This study focuses on the flow of stagnation region and heat transfer of carbon nanotubes (CNTs) over an exponentially stretching/shrinked sheet in the presence of homogeneous–heterogeneous reactions. Kerosene and water are considered base fluids in both single-wall and multi-wall carbon nanotubes. After employing the appropriate similarity variables, the system of partial differential equations is transformed to a system of nonlinear ordinary differential equations. Solution of the problems is obtained numerically using the bvp4c solver in MATLAB software. The impact of physical parameters, such as solid volume fraction, stretching/shrinking parameter, homogeneous and heterogeneous reaction rate, Schmidt number on the velocity, temperature and concentration profiles, skin friction, and heat transfer rate are discussed graphically and interpreted physically. The results indicate that for an exponentially shrinking sheet, dual solutions exist for a certain range. It is clear from figures that the concentration profile increases for increasing values of heterogeneous parameter and decreasing values of homogeneous parameter. Heat transfer and skin friction were observed to have a greater impact for single-wall carbon nanotubes (SWCNTs) compared to multi-wall carbon nanotubes (MWCNTs). A stability analysis has been performed to show which solutions are linearly stable.

A model of unsteady stagnation-point flow and heat transfer over a permeable exponential stretching/shrinking sheet with the presence of velocity slip is considered in this paper. The nanofluid model proposed by Tiwari and Das is applied where water with Prandtl number 6.2 has been chosen as the base fluid, while three different nanoparticles are taken into consideration, namely Copper, Alumina, and Titania. The ordinary differential equations are solved using boundary value problem with fourth order accuracy (bvp4c) program in Matlab to find the numerical solutions of the skin friction and heat transfer coefficients for different parameters such as stretching/shrinking, velocity slip, nanoparticle volume fraction, suction/injection, and also different nanoparticles, for which the obtained results (dual solutions) are presented graphically. The velocity and temperature profiles are presented to show that the far field boundary conditions are asymptotically fulfilled, and validate the findings of dual solutions as displayed in the variations of the skin friction and heat transfer coefficients. The last part is to perform the stability analysis to determine a stable and physically-realizable solution.

In this research, a six-order, fully implicit Block Backward Differentiation Formula with two off-step points (BBDFO(6)), for the integration of first-order ordinary differential equations (ODEs) that exhibit stiffness, is proposed. The order, consistency and stability properties of the method are discussed, and the method is found to be zero stable and consistent. Hence, the method is convergent. The numerical comparisons with the existing methods of a similar type are given to demonstrate the accuracy of the derived method.

Newtonian fluid is ideal for lubrication purposes because the viscosity of this fluid remains as a function of the shear. Besides, the heat and mass transfer are an important study area in fluid dynamics due to its vast applications in industrial processes. The heat-mass transfer can be defined as the Soret and Dufour effect, which implemented in many industrial applications such as in chemical engineering and geosciences field. In addition, the fluid flow over an extending/compressing sheet has significant industrial applications such as the cooling of continuous strips, glass fibre production, the extrusion of plastic sheets from a die, etc. As a response, this study aims to investigate the impacts of Soret and Dufour parameters on the Newtonian fluid flow over an inclined stretching/shrinking sheet. The methodology of this mathematical model are stated as follow: 1) the transformation of partial differential equations (PDEs) to the ordinary differential equations (ODEs), and 2) The ODEs are solved using bvp4c solver in MATLAB software. The bvp4c solver is a MATLAB program directory that solves general form and multi-point boundary layer problems. The main sections of bvp4c are: 1) The solution of the ODEs, 2) The related boundary conditions that can produce the expected results, and 3) An initial guess to run the bvp4c solver. As a result, the numerical and graphical results show that the Soret effect increases the concentration profile whereas decreases the temperature profile. The vice versa occurrence is true for Dufour effect. The convective mass transfer caused by a temperature gradient is known as thermal-diffusion (Soret) effect. The convective heat transfer produced by concentration differences is known as diffusion-thermo (Dufour) effect. However, since the process of heat and mass transfers are related to each other, the Soret and Dufour effects are able to influence both of this process simultaneously. In conclusion, the convective heat transfer is enhanced by increasing Soret and Dufour number while the convective mass transfer is declined by increasing the two numbers.

The discovery of hybrid carbon nanotubes shows the tendency toward the improvement of heat transfer performance in comparison to various classical fluids. This paper expands the novelty in utilizing the hybrid carbon nanotubes over vertical stretching/shrinking cylinder in presence of hydromagnetic and thermal radiation. It is essential to analyze the hydromagnetic due to its high potential capability especially in drug and gene release, hyperthermia effects as well as cell separation and manipulation in bio-medical field. The investigation on thermal radiation effect is added in this current study as it enhances the rate of heat transfer. To initiate this problem, partial differential equations (PDE) for the hybrid nanofluid flow with relevant boundary conditions (BCs) is set up and transformed into an ordinary differential equation (ODE). Adopting the similarity solutions and numerically solved using bvp4c (MATLAB). Findings on the variation of local Nusselt number, skin friction coefficient, shear stress and local heat flux having the effects of magnetic, $M,$curvature, ϒ, thermal radiation, Nr, mixed convection parameter, $\lambda$ as well as volume fraction of nanoparticles, $\phi$ are demonstrated and elaborated in detail. Moreover, the research reveals that duality of solutions occurs when the buoyance force is in opposing flow with respect to the fluid motion, $\lambda <0$, as well as shrinking area, $\epsilon <0$. The occurrence of magnetic reduces the heat transfer as well as skin friction coefficient. In addition, the skin friction coefficient and local Nusselt number tend to improve as volume fraction of nanoparticles and curvature are increased. In contrast, the low of thermal radiation enhance the heat transfer. Indeed, the consequences of using hybrid carbon nanotubes help intensify the skin friction coefficient and Nusselt number compared to SWCNT nanofluid and MWCNT nanofluid. These crucial findings may benefit the scientists and academicians hence giving an add-on value to their expertise. A stability analysis must be performed since there exists a non-unique solution throughout the computation.