Boundary layer flow and heat transfer towards a stretching or shrinking cylinder within carbon nanotubes with hydromagnetic effects

The numerical investigation of stagnation point flow past a stretching or shrinking cylinder in carbon nanotubes with the presence of hydromagnetic effects is examined.  This study has been solved by ordinary differential equations obtained using the similarity transformation that transformed from the governing equations along with the boundary conditions, then implemented the bvp4c solver in MATLAB software platform, ensuring accurate results.  Considering two types of base fluids, namely water and kerosene.  Also, single-wall and multi-wall types of carbon nanotubes are used in this study, with single-wall and multi-walled consisting of different values of density.  The study is investigated by generating the graphical result of the velocity, temperature, skin friction coefficient and heat transfer rate.  Several parameters are considered, such as magnetic field $M$, curvature parameter $\gamma$, stretching or shrinking parameter $\varepsilon$ and nanoparticle volume fraction $\varphi$.  From the numerical results, it is demonstrated that a non-unique solution exists for the shrinking parameter, while a unique solution is obtained for stretching parameter.  Furthermore, single-walled carbon nanotubes and kerosene-based nanofluid are found to be more effective for heat transfer than multi-walled carbon nanotubes and water-based nanofluid.  Response surface methodology (RSM) is utilized to optimize the heat transfer, with the heat transfer maximized for water-SWCNT estimated at $0.880135$ and a desirability of $99.98\%$ which is higher than water-MWCNT, indicating water-SWCNT is more effective.