This paper presents the effect of thermal radiation on the boundary layer over a flat plate. The convective boundary condition is applied at the surface of the flat plate. The solution to the coupled non-linear transport equations is obtained using the Runge-Kutta-Fehlberg fourth-fifth order (RKF45) method. The impact of thermal radiation on mercury, air, sulphur oxide and water whose Prandtl numbers ( Pr ) are 0.044, 0.72, 2, and 7 respectively are depicted using line graphs and tables. The impact of Prandtl number ( Pr ), local convective heat transfer (a) and temperature difference ( C T ) on temperature distribution are also presented. The results indicated that the boundary layer thickness decreases with Pr augment but increases with increasing values R . Furthermore, R augment is inversely proportional to the temperature gradient near the plate while an opposite trend is observed away from the plate. The results also indicated that R augment reduces heat transfer but an opposite trend is observed with Pr augment. Pr augment has a decreasing effect on boundary layer thickness and a augment has an increasing effect on temperature.

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A non-linear approximation for natural convection boundary layer flow near a vertical wall under the influence of thermal radiation is analysed. The governing equation comprises of the set of non-linear partial differential equations is transformed into ordinary differential equations via the similarity transformation. The final dimensionless equations are solved numerically using the Runge Kutta Ferlberg fourth-fifth order (RKF45) method. The effects of the embedded parameters affecting the flow formation, temperature distribution, Nusselt number and the Skin friction are thoroughly examined. It is found that the temperature gradient is proportional to the thermal radiation near the plate whereas inversely proportional to the temperature gradient far away from the plate. The flow formation in the boundary enhanced near the vertical wall with thermal radiation parameter increase but remain constant in the free stream region. The rate of heat transfer enhanced with the thermal radiation whereas decreases with other embedded parameters under consideration.

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This article explored the mixed convection flow from a convectively heated vertical porous plate influenced by nonlinear thermal radiation with suction/injection. The significant effect of the internal heat generation is also taken into account. The boundary layer approximation equations responsible for the flow characteristics are formulated and translated to ordinary differential equations (ODEs) with the help of the similarity variable. The shooting technique is then employed to reduce the second-order ODEs to an initial value problem which is solved numerically by the Runge-Kutta method in maple software. Some of the results obtained are: that for weak mixed convection, the velocity and fluid temperature decay exponentially with suction whereas appreciating with fluid injection. The radiative heat flux boosts the fluid temperature and in turn propagates the fluid flow. The internal heat generation ([Formula: see text]) serves as a barrier for the heat flow from the left plate surface to the right plate surface except the mixed convection is resiliently sufficient to counter the generated heat and heat conducted via the plate from its left surface. However, for a weak [Formula: see text], the heat transfer from the left to right plate surface is feasible even with weak mixed convection. For [Formula: see text], the wall temperature is less than 1 and the heat transfer flows from the wall into the free stream. However, for [Formula: see text], the heat transfer flow back into the wall. With weak convection and buoyancy effect, the heat transfer could be increased by increasing the internal heat generation but the contrary is true for nonlinear thermal radiation.

An attempt to examine the relevance of heat source/sink on magnetohydrodynamics free convection flow in a vertical channel with an induced magnetic field is achieved. The analytical solution to the set of the differential equation is obtained by perturbation method for small thermophoresis and Brownian diffusion parameters under the unified thermal boundary condition (isothermal and isoflux boundary condition) for the energy equation. Numerical solution to the flow equation is also obtained by incorporating RKF45 in Maple software. The influence of active parameters such as Hartman number (Ha), magnetic Prandtl number (Pm), heat source/sink parameter (± S), Buoyancy ratio (Br), Brownian motion (Nb) and thermophoretic parameter (Nt) on velocity, induced magnetic field, induced current density, nanoparticles concentration, temperature and skin friction are depicted and discussed in detail. Results reveal that the Brownian motion parameter (Nb) and Buoyancy ratio (Br) augment enhances the shear stress whereas the contrast is observed with Hartman number (Ha) and thermophoretic parameter (Nt). Results also reveal that Hartman number (Ha) and thermophoretic parameter (Nt) enhances the induced current density while the contrast is true for heat sink parameter (−S). Finally, the temperature of the nanofluid could be enhanced with increase in Brownian motion parameter (Nb) and heat source parameter (+ S).

This article presents a close form of solution to the magnetohydrodynamics free convection between two parallel vertical plates filled with nanoparticles with induced magnetic field effect. The surface of the channel is maintained at constant heat flux or constant temperature.The exact and the numerical solution is obtained through the method of undetermined coefficient and RKF45 in maple software respectively while the analytic solution is obtained through perturbation. The governing equations include the significant effects of the thermophoretic and Brownian motion parameters due to the presence of nanofluid. The role of the active parameters on the velocity, temperature, induced current density, concentration and induced magnetic field is illustrated using graphs. The results of the study showed that Brownian motion parameter ðNbÞ, Buoyancy ratio ðBrÞ, and the heat source parameter ðSÞ plays a supportive role on the velocity whereas other active parameters found to have decreasing effect on the velocity profile. Regarding the temperature distribution, the heat source parameter ðSÞ, Brownian motion parameter ðNbÞ, and the thermophoretic parameter ðNtÞ augment enhance the nanofluid temperature. The skin friction decreases with Hartmann number ðHaÞ and magnetic Prandtl number ðPmÞ augment but increases with Buoyancy ratio ðBrÞ and heat source parameter ðSÞ:

This report presents a similarity solution for the buoyancy-driven flow of viscous incompressible fluid past an inclined porous plate influenced by nonlinear thermal radiation and thermophoresis. The boundary layer equations are reduced to some set of ODEs through similarity variables. Furthermore, the ODEs are converted to IVP through the shooting technique. The numerical solution is obtained through the Runge-Kutta algorithm in Maple software. The impact of the emergence parameters present in the mathematical model is explained through graphs and tables. Results obtained showed that with combined effects of suction/injection and nonlinear thermal radiation, the heat transfer rate is directly proportional to the angle of inclination but inversely proportional to wall shear stress and mass transfer rate. Furthermore, it was observed that the heat transfer rate declines with higher buoyancy force but enhances the wall shear stress. Also, the mass transfer rate could be enhanced with a higher thermophoresis effect. Suction propagates the velocity and temperature profiles whereas it decreases the rate of particle concentration, while the contrast is true for injection. In addition, nonlinear thermal radiation complements the fluid temperature, particle concentration and fluid transport.