Purpose This study aims to investigate the flow and heat transfer of a hybrid nanofluid through an exponentially stretching/shrinking sheet along with mixed convection and Joule heating. The nanoparticles alumina (Al2O3) and copper (Cu) are suspended into a base fluid (water) to form a new kind of hybrid nanofluid (Al2O3-Cu/water). Also, the effects of constant mixed convection parameter and Joule heating are considered. Design/methodology/approach The governing partial differential equations are transformed into ordinary differential equations (ODEs) using appropriate similarity transformations. The transformed nonlinear ODEs are solves using the bvp4c solver available in MATLAB software. A comparison of the present results shows a good agreement with the published results. Findings Dual solutions for hybrid nanofluid flow obtained for a specific range of the stretching/shrinking parameter values. The values of the skin friction coefficient increases but the local Nusselt number decreases for the first solution with the increasing of the magnetic parameter. Enhancing copper volume fraction and Eckert number reduces the surface temperature, which intimates the decrement of heat transfer rate for the first and second solutions for the stretching/shrinking sheet. In detail, the first solution results show that when the Eckert number increases as 0.1, 0.4 and 0.7 at λ = 1.5, the temperature variations reduced to 10.686840, 10.671419 and 10.655996. While in the second solution, keeping the same parameters temperature variation reduced to 9.750777, 9.557349 and 9.364489, respectively. On the other hand, the results indicate that the skin friction coefficient increases with copper volume fraction. This study shows that the thermal boundary layer thickness rises due to the rise in the solid volume fraction. It is also observed that the magnetic parameter, copper volume fraction and Eckert number widen the range of the stretching/shrinking parameter for which the solution exists. Practical implications In practice, the investigation on the flow and heat transfer of a hybrid nanofluid past an exponentially stretching/shrinking sheet with mixed convection and Joule heating is crucial and useful. The problems related to hybrid nanofluid have numerous real-life and industrial applications, such as microelectronics, manufacturing, naval structures, nuclear system cooling, biomedical and drug reduction. Originality/value In specific, this study focuses on increasing thermal conductivity using a hybrid nanofluid mathematical model. The novelty of this study is the use of natural mixed convection and Joule heating in a hybrid nanofluid. This paper can obtain dual solutions. The authors declare that this study is new, and there is no previous published work similar to the present study.
In this paper, we considered the stagnation point flow and heat transfer of nanofluid over the stretching/shrinking surface by utilizing of Tiwari and Das nanofluid model. Additionally, the impact of suction and the first order slip likewise have been taken into the account. The system of governing partial differential equations (PDEs) is changed into the system of non-linear ordinary differential equations (ODEs) by means of similarity transformation. The resultant ODEs are solved by using BVP solver (bvp4c) in MATLAB software. The impact of some physical parameters, for example the suction parameter and the slip parameter on the skin friction coefficients and the local Nusselt number as well as the temperature and velocity profiles have been investigated, tabulated and graphically presented. These profiles and variations demonstrate that there exist dual solutions for a specific range of the stretching/shrinking parameter. Both suction and slip effects has enhance the local Nusselt number which represent heat transfer rate at the surface. It is also found that inclusion of both suction and slip effects expands the range of the dual solutions exist. The existence of the dual solutions only occurs in in the shrinking region. The flow separation in the boundary layer delay due to suction and slip effects imposed in the boundary condition.
Following to improved thermal impact of hybrid nanomaterials, wide range applications of such materials is observed in the thermal engineering, extrusion systems, solar energy, power generation, heat transfer devices etc. The hybrid nanofluid is a modified form of nanofluid which is beneficial for improving energy transfer efficiency. In current analysis, the solid nanoparticles aluminium ($$\phi_{{{\text{Al}}_{2} {\text{O}}_{3} }}$$ ϕ Al 2 O 3 ) and copper ($$\phi_{{{\text{Cu}}}}$$ ϕ Cu ) have been mixed with water to produce a new hybrid nanofluid. The investigation of a steady two-dimensional mixed convection boundary layer flow of the resultant hybrid nanofluid on a vertical exponential shrunk surface in the existence of porous, magnetic, thermal radiation, velocity, and thermal slip conditions is carried out. Exponential similarity variables are adopted to transform the nonlinear partial differential equation into a system of ordinary differential equations which has been then solved by employing the shooting method in Maple software. The obtained numerical results such as coefficient of skin friction $$f^{\prime \prime } \left( 0 \right)$$ f ″ 0 , heat transfer rate $$- \theta^{\prime } \left( 0 \right)$$ - θ ′ 0 , velocity $$f^{\prime } \left( \eta \right)$$ f ′ η and temperature $$\left( {\theta \left( \eta \right)} \right)$$ θ η distributions are presented in the form of different graphs. The results revealed that duality exists in solution when the suction parameter $$S \ge S_{ci}$$ S ≥ S ci in assisting flow case. Due to non-uniqueness of solutions, a temporal stability analysis needs to be performed and the result indicates that the upper branch is stable and realizable compared to the lower branch.
One of the most pressing issues in contemporary applied mathematics is the regulation of energy transfer via the application of external forces. The processes of heat transfer are affected by magnetic force, which has many practical uses in industry, engineering, and medicine. This research explores the magnetohydrodynamics (MHD) three-dimensional stable axisymmetric boundary layer over a permeable moving plate, which consist of water as a base liquid and binary distinct nanoparticles to generate a hybrid nanofluid. In all of these, flow beyond the boundary layer area might be calculated by a small crosswise velocity. As a result of its high thermal conductivity, a pair of distinct kinds of nanoparticles have been considered, namely alumina and copper, which are integrated into the base water. The mathematical model is built within a boundary of specified geometry and then converted into a set of ordinary differential equations (ODEs). Resultant ODEs are solved numerically using the technique of three-stage Lobatto IIIa in bvp4c solver in 2017, MATLAB software. Results revealed that two branches exist in certain ranges of moving parameter. Impacts of an increasing physical parameter on profiles of velocities and temperature with skin friction as well as with heat transfer rate are represented in graphs. The effect of viscous dissipation on the temperature profile in the -direction has the same rising results as observed in the -direction. According to the results of the temporal stability analysis, the upper branch is realizable and stable.
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