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This work explains the flow of a G-H2O nanofluid under the special case of MHD stagnation point flow, and the detailed investigation of the Navier's stokes equations extracted analytically. The main methodology is given work of PDEs is converted into ODEs using the appropriate similarity transformations. The momentum equations solved analytically to derive the solution domain, the impact of thermal radiation is seen in energy equation, four different scenarios are used to solve the energy equation. Aim of the present work is to study the theoretical analysis and it can be discussed for dual nature behavior by providing different physical parameters, these parameters control the domain, momentum and heat transpiration. In the heat transfer analysis solutions are derived in terms of incomplete gamma function and confluent hypergeometric form. The current work is examined using graphene nanoparticles, and the value of Pr is fixed at 6.2. The present problem is the benchmark solution for the results and it is significance in industrial and technological applications in fluid-based systems involving shrinkable/stretchable materials. At the end we get Velocity decreases with increases of VC for upper branch of solution and increases with increases of VC for lower branch of solution in the case shrinking sheet.

The current work employs the single wall and multiwall carbon nanotube models to investigate the effect of suction and injection over a Marangoni boundary layer flow on magnetohydrodynamics (MHD) Casson fluid and thermal radiation on a permeable surface. The nonlinear partial differential equations (PDEs) produced by the governing equations for the prescribed flow are translated using the proper similarity transformations into a system of nonlinear ordinary differential equations (ODEs). The consequential nonlinear equation is solved analytically by using incomplete Gamma function. The effects of suction and injection over velocity profile and temperature distribution are presented graphically. It was found that the heat transfer rate increased due to increases in the volume fraction. In addition, the applied magnetics energy, on the other hand, causes it to decrease. Further, single wall carbon nanoliquid has a higher temperature than multiwall nanoliquid. The problem under consideration has a wide range of commercial and scientific applications in various industries like aviation and medicine field. The impacts of MHD and suction/injection on the flow and heat transfer properties of the fluid are examined under the influence of radiation, and the solution of each profile is shown in the form of figures along with the outcomes of the interface velocity and heat transfer rate at the surface.

Due to its extensive applicability and employment in different systems, the study of thermal transport is a sig-nificant area of research. The main goal of this investigation is to analyse the impact of thermal radiation as well as the Navier-Stokes condition upon the two-dimensional flow of second-grade as well as Walter's B ternary nanoliquids through a permeable shrinking flat surface. The phrase "ternary nanofluid" refers to a suspension of three different types of nanoparticles, notably silver (Ag), SWCNTs, and graphene particles, in water as the base fluid. The inverse Darcy phenomenon has an effect on the momentum equation as well. The energy equation also accounted for nonlinear thermal radiations as well as heat source and sink components. The phenomenon is portrayed as a nonlinear system of partial differential equations. The model equations are transformed into a non-dimensional series of ordinary differential equations by replacing similarity. Afterwards, the transformed ODEs were tackled analytically in order to establish an analytical solution of the energy equation in terms of a confluent hyper-geometric function by utilizing similarity conversion. This research reveals that the velocity field is reduced significantly by the slip parameter. The benefits of a few emerging factors, such as the viscoelastic parameter, inverse Darcy number, slip parameter, solid volume fraction, radiation, Prandtl number, and skin friction coefficient effects on solution, have been displayed through plots and discussed. Additionally, it has been concluded from this analysis that ternary nanofluids have a higher thermal flow rate than hybrid or standard nanofluids.