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In this paper, a steady solution is presented for the equations that represent the MHD rarefied gas fluid flow and heat transfer due to a permeable stretching sheet with second-order velocity slip and thermal slip phenomenon. By using nondimensional transformations, the system of partial differential equations governing the problem is transformed into another system of nonlinear ordinary differential another equations. Novel solutions are investigated for the resulting ordinary differential equation which describe the momentum equation. The numerical results obtained agreed very well with previously reported cases available in the literature. Additionally, the effects of the magnetic parameter, first- and second-order velocity slip parameter, conductivity parameter, thermal slip parameter and the suction (injection) parameter on both the velocity and temperature profiles and on the local skin-friction coefficient are discussed and presented through tables and graphs.

Stagnation point flow of viscoelastic second grade fluid over a stretching cylinder under the thermal slip and magnetic hydrodynamics effects are studied. The mathematical model has been developed under the assumption of non-Newtonian viscoelastic fluid flow over a stretching cylinder by means of the boundary layer approximations. The developed model further reduced through the similarity transformations and constructs the model of nonlinear ordinary differential equations. The system of nonlinear differential equations is dimensionless and solved through the numerical technique bvp5c methods. The results of the physical parameters are found and interpreted in the form of tables and graphs. The velocity shows that the graph of curves enhances away from the surface when the values material parameter ${\beta }^{\ast }$ increase, which means the momentum boundary layer increases for enhancing the material parameter ${\beta }^{\ast }$. The temperature gradient reduced due enhancing the values of material parameter ${\beta }^{\ast }$ because thermal boundary layer reduced for higher values of material parameter ${\beta }^{\ast }$.

This thermal case pronounced the stability framework for stagnation point flow of magnetized alumina and copper nanoparticles with due exponentially shrinking permeable surface. The thermal stability and enhancement of water base liquid had been taken into account with uniform impulsion of hybrid nanomaterials. The induced flow results via exponentially shrinking permeable surface. The similarity transformation simplifies the mathematical model where governing formulated system for hybrid nanofluid is altered into the nondimensional form. A numerical solver called bvp4c is employed in MATLAB software to aid in the problem-solving process, and dual branches have been found. The significance of pertaining parameters associated to the flow model is inspected in view of thermal properties. The findings show that there are two branches for suction strength $(S>S_c)$ and magnetic strength $(M>M_c)$. The bifurcation values $S_c$ and $M_c$ reduce for the occurrence of dual branches as the solid volume percentages of copper increase. Furthermore, for the upper branch solutions, the skin friction and heat transfer rate rise as ${\varnothing }_{\mathrm{\text{Cu}}}$ increases. The temporal stability analysis determines the stability of the dual branches, and it is discovered that only one of them is stable and physically applicable. The presence of suction parameter effectively controls the thermal transportation phenomenon.

Advances in nanoscience and technology acquired the significance of the nanofluid in novel functional polymers like fibre insulation, geothermal system and chemical catalytic reactors. Inspired by the above applications, an innovative mathematical model is established for radiative nanoliquid flow and is engendered due to stretching sheet with inclined magnetic field which is immersed with nanoparticles. Joule dissipation and exponentially-based heat source/sink effects are employed in the present phenomenon under the heat constraints. The governing equations, which describe the flowing nanofluid, are transformed into invariant dimensionless equations with suitable similarity quantities. With the adoption of a shooting scheme with Runge–Kutta-45, the resultant equations are numerically simplified. The impact of several converted dimensionless elements on physically interesting values is depicted visually. The current analysis is validated through comparison with some selected related literature, which shows a positive correlation. The nanoparticle thermal conductivity is raised for an increased value of the thermal radiation, thermal viscosity and heat source to propel temperature profiles. The heat flux gradient significantly affects the heat propagation all over the flow regime.

In the recent decades, the increasing energy demands and its applications have seen the focus shifting to the hybrid nanofluid flows but so much is still left to be investigated. This analysis is executed to explore the hydro-magnetic flow to investigate the incompressible flow and heat transfer towards a stretching surface with velocity and thermal slips. The scaling similarity transformations are created using Lie group analysis and employing these to convert nonlinear partial differential equations to the nonlinear ordinary differential equations. Here, after converting equations from dimensional to non-dimensional, we will use the BVP4C solver (MATLAB) for plotting the graphs to analyze how distinct non-dimensional parameters affect the skin friction and Nusselt number transfer rate, case 1 graphene + CNT + aluminum oxide with base fluid as water and case 2 magnesium oxide + zirconium oxide + copper oxide with water as base fluid, here taking nanoparticles without different shapes. The hybrid nanofluid temperature profile has mixed behavior, and the velocity profile increases when M rises. The hybrid nanofluid temperature profile curvature has composite behavior when $Pr$ rises. The link between several independent or predictor variables and one dependent or criterion variable has been examined using multilinear regression analysis (MLR). When coefficient values for many variables are subject to change, it can forecast a wide range of outcomes.