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The fractional model has been developed for the thermal flow of hybrid nanofluid due to the inclined surface. The thermal investigation of the hybrid nanomaterial is predicted by utilizing the molybdenum disulphide nanoparticles and graphene oxide nanomaterials. The flow computations for mixed convection flow of nanoparticles and base fluids are performed due to vertical oscillating plate. The simulations for the formulated model have been done ρ-Laplace transform technique for Caputo fractional simulations. Definitions of Mittage–Leffler function and ρ-Laplace transform are also presented for the governing model. The application of updated definitions of ρ-Laplace transform for the Caputo fractional model is quite interesting unlike traditional Laplace transforms. The comparative investigation for both types of nanoparticles is performed for heat and mass transfer rates. It is observed that the heat enhancement rate due to water-based nanoparticles is relatively impressive compared to the kerosene oil-based nanomaterials.

The impact of Marangoni convection has noteworthy applications in nanotechnology, atomic reactor, silicon wafers, semiconductor processing, soap films, materials sciences, thin-film stretching, crystal growth, and melting and welding processes. On the other hand, thermophoretic particle deposition (TPD) has a significant application in building ventilation systems, crushed coal burners, thermal exchangers, and air cleaners. Inspired by these applications, the present work mainly concentrates on the Marangoni convection flow of Al2O3/water-based nanofluid over a stretching sheet in a porous medium with TPD in the presence of Newtonian heating. Additionally, heat absorption/generation in energy expression is considered. A suitable similarity variable is applied to simplify the partial differential equations into a set of ordinary differential equations (ODEs). Furthermore, Runge Kutta Fehlberg fourth fifth order method along with the shooting technique is implemented to solve the reduced ODEs. Furthermore, mathematical computational software helps to acquire a numerical solution. The velocity of nanofluid increases for enhancement of Marangoni number and diminishes for porosity parameter. The heat absorption/generation parameter improves thermal dispersion in both common wall temperature and Newtonian heating cases. For the upgradation in the thermophoretic parameter, the concentration decreases and the rate of mass transfer increases. The rate of heat transfer increases as the heat source parameter grows and decreases as the heat sink parameter decreases. In all of the profiles, nanofluid outperforms viscous fluid.

In the current study, we numerically analyze the significance of motile microbes on the magnetohydrodynamic steady convective streams of tangent hyperbolic (TH) nanofluid flow across an elastic nonlinearly stretching surface of an irregular thickness. The consequences of an external magnetic field, thermal radiation, and thermal conductivity are also examined on the TH nanofluid. The governing system of equations (nonlinear set of partial differential equations) is transfigured into a system of ordinary differential equations (ODEs) by using the similarity variable conversions. Furthermore, the reduced form of nonlinear ODEs is numerically computed through the parametric continuation method (PCM) using MATLAB software. The relative evaluation is carried out to authenticate the numerical outcomes. It has been observed that the energy field accelerates with the Rayleigh number, Weissenberg number, and Brownian motion. The mass propagation ratio improves with the effect of activation energy and decreases with the influence of chemical reactions. Furthermore, the motile microbes’ profile declined with the outcome of the Peclet and Lewis numbers. The skin friction increases up to 7.3% with various magnetic values ranging from 0.5 to 1.5. However, the energy transfer rate declines to 5.92%. The thermal radiation boosts the energy propagation rate and flow velocity by up to 11.23 and 8.17%, respectively.