This study is motivated by the need to understand complex thermal and hydrodynamic behaviors of couple stress fluids, which commonly occur in lubrication systems, microfluidic devices, and polymeric material processing. Its significance lies in modeling non-isothermal couple stress fluid flow through an inclined Poiseuille channel bounded by two heated parallel plates, a configuration relevant to advanced heat and mass transfer applications. The aim is to determine the velocity profile, temperature distribution, volumetric flow rate, average velocity, and shear stress for the incompressible fluid. To achieve this, the highly nonlinear coupled ordinary differential equations governing the system are solved using the Optimal Homotopy Asymptotic Method and the Homotopy Perturbation Method, which provide accurate approximate solutions without linearization. The major findings show excellent agreement between the two approaches, confirming their validity, while parametric studies reveal how physical factors such as couple stress effects, plate inclination, and thermal gradients influence the flow. The specific applications of this work include lubrication processes, thermal energy devices, and fluid transport systems requiring precise control of flow and heat transfer.

In this article, we introduce the concept of a controlled fuzzy 2-metric space, formulated by incorporating three control functions that flexibly regulate the fuzzy distance relationships among triplets of points. This structure provides a flexible analytical tool for modeling systems influenced by uncertainty, interdependence, and approximate reasoning. We establish several fundamental properties of this structure and derive fixed-point results. To demonstrate its practical relevance, we apply the proposed framework to a dynamic market-equilibrium problem, in which agents’ interactions are governed by fuzzy relations and control-dependent adjustments. The study also discusses implications for soft computing and decision-making systems, highlighting the framework’s potential in modeling adaptive and uncertain environments.