Numerical simulation of unsteady MHD bio-convective flow with Cattaneo-Christov heat flux over a stretching surface

The unsteady, two-dimensional, laminar boundary-layer flow of a viscous nanofluid containing gyrotactic microorganisms is examined over a permeable, radiative stretching sheet subject to an inclined magnetic field, Darcy resistance, internal heat generation, wall mass suction, velocity slip and convective thermal/solutal wall conditions, with heat transport described by the Cattaneo–Christov constitutive law. The governing equations are reduced by similarity transformations to a coupled nonlinear ODE system and solved by a shooting procedure (fourth-order Runge–Kutta with Newton–Raphson correction of the wall gradients), with every case cross-checked against the MATLAB solver bvp4c and validated against previously published limiting cases (Tables 1–3). The axial velocity is reduced by both the Hartmann number and the field-inclination angle, since the Lorentz force enters the momentum balance through the single group M 2 sin 2 β ; wall suction and a larger stretching ratio thin the momentum layer, while velocity slip reduces the wall shear. The temperature falls as the thermal relaxation parameter δ 1 increases and rises with the thermal Biot number and the radiation parameter. The density of motile microorganisms is reduced by the bioconvection Péclet and Schmidt numbers. The skin-friction coefficient and the reduced Nusselt number respond in opposite senses to the sign of the unsteadiness parameter A, a coupling absents from the corresponding steady-state analysis. The principal contribution of this work is a corrected and internally consistent similarity reduction of the momentum equation for a stretching sheet with a genuinely moving free stream – the porous, magnetic and unsteady terms are shown to act on the velocity deficit ( f ′−1) rather than on f ′ alone, a distinction that is immaterial in most of the literature (where the free stream is quiescent) but is required here for the far-field boundary condition to be satisfied identically, and which we verify both analytically and by direct numerical solution.