3934/math.20261395
Wire coating, fiber drawing, and the extrusion of filled thermoplastics all draw a slender continuous filament through a hot, particle-laden viscoelastic melt, often in a packed die and under an imposed magnetic field. This paper formulates that configuration as a steady two-phase boundary layer flow of a radiative magnetohydrodynamic (MHD) dusty Maxwell nanofluid over a stretching or shrinking cylinder in a Darcy–Forchheimer porous medium, retaining a micrometer-scale dust phase with velocity and thermal slip alongside a Buongiorno nanoparticle description, nonlinear Rosseland radiation, and an Arrhenius reaction. Similarity reduction yields a tenth-order system in which transverse curvature enters through the parameter $ \gamma $, multiplying every diffusion operator by $ (1+2\gamma\eta) $ and adding a viscoelastic term with no plane-sheet analog. A collocation solver with fold-following continuation, verified against published stretching cylinder data to within $ 0.004\% $ at matched truncation, resolves both solution branches for the decelerating suction-assisted regime. Curvature is found to govern the problem principally through feasibility: Because the coefficient of the highest derivative is $ (1+2\gamma\eta)-\beta f^2 $, curvature relaxes the condition for a physically admissible solution, enlarging the admissible region of the suction–Deborah plane by about two-thirds at $ \gamma = 0.5 $ and doubling or tripling the largest attainable Deborah number at fixed suction. It also contracts the range of shrinking rates supporting dual solutions, moving the critical value from $ \lambda_c = -1.8336 $ on a plane sheet to $ -1.6840 $ at $ \gamma = 0.5 $, with the contraction strengthening as suction increases. A linear temporal stability analysis of the complete disturbance system confirms the upper branch to be stable and the lower to be unstable; the linearized operator is shown to be block lower-triangular, so the customary restriction to the momentum–dust subsystem is exact rather than assumed. Finally, a machine learning surrogate is assessed against classical alternatives on equal data. A Gaussian process built on 10 reference solutions attains $ 0.18\% $ accuracy for under three seconds of construction and repays itself after 11 evaluations, against roughly 875 for a network trained on the differential residual. The same Gaussian process degrades by nearly three orders of magnitude approaching the turning point, erring by $ 12.3\% $ at a distance of $ 0.036 $ from the fold, where the branch has a vertical tangent and smooth interpolation is the wrong hypothesis class. The case for embedding the governing equations in a surrogate rests on that neighborhood, and not on the cost.
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Figures(19) / Tables(18)
Bilal Ahmad, Muhammad Abubakar, Amber Sultan, Syed Baqar Hussain, Yazeed Yasin Ghadi. Physics-informed machine learning approach for radiative MHD dusty Maxwell nanofluid flow over a stretching/shrinking surface in a non-Darcy porous medium: A case of dual solutions[J]. AIMS Mathematics, 2026, 11(11): 36026-36064. doi: 10.3934/math.20261395
Stretching/shrinking cylinder of radius $ R $ in a non-Darcy porous medium, carrying a dusty Maxwell nanofluid under a transverse magnetic field with radial suction
The numerical procedure. Stage 1 marches $ \lambda $ downwards until the Newton solver fails near the fold; Stage 2 exchanges the roles of $ \lambda $ and $ f''(0) $ so that the turning point becomes an ordinary point of the solution curve. Every accepted solution is screened on the Maxwell admissibility margin before entering the stability analysis
Surrogate architecture. A single input $ \eta $ maps through five $ \tanh $ layers to the five primitive fields; derivatives are obtained by automatic differentiation, so the five ODE residuals are evaluated exactly rather than by finite differences and combined with the boundary residuals into the training objective
Left: Skin friction against curvature, comparing the present solver at matched truncation and at convergence with the published values. Right: Convergence of $ f''(0) $ with the truncation $ \eta_\infty $ for the full tenth-order system
Left: Largest admissible Maxwell parameter against suction, at four curvatures. Right: The number of admissible $ (S, \beta) $ combinations out of the 30 tested; the dotted line marks the plane sheet value
Critical shrinking rate $ \lambda_c $ (left) and the skin friction at the turning point (right) against the curvature parameter
Solution curves $ f''(0) $ against $ \lambda $ at two curvatures. Open circles mark the turning points. The upper portion of each curve is the first (stable) solution; the lower portion is the second (unstable) solution
Velocity, temperature, and concentration profiles on both branches at $ \gamma = 0.3 $. The two solutions are taken at the traced points nearest $ 0.92\lambda_c $ on their respective branches, with $ \lambda = -1.600 $ on the upper and $ \lambda = -1.707 $ on the lower; the lower branch occupies only a narrow interval above $ \lambda_c $, so an exact match in $ \lambda $ is not available. The second solution carries a markedly thicker velocity and thermal layer
Critical shrinking rate against suction at two curvatures
Left: Smallest eigenvalue against $ \lambda $ on both branches at two curvatures; the dotted line is $ \omega = 0 $. Right: The leading eigenvalues of the momentum and thermal blocks at $ \gamma = 0.3 $, $ \lambda = -1.0 $, showing that the thermal block is bounded away from zero
Fluid axial velocity for the curvature, resistance, Forchheimer, Maxwell, and dust loading groups
Fluid and dust fields at several loadings; dust quantities are shown faded. The particulate phase lags the carrier in both velocity and temperature
Fluid temperature for the curvature, radiation, Brownian, thermophoresis, and heat source groups
Nanoparticle concentration for the curvature, Schmidt, reaction rate, and activation energy groups
Streamlines (solid) and isotherms (dotted, with red shading) in the physical plane at $ \lambda = 1 $, $ S = 2 $, and $ \beta = 0.1 $. Left: Plane sheet. Right: Cylinder at $ \gamma = 0.3 $. The vertical coordinate is the wall-normal distance $ \hat r-R $
Left: Relative $ L_2 $ error for each state variable over 10 seeds; horizontal bars mark the means. Right: The final ODE residual loss by seed
Left: Accuracy of the classical surrogates for $ f''(0) $ against the number of reference solutions. Right: Cumulative cost of repeated direct solution compared with building a Gaussian process on 10 solutions and querying it; the surrogate repays its construction after about 11 evaluations
Left: Relative $ L_2 $ error in $ f $ over the depth-by-width grid. Right: Collocation density convergence for $ f $ and $ \theta_p $
Standardized linear coefficients for each response, ordered by magnitude. Bars are a sensitivity ranking, not effect estimates with confidence intervals
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