Heat transfer improves significantly when the working fluid has high thermal conductivity. Heat transfer can be found in fields such as food processing, solar through collectors, and drug delivery. Considering this notable fact, this work is focused on investigating the bio-convection-enhanced heat transfer in the existence of convective boundary conditions in the flow of hybrid nanofluid across a stretching surface. Buongiorno fluid model with hybrid nanoparticles has been employed along the swimming microorganisms to investigate the mixture base working fluid. The developed nonlinear flow governing equations have been tackled numerically with the help of the bvp4c. The effects of relevant parameters on the flow dynamic have been portrayed in a graphical representation. The velocity profile decreases by raising the levels of buoyancy ratio and mixed convection in the range of $0.1<\mathrm{\lambda }\le 0.3$. It has been discovered that when bioconvection levels rise, motile microbe migration abruptly slows, which results in a decrease in fluid acceleration. The concentration of fluid flow declined for the Lewis number, but the opposite trend has been observed for the elastic parameter, thermophoresis parameter, and buoyancy ratio. With rising values of Brownian motion and thermophoretic diffusion, the surface drag and Nusselt number decrease significantly. Whereas, the opposite trend has been observed when the values of the thermal Biot number, Prandtl number and buoyancy ratio are enhanced. Additionally, data from this study have been validated by comparison with those that have previously been published, and an appropriate rate of agreement has been observed.

This study intends to evaluate the influence of temperature stratification on an unsteady fluid flow past an accelerated vertical plate in the existence of viscous dissipation. It is assumed that the medium under study is a grey, non-scattered fluid that both fascinates and transmits radiation. The leading equations are discretized using the finite difference method (FDM). Using MATLAB software, the impacts of flow factors on flow fields are revealed with particular examples in graphs and a table. In this regard, FDM results show that the velocity and temperature gradients increase with an increase of Eckert number. Furthermore, tables of the data indicate the influence of flow-contributing factors on the skin friction coefficients, and Nusselt numbers. When comparing constant and variable flow regimes, the constant flow regime has greater values for the nondimensional skin friction coefficient. This research is both innovative and fascinating since it has the potential to expand our understanding of fluid dynamics and to improve many different sectors.

In the current work, inclined magnetic field, thermal radiation, and the Cattaneo-Christov heat flux are taken into account as we analyze the impact of chemical reaction on magneto-hydrodynamic Casson nanofluid flow on a stretching sheet. Modified Buongiorno’s nanofluid model has been used to model the flow governing equations. The stretching surface is embedded in a porous medium. By using similarity transformations, the nonlinear partial differential equations are transformed into a set of dimensionless ordinary differential equations. The numerical solution of transformed dimensionless equations is achieved by applying the shooting procedure together with Rung-Kutta 4th-order method employing MATLAB. The impact of significant parameters on the velocity profile $f(\zeta )$, temperature distribution $\theta (\zeta )$, concentration profile $\phi (\zeta )$, skin friction coefficient $(C_f)$, Nusselt number $(N{u}_x)$ and Sherwood number $(S{h}_x)$ are analyzed and displayed in graphical and tabular formats. With an increase in Casson fluid $0.5<\beta <2$, the motion of the Casson fluid decelerates whereas the temperature profile increases. As the thermal relation factor expands $0.1<{\gamma }_1<0.4$, the temperature reduces, and consequently thermal boundary layer shrinks. Additionally, by raising the level of thermal radiation $1<Rd<7,$ the temperature profile significantly improves, and an abrupt expansion has also been observed in the associated thermal boundary with raise thermal radiation strength. It was observed that higher permeability $0<K<4$ hinders the acceleration of Casson fluid. Higher Brownian motion levels $0.2<Nb<0.6$ correspond to lower levels of the Casson fluid concentration profile. Moreover, it is observed that chemical reaction $0.2<{\gamma }_2<0.5$ has an inverse relation with the concentration level of Casson fluid. The current model’s significant uses include heat energy enhancement, petroleum recovery, energy devices, food manufacturing processes, and cooling device adjustment, among others. Furthermore, present outcomes have been found in great agreement with already published work.