A mathematical analysis is communicated to the thermal radiative and heat absorption effects on 3D MHD Williamson nanoliquid motion via stretching sheet. The convective heat and mass boundary conditions are taken in sheet when liquid is motion. The suitable similarity transformations, non-dimensionless has been utilized for reduce basic governing PDE’s into coupled system of ODE’s, also compute numerical solutions with the help of 4th order R-K-F procedure with shooting technique in MATLAB programming. The various physical parameters analysed numerically on \(f'(\eta )\)(“Velocity profile”),\(\operatorname{Re} _{x}^{{ - 1/2}}N{u_x}\) (“Heat Transfer Rate”) and \(\operatorname{Re} _{x}^{{ - 1/2}}Sh\) (“Mass Transfer Rate”). We noticed that, the \({\operatorname{Re} _x}^{{1/2}}{C_{fx}}\)(“Skin friction coefficient along \({x^*}\)-axis”), \({\operatorname{Re} _x}^{{1/2}}{C_{fy}}\)(“Skin friction coefficient along \({y^*}\)-axis”) and also compared with precious results and present results for various conditions. Finally, in the present result is good invention of the previous results.

A mathematical analysis is communicated to the thermal radiation and heat absorption effects on 3D MHD Williamson nanoliquid (NFs) motion via stretching sheet. The convective heat and mass boundary conditions are taken in sheet when liquid is motion. As a novelty, the effects of thermal radiation, heat absorption and heat and mass convection are incorporated. The aim is to develop heat transfer. Williamson NFs are most important source of heat absorption, it having many significant applications in “energy generation, HT, aircraft, missiles, electronic cooling systems, gas turbines” etc. The suitable similarity transformations have been utilized for reduce basic governing P.D. E’s into coupled nonlinear system of O.D. E’s. Obtained O.D. Es are calculated by help of R–K–F (“Runge–Kutta–Fehlberg”)4th order procedure with shooting technique in MATLAB programming. We noticed that, the skin friction coefficient is more effective in Williamson liquid motion when compared with NFs motion with higher numerical values of stretching ratio parameter, Williamson liquid motion is high when compared to NFs motion for large values of magnetic field. We compared with present results into previous results for various conditions. Finally, in the present result is good invention of previous results.

A numerical technique for the nonlinear thermal radiation effect on 3D (“Three Dimensional”) nanofluid (NFs) motion through shrinking or stretching surface with convective boundary condition is examined. In this investigation we use the convective and velocity slip conditions. The governing equations were converted into a set of couple non-linear ODE’s by suitable similarity transformations. The converted nonlinear equations are obtained by applying R-K-F (“Range-Kutta-Fehlberg”) procedure along with shooting technique. The physical parameters are explained graphically on velocity, temperature and concentration. Moreover, we found the coefficient of skin friction, rate of heat transfer with various nanofluid parameters. It is very good agreement when compared with previous study.

This study involvesthe numerical modeling of steady thermal radiation and chemical reaction on non‐Newtonian fluid motion via a bidirectional stretching surface. We have taken convective boundary conditions, and heat sources on the stretching surface. The working fluid of the present study is Casson fluid (“non‐Newtonian”) with couple stress. The self‐similarity forms of the nonlinear thermal radiative flow model are obtained by using similarity variables. Furthermore, the numerical results are computed with the help of fourth‐order Runge–Kutta–Fehlberg method with a shooting algorithm after reducing nonlinear partial differential equations have been translated into strong ordinary differential equations (ODEs). Impacts of the various flow physical parameters especially Biot number, nonlinear thermal radiation, and heat source parameters containing nonlinear ODEs are discussed in detail for distinct numerical values. A comparison of calculated results with the known numerical results made with the previously published literature is mentioned and obtained a good agreement. Finally, we found that the R e x 1 / 2 C f x $R{e}_{x}^{1/2}{C}_{fx}$ (“coefficient of skin friction”) declines along x * ,0.35em y * $x* ,\,y* $ directions, respectively, with β $\beta $ via λ $\lambda $ while the opposite direction follows M $M$ with respect to λ $\lambda $ and the R e x − 1 / 2 N u x $R{e}_{x}^{-1/2}N{u}_{x}$ (“heat transfer rate”), R e x − 1 / 2 S h $R{e}_{x}^{-1/2}Sh$ (“mass transfer rate”) increase with Γ $\Gamma $ via γ 1 ${\gamma }_{1}$ while opposite direction follows γ 1 ${\gamma }_{1}$ with respect to γ 2 ${\gamma }_{2}$.