Simulation of time-dependent radiative heat motion over a stretching/shrinking sheet of hybrid Nanofluid: Stability analysis for dual solutions

The heat transfer characteristics for the flow of a time-dependent hybrid nanofluid with thermal radiation and source/sink over a stretching/shrinking sheet are examined in the current investigation. We have transformed the governing equations of the presented study into the similarity equations utilizing similarity variables. However, a numerical solution is obtained by using in-build MATLAB code bvp5c. The mass and energy profiles for diverse values of thermophysical parameters are studied together with their physical quantities. It is observed that dual solutions exist, that is, one is upper, and the other is lower branch solution for a definite choice of the unsteadiness parameter. Also, stability analysis is executed to determine the long-term stability of dual solutions, indicating that out of the two, only one is stable and the other is unstable. It is revealed that comparatively, the first solution shows stability, while the second solution shows instability. There is a considerable influence of second-order slip on the problem’s respective flow and heat transfer characteristics. Further, major outcomes also show the dimensionless frictional stress and the magnitude of conventional heat transfer enhancement with growing suction parameter values.

Stability analysis of heat transfer in nanomaterial flow of boundary layer towards a shrinking surface: Hybrid nanofluid versus nanofluid

Many boundary value problems (BVPs) have dual solutions in some cases containing one stable solution (upper branch) while other unstable (lower branch). In this paper, MHD flow and heat transfer past a shrinking sheet is studied for three distinct fluids: kerosene hybrid nanofluid, kerosene nanofluid, and kerosene nanofluid. The partial differential equations (PDEs) are turned into ordinary differential equations (ODEs) using an appropriate transformation and then dual solutions are obtained analytically by employing the Least Square method (LSM). Moreover, stability analysis is implemented on the time-dependent case by calculating the least eigenvalues using Matlab routine bvp4c. It is noticed that negative eigenvalue is related to unstable solution i.e., it provides initial progress of disturbance and positive eigenvalue is related to stable solution i.e., the disturbance in solution decline initially. The impacts of various parameters, skin friction coefficient, and local Nusselt number for dual solutions are presented graphically. It is also noted that the results obtained for hybrid nanofluids are better than ordinary nanofluids.

Hybrid Nanofluid Flow and Heat Transfer Past an Inclined Surface

This paper examines the hybrid nanoparticles and the magnetic field impacts on the mixed convection boundary layer flow and heat transfer caused by an inclined shrinking–stretching surface in a hybrid nanofluid. Silver (Ag) is added into a MgO–water nanofluid to form Ag-MgO–water hybrid nanofluid. By making use of proper similarity transformations, the governing equations are transformed to ordinary differential equations. The problem is numerically solved with the help of the MATLAB function bvp4c. The influences of the chosen parameters on the temperature, velocity, heat transfer rate and the skin friction coefficient are addressed and graphically illustrated. The results show that increasing the magnetic parameter substantially improves the heat transfer rate and increases the skin friction coefficient. The findings also suggest that increasing the nanoparticle volume fraction φ2 (Ag) improves the skin friction coefficient while decreasing the heat transfer rate. For both stretching and shrinking instances, non-unique (dual) solutions are discovered. Only the first solution is stable, according to the temporal stability analysis of the dual solutions.

Influence of a Darcy-Forchheimer porous medium on the flow of a radiative magnetized rotating hybrid nanofluid over a shrinking surface

In this paper, the heat transfer properties in the three-dimensional (3D) magnetized with the Darcy-Forchheimer flow over a shrinking surface of the $$Cu + Al_{2} O_{3} /$$ C u + A l 2 O 3 / water hybrid nanofluid with radiation effect were studied. Valid linear similarity variables convert the partial differential equations (PDEs) into the ordinary differential equations (ODEs). With the help of the shootlib function in the Maple software, the generalized model in the form of ODEs is numerically solved by the shooting method. Shooting method can produce non-unique solutions when correct initial assumptions are suggested. The findings are found to have two solutions, thereby contributing to the introduction of a stability analysis that validates the attainability of first solution. Stability analysis is performed by employing if bvp4c method in MATLAB software. The results show limitless values of dual solutions at many calculated parameters allowing the turning points and essential values to not exist. Results reveal that the presence of dual solutions relies on the values of the porosity, coefficient of inertia, magnetic, and suction parameters for the specific values of the other applied parameters. Moreover, it has been noted that dual solutions exist in the ranges of $$F_{s} \le F_{sc}$$ F s ≤ F sc , $$M \ge M_{C}$$ M ≥ M C ,$$S \ge S_{C} ,$$ S ≥ S C , and $$K_{C} \le K$$ K C ≤ K whereas no solution exists in the ranges of $$F_{s} > F_{sc}$$ F s > F sc , $$M < M_{c}$$ M < M c , $$S < S_{c}$$ S < S c , and $$K_{C} > K$$ K C > K . Further, a reduction in the rate of heat transfer is noticed with a rise in the parameter of the copper solid volume fraction.

Effect of convective boundary condition on unsteady flow of CNT-H2O nanofluid towards a stagnation-point on a shrinking/expanding flat sheet

A model study of unsteady stagnation-point flow of most important nanoparticles, that is, carbon nanotubes suspended nanofluid towards shrinking/expanding sheet with convective boundary condition is demonstrated. Two types of carbon nanotubes, namely, single-wall and multi-wall nanotubes are carefully considered. Numerical solutions of converted equations from governing equation of the problem are obtained and those are graphically presented. Similar to without carbon nanotubes case, dual and unique solutions in specific ranges of velocity ratio parameter are achieved. Analysis disclosures that the condition on range where dual solutions exist is unaltered with solid-volume fraction and type of carbon nanotubes. The surface drag-force and heat transfer rate from wall are larger for single-walled carbon nanotubes nanofluid than multi-walled carbon nanotubes nanofluid. An increment in the parameter related to convective boundary condition generates high rate of heat transfer. After stability analysis, it is identified that in case of dual solutions, upper branch is stable and lower branch is unstable, while unique solution is always stable.

Mixed Convection Stagnation Point Flow of a Hybrid Nanofluid Past a Permeable Flat Plate with Radiation Effect

This article focuses on the stagnation point flow of hybrid nanofluid towards a flat plate. The cases when the buoyancy forces and the flow are in the opposite direction and the same direction are discussed. The effect of radiation and suction is also taken into account. The similarity transformations are used to convert the partial differential equations into nonlinear ordinary differential equations. These equations are computed numerically via the bvp4c function in MATLAB software. A comparison with the previously published articles is carried out, where an outstanding agreement is observed. The dual solutions exist in the case of opposing flow (λ<0) and the suction parameter S>0.6688. Meanwhile, only unique solutions exist in the case of assisting flow (λ>0). The existence of dual solutions leads to stability analysis. From the analysis, the first solution is confirmed as a stable solution. Furthermore, the heat transmission rate increases, while the skin friction coefficient decreases as the radiation rate increases. An increase in the radiation rate from 0 (no radiation) to 1.0 increases the heat transmission rate by 5.01% for water, 4.96% for nanofluid, and 4.80% for hybrid nanofluid. Finally, it is worth mentioning that the present study yields new and original results. This study has also not been done by other researchers, indicating its novelty.