In this paper, the steady flow of an incompressible hybrid Casson nanofluid over a vertical permeable exponential stretching sheet is considered. The influence of the induced magnetic field is investigated. The influence of heat production and nonlinear radiation on slip effects is studied. Typically, three hybrid nanofluidic models are presented in this paper, namely: Xue, Yamada-Ota, and Tiwari Das. A study of a single-walled carbon nanotube and a multi-walled carbon nanotube with base fluid water is also provided. The governing equations are developed under flow assumptions in the form of partial differential equations by using boundary layer approximations. Using the appropriate transformations, partial differential equations are converted into ordinary differential equations. The ordinary differential equations are solved by the fifth-order Runge-Kutta-Fehlberg approach. Impacts concerning physical parameters are revealed by graphs and numerical values through tables. Temperature profile increases as concentration of solid nanoparticles increases. Because the thermal conductivity of the fluid is enhanced due to an increment in solid nanoparticles, which enhanced the temperature of the magneto-Casson hybrid nanofluid. The skin friction achieved higher values in the Yamada-Ota model of hybrid nanofluid as compared to the Xue model and Tiwari Das model. The results of this study show the Yamada-Ota model achieved a higher heat transfer rate than the Xue and Tiwari Das models of hybrid nanofluid.

The stagnation point flow of unsteady compressible Casson hybrid nanofluid flow over a vertical stretching sheet was analyzed. The comparative study of Yamada Ota, Tiwari Das, and Xue hybrid nanofluid models was performed. The Lorentz force was applied normal to flow directions. The effect of nonlinear radiation was studied. We considered the SWCNT (signal wall carbon nanotube) and MWCNT (multi-wall carbon nanotube) with base liquid (water). Under the flow suppositions, a mathematical model was settled by means of boundary layer approximations in terms of partial differential equations. The suitable transformation was developed by using the lie symmetry method. Partial differential equations were transformed into ordinary differential equations by suitable transformations. The dimensionless system was elucidated through a numerical technique named bvp4c. The impacts of pertinent flow parameters on skin friction, Nusselt number, and temperature and velocity distributions were depicted through tabular form as well as graphical form. In this study, the Yamada Ota model achieved a higher heat transfer rate compared to the Tiwari Das and Xue hybrid nanofluid models. The skin friction (${\mathrm{C}}_{\mathrm{fx}}{\mathrm{Re}}^{-1/2}$) increased and temperature gradient (${\mathrm{Nu}}_{\mathrm{x}}{\mathrm{Re}}^{-1/2}$) declined due to the increment of solid nanoparticle concentration (${\mathsf{\varphi }}_2$). Physically, skin friction increased because the higher values of the solid nanoparticles increased resistance to the fluid motion.

Heat and mass transfer study of hybrid nanomaterial Casson fluid with time-dependent flow over a vertical Riga sheet was deliberated under the stagnation region. In the presence of the Riga sheet in fluid flow models, this formulation was utilized to introduce Lorentz forces into the system. We considered the three models of hybrid nanomaterial fluid flow: namely, Yamada Ota, Tiwari Das, and Xue models. Two different nanoparticles, namely, SWCNT and MWCNT under base fluid (water) were studied. Under the flow suppositions, a mathematical model was settled using boundary layer approximations in terms of PDEs (partial differential equations). The system of PDEs (partial differential equations) was reduced into ODEs (ordinary differential equations) after applying suitable transformations. The reduced system, in terms of ODEs (ordinary differential equations), was solved by a numerical scheme, namely, the bvp4c method. The inspiration of the physical parameters is presented through graphs and tables. The curves of the velocity function deteriorated due to higher values of $M$. The Hartmann number is a ratio of electric force to viscous force. The electric forces increased due to higher values of the modified Hartmann number, ultimately declining the velocity function. The skin friction was reduced due to an incremental in $\varpi$, while the Nusselt number raised with higher values of $\varpi$. Physically, the Eckert number increased, which improved kinetic energy and, as a result, skin friction declined. The heat transfer rate increased as kinetic energy increased, and the Eckert number increased. The skin friction reduced due to physical enhancement of ${\beta }_1$, the shear thinning was enhanced which reduced the skin friction.

In this analysis, we considered a comparative study of micropolar Casson nanofluid flow on a vertical nonlinear Riga stretching sheet. Effects of thermal and velocity slip are considered under thermophoresis and Brownian motions. Select nonlinear PDEs transformed into nonlinear coupled ODEs using the set of suitable transformations. The nonlinear coupled ODEs are solved through a numerical technique along with the Runge–Kutta 4th-order scheme. The impacts of pertinent flow parameters on skin friction, Nusselt number, temperature, and velocity distributions are depicted through tabular and graphical form. Brownian motion and the magnitude of the Sherwood number have opposite performances; likewise, the Nusselt number and Brownian motion also have opposite performances. The Sherwood number and Nusselt number succeeded with higher values. The increment of the Casson fluid parameter declined with fluid velocity, which shows that thickness is reduced due to the increment of the Casson fluid parameter. Fluid velocity distribution curves show increasing behavior due to increments of the micropolar parameter.