Entropy production minimization and heat transfer enhancement in a cavity filled with micropolar hybrid nanofluid under an influence of discrete heaters and uniform magnetic field

Abstract

This study numerically simulates optimal thermal convection and entropy production minimization in a steady magnetohydrodynamic micropolar hybrid nanofluid (Ag–Al₂O₃/H₂O) flow through an annulus region bounded between an elliptical cylinder (inner) and a square cylinder (outer) with partially insulated and partially heated walls. Four different configurations with heater locations were considered when the heater was placed continuously or discretely at different locations along the annulus’ boundaries. The governing equations were derived from the Navier–Stokes equations along with the angular momentum equation of microrotation. The flow domain, bounded between two dissimilar geometrical surfaces, was transformed into a computational domain. The modeled set of equations was solved using an in-house developed MATLAB program code based on the finite difference method with the succesive over relaxation, succesive under relaxation, and Gauss–Seidel iteration techniques. The impacts of the Hartmann number, Rayleigh number, volume fraction of nanoparticles, and vortex viscosity parameter on the streamlines, isotherms, average Nusselt number, average Bejan number, and entropy generation number were computed to analyze the flow dynamics and convective heat transfer mechanism. The average Nusselt number increased linearly with ϕ_hnf, higher for the inner elliptic cylinder. As K₀ rises, heat convection declines. In contrast to Cases II and I, Cases III and IV exhibit a greater attenuation of heat convection at the inner elliptical cylinder. The Nu_o declines rapidly for Cases I and III compared to Cases II and IV. Increasing the concentration of nanoparticles improved heat convection. Both Nu_i and Nu_o increase linearly with ϕ_hnf, but the rise is more pronounced from the inner elliptical cylinder than the outer square cylinder. It is found that Cases II and III have the highest and lowest Be_avg values, while Cases III and I have the highest and lowest N_s values. Be_avg diminishes and the total entropy generation N_s increases with Ha, Ra, and ϕ_hnf, while reverse effect of vortex viscosity K_o.

Appendix-A (for reviewer’s response, not to include in publication)

The governing Eqs. (1)–(4) with corresponding boundary conditions in physical domain are solved numerically by using Gauss–Seidel method followed by successive over-relaxation and successive under-relaxation techniques. The current investigation takes place within a fluid flow domain that is confined by two dissimilar geometric surfaces. As a result, it was not possible to generate a rectangular grid directly over the annulus. Hence, it is suitable to use a body-fitted coordinate system (BFCS) for transforming the whole set of governing equations into curvilinear coordinates (ξ, η), which will be the computational domain for the problem. Therefore, for this particular geometry, the following defined transforming function is used

$$ \begin{aligned} X\left( {\xi ,\eta } \right) & = \sin \left( {2\pi \xi } \right)\left[ {r_{{\text{i}}} + \left( {r_{0} - r_{{\text{i}}} } \right)\eta } \right] \\ Y\left( {\xi ,\eta } \right) & = - \cos \left( {2\pi \xi } \right)\left[ {r_{{\text{i}}} + \left( {r_{0} - r_{{\text{i}}} } \right)\eta } \right] \\ \end{aligned} $$

Here $r_{{\text{i}}} = b_{{\text{i}}} \left[ {\left( {\frac{{b_{{\text{i}}} }}{{a_{{\text{i}}} }}} \right)^{2} \sin^{2} \left( {2\pi \xi } \right) + \cos^{2} \left( {2\pi \xi } \right)} \right]^{ - 0.5}$ and $r_{0} = \left[ {\cos^{200} \left( {2\pi \xi } \right) + \sin^{200} \left( {2\pi \xi } \right)} \right]^{ - 0.005}$. Where a_i and b_i are the lengths of semi-major and semi-minor axes of inner elliptic cross section, respectively. Applying these transformations as mentioned in Roy [35], the first-order derivatives become like as follows:

$$\frac{\partial }{\partial X} = \frac{1}{J}\left( {\frac{\partial }{\partial \xi }\frac{\partial y}{{\partial \eta }} - \frac{\partial }{\partial \eta }\frac{\partial y}{{\partial \xi }}} \right), \, \frac{\partial }{\partial Y} = \frac{1}{J}\left( {\frac{\partial }{\partial \eta }\frac{\partial x}{{\partial \xi }} - \frac{\partial }{\partial \xi }\frac{\partial x}{{\partial \eta }}} \right)$$

, where $J = \frac{\partial y}{{\partial \eta }}\frac{\partial x}{{\partial \xi }} - \frac{\partial y}{{\partial \xi }}\frac{\partial x}{{\partial \eta }}$.

By using the transformation given in Roy [35], the physical region of the problem was changed into a computational domain bounded in the region, 0 ≤ ξ ≤ 1 and 0 ≤ η ≤ 1 and now the orthogonal grids are possible. Equations (1)–(4) in computational domain can be written as given in Eqs. (23)–(26) along with the boundary conditions given in Eqs. (27)–(31). The diffusion and convection terms in Eqs. (23)–(31) were discretized employing the second-order central and upwind difference schemes, respectively, and then in-house MATLAB codes are designed for the computation of ψ, ω, N and θ from the system of discretized equations. Convergence criteria for ψ, ω, N and θ are given by $\frac{{\sum\nolimits_{{{\text{ij}}}} {\left| {\Phi_{{{\text{ij}}}}^\text{k + 1} - \Phi_{{{\text{ij}}}}^\text{k} } \right|} }}{{\sum\nolimits_{{{\text{ij}}}} {\left| {\Phi_{{{\text{ij}}}}^\text{k + 1} } \right|} }} \le 10^{ - 6}$, where Φ demonstrates any of the computed magnitudes of ψ, ω, N and θ.