Heat transfer analysis in MHD curved channel flow with ciliated walls: Numerical solutions with error control

This work presents a detailed analysis of heat and momentum transport driven by the coordinated action of metachronal waves along ciliated channel walls, in the presence of a magnetic field acting on a micropolar fluid within a curved geometry. The rhythmic beating of the cilia is modeled as a peristaltic motion, generating a low-Reynolds-number flow environment, allowing for the application of the lubrication theory to simplify the governing equations formulated in curvilinear coordinates. These equations capture the complex interplay between microrotation effects, magnetic body forces, and channel curvature. The nonlinear coupled system is solved numerically using Wolfram Mathematica's NDSolve, ensuring high-resolution solutions across parameter spaces of practical interest. Key dimensionless quantities such as the Hartmann number (quantifying Lorentz force effects), the curvature ratio (governing geometric influence), the coupling number (capturing micropolar-fluid spin viscosity), and the Hall parameter (representing current deviation due to magnetic interactions) are systematically varied. Their impacts on axial velocity, temperature gradients, microrotation profiles, pressure distribution, heat transfer coefficients, and stream function structures are thoroughly examined. The role of metachronal coordination is emphasized, demonstrating its capacity to enhance convective transport by modifying near-wall vorticity and improving mixing efficiency. Flow visualization results reveal that the synergy between wave-induced boundary motion and geometric curvature fosters complex secondary flows and thermal patterns, which have direct implications for optimizing biomedical microdevices and microfluidic heat exchangers. Furthermore, a numerical error analysis validates the reliability of the computational approach.