Natural convection of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathrm {Al}_{2}\mathrm {O}_{3}$$\end{document}Al2O3-water nanofluid in a non-Darcian wavy porous cavity under the local thermal non-equilibrium condition

This study investigates thermal natural convective heat transfer in a nanofluid filled-non-Darcian porous and wavy-walled domain under the local thermal non-equilibrium condition. The considered cavity has corrugated and cold vertical walls and insulated horizontal walls except the heated part positioned at the bottom wall. The transport equations in their non-dimensional model are numerically solved based on the Galerkin finite-element discretization technique. The dimensionless governing parameters of the present work are the nanoparticle in volume concentration, the Darcy number, number of undulations, modified heat conductivity ratio, dimensionless heated part length, and location. Comparisons with other published theoretical and experimental results show good agreement with the present outcomes. The findings indicate that the heater length, its position, and the waves number on the side vertical walls as well as the nanoparticles concentration can be the control parameters for free convective motion and heat transport within the wavy cavity.

www.nature.com/scientificreports/ sinusoidally-walled cavity were perused by Sheikholeslami et al. 5 . Hu et al. 6 concluded that the thermal free convection of TiO 2 -H 2 O based nanoliquid was enhanced with the addition of the nano-sized solid particles. Alsabery et al. 7 studied transient natural-convective flow and heat exchange characteristics under the effects of non-uniform thermal boundary conditions of a nanofluid-filled trapezoidal chamber. They reported that the best heat exchange enhancement was obtained by the larger side wall inclination angle and a higher concentration in volume of the used solid nanomaterials. For good reviews on the fundamental theory and applications of nanofluid flows and heat transfer the readers can refer the works of Mahian et al. 8,9 . Most often, the overall nanoliquid heat conductivity and viscosity of are evaluated utilizing the Maxwell-Garnett and Brinkman theoretical models, respectively. A new model inspired by experimental data has been proposed by Corcione 10 to evaluate these properties. Motlagh and Soltanipour 11 investigated arithmetically using the Corcione's nanofluid model 10 the thermal natural convection characteristics in a nanofluid-filled square domain using the two-phase approach. The results revealed that the nanofluid enhances the heat exchange rate and positively depends on the increment of solid particle volume fraction. Tayebi and Chamkha 12,13 described the thermal-free-convective motion, entropy production behaviour and heat exchange rate of a hybrid nanoliquidfilled square domain with different inserted conductive bodies employing Corcione's nanofluid model. Using the same nanofluid model 10 , Alsabery et al. 14 examined through a numerical code the conjugate heat transfer in a square cavity having a concentric solid insert. The Al 2 O 3 -water nanoliquid with different values of volume fraction is used.
Thermal convection within porous mediums has many practical applications, such as petroleum processing, groundwater flows, filtering, food processing, and fibrous insulation 15 . There are two different models for the heat equation related to the two phases of the porous matrix. The first one is the so-called local thermal equilibrium (LTE) model which assumes that both the solid and fluid temperatures are the same. The literature on this model is abundant. Basak et al. 16 have perused the effects of several types of thermal boundary conditions on thermal free convection in a porous domain. Zahmatkesh 17 also investigated the significance of thermal boundary conditions in heat transfer to cause natural convection and entropy generation in a square porous domain heated from the bottom. He found that the heat exchange boosted towards the center of the cavity. Simultaneous thermal free convection and entropy production behavior in porous rhombic cavities were investigated by Anandalakshmi and Basak 18 . Sheikholeslami 19 considered the problem of CuO-water nanoliquid natural convection in a porous cavity using the Darcy law model. In the research work of Rashad et al. 20 , the impacts of the sink/source position and enlargement on the entropy production and free convective motion under a magnetic excitation in a titled porous domain saturated with a Copper-water nanoliquid are interpreted. Since the LTE approach is not feasible under some conditions in which the solid and fluid temperatures are different, a second model so-called the local thermal non-equilibrium (LTNE) was introduced. The LTNE model is applicable in many applications such as food drying/freezing, microwave heating, and cooling of microelectronics devices 21 .
Theoretical and experimental studies adopting the LTNE approach in porous mediums are still limited. The problem of thermal-natural convection in a porous enclosure taking into consideration the LTNE approach was studied numerically by Baytaş and Pop 22 . Baytaş 23 used the LTNE model to investigate natural convection in a non-Darcian square porous cavity having a heat-generating solid phase. They concluded that when the heat transfer coefficient is small, then the temperature of the fluid phase was lower than that of the solid phase. Badruddin et al. 24 also conducted a study on the effect of thermal-natural convection in a square porous cavity using LTNE approach. The onset of convection in a nanofluid saturated porous layer employing the LTNE approach was studied by Kuznetsov and Nield 25 and Bhadauria and Agarwal 26 . Sheremet et al. 27 examined numerically the effect of the LTNE approach on thermal-natural convection in a nanoliquid saturated-porous enclosure. The study concluded that lower inter-phase heat transfer coefficient can decrease the heat exchange rate. Considering the LTNE condition and Buongiorno's model, Zargartalebi et al. 28 have investigated numerically the unsteady-state thermal free convection in a nanoliquid-saturated porous enclosure. It was found that the average heat exchange can be enhanced by reducing the buoyancy ratio and increasing the Rayleigh number. Baytaş and Baytaş 29 perused the effect of LTNE model on thermal free convection in a square domain in the presence of internal heat generation of a porous layer. Alsabery et al. 30 investigated the non-uniform heating and wall conduction effects on thermal free convection in a square porous cavity based on the LTNE approach. Tahmasebi et al. 31 explained the effect of LTNE model on the natural convective heat transfer in a cavity that partially filled with porous media with the use of Buongiorno's nanofluid model. They observed that with the increasing of the interface heat transfer parameter, the overall exchange rate of the nanoliquid phase was reduced. Sheremet and Pop 32 applied the finite difference method to investigate the effects of the size and position of a local heater on the natural convection in a square porous cavity based on the LTNE and Buongiorno's models. Izadi et al. 33 perused the effect of the LTNE condition on convective heat transfer in a square porous cavity filled with CuO-water micropolar nanofluids.
The complex geometry of cavities is an important aspect that can reflect its existence in many engineering applications such as heat exchangers, devices of food treatment, refineries, refrigeration systems, condensates and so on. Therefore, several researches have been conducted involving thermal convection problems in complex cavities. Kumar 34 computationally explained the thermal free convection and fluid motions within a wavy cavity that filled with a porous medium. The results indicated that convective exchanges were clearly affected by Rayleigh number, undulations 'amplitude, phase of wave, and undulations number of the vertical-cavity walls. Das and Mahmud 35 have examined the thermal natural convection within a double wavy-walled domain. Their results suggested that heat exchange was highly influenced by the amplitude of the wavy wall and the number of undulations. Misirlioglu et al. 36 analyzed the effect of internal heat generation on thermal free convection in a porous square cavity using the Darcy model. Thermal free convection in a porous-undulated enclosure based on the Darcy-Brinkman-Forchheimer extended model is computationally studied by Chen et al. 37 . It was found that local heat exchange dependent on the Darcy number and the porosity of the media. Fluid flow due to thermal free convection in a non-Darcian wavy-walled porous enclosure was considered by Khanafer et al. 38 .

Scientific Reports
| (2020) 10:18048 | https://doi.org/10.1038/s41598-020-75095-5 www.nature.com/scientificreports/ The results depicted that the convective flow circulation within the examined enclosure was boosted with rising Rayleigh number. Using the finite volume method, Abu-Nada and Öztop 39 investigated the thermal convection within a wavy-walled enclosure occupied with Al 2 O 3 -H 2 O nanoliquids. They noticed that the insertion of solid nanoparticles into the water (based fluid) makes the convective heat exchange in the domain enhances. Combined free and forced thermal convection in a nanofluid-filled lid-driven wavy-walled enclosure was carried out by Abu-Nada and Chamkha 40 . They explained that the convective exchange was positively increases with increasing nanoparticles concentration (in volume) for all Richardson numbers. Sheremetet al. 41 have worked on an open nanolliquid-filled wavy porous enclosure to investigate the unsteady-state free convection using Buongiorno's model. Using the same model of nanofluid, Sheremet and Pop 42 investigated the influence of nonuniform temperature conditions on thermal natural convection in a wavy-walled porous enclosure. Additionally, Alsabery et al. 43 examined numerically the entropy production and convective heat exchange in a wavy-walled porous enclosure having a rotating rigid cylinder and heated from below. They concluded that an increment in the porosity of the medium caused an enhancement in the convection heat transfer. Alsabery et al. 44 studied the impact of the two-phase nanofluid technique and the role of the heat source or sink on thermal natural convection in a square cavity with a rigid cylinder. According to the studies discussed above and to our best knowledge, it can be noticed that there is no research about the problem of thermal free convection in a wavy-porous cavity that filled with regular fluids or nanofluids considering the local thermal non-equilibrium condition. Thus, this is an original piece of work and potentially a valuable source. Consequently, the aim of this study is to evaluate the flow motions and thermal natural convective heat exchanges of Al 2 O 3 -H 2 O-based nanoliquid in a non-Darcian wavy-porous cavity under the local thermal non-equilibrium condition. The findings of such a work may have possible applications in many practical systems such as heat exchangers, solar-energy devices, food treatment systems, refineries, refrigeration systems, condensates, and so on. Figure 1 shows the problem geometry that will be considered in this work. An isothermal heater is put in place within the horizontal bottom wall with length d, whereas the vertical wavy walls are cold ( T c ). The rest of the walls are adiabatic. The porous cavity is filled with Al 2 O 3 -water nanofluid. The Forchheimer-Brinkman-extended Darcy model are adopted together with the Boussinesq approximation. The type of the porous media used is glass balls. Considering the local thermal non-equilibrium (LTNE) model, the governing equations are as follows www.nature.com/scientificreports/ where |u| = √ u 2 + v 2 represents the Darcy velocity, g is the gravitational acceleration, ε the medium's porosity,

Mathematical formulation
150(1−ε) 2 is the medium's permeability, with d m representing the porous bed's average particle size. The following nanofluid thermophysical properties are utilized: where φ is the nanoparticle volume fractions. The nanofluid's ratios of thermal conductivity and dynamic viscosity are 10 where . Further details can be found in Corcione 10 . Next the following dimensionless variables are utilized: As a results we have where the modified conductivity ratio γ = εk f (1−ε)k s and the inter-phase heat transfer coefficient H = hL 2 εk f . The boundary conditions now become where i is the iteration number and η is set at η = 10 −6 in the present work. Based on the grid sensitivity checks presented in Table 1, for all computations in this paper, the G6 uniform grid is employed.
To validate the numerical data of the current work, the present figures were compared with the previous published numerical and experimental results presented by Calcagni et al. 46 for the natural convection heat transfer in a square cavity heated from below, as shown in Fig. 2. In addition, Fig. 3 presents a comparison of the average Nusselt numbers between the existing work and the previous published results described by Calcagni et al. 46 . Moreover, a comparison is provided between the figures of the current work and the one indicated in Kaluri and Basak 47 for the problem of natural convection in a square porous cavity heated from below, as described in Fig. 4. In Fig. 5, the predictions of the thermal conductivity enhancement are in agreement with the results of Chon et al. 48 , Corcione et al. 49 and Ho et al. 50 . Except at some nanoparticle volume fractions, . The study outcomes appear to establish the validity and accuracy of the numerical approach.

Results and discussion
This section presents numerical results for the streamlines, isotherms of the nanofluid phase, and isotherms of the solid phase for the following six parameters. These parameters are the Darcy number (  Table 2. Effect of Darcy number.   www.nature.com/scientificreports/  www.nature.com/scientificreports/ less permeable so the speed of the flow becomes low. As a result, the porous matrix causes the flow to cease in the porous region. For the streamlines, we noticed the formation of two counter-rotating cells, one of which rotates clockwise while the second rotates anti-clockwise. The fluid that is heated by the bottom heater moves to the upper part of the cavity where it divides into two streams, one moves towards the left vertical cold wavy wall and the other moves towards the right one. When Darcy exceeds the value of 10 −4 , convective heat transfer begins to dominate, the isotherms are destroyed and a natural convection plume appears and the two cells take an elliptical form. The flow velocity increases because the permeability increases with increasing Darcy number, so fluid displacement in the pores is freer and the convection mode is dominant. It can be seen from this figure also that the isotherms contours within the solid phase are less destroyed due to the relatively low modified conductivity ratio ( γ ). It should be noted that the isotherms of the nanofluid phase and those of the solid phase are almost the same for the low Darcy number. This allows us to conclude that the local thermal equilibrium (LTE) state in the porous cavity is verified at this situation when the heat transfer mode is dominated by conduction. The variation of the local Nusselt number for the nanofluid and the solid phase along the heated part of the bottom horizontal wall of the cavity show that the Darcy number affects the distribution of the local Nusselt number. Minimum values of the local Nusselt number are obtained in the middle of the heated part since this is where the temperature gradient is low in view of the isotherms of Fig. 6. On the other hand, the two peaks of the distribution of Nusselt numbers on the edges of the heater part correspond to the contact of the cold fluid from the side walls with the ends of the heated source, which will give rise to the highest temperature gradients. It is seen also that the values of Nusselt for the nanofluid phase are always higher than that of the solid phase (see Fig. 7a,b).   www.nature.com/scientificreports/ Effect of nanofluid loading. Figure 9 shows the effect of addition of nanoparticles on the streamlines, isotherms of the nanofluid phase and isotherms of the solid phase at Da = 10 −3 , N = 2 , γ = 10 , B = 0.5 and D = 0.5 . The first remark is that the addition of nanoparticles leads the intensity of streamlines to decreases due to the increase in the effective viscosity of the mixture. The thermal contours depict that the distribution of the isotherms within the nanofluid phase approaching to be similar to that of the solid phase by increasing the volume fraction of nanoparticles. This is well illustrated for weak Darcy numbers where the Nusselt numbers for the two phases approach each other for Da ≤ 10 −4 in the stage where heat transfer by conduction dominates (See Fig. 11). In addition, it can be seen from this figure that with increase of the volume fraction from 0.01 to 0.05, the Nusselt numbers of the nanofluid phase and the solid phase increase. Furthermore, the addition of the www.nature.com/scientificreports/ nanoparticles greatly affects the solid phase Nusselt numbers in comparison to those of the nanofluid phase. This is because nanoparticles increase the thermal conductivity in particular which increases the rate of heat transfer when the conductive heat transfer is dominant. From Fig. 10, we clearly observe that the nanoparticles concentration affects much more the distribution of the solid phase local Nusselt numbers, Nu s compared to their effect on the local Nusselt numbers for the nanofluid phase, Nu nf this is due to the augmentation of the thermal conductivity of the solid nanoparticles which is directly related to the solid phase thermal conductivity. Figure 12a, b shows that when the convective flow is strong at a relatively high Da value ( Da = 10 −3 ), the increase of the dimensionless length of the heat source causes an increase in the nanofluid phase and solid phase average Nusselt numbers. At this stage for a given value of D, the addition of nanoparticles increases the mean Nusselt number and this rate of increase is greater for low values of φ.

Effect of undulations.
In order to investigate the effect of the of oscillations N (from 0 to 3) on the thermal and dynamic characteristics within the enclosure, we illustrate in Fig. 13 the streamlines, isotherms of the nanofluid phase and the isotherms of the solid phase at Da = 10 −3 , φ = 0.02 , γ = 10 , B = 0.5 and D = 0.5 . It appears from this figure that the presence of undulations on the vertical walls influences the geometric shape of the flow cells, as well as the distribution of isotherms near the corrugated walls. The streamlines are serried at the crests of the wave leading to an acceleration of the fluid in these areas. In addition, by increasing the oscillation number, the temperature gradient increases near the cold vertical walls especially in the upper part of these walls, so a significant heat transfer rate occurs close to these areas.
Because the corrugations on the vertical walls contribute to the reduction of natural convective flow by blocking the circulation of fluid inside the cavity, and therefore promote the thermal transfer by conduction, the solid phase Nusselt numbers at the heated part of the bottom horizontal wall increase by increasing the undulation number for all values of Da with a significant effect at low Darcy number. While, the increase in the number of waves, N enhances the nanofluid Nusselt numbers only for low Darcy numbers when the conduction is the dominant heat transfer (See Figs. 14 and 15).
As is expected, for a relatively high Darcy number, the mean Nusselt number for the nanofluid is decreased with the increase of N while the corresponding one for the solid is increased as can be observed from Fig. 16.
Effect of modified conductivity ratio. In Fig. 17, we show the effect of the modified conductivity ratio ( γ ) on the streamlines, isotherms of the nanofluid phase and the isotherms of the solid phase for Da = 10 −3 , φ = 0.02 , N = 2 , B = 0.5 and D = 0.5 . At this Darcy number, the modified conductivity ratio does not significantly affect the streamlines and the isotherms for the nanofluid phase, whilst the isotherms of the solid phase are considerably affected. Thus, it is seen from the isothermal contours that the local thermal equilibrium state is verified for large values of the modified conductivity ratio, which is indicated by the temperature fields for the two phases that are almost identical. www.nature.com/scientificreports/ It can be seen from Figs. 18, 19 and 20 that when the modified conductivity ratio is weak, there is a very remarkable difference between the heat transfer rates of the nanofluid and solid phases, indicating that nonequilibrium states are considerable when the modified conductivity ratio is weak. Thus, high values of the modified conductivity ratio reduce the non-equilibrium state between the porous matrix and the saturating nanofluid.
In addition, Figs. 19 and 20 show that at a given value of the modified conductivity ratio, the Nusselt numbers increase by increasing both φ and D and this increase is very sensitive for the solid phase, as was reported earlier.    www.nature.com/scientificreports/ As seen from the Fig. 22, the variation of the local Nusselt number along of the heat source shows that the position of the heater affects the distribution of the solid local Nusselt numbers on the heater surface by producing a corresponding number of peaks and valleys that correspond to the maximum or the minimum heat transfer rate. Indeed, the maximum values of the solid Nusselt numbers move to the right edge of the heater when displacing it to the left and to the left edge of it when displacing it to the right (where the temperature gradient is important). On the contrary, the position of the heater does not affected much the distribution of nanofluid Nusselt numbers.
As expected, when displacing the heat source, the convection flow is reduced due to the unsymmetrical flow. Thus, we note a decrease in the average Nusselt numbers of the nanofluid phase while the average Nusselt   www.nature.com/scientificreports/ numbers of the solid phase increase. It is to be noted that moving the hot source (to the left or to the right with the same degree) only affects the distribution of the local Nusselt numbers while the mean Nusselt numbers are not affected (Fig. 23). This is due to the fact that the hot source is very close in this case which makes the heat transfer so fast. Figure 24 illustrates symmetrical profiles of the nanofluid and the solid local Nusselt numbers along the heater with different lengths. It can be seen that in the convection dominated situation, increasing the heater size causes the local Nusselt numbers profile to decrease. This is because as the heater size increases, the temperature of the heater increases due to the higher heat flux generated and consequently, the corresponding Nusselt number decreases.   www.nature.com/scientificreports/ Evolution of the average solid and nanofluid Nusselt numbers as a function of the volume fraction of the nanoparticles for different values of D is represented on the graphs of Fig. 25a,b at Da = 10 −3 , N = 2 , γ = 10 and B = 0.5 . In general, for given values of D, the average Nusselt number, whether of the solid phase or of the nanofluid phase, increases as φ increases. We also note that the two mean Nusselt numbers increase with the increase in the heater length. This can be explained by the higher heat generation rates as the heater size increases which clearly increases the area exposed to the heat transfer. www.nature.com/scientificreports/

Conclusions
Several significant conclusions drawn from this study are listed in the following: 1. When the Darcy's numbers are small, the conduction heat transfer takes place. Once the Darcy number exceeds the value of 10 −4 , the convective heat transfer begins to dominate. 2. The local thermal equilibrium (LTE) state in the porous cavity is verified at low Darcy numbers when the heat transfer mechanism is dominated by conduction. 3. High values of the modified conductivity ratio, γ reduces the non-equilibrium state between the porous matrix and the saturating nanofluid. 4. Displacing the heater from the side walls towards the middle of the bottom wall, enhances the strength of natural convection flow within the enclosure. Also, when the heater remains in the middle of the bottom wall, symmetrical flow cells are generated regardless of the heater length. 5. The presence of waves on the vertical walls leads to the reduction of the overall heat transfer when the convection is the dominant heat transfer mechanism whereas it enhances when the heat transfer mode is dominated by conduction. 6. It is found that the heater length, its position, and the number of waves on the side vertical walls as well as the nanoparticles concentration can be control parameters for heat transfer and fluid flow within the cavity.   www.nature.com/scientificreports/