Mathematical modeling of entropy generation in MHD mixed convective Cu–Ag-Al₂O₃/H₂O tri-hybrid nanofluid over an exponential permeable shrinking surface with radiation and slip impacts: Multiple solutions with stability analysis

Abstract

The present investigation aims to determine the existence of a dual solution in magnetohydrodynamic (MHD) mixed convective radiative flow of Cu-Ag-$A{l}_2{O}_3$/$H_{2}O$ tri-hybrid nanofluid in a permeable Darcy–Forchheimer porous medium over an exponentially shrinking surface by considering velocity, thermal slips, and suction effects at the surface. This study focuses on assessing entropy production in novel coolant applications. To streamline the analysis, the complex nonlinear partial differential equations (PDEs) are simplified by converting them into a set of ordinary differential equations (ODEs) through the utilization of the similarity transformation technique. These ODEs are then solved utilizing the bvp4c numerical MATLAB function. Graphical and tabular analyses are conducted to investigate the impact of emerging variables on entropy generation, as well as velocity, temperature, skin friction coefficient, and Nusselt number. Due to the contraction of the surface, dual solutions are found, but dual solutions cannot be found beyond the critical values. The critical values of $S_C$ are 1.58928, 1.48700, and 1.66058, but ${\xi }_C$ are $-1.239499,$ $-1.416999,$ and $-1.130030$ for $\mathit{\text{Cu}}/H_{2}O$ nanofluid, $\mathit{\text{Cu}}-\mathit{\text{Ag}}/H_{2}O$ hybrid nanofluid, and $\mathit{\text{Cu}}-\mathit{\text{Ag}}-A{l}_2{O}_3/H_{2}O$ tri-hybrid nanofluid, respectively. A positive minimum eigenvalue ${\gamma }_1$ signifies the existence of the upper stable solution branch, while a negative minimal eigenvalue indicates the presence of the lower unstable solution branch. The tri-hybrid nanofluid exhibits superior thermal properties compared to nanofluid and hybrid fluids. Adding nanoparticles to traditional fluids is perceived as improving their ability to transmit heat. The analysis has demonstrated that the thermal radiation and temperature slip parameters significantly influence the heat transfer rate. Thermal radiation is found to speed up the creation of entropy. Additionally, in the case of a stable solution, an increase in the Forchheimer number results in a deceleration of the liquid flow, an effect which is further amplified by the presence of the magnetic field and velocity slip parameter.

Keywords:

• Darcy–Forchheimer
• dual solutions
• entropy generation
• Grashof number
• magnetic field
• ternary radiative heat flux hybrid nanofluid
• thermal radiation
• stability analysis

Disclosure statement

The authors declare no conflict of interest.

Data availability statement

The data used to support the findings are included in the article.

Additional information

Funding

This study has no funding source.