Thermal radiation and magnetohydrodynamic effects on nanofluid boundary-layer flow towards a moving plate: an adaptive residual-controlled collocation solution

Abstract This paper re-examines steady, two-dimensional magnetohydrodynamic (MHD) boundary-layer flow of an incompressible, electrically conducting nanofluid towards a moving plate under non-linear thermal radiation, Brownian motion and thermophoresis. The flow model itself is unchanged from the earlier treatments of Roşca and Pop and of Mohamed et al., extended only by the inclusion of the magnetic parameter M and the radiation parameter Rd; the contribution of this paper is confined to the numerical method used to solve the resulting boundary-value problem, not to the physical formulation. The governing partial differential equations are transformed into a system of coupled nonlinear ordinary differential equations via a similarity transformation. In contrast to earlier treatments of this problem, which relied on a fixed-step fourth-order Runge–Kutta scheme coupled with a manually tuned shooting procedure, the resulting two-point boundary-value problem is solved here with an adaptive, residual-controlled collocation method (a bvp4c-type solver; the method is a 4th-order finite collocation scheme with adaptive mesh refinement, not a spectral method) that automatically refines the computational mesh until a prescribed residual tolerance is met. The proposed method removes the need for hand-tuned missing initial conditions, converges from a single generic initial guess across the entire parameter space explored (six of six representative cases at a consistently applied far-field truncation length, against one of six for classical shooting from the same guess), and reaches a residual below 10 −9 with substantially less manual tuning than shooting. Directly against the published benchmark solutions of Roşca and Pop and of Mohamed et al., the present method matches the reference Nusselt numbers to within 10 −5 –10 −4 in four of five test cases, one to two orders of magnitude closer than the shooting-RK4 results reported in the authors’ earlier shooting-RK4 study of this configuration, providing direct, published-benchmark evidence that the proposed method is correct. The demonstrated advantage of the adaptive method is confined to robustness and residual control, not raw computational speed: shooting, when it converges, is typically faster in CPU time. Parametric results for the skin-friction coefficient, local Nusselt number and local Sherwood number are tabulated for the magnetic field parameter ( M ), radiation parameter (Rd), Brownian motion parameter (Nb), thermophoresis parameter (Nt), Lewis number (Le), Prandtl number (Pr) and plate-velocity parameter ( λ ), and the corresponding velocity, temperature and concentration profiles are presented graphically and discussed physically.