MHD mixed convection of hybrid nanofluid in a wavy porous cavity employing local thermal non-equilibrium condition

The current study treats the magnetic field impacts on the mixed convection flow within an undulating cavity filled by hybrid nanofluids and porous media. The local thermal non-equilibrium condition below the implications of heat generation and thermal radiation is conducted. The corrugated vertical walls of an involved cavity have \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${T}_{c}$$\end{document}Tc and the plane walls are adiabatic. The heated part is put in the bottom wall and the left-top walls have lid velocities. The controlling dimensionless equations are numerically solved by the finite volume method through the SIMPLE technique. The varied parameters are scaled as a partial heat length (B: 0.2 to 0.8), heat generation/absorption coefficient (Q: − 2 to 2), thermal radiation parameter (Rd: 0–5), Hartmann number (Ha: 0–50), the porosity parameter (ε: 0.4–0.9), inter-phase heat transfer coefficient (H*: 0–5000), the volume fraction of a hybrid nanofluid (ϕ: 0–0.1), modified conductivity ratio (kr: 0.01–100), Darcy parameter \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\left(Da: 1{0}^{-1}\,\mathrm{ to }\,1{0}^{-5}\right)$$\end{document}Da:10-1to10-5, and the position of a heat source (D: 0.3–0.7). The major findings reveal that the length and position of the heater are effective in improving the nanofluid movements and heat transfer within a wavy cavity. The isotherms of a solid part are significantly altered by the variations on \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Q$$\end{document}Q, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${R}_{d}$$\end{document}Rd, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${H}^{*}$$\end{document}H∗ and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${k}_{r}$$\end{document}kr. Increasing the heat generation/absorption coefficient and thermal radiation parameter is improving the isotherms of a solid phase. Expanding in the porous parameter \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon$$\end{document}ε enhances the heat transfer of the fluid/solid phases.


Mathematical formulation
shows the preliminary geometry of an inclined undulating cavity. The involved wavy cavity is filled by porous media and hybrid nanofluids. A partial heat source is laid in the bottom wall with a variable-length B and the other horizontal walls are adiabatic. The vertical sidewalls are cold ( T c ), the left wavy and top walls have lid velocities. The magnetic field has an inclination angle . The undulating cavity is inclined by an inclination angle α . The hybrid nanofluid convection is not in a local thermodynamic equilibrium condition. The normal direction and constant value are considered for the gravity acceleration. Dirichlet type applied on all boundaries (no-slip condition). Considering the earlier specified hypotheses, the continuity, momentum, and energy equations concerning the hybrid nanofluid, incompressible, laminar, single-phase, and steady-state flow are formulated as follows 40,41 : where u and v are the velocity components, T is a temperature, ρ hnf is the density, ν hnf is kinematic viscosity. g is a gravity, p is a pressure, µ hnf is a dynamic viscosity. Q 0 is the heat generation (Q 0 > 0) or absorption (Q 0 < 0) coefficient. Introducing the dimensionless set as: (1)  And average Nusselt number is:

Hybrid nanofluid
The effective thermal diffusion and conductivity are: Thermal diffusivity is: Effective density is: The heat capacitance is: The thermal expansion is: The thermal conductivity is: where The effective dynamic viscosity is: where

Numerical method
In this study, the finite volume method (FVM) based on the SIMPLE algorithm 44 is applied to solve the governing equations. The system of the governing Eqs. (7)- (11) corresponding to the boundary conditions (12)- (15) is written in the following form (Table 1): where φ refers to U, V , θ f and θ s and refers to the control volume. The first upwind scheme is used for the advection term and the central differences approach is applied for the diffusive fluxes; then the following algebraic system is obtained: where A is the area of the cell and f refers to the faces. Here the convergence criteria are 10 −6 .  Table 2 and it is found that the grid size of 141 × 141 is appropriate for all the computations. Moreover, no special treatment on the grids of the curved walls is considered. Almost uniform grids are adopted for all the geometry. Also, Table 3 shows the comparison of the average Nusselt Number for the different values Hartmann number Ha . From this comparison, it is seen that the present results from the finite volume method agree well with the results from Biswas and Manna 45 .
An in-house code of the FVM with SIMPLE scheme is written in FORTRAN-90. The calculations are performed by SHAHEEN-II Cluster managed by King Abdullah University of Science and Technology (KAUST), Jeddah, Saudi Arabia.

Results and discussion
The research treats the numerical flow of hybrid nanofluids motivated by mixed convection in a wavy inclined cavity with LTNE condition and saturated by porous media. The scales of the varied parameters are partial heat length B = 0.2 − 0.8 , heat generation/absorption coefficient Q = −2 − 2 , thermal radiation parameter R d = 0 − 5 , Hartmann number Ha = 0 − 50 , the porosity parameter ε = 0.4 − 0.9 , inter-phase heat transfer coefficient H * = 0 − 5000 , the volume fraction of a hybrid nanofluid φ = 0 − 0.1 , modified conductivity ratio k r = 0.01 − 100 , Darcy parameter Da = 10 −1 − 10 −5 , and the position of a heat source D = 0.3 − 0.7 . The fixed parameters are the Grashof parameter Gr = 10 3 , an inclination angle of a cavity α = π/4, amplitude parameter A = 0.1 , angle of a magnetic field � = π/3 , Richardson parameter Ri = 1 , lid-velocity t = l = 1 , and a phase deviation = 2 . The physical attributes of the water, copper, and titanium dioxide are tabulated in Table 1. The ranges of the pertinent parameters are relevant to the references 37,46,47 . Figure 2 indicates the contours of the streamlines, isotherms of the fluid/solid phases (two phases) below changes on a partial heat length B for a hybrid nanofluid at φ Cu = φ TiO2 = φ/2, φ = 0.05, Q = 1, R d = 0.5, ε = 0.5, Da = 10 −3 , Ha = 10, k fs = 1, D = 0.5, k r = 1, H * = 10 . In Fig. 2a, there is a little change in the intensity of the streamlines below the changes on a partial heat length B. In Fig. 2b,c, the isotherms in the two phases are expanded across a wavy cavity as the partial heat length B expanded. Figure 3 shows the sketches of local Nusselt number along with the heater of a fluid phase Nu fs and of a solid phase Nu ss , below the changes on a partial heat length B for a hybrid nanofluid at . It is noted that the values of Nu fs and Nu ss are strongly depending on the distance of a heater. An expansion in a heater length B raises the values of Nu fs and Nu ss . Physically, an increase in the partial heat length B powers the buoyancy force, and consequently the temperature distributions are expanded across a cavity. Figure 4 introduces the contours of the streamlines, isotherms of the two phases below changes on heat generation/absorption coefficient Q for a hybrid nanofluid at φ Cu = φ TiO2 = φ/2, φ = 0.05, B = 0.5, R d = 0.5, ε = 0.5, Da = 10 −3 , Ha = 10, k fs = 1, D = 0.5, k r = 1, H * = 10 . In Fig. 4a, since the isotherms are formed from the lid velocities in the top and left cavity walls, an increment in the heat generation coefficient Q has little impact on the streamline contours. In Fig. 4b,c, an increase in Q raises the isothermal lines of the two phases within a wavy cavity. The impacts of Q on the Nu fs , and Nu ss , along with a heat source as well as the Nu mf and Nu ms are shown in Figs. 5 and 6. The first remark is that an increment in Q declines Nu fs and Nu ss . Moreover, the values of Nu mf and Nu ms are decreasing as Q increases. Growing the concentration of the nanoparticles powers the values of Nu mf and it has slight influences on Nu ms . The physical reason is returning to the power of a heat generation coefficient Q that enhances the heat transfer in a cavity. Figure 7 gives the contours of streamlines, isotherms of the two phases below the changes on thermal radiation parameter R d for a hybrid nanofluid at φ Cu = φ TiO2 = φ/2, φ = 0.05, B = 0.5, Q = 1, ε = 0.5, Da = 10 −3 , Ha = 10, k fs = 1, D = 0.5, k r = 1, H * = 10. It is remarked that the thermal radiation parameter R d has a minor significance on the contours of the streamlines and isothermal lines of a fluid phase within a wavy cavity. Besides, an increment in the thermal radiation parameter R d improves the isotherms of a solid phase within a wavy cavity. In general, an increment in the thermal radiation parameter enhances the isotherms.    . The values of the Nu ss and Nu ms are enhancing as R d increases. Figure 10 introduces the contours of the streamlines, isotherms of the two phases below the changes on the porosity parameter ε for a hybrid nanofluid at φ Cu = φ TiO2 = φ/2, ϕ = 0.05, B = 0.5, Q = 1, R d = 0.5, Da = 10 −3 , Ha = 10, k fs = 1, D = 0.5, k r = 1, H * = 10. An increment in a porous parameter ε from 0.4 to 0.9 declines the absolute of streamlines' maximum by 33.33%. Moreover, the contours of the isotherms of the two phases are enhanced as a porosity parameter raises. Figure 11 introduces the contours of the streamlines, isotherms of the two phases below the changes on the Hartmann number Ha for hybrid nanofluid at φ Cu = φ TiO2 = φ/2, φ = 0.05, B = 0.5, Q = 1, R d = 0.5, Da = 10 −3 , ε = 0.5, k fs = 1, D = 0.5, k r = 1, H * = 10 . Physically, an extra Ha generates more Lorentz forces that suppress the flow speed. In Fig. 11a, the absolute of the streamlines' maximum is decreasing by 26.32% as Ha increases from 0 to 50. In Fig. 11b,c, the isotherms of the two phases are enhanced as the Hartmann number powers. The influences of the Hartmann number on the Nu fs along with the heat source, Nu mf and Nu ms along φ have been shown in Figs. 12 and 13. An extension in the Hartmann number reduces Nu fs , Nu mf and Nu ms . Further, at any value of the Hartmann number, an increase on φ enhances Nu mf . Figure 14 shows the contours of the streamlines, isotherms of the two phases below the changes on the interphase heat transfer coefficient H * for a hybrid nanofluid at k r = 1, φ Cu = φ TiO2 = φ/2, φ = 0.05, B = 0.5, Q = 1,

Conclusion
This study is introducing the first attempt in solving the mixed convection of hybrid nanofluids within an undulating porous cavity under the LTNE condition. The contours of the streamlines, isotherms of fluid/solid phases as well as the profiles of local and average Nusselt number on the fluid/solid phases under the variations of the key parameters like partial heat length (B)-position (D), modified conductivity ratio k r , coefficient of heat generation/absorption Q , thermal radiation parameter R d , Hartmann number Ha , porosity parameter ε , an inter-phase heat transfer coefficient H * , Darcy parameter Da , and hybrid nanofluid parameter φ have been obtained. The remarkable points could be concluded as: