Thermal boundary layer analysis of MHD nanofluids across a thin needle using non-linear thermal radiation

An analysis of steady two-dimensional boundary layer MHD (magnetohydrodynamic) nanofluid flow with nonlinear thermal radiation across a horizontally moving thin needle was performed in this study. The flow along a thin needle is considered to be laminar and viscous. The Rosseland estimate is utilized to portray the radiation heat transition under the energy condition. Titanium dioxide (TiO$ _2 $) is applied as the nanofluid and water as the base fluid. The objective of this work was to study the effects of a magnetic field, thermal radiation, variable viscosity and thermal conductivity on MHD flow toward a porous thin needle. By using a suitable similarity transformation, the nonlinear governing PDEs are turned into a set of nonlinear ODEs which are then successfully solved by means of the homotopy analysis method using Mathematica software. The comparison result for some limited cases was achieved with earlier published data. The governing parameters were fixed values throughout the study, i.e., $ k_1 $ = 0.3, $ M $ = 0.6, $ F_r $ = 0.1, $ \delta_\mu $ = 0.3, $ \chi $ = 0.001, $ Pr $ = 0.7, $ Ec $ = 0.5, $ \theta_r $ = 0.1, $ \epsilon $ = 0.2, $ Rd $ = 0.4 and $ \delta_k $ = 0.1. After detailed analysis of the present work, it was discovered that the nanofluid flow diminishes with growth in the porosity parameter, variable viscosity parameter and magnetic parameter, while it upsurges when the rate of inertia increases. The thermal property enhances with the thermal conductivity parameter, radiation parameter, temperature ratio parameter and Eckert number, while it reduces with the Prandtl number and size of the needle. Moreover, skin friction of the nanofluid increases with corresponding growth in the magnetic parameter, porosity parameter and inertial parameter, while it reduces with growth in the velocity ratio parameter. The Nusselt number increases with increases in the values of the inertia parameter and Eckert number, while it decliens against a higher estimation of the Prandtl number and magnetic parameter. This study has a multiplicity of applications like petroleum products, nuclear waste disposal, magnetic cell separation, extrusion of a plastic sheet, cross-breed powered machines, grain storage, materials production, polymeric sheet, energy generation, drilling processes, continuous casting, submarines, wire coating, building design, geothermal power generations, lubrication, space equipment, biomedicine and cancer treatment.