Dual Solutions and Stability Analysis of Magnetized Hybrid Nanofluid with Joule Heating and Multiple Slip Conditions

Abstract

This paper investigates the steady, two dimensional, and magnetohydrodynamic flow of copper and alumina/water hybrid nanofluid on a permeable exponentially shrinking surface in the presence of Joule heating, velocity slip, and thermal slip parameters. Adopting the model of Tiwari and Das, the mathematical formulation of governing partial differential equations was constructed, which was then transformed into the equivalent system of non-linear ordinary differential equations by employing exponential similarity transformation variables. The resultant system was solved numerically using the BVP4C solver in the MATLAB software. For validation purposes, the obtained numerical results were compared graphically with those in previous studies, and found to be in good agreement, as the critical points are the same up to three decimal points. Based on the numerical results, it was revealed that dual solutions exist within specific ranges of the suction and magnetic parameters. Stability analysis was performed on both solutions in order to determine which solution(s) is/are stable. The analysis indicated that only the first solution is stable. Furthermore, it was also found that the temperature increases in both solutions when the magnetic parameter and Eckert number are increased, while it reduces as the thermal slip parameter rises. Furthermore, the coefficient of skin friction and the heat transfer rate increase for the first solution when the magnetic and the suction parameters are increased. Meanwhile, no change is noticed in the boundary layer separation for the various values of the Eckert number in the heat transfer rate.

Rotating 3D Flow of Hybrid Nanofluid on Exponentially Shrinking Sheet: Symmetrical Solution and Duality

“…The same state of heat transfer rate was obtained by Lund et al [19,20] during the examination of a hybrid nanofluid. Yan et al [21] also found similar results when they analyzed a hybrid nanofluid over the exponential surface with joule heating effects. Further, it is stated that "the skin friction coefficient, f (0), enhances for the first solution when the suction S and magnetic M parameters are increased, while f (0) reduces for the higher effect of the velocity slip factor, δ".…”

Hybrid Nanofluid Slip Flow over an Exponentially Stretching/Shrinking Permeable Sheet with Heat Generation

“…Referencing the work reported by Merkin [55], Weidman et al [56], Waini et al [16] and Yan et al [38], the dimensionless time-dependent variable transformation for Equations (11) and ( 12) is…”

“…where γ is an unidentified eigenvalue, while F(η) and G(η) are small relative to f 0 (η) and θ 0 (η), correspondingly (Weidman et al [56], Waini et al [16], Yan et al [38]). Substituting Equation ( 17) into Equations ( 14)-( 16) and setting the value of τ to zero, the initial decay or growth of disturbance in Equation ( 17) can be identified and the subsequent linearized equations are attained…”

“…A comparative investigation was made by Hayat et al [37] on the hybrid and mono nanofluid of multi-walled carbon nanotubes with the slip flow. Yan et al [38] elucidated the multiple slip conditions on the magnetized hybrid nanofluid with the consideration of a few other parameters. It was found out that thermal slip diminishes the heat transfer rate.…”

MHD stagnation point flow of a Maxwell nanofluid over a shrinking sheet (multiple solution)

“…Ismail et al 20 considered a moving flat plate in a parallel stream to analyze the heat transfer with constant heat flux. The characteristics of an axisymmetric mixed convection flow of a copper nanofluid and slip effects were determined by Soomro et al 21 Some latest studies about dual solutions are found in References 22‐25 …”