Insight of thermal transport in Maxwell fluid flow on a porous contracting flat surface inside a porous material with non-uniform heat source/sink: Multiple solutions and stability treatment

The thermal behaviour of viscoelastic fluids, particularly those described by the Maxwell model capable of capturing stress-relaxation effects, exhibits several intriguing and practically relevant characteristics. Likewise, flow induced by a contracting surface presents distinctive and non-classical boundary layer features. Motivated by these aspects, the present theoretical study investigates heat transfer in Maxwell fluid flow over a porous contracting flat surface embedded in a porous medium, incorporating a temperature-dependent and spatially-dependent non-uniform heat source/sink. The governing partial differential equations are reduced to self-similar ordinary differential equations using appropriate similarity transformations, and the resulting nonlinear system is solved with high accuracy using the shooting method along with fourth-order Runge–Kutta (RK-4) technique and Secant method. For sufficiently strong suction, dual solutions of the transformed equations are obtained, and a stability assessment confirms that the upper branch solutions are stable, while the lower branch solutions are unstable. The comprehensive analysis uncovers several notable physical insights. The analysis shows that viscoelasticity of Maxwell fluids plays a key stabilizing role: increasing the Deborah number weakens vorticity production, lowers the suction required to maintain an attached boundary layer, and broadens the parameter range in which similarity solutions exist. Permeability of the porous medium also strongly influences boundary layer behaviour; higher resistance suppresses vorticity generated by sheet contraction, enhancing skin friction and heat transfer in the upper branch but diminishing both in the lower branch. Both temperature- and space-dependent heat sources reduce heat transfer rates, while their sink counterparts enhance cooling, though spatial heat generation/absorption produces a far more pronounced thermal response. Suction strengthens and compresses the boundary layer in the stable upper branch but weakens the flow in the unstable lower branch. Overall, the study clarifies how elasticity, permeability, heat generation mechanisms, and mass transfer collectively shape the dual solution structure of Maxwell fluid boundary layers, offering detailed insights relevant to thermal control in viscoelastic–porous systems.