Magnetohydrodynamic mixed convection of TiO₂–Cu/water between the double lid-driven cavity and a central heat source surrounding by a wavy tilted domain of porous medium under local thermal non-equilibrium

Abstract

The magnetohydrodynamic (MHD) mixed convection of heat and mass transfer is carried out using finite difference method applied inside a tilted porous cavity saturated with a hybrid nanofluid due to the presence of the double-moving lid and the heat sources. In contrast to the earlier research, various effects which are recognized by heat generation in the local thermal non-equilibrium case at the extended Brinkman Darcy model subjected to inclined magnetic field are thoroughly examined numerically. For instance, unusual observations of the cold mass surrounding the heat source emphasize that the maximum fluid temperature highly depends on the forced convection. Additionally, solid-phase temperature acts in accordance to the heat source location while fluid temperature is agitated by the moveable sides which points up the disparity at the thermal energy transportation. However, the transfer of heat and mass at the model requires a specific conduct due to the existence of damping factors. The magnetic field, for example, suppresses the fluid flow. Moreover, the thermal non-equilibrium condition deteriorates the global heat generation.

Article Highlights

• The flow shear at the proposed model is more feasible for the heat and mass transfer than the buoyancy force.

• Local thermal non-equilibrium case creates inaccessible regions that disapprove the harmony in the flow behavior.

• The ratio of natural to forced convection is the most pertinent to the thermal nonequilibrium scenario.

1 Introduction

Mixed convection with magnetic fields is successfully used in polymers, metallurgy, geothermal energy extraction, and fusion reactors. Due to its properties, it is utilized in a wide range of industries, including mechanical engineering, the petroleum industry, chemical engineering, the medical profession, and other physiological frameworks [1,2,3,4,5]. This idea is especially helpful for applications like magnetic resonance imaging, blood pumping, radiofrequency ablation, cancer tumor treatment, nuclear reactor cooling, magnetic electro-catalysis, metallurgical processes, thermal protection, receiving and transmitting antennas, magnetic electro-catalysis [6,7,8,9,10], etc.

Additionally, obstructions existence at the convection have been compared by Scott et al. [11] to the opening function, where the blockages are like flaps. They split motion into two main categories: bulk density variations and motion pressure gradient, which may both be utilized to explain natural convection flow in multizone enclosures. Partitions should span the ground and roof of a square cavity, according to Chang et al. [12], such that heat transfer reduces as the thickness of the partition climbs. Moreover, Eshaghi et al. [13] have studied the double-diffusive natural convection in an H-shaped cavity with a baffle. They reported that the best Nusselt number and Sherwood number have been grasped. According to Nansteel and Greif [14] and Lin and Bejan [15], impediments between zones tend to keep the boundary layer's thermal equilibrium condition stable, which suggests a general drop in convective heat transmission. As per Sahoo et al. [16], the rectangular blockage at the vertical channel causes a "fluid flow step" that is "backward-facing" and can alter the maximum average temperature. In addition, heat transmission in vertical channels with semi-circular barriers has been studied by Said and Krane [17]. Depending on the Rayleigh number, the obstruction resulted in a drop in the average Nusselt number at a uniform wall temperature (it increases as Ra decreases). In the same context, Seyyedi et al. [18] have investigated heat transfer characteristics as well as entropy generation in a hexagonal enclosure. While, An economic analysis was carried out [19] for heat transfer between two non-similar circular cylinders. However, Dogonchi et al. [20] have proved that the suction process has a dual effect on heat transfer and fluid flow using the Duan-Rach approach.

Shahriari et al. [21] reported their findings in a research on Magnetohydrodynamic natural convection within the angled undulating cavity packed with CuO-water nanofluid. They observed that the magnetic field's inhibitory impact causes the mean Nusselt number and Bejan number to become decreasing functions as the Hartmann number increases, while entropy formation is enhanced. Another illustration is Sheremet et al.'s [22] study of the Magnetohydrodynamic non-driven convection of a nano-fluid in an open porous "high cavity" with a "corner heater". Moreover, Dogonchi et al. [23] reported that “Rayleigh number could overshoot the velocity gradient” for the natural convection at a porous medium saturated between two square cylinders filled with magnetic nanofluids. In addition, Mandal et al. [24] studied the magnetohydrodynamic mixed bioconvection.

On the other hand, the larger amplitude and higher frequency of the moving wall can effectively mix the viscous fluid, according to a significant study by Li et al. [25]. Moreover, Zou et al. [26] concluded that the flow direction should be taken into account to promote convective mixing. To examine the effects of the Lorentz force on the free convection inside the wavy triangular cavity, Shekaramiz et al. [27] carried out a numerical research. They discovered that entropy generation first rises with rising Hartmann numbers before remaining constant.

For instance, in a square cavity containing Cu-water nanofluids, Rashad et al. team 's [28] numerically examined the MHD mixed convection flow. In a porous cavity with a lid with a “Cattaneo-Christov” heat flux pattern by mixed convection, Jakeer et al. [29] numerically investigated the effect of the heated barrier location on the magneto-hybrid nanofluid flow. For MHD natural convection, Sheikholeslami and Sadoughi [30] investigated the “lattice Boltzmann” technique inside a porous cavity that was filled with a nanofluid enclosing that contained 4 square-heaters. In addition, CuO-MWCNT-oil hybrid nanofluid's turbulent flow field, heat transfer, and entropy formation in a trapezoidal container under the influence of a magnetic field were all quantitatively investigated by Aghaei et al. [31] using the finite volume approach. According to their research, the average Nusselt numbers for different Rayleigh numbers and Hartmann numbers decrease with the increased "volume fraction" of nanoparticles. Additionally, the efficiency of heat exchange is dramatically altered by the nanoparticles dispersed, especially those made of metals (Cu, Al₂O₃, ZnO, Ag, and SiO₂) inside the fluid matrix as compared to regular liquid flows (Khanafer et al., Chamkha et al., and Aly et al., [32,33,34,35]). Tiwari and Das [36] looked into the supplementary effect for a lid-driven cavity, while Billah et al. [37] looked into the buoyancy-driven force at a slanted triangular compartment. Moreover, The Nusselt number greatly change based on the shape of the used nanoparticles [38].

In accordance with the size, form, and composition of the nanoparticles that are dispersed inside them, nanofluids have different thermal conductivities (Suresh et al. [39]). For instance, metallic particles have a "higher thermal conductivity than non-metallic particles". Nevertheless, as a result of their size, cylindrical nanoparticles function better thermally than spherical nanoparticles ones. Consequently, when the atomic structure of the fluid matrix is precisely treated, significant changes take place. For businesses looking for a product that exhibits excellent stability and superior chemical inertness, Al₂O₃ is the solution. However, this will oftentimes be inadequate for nanofluid applications because they will lose their benefit of heat conductivity. This problem can be handled by adding a few metal particles to an aluminum framework to improve its thermal characteristics. By chemically preparing CuO–Al₂O₃ combinations and hydrogen-reducing them, Jeena et al. [40] created the Cu–Al₂O₃ nanocomposites. Biswas et al. [41] were able to rise the thermal performance at a tilted porous cavity using Cu–Al₂O₃–water nanofluid subjected to the partially active magnetic field. Sung-Tag Oh et al. [42] demonstrated the fabrication of an Al₂O₃–Cu nanocomposite using powder mixtures of fine Al₂O₃ and copper oxide sizes. As an Al₂O₃–Cu/water hybrid nanofluid flowed through a circular tube with a constant heat flux, Suresh et al. [39] assessed the "Nusselt number and friction factor". Davood Toghraie et al. [43] concluded from their experimental studies on ZnO–TiO₂/EG hybrid nanofluids that the significant increase in heat transmission occurs either at a warmer temperature proportionate to solid volume fraction or vice versa.

However, it is reported that when carbon nanotubes (CNTs) were attached to the surface of spherical oxide nanoparticles (Huang et al. [44]) then the thermal conductivity was significantly enforced (Han et al. [45]) due to the strong thermal action among the CNTs attached to the alumina/iron oxide particle. On the other hand, nano-encapsulated phase change material has been utilized [46] for free convection in a porous, wavy enclosure. For a better comprehension, Suresh et al. [39] discovered that for a hybrid Al₂O₃–Cu/deionized water nanofluid with a "0.1% volume concentration", the Nusselt number "was 13.56% greater than that of water at Reynold number (Re) = 1730". Furthermore, Labib et al. [47] reported that improved heat transmission in mixes of Al₂O₃ nanoparticles in CNT/water nanofluids is enabled by the shear shining features of CNT nanofluids. We contend that because the increment is independent of any discernible chemical change in the research area, qualitative advancement can happen at the heat transfer. The distribution of Nusselt numbers at cavities, on the other hand, is said to follow the geometric pattern of the wall by Arefmanesh et al. [48], which is one of the key purposes of wavy walls at cavities. Engineers can create thermal access points depending on flexible usage at various locations using these models. Additionally, it reduced the development of entropy (Alsabery et al. [49]). Alongside, Das and Mahmud [50] discovered that the heat exchange rate is significantly influenced by the amplitude of the wavy wall and the amount of undulations, particularly in the convective mode (Kumar [51]). A further important aspect is that when the hollow wall is wave shaped, the fluid domain is vulnerable to some disruption, as is the case in storm courses, earthquake systems, and maritime transportation.

Nevertheless, because various mediums have varying temperatures, heat transportation to trace energy dynamics follows a path along a temperature gradient. The viewpoint of thermal equilibrium is used to analyze the above phenomenon. In contrast, the term "local thermal non-equilibrium" (LTNE) is used to describe situations in which there is a “significant seepage velocity, a significant temperature difference between the fluid and solid matrix of a porous medium, or when a heat source exists in either a solid domain or a pure fluid domain within the porous medium” as was stated by Alsabery et al. [52]. Accordingly, when the heat transmission coefficient is low, the temperature of the solid phase is lower than the temperature of the fluid phase as a result of Baytaş's [53] application of a heat source to the solid phase. Aside from that, Sheremet et al. [54] show that a lower interface heat transfer coefficient implies to decrease in the heat exchange rate, which inhibits the LTNE. In addition, Zargartalebi et al. [55] found that the buoyancy ratio has a negative impact on the heat exchange ratio while positively affecting the Rayleigh number. This finding indicates that the Rayleigh number enhances the thermal equilibrium because the thermal transport via diffusion is greater than that via convection. The system's increasing entropy state, however, causes the buoyancy ratio to induce a non-equilibrium state to rise.

In light of the influence of Buongiorno's concept, Tahmasebi et al. [56] discovered that speeding up the exchange of heat at the interface entails slowing down the interchange of the micro liquid phase. Regarding Brownian motion, this model. On the basis of our knowledge, Brownian motion prevents heat exchange at the contact. However, Nield et al. [57] found that the local Nusselt number is poorly dependent on the Darcy number or the fluid–solid heat exchange parameter and relies entirely on the Peclet number and the solid/fluid conductivity. This could be due the advection/diffusion regimes are so effective since the velocity profile overtakes heat transfer in boundary layer flows. Nonetheless, we may conclude that there is a connection between the Darcy number and the heat exchange parameter in that scenario to understand why the heat exchange is ineffective. The reliability of the local thermal equilibrium for all porous media when the conduction mode is predominate has been shown by Kim and Jang [58].

Furthermore, when the interstitial heat transfer coefficient is proportional to Re^exponent>1 (such as sintered metals and cellular ceramics) and the convection mode is dominant, the local thermal equilibrium is applicable. For the purpose of choosing an appropriate analytical interpretation, Amiri et al. [59] made the supposition that the constant heat flux at boundaries is satisfied when either heat propagation is based on temperature gradients (effective conductivities) or each phase receives the same amount of the flux at the interface. In addition, Mahmoudi and Karimi [60] have shown that the Nusselt number for the first method is higher than the other one for the second. The Darcy number was negatively associated with the thickness of the local thermal non-equilibrium porous zone, which they were able to measure. Aside from that, Izadi et al. [61] have observed that the thermal nonequilibrium case deteriorates as the thermal conductivity ratio and heat transfer parameter increase. Moreover, the reduction in the thermal resistance [76] of fluid phase can enhance the heat transmission rate due to the increase in the thermal conductivity ratio. In addition, permeability influences the Nu number (the dimensionless quantity) of both solid/fluid phases due to nanoparticles rotating around the center of gravity. Sivasankaran et al. [62] have found that exchange speed of thermal energy is enhanced by increasing values of the modified conductivity ratio and the porosity of the media. Entirely, Tahmasebi et al. [63] have found that the average Nu is supported by Rayleigh number, Darcy number, and thermal conductivity ratios at solid and fluid phases. Furthermore, the convection interaction regime at pores increases the Nu for the solid phase whereas decreases it for the nano-fluid.

Regarding the running study, the control parameters of the model encompasses various heat and mass dynamics have been analyzed numerically in line with the MHD mixed convection around a central circular heat source and the other cold wavy boundaries. The manuscript is organized as following; first section is the introduction where we shed light on the different styles of heat transfer mechanisms due to fluid flow within miscellaneous media under different conditions. In the second section, the mathematical formulation for the problem is designated. Then the fundamentals of the numerical investigation are demonstrated in the third section. Therefore, at section four, the results are displayed in the form of the contours of the temperature and fluid flow which disclose the active regions within the cavity while the Nusselt number profiles monitor the heat flux behavior at the boundaries. Ultimately, the fifth section sum up the findings of the proposed work.

2 Mathematical modeling

Figure 1 shows the preliminary geometry of an undulating inclined cavity. The involved wavy cavity is filled by porous media and hybrid nanofluids. The flow of porous medium is subjected to the extension of Darcy’s law using Dupuit-Forchheimer relationship. The right and left side wavy-walls are cold (${T}_{c}$), the bottom and top sides are adiabatic. Moreover, the fluid domain is inclined with angle α and the magnetic field has an inclination angle $\Phi$. 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). According to the previously specified hypotheses, continuity, momentum, concentration, and energy equations are formulated for hybrid nanofluids, incompressible, laminar, and single-phase flows in steady-state as follows [64, 65]:

$$\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}=0$$

(1)

$$\begin{aligned} \frac{1}{{\varepsilon^{2} }}\left( {u\frac{\partial u}{{\partial x}} + v\frac{\partial u}{{\partial y}}} \right) & = - \frac{1}{{\rho_{hnf} }}\frac{\partial p}{{\partial x}} + \frac{1}{\varepsilon }.\frac{{\mu_{hnf} }}{{\rho_{hnf} }}.\nabla^{2} u - \frac{{\mu_{hnf} }}{{\rho_{hnf} }}.\frac{u}{K} + \frac{{(\rho \beta )_{hnf} }}{{\rho_{hnf} }}g(T_{f} - T_{c} )\sin \alpha \\ & \quad + \frac{{(\rho \beta^{*} )_{hnf} }}{{\rho_{hnf} }}g(C - C_{c} )\sin \alpha + \frac{{\sigma_{hnf} B_{0}^{2} }}{{\rho_{hnf} }}\left( {v\sin \Phi \cos \Phi - u\sin^{2} \Phi } \right) \\ \end{aligned}$$

(2)

$$\begin{aligned} \frac{1}{{\varepsilon^{2} }}\left( {u\frac{\partial v}{{\partial x}} + v\frac{\partial v}{{\partial y}}} \right) & = - \frac{1}{{\rho_{hnf} }}\frac{\partial p}{{\partial y}} + \frac{1}{\varepsilon }.\frac{{\mu_{hnf} }}{{\rho_{hnf} }}.\nabla^{2} v - \frac{{\mu_{hnf} }}{{\rho_{hnf} }}.\frac{v}{K} + \frac{{(\rho \beta )_{hnf} }}{{\rho_{hnf} }}g(T_{f} - T_{c} )\cos \alpha \\ & \quad + \frac{{(\rho \beta^{*} )_{hnf} }}{{\rho_{hnf} }}g(C - C_{c} )\cos \alpha + \frac{{\sigma_{hnf} B_{0}^{2} }}{{\rho_{hnf} }}\left( {u\sin \Phi \cos \Phi - v\cos^{2} \Phi } \right) \\ \end{aligned}$$

(3)

Fig. 1

Schematic diagram of the model

$$\frac{1}{\varepsilon }(u\frac{\partial {T}_{f}}{\partial x}+v\frac{\partial {T}_{f}}{\partial y})={\alpha }_{eff.hnf}.{\nabla }^{2}{T}_{f}+\frac{{h}_{hnfs}({T}_{s}-{T}_{f})}{\varepsilon (\rho {c}_{p}{)}_{hnf}}+\frac{{Q}_{0}}{\varepsilon (\rho {c}_{p}{)}_{hnf}}$$

(4)

$$0=\left(1-\varepsilon \right)[{k}_{s}+\frac{16{\sigma }^{*}{T}_{c}^{3}}{3{k}^{*}}].{\nabla }^{2}{T}_{s}+{h}_{nfs}({T}_{f}-{T}_{s})+(1-\varepsilon ){Q}_{0}$$

(5)

$$u\frac{\partial C}{\partial x}+v\frac{\partial C}{\partial y}={D}_{B}.{\nabla }^{2}C$$

(6)

The boundary conditions imposed on the flow field are taken as:

• On the left wall

  $$u=v=0,{T}_{f}={T}_{s}={T}_{c},C={C}_{c},x=AH\left[1-\mathit{cos}\left(2\pi \lambda y/H\right)\right],0\le y\le H$$

  (7.1)

• On the right wall

  $$u=v=0,{T}_{f}={T}_{s}={T}_{c},C={C}_{c},x=H-AH\left[1-\mathit{cos}\left(2\pi \lambda y/H\right)\right],0\le y\le H$$

  (7.2)

• On the top wall

  $$u=\pm {\lambda }_{t}{U}_{0},v=0,{T}_{f}={T}_{s}={T}_{c},C={C}_{c},0\le x\le H$$

  (7.3)

• On the bottom wall

  $$u=\pm {\lambda }_{d}{U}_{0},v=n=0,{T}_{f}={T}_{s}={T}_{c},C={C}_{c},0\le x\le H$$

  (7.4)

• On the inner circular cavity

  $${T}_{f}={T}_{s}={T}_{h}$$

  (8)

  where $u$ and $v$ are the velocity components, $T$ is temperature, s-index refers to solid phase, f-index refers to fluid phase, C stands for the concentration of nanoparticles, ${\rho }_{hnf}$ is the density of the hybrid nanofluid, ${\nu }_{hnf}$ is kinematic viscosity. $g$ is a gravity, $p$ is a pressure, ${\mu }_{hnf}$ is a dynamic viscosity, β* = (− 1/ρ)(dρ/dC)_T,p, the buoyancy rate due to nanoparticle concentration currents, and ${Q}_{0}$ is the heat generation.

2.1 Dimensionless forms of equations

Presenting the next non-dimensional variables (Eq. 9) into Eqs. (1)–(7) yields the dimensionless equations (Eqs. 10–15):

$$X,Y=\frac{x,y}{H},U,V=\frac{u,v}{{U}_{0}},P=\frac{p}{{\rho }_{nf}{U}_{0}^{2}},{\theta }_{f}=\frac{\left({T}_{f}-{T}_{c}\right)}{\left({T}_{h}-{T}_{c}\right)}, {\theta }_{s}=\frac{\left({T}_{s}-{T}_{c}\right)}{\left({T}_{s}-{T}_{c}\right)}, \varphi =\frac{\left(\mathrm{C}-{C}_{c}\right)}{\left({C}_{h}-{C}_{c}\right)},Ri=\frac{Gr}{{Re}^{2}},\mathit{Re}=\frac{{U}_{0}H}{{\nu }_{f}}$$

(9)

$$\frac{\partial U}{\partial X}+\frac{\partial V}{\partial Y}=0$$

(10)

$$\begin{aligned} \frac{1}{{\varepsilon^{2} }}\left( {U\frac{\partial U}{{\partial X}} + V\frac{\partial U}{{\partial Y}}} \right) & = - \frac{\partial P}{{\partial X}} + \frac{1}{\varepsilon \cdot Re} \cdot \left( {\frac{{\rho_{f} }}{{\rho_{hnf} }}} \right)\left( {\frac{{\mu_{hnf} }}{{\mu_{f} }}} \right) \cdot \nabla^{2} U \\ & \quad - \frac{1}{Da \cdot Re} \cdot \left( {\frac{{\rho_{f} }}{{\rho_{hnf} }}} \right)\left( {\frac{{\mu_{hnf} }}{{\mu_{f} }}} \right) \cdot U \\ & \quad + Ri \cdot \frac{{(\rho \beta )_{hnf} }}{{\rho_{hnf} \cdot \beta_{f} }}\sin \alpha \cdot \theta_{f} + Gr_{c} \cdot \sin \alpha \cdot \phi \\ & \quad + \left( {\frac{{\rho_{f} }}{{\rho_{hnf} }}} \right)\left( {\frac{{\sigma_{hnf} }}{{\sigma_{f} }}} \right) \cdot \frac{{Ha^{2} }}{Re}\left( {V\sin \Phi \cos \Phi - U\sin^{2} \Phi } \right) \\ \end{aligned}$$

(11)

$$\begin{aligned} \frac{1}{{\varepsilon^{2} }}\left( {U\frac{\partial V}{{\partial X}} + V\frac{\partial V}{{\partial Y}}} \right) & = - \frac{\partial P}{{\partial Y}} + \frac{1}{\varepsilon \cdot Re} \cdot \left( {\frac{{\rho_{f} }}{{\rho_{hnf} }}} \right)\left( {\frac{{\mu_{hnf} }}{{\mu_{f} }}} \right) \cdot \nabla^{2} V \\ & \quad - \frac{1}{Da \cdot Re} \cdot \left( {\frac{{\rho_{f} }}{{\rho_{hnf} }}} \right)\left( {\frac{{\mu_{hnf} }}{{\mu_{f} }}} \right) \cdot V \\ & \quad + Ri \cdot \frac{{(\rho \beta )_{hnf} }}{{\rho_{hnf} \cdot \beta_{f} }}\cos \alpha \cdot \theta_{f} + Gr_{c} \cdot \cos \alpha \cdot \phi \\ & \quad + \left( {\frac{{\rho_{f} }}{{\rho_{hnf} }}} \right)\left( {\frac{{\sigma_{hnf} }}{{\sigma_{f} }}} \right)\frac{{Ha^{2} }}{Re}\left( {U\sin \Phi \cos \Phi - V\cos^{2} \Phi } \right) \\ \end{aligned}$$

(12)

$$\begin{aligned} \frac{1}{\varepsilon }\left( {U\frac{{\partial \theta_{f} }}{\partial X} + V\frac{{\partial \theta_{f} }}{\partial Y}} \right) & = \left( {\frac{1}{Re Pr}} \right)\frac{{\alpha_{eff.hnf} }}{{\alpha_{f} }} \cdot \nabla^{2} \theta_{f} + \frac{1}{\varepsilon \cdot Re Pr}\frac{{(\rho c_{p} )_{f} }}{{(\rho c_{p} )_{hnf} }} \\ & \quad \cdot H^{*} (\theta_{s} - \theta_{f} ) + \frac{1}{\varepsilon \cdot Re Pr}\frac{{(\rho c_{p} )_{f} }}{{(\rho c_{p} )_{hnf} }} \cdot Q \\ \end{aligned}$$

(13)

$$0={\nabla }^{2}{\theta }_{s}+{K}_{r}.{H}^{*}({\theta }_{f}-{\theta }_{s})+{k}_{fs}.Q$$

(14)

$$U\frac{\partial \varphi }{\partial X}+V\frac{\partial \varphi }{\partial Y}=\frac{1}{\mathit{Re}.Sc}(\frac{{\partial }^{2}\varphi }{\partial {X}^{2}}+\frac{{\partial }^{2}\varphi }{\partial {Y}^{2}})$$

(15)

where $\mathit{Pr}=\frac{{\nu }_{f}}{{\alpha }_{f}},\mathit{Re}=\frac{{U}_{0}H}{{\nu }_{f}},Gr=\frac{g{\beta }_{f}{H}^{3}\Delta T}{{{\nu }^{2}}_{f}},\mathit{Ha}={B}_{0}H\sqrt{\frac{{\sigma }_{f}}{{\mu }_{f}}}, Da=\frac{K}{{H}^{2}},{R}_{d}=\frac{16{\sigma }^{*}{T}_{c}^{3}}{3{k}^{*}{k}_{s}},$ the inter-phase heat transfer coefficient;

$${H}^{*}={h}_{nfs}.\frac{{H}^{2}}{{k}_{f}},Q={Q}_{0}.\frac{{H}^{2}}{{k}_{f}},{k}_{\mathit{fs}}=\frac{{k}_{f}}{{k}_{s}},{K}_{r}=\frac{{k}_{f}}{(1-\varepsilon ){k}_{s}},$$

(16)

And the corresponding dimensionless boundary @@conditions are;

• On the left wall:

  $$U=V=0,{\theta }_{f}={\theta }_{s}=\varphi =0,X=A\left[1-\mathit{cos}\left(2\pi \lambda Y\right)\right],0\le Y\le 1$$

  (17.1)

• On the right wall:

  $$U=V=0,{\theta }_{f}={\theta }_{s}=\varphi =0,X=1-A\left[1-\mathit{cos}\left(2\pi \lambda Y\right)\right],0\le Y\le 1$$

  (17.2)

• On the top wall

  $$U=\pm {\lambda }_{t},V=0,{\theta }_{f}={\theta }_{s}=0,\varphi =0,0\le X\le 1,$$

  (17.3)

• On the bottom wall:

  $$U=\pm {\lambda }_{d},V=0,{\theta }_{f}={\theta }_{s}=0,\varphi =0,0\le X\le 1$$

  (17.4)

• On the inner circular:

  $${\theta }_{f}={\theta }_{s}=1$$

  (18)

The local Nusselt number of the fluid can be formulated as:

$${Nu}_{fs(left,right)}=-\frac{{k}_{hnf}}{{k}_{f}}{\left(\frac{\partial {\theta }_{f}}{\partial X}\right)}_{left,right}, {Nu}_{fs(Y=0,1)}=-\frac{{k}_{hnf}}{{k}_{f}}{\left(\frac{\partial {\theta }_{f}}{\partial Y}\right)}_{Y=0,1}$$

(19)

The local Nusselt number of the solid can be written as:

$${Nu}_{ss(left,right)}=-\frac{{k}_{hnf}}{{k}_{f}}(1+{R}_{d}){\left(\frac{\partial {\theta }_{s}}{\partial X}\right)}_{left,right}, {Nu}_{ss(Y=0,1)}=-\frac{{k}_{hnf}}{{k}_{f}}(1+{R}_{d}){\left(\frac{\partial {\theta }_{s}}{\partial Y}\right)}_{Y=0,1}$$

(20)

The average Nusselt number of the fluid and solid are

$$N{u}_{mf}={\int }_{0}^{H}N{u}_{fs}dX,N{u}_{ms}={\int }_{0}^{H}N{u}_{ss}dX$$

(21)

in the above Eq. (13) ${\alpha }_{eff,hnf}$ is defined as

$${\alpha }_{eff,hnf}=\frac{{k}_{eff,hnf}}{{\left(\rho {c}_{p}\right)}_{hnf}}$$

(22)

where

$${k}_{eff,hnf}=\varepsilon {k}_{hnf}+\left(1-\varepsilon \right){k}_{s}$$

(23)

2.2 Thermophysical properties of nanofluid and hybrid nanofluid

While some research has looked at determining the thermophysical properties of nanoparticles (Table 1), classical models do not necessarily apply to nanofluids. Experimental results facilitate the selection of an appropriate model for a particular property. Defining the effective properties of TiO₂/water nanofluids and TiO₂–Cu/water hybrid nanofluids (Table 1), respectively, are as follows:

$${\rho }_{nf}=(1-\phi ){\rho }_{bf}+\phi {\rho }_{p}$$

(24)

which determine density of nanofluid, the density of hybrid nanofluid is specified as [66]:

$${\rho }_{hnf}={\phi }_{{TiO}_{2}}{\rho }_{{TiO}_{2}}+{\phi }_{Cu}{\rho }_{Cu}+\left(1-\phi \right){\rho }_{bf}$$

(25)

where ϕ is the overall volume concentration of two different types of nanoparticles dispersed in hybrid nanofluid and is calculated as; $\phi ={\phi }_{{TiO}_{2}}+{\phi }_{Cu}$, and the heat capacitance of the nanofluid given is as,

Table 1 Thermophysical properties of water, Copper, and Titanium [69,70,71]

$$(\rho {C}_{p}{)}_{nf}=\phi (\rho {C}_{p}{)}_{p}+(1-\phi )(\rho {C}_{p}{)}_{bf}$$

(26)

According to (6), heat capacity of hybrid nanofluid can be determined as follows:

$$(\rho {C}_{p}{)}_{hnf}={\phi }_{{TiO}_{2}}(\rho {C}_{p}{)}_{{TiO}_{2}}+{\phi }_{Cu}(\rho {C}_{p}{)}_{Cu}+(1-\phi )(\rho {C}_{p}{)}_{bf}$$

(27)

The thermal expansion coefficient of the nanofluid can be determined by:

$$(\rho \beta {)}_{nf}=\phi (\rho \beta {)}_{p}+(1-\phi )(\rho \beta {)}_{bf}$$

(28)

where ${\beta }_{bf}$ and ${\beta }_{p}$ are the coefficients of thermal expansion of the fluid and of thesolid fractions respectively.

Hence, for hybrid nanofluid, thermal expansion can be defined as follows:

$$(\rho \beta {)}_{hnf}={\phi }_{{TiO}_{2}}(\rho \beta {)}_{{TiO}_{2}}+{\phi }_{Cu}(\rho \beta {)}_{Cu}+(1-\phi )(\rho \beta {)}_{bf}$$

(29)

Thermal diffusivity,${\alpha }_{nf}$ of the nanofluid isdefined as:

$${\alpha }_{nf}=\frac{{k}_{nf}}{(\rho {c}_{p}{)}_{nf}}$$

(30)

${k}_{nf}$ is the thermal conductivity of the nanofluid which for spherical nanoparticles, according to the Maxwell–Garnetts model [67], is:

$$\frac{{k}_{nf}}{{k}_{bf}}=\frac{({k}_{p}+2{k}_{bf})-2\phi ({k}_{bf}-{k}_{p})}{({k}_{p}+2{k}_{bf})+\phi ({k}_{bf}-{k}_{p})},$$

(31)

Thus, thermal diffusivity, ${\alpha }_{hnf}$ of the hybrid nanofluid can be defined as;

$${\alpha }_{hnf}=\frac{{k}_{hnf}}{(\rho {C}_{p}{)}_{hnf}},$$

(32)

If the thermal conductivity of hybrid nanofluid is defined according to Maxwell model,

$$\begin{aligned} \frac{{k_{hnf} }}{{k_{bf} }} & = \left( {\frac{{\left( {\phi_{{TiO_{2} }} k_{{TiO_{2} }} + \phi_{Cu} k_{Cu} } \right)}}{\phi } + 2k_{bf} + 2\left( {\phi_{{TiO_{2} }} k_{{TiO_{2} }} + \phi_{Cu} k_{Cu} } \right) - 2\phi k_{bf} } \right) \\ & \quad \times \left( {\frac{{\left( {\phi_{{TiO_{2} }} k_{{TiO_{2} }} + \phi_{Cu} k_{Cu} } \right)}}{\phi } + 2k_{bf} - \left( {\phi_{{TiO_{2} }} k_{{TiO_{2} }} + \phi_{Cu} k_{Cu} } \right) + \phi k_{bf} } \right)^{ - 1} \\ \end{aligned}$$

(33)

The effective dynamic viscosity of the nanofluid based on the Brinkman model [68] is given by

$${\mu }_{nf}=\frac{{\mu }_{bf}}{(1-\phi {)}^{2.5}}$$

(34)

where μ_bf is the viscosity of the fluid fraction, then the effective dynamic viscosity of the hybrid nanofluid is;

$${\mu }_{hnf}=\frac{{\mu }_{bf}}{{\left(1-\left({\phi }_{{TiO}_{2}}+{\phi }_{Cu}\right)\right)}^{2.5}},$$

(35)

and the effective electrical conductivity of nanofluid was presented by Maxwell as

$$\frac{{\sigma }_{nf}}{{\sigma }_{bf}}=1+\frac{3\left(\frac{{\sigma }_{p}}{{\sigma }_{bf}}-1\right)\phi }{\left(\frac{{\sigma }_{p}}{{\sigma }_{bf}}+2\right)-\left(\frac{{\sigma }_{p}}{{\sigma }_{bf}}-1\right)\phi }$$

(36)

and the effective electrical conductivity of hybrid nanofluid is;

$$\frac{{\sigma }_{hnf}}{{\sigma }_{bf}}=1+\frac{3\left(\frac{\left({\phi }_{{TiO}_{2}}{\sigma }_{{TiO}_{2}}+{\phi }_{Cu}{\sigma }_{Cu}\right)}{{\sigma }_{bf}}-\left({\phi }_{{TiO}_{2}}+{\phi }_{Cu}\right)\right)}{\left(\frac{\left({\phi }_{{TiO}_{2}}{\sigma }_{{TiO}_{2}}+{\phi }_{Cu}{\sigma }_{Cu}\right)}{\phi {\sigma }_{bf}}+2\right)-\left(\frac{\left({\phi }_{{TiO}_{2}}{\sigma }_{{TiO}_{2}}+{\phi }_{Cu}{\sigma }_{Cu}\right)}{{\sigma }_{bf}}-\left({\phi }_{{TiO}_{2}}+{\phi }_{Cu}\right)\right)}$$

(37)

3 Numerical method and validation

A grid transformation is accomplished to perform the numerical contributions at an appropriate computational domain which accelerates the convergence of the approximated solutions and averts the numerical errors yielded sometimes from the accumulation of the digital truncations during the improvement of numerical iterations. The variables domain needs to abstain from the geometrical complexity of wavy boundaries, stagnation points, and critical points which often cause disturbances in the approximation solution. This issue could be more realistic from the physical domain view but it sacrifices the stability of the solution environment. Therefore, such an approved computational domain is used with the finite volume method presented in [72, 73] which is extended here to the non-orthogonal grids. This technique starts with definitions of the new coordinates (ξ, η) as following:

$$X_{1} = A\left( {1 - \cos (2\pi \lambda Y} \right),\quad X_{2} = 1 - A\left( {1 - \cos (2\pi \lambda Y} \right),\quad \xi = \frac{{X - X_{1} }}{{X_{2} - X_{1} }},\quad \eta = Y$$

(38)

Using Eq. (38), the irregular physical model is mapping to a rectangular computational model as it is shown in Fig. 2. The resulting transformed system is treated using the finite volume method. Here, the advection terms are evaluated using "the second upwind scheme", whilst the Laplace operators are handled using the central difference methods. Iteratively utilizing ADI (alternating direction implicit), the resultant algebraic structure is solved. This study's convergence criteria were selected to be systematic. Additionally, the size of the grid is discovered to be appropriate for all computations after doing several grid tests. Additionally, Fig. 3 compares the outcomes of the current simulation to those attained by Cheong et al. [74] in certain instances (ϕ = φ = Ha = Q = 0). It is discovered that the outcomes are in good accordance with one another. Furthermore, another comparison test is performed with the results of Khanafer and Chamkha [75] where the mixed convection due to the moving of the upper wall of a square enclosure has been examined. Table 2 shows an excellent agreement between the results are obtained.

Fig. 2

Mapping between physical and computational models

Fig. 3

Comparison between the present results (right) and those obtained by Cheong et al. [74] (left)

Table 2 Comparisons of the horizontal velocity with those of Khanafer and Chamkha [75] at Gr = 100 and Pr = 0.71, $\phi =0\%.$

4 Results and discussion

In this section, the obtained results are presented step-by-step in terms of key parameters such as the heat generation parameter (Q) is varied from 0.1 to 1, the effects of Hartmann number (Ha) is varied from 0 to 150, the radiation parameter 0–5, the porosity parameter (ε) varied as 0.4 to 0.9, the inter-phase heat transfer coefficient (H^*) is varied from 0.005 to 50, the Darcy number ($Da$) is varied from 10^–1 to 10^–5, Prandtl number Pr = 6.2, and the nanoparticle volume fraction (ϕ) is varied from 0.01 to 0.05. The obtained results have been illustrated using streamlines, isotherms of fluid phase, isotherms of solid phase, isoconcentrations and Local and average Nusselt and Sherword numbers. Except for the variations in the respective figures, these values are considered to be fixed on the entire computation; Ha = 10, ϕ_Cu = ϕ_TiO2 = ϕ/2, ϕ = 0.05, Q = 1, λ = 2, α = Φ = π/3, Da = 10^–3, k_r = l, λ_d = λ_t = 1, and γ = 0.5.

It is important to glance on the prominent terms at the governing equations. For example for momentum conservation equations, the surface forces are represented by the pressure components at the porous medium, while the body forces come from gravitational term which the linear combination of both heat and nanoparticles concentration and the magnetic field. The term H^*(θ_s − θ_f) in energy equation points to the internal heat exchange between the solid and fluid phases which characterizes the non-thermal equilibrium at porous medium.

For streamlines as in Fig. 4, there are two vortices of the utmost stream function that are concentrated adjacent to the mobile surfaces (top and bottom). The unnormal issue is the stagnant fluid around the heat source which means that the forced convection dominates more regions than the natural convection. It could be back to the value of Richardson number where “Ri (→ 1)” in this study which is relatively small. The effect of heat generation parameter “Q” is weak due to the non-thermal equilibrium. For the fluid isotherms, two vortices are established near to those of streamlines. Still the amazing event is the cold fluid around the heat source region meaning that the supremum fluid temperature is created basically from forced convection. The solid isotherms are different than fluid isotherms at the location. The isotherms are two adjoining vortices and their centers (maximum solid temperature values) are away from those of streamlines because the solid isotherms approaches to the heat source. The two vortices are united at the left half may be due to the numerical calculations or the effect of the cavity inclination angle. The two solid vortices appeal to make a circle around the heat source. For the interpretation; the non-thermal equilibrium enhances the inconsistency between the fluid and solid isotherms distributions. We can see that fluid temperature is greatly affected by the movable sides while the solid temperature is influenced in the heat source position. the mutual effect between solid and fluid phase could not be neglected but it doesn’t represents the basic factor of heat transmission here. Moreover, the isoconcentrations contours are concentrated at the bottom-left corner due to the effect of inclination angle “α”.

Fig. 4

Effect of heat generation variation (Q = {0.1,0.2,0.3,0.4,0.5, 1}from left to right) on contours of streamlines (1st row), isotherms of fluid phase (2nd row), isotherms of solid phase (3rd row), isoconcentrations (4th row), for TiO₂–Cu/water hybrid nanofluid at the moderate values; Ha = 10, Da = 10^–3, Ri = 1, Gr_c = 1, H^* = 10, R_d = 0.5, ε = 0.5, γ = 0.5, Φ = π/3, ϕ_Cu = ϕ_TiO2 = ϕ/2, ϕ = 0.05

The fluid Nusselt number along the bottom side (Fig. 5) scores nonzero values at the origin point of the cavity which is the conjunction point of movable bottom side and wavy left side. Besides, Nusselt number completely vanishes when arrives the right side so the left side is so active region while the right corner is semi isolated region at the cavity may be due to the effect of the inclination angle. Normally, As “Q” increases the Nusselt number increases (Fig. 6). On the other hand, the solid Nusselt number along the bottom side is represented by a curve of the second order which is symmetric around the heat source location. Consequently, this represents an evidence of the robust dependency between the heat source position and the heat transfer at the solid phase which has been mentioned above at the contours discussion. In addition, the average Nusselt number (Fig. 7) for the fluid phase is larger (~ 100%) than that of the solid phase, which reflex the strong fluid response to the Q parameter more than the solid phase.

Fig. 5

Heat generation heat transfer parameter profiles of the local Nusselt number of fluid phase for TiO₂–Cu/water Hybrid Nanofluid at Ha = 10,Da = 10^–3, Ri = 1, Gr_c = 1, H^* = 10, R_d = 0.5, ε = 0.5, γ = 0.5, Φ = π/3, λ = , α = , ϕ_Cu = ϕ_TiO2 = ϕ/2, ϕ = 0.05

Fig. 6

Heat generation heat transfer parameter profiles of the local Nusselt number of solid phase for TiO₂–Cu/water Hybrid Nanofluid at Ha = 10, Da = 10^–3, Ri = 1, Gr_c = 1, H^* = 10, R_d = 0.5, ε = 0.5, γ = 0.5, Φ = π/3, λ = , α = , ϕ_Cu = ϕ_TiO2 = ϕ/2, ϕ = 0.05

Fig. 7

Heat generation heat transfer parameter profiles of the average Nusselt number of fluid phase (top) and solid phase (bottom) for TiO₂–Cu/water Hybrid Nanofluid at Ha = 10, Da = 10^–3, Ri = 1, Gr_c = 1, H^* = 10, R_d = 0.5, ε = 0.5, γ = 0.5, Φ = π/3, λ = , α = , ϕ_Cu = ϕ_TiO2 = ϕ/2, ϕ = 0.05

The effect of Hartmann number appears as a feedback of the inclination angle of the magnetic field used in the model as it is shown at Fig. 8 for the streamlines. Consequently, the magnetic field enhances the contours deformation along the second diagonal of the square cavity. Accordingly, as “Ha” increases, the supreme isotherms extend through the left side of the cavity (because of the cavity inclination angle effect too) and join to each other. Therefore, the magnetic field supports the thermal equilibrium case where the fluid and solid isotherms are congruent. Iso-concentrations contours show a slightly reasonable response to the isotherm’s dynamics may be because both of them is added to momentum conservation equations as a linear component. Fluid Nusselt number along the bottom boundary (Fig. 9) shows variations at the left side of the cavity as “Ha” changes. By the way, the difference between left and right halves is discussed above (Fig. 6). The noticeable point here is that Hartmann number suppresses the heat transfer (more than ~ 18%); meaning that the magnetic field has a negative effect on the forced convection which clearly appears at Fig. 9 for the average fluid Nusselt number. Therefore, the magnetic field works for the equilibrium between the forced and natural convection regimes.

Fig.8

Effect of Hartmann number variation (Ha = {0, 50, 75, 100, 150}from left to right) on contours of streamlines (1st row), isotherms of fluid phase (2nd row), isotherms of solid phase (3rd row), isoconcentrations (4th row), for TiO₂–Cu/water hybrid nanofluid at the moderate values; Q = 1, Da = 10^–3, Ri = 1, Gr_c = 1, H^* = 10, R_d = 0.5, ε = 0.5, γ = 0.5, Φ = π/3, λ = , α = , ϕ_Cu = ϕ_TiO2 = ϕ/2, ϕ = 0.05

Fig. 9

Hartmann number for heat transfer profiles of the local Nusselt number of fluid phase (left) and average values (right) for TiO₂–Cu/water Hybrid Nanofluid at Q = 1, Da = 10^–3, Ri = 1, Gr_c = 1, H^* = 10, R_d = 0.5, ε = 0.5, γ = 0.5, Φ = π/3, λ = , α = , ϕ_Cu = ϕ_TiO2 = ϕ/2, ϕ = 0.05

Solid Nusselt number (Fig. 10) along the bottom shows a somewhat change near to the midpoint as the radiation parameter increases while it keeps as a latent at the limits. The fluid average Nusselt number (Fig. 11) discovers the inhibitory role of the radiation which could be due to the Titanium nanoparticles characters while for the solid mean Nusselt number is vice versa. As we can see, the fluid Nusselt number decreases about 10% while the solid Nusselt number increases about ~ 13%. Figure 12 reflects again the opposite behavior of heat transfer at the solid and fluid phases. It’s found that the solid average Nusselt number is enhanced by “H^*” while it decays for the fluid phase. The reason could be referred to some factors such as the thermophysical properties of the used nanoparticles, the dominant convection regime, or the boundary condition effects.

Fig. 10

Radiation heat transfer parameter profiles of the local Nusselt number of solid phase along the bottom (left figure) and the left side (right figure) for TiO₂–Cu/water Hybrid Nanofluid at Ha = 10, Q = 1, Da = 10^–3, Ri = 1, Gr_c = 1, H^* = 10, ε = 0.5, γ = 0.5, Φ = π/3, λ = , α = , ϕ_Cu = ϕ _TiO2 = ϕ/2, ϕ = 0.05

Fig. 11

Radiation heat transfer parameter profiles of the average Nusselt number of fluid phase (a) and solid phase (b) for TiO₂–Cu/water nanofluid at Ha = 10, Q = 1, Da = 10^–3, Ri = 1, Gr_c = 1, H^* = 10, ε = 0.5, γ = 0.5, Φ = π/3, λ = , α = , ϕ_Cu = ϕ _TiO2 = ϕ/2, ϕ = 0.05

Fig. 12

Internal heat transfer parameter profiles of the average Nusselt number of fluid phase (left) and solid phase (right) for TiO₂–Cu/water nanofluid at Ha = 10, Q = 1, Da = 10^–3, Ri = 1, Gr_c = 1, R_d = 0.5, ε = 0.5, γ = 0.5, Φ = π/3, λ = , α = , ϕ_Cu = ϕ _TiO2 = ϕ/2, ϕ = 0.05

Streamlines increase as the porosity increases (Fig. 13) which agrees with the natural behavior. On the other hand, Fig. 14 shows that the average Nusselt number decreases as the porosity increases, because of the forced convection decay. Exhaustively, the average fluid Nusselt falls down ~ 60% while the average solid Nusselt descends only ~ 6%. In the same context, Fig. 15 shows the Darcy number effect, where “Da” tracks the permeability at the unit area intersection. Decreasing “Da” means weak forced convection which clearly appears from streamlines. Consequently, the supreme isotherms vortices transfer from the top/bottom boundaries to the circular source at the center. The amazing phenomenon is that the concentration descends all over the boundaries while grows up around the circle heat source.

Fig. 13

Effect of the porosity (ε = {0.4, 0.5, 0.6, 0.7, 0.9}from left to right) the isotherms of fluid phase for TiO₂–Cu/water nanofluid at the moderate values; Ha = 10, Q = 1, Da = 10^–3, Ri = 1, Gr_c = 1, H.^* = 10, R_d = 0.5, γ = 0.5, Φ = π/3, λ = , α = , ϕ_Cu = ϕ _TiO2 = ϕ/2, ϕ = 0.05

Fig. 14

Porosity parameter profiles of the average Nusselt number of fluid phase (left) and solid phase (right) for TiO₂–Cu/water Hybrid Nanofluid at Ha = 10, Q = 1, Da = 10^–3, Ri = 1, Gr_c = 1, H^* = 10, R_d = 0.5, γ = 0.5, Φ = π/3, λ = , α = , ϕ_Cu = ϕ _TiO2 = ϕ/2, ϕ = 0.05

Fig.15

Effect of Darcy number variation (Da = {10^–1, 10^–2, 10^–3, 10^–4, 10^–5}from left to right) on contours of streamlines (1st row), isotherms of fluid phase (2nd row), isoconcentrations (3rd row), for TiO₂–Cu/water hybrid nanofluid at the moderate values; Ha = 10, Q = 1, Ri = 1, Gr_c = 1, H^* = 10, R_d = 0.5, ε = 0.5, γ = 0.5, Φ = π/3, λ = , α = , ϕ_Cu = ϕ _TiO2 = ϕ/2, ϕ = 0.05

It’s known that the augmentation for the Richardson number supports the natural convection [76], therefore the streamlines and isotherms are concentrated around the heat source (Fig. 16). As a consequence the fluid Nusselt number is depressed ~ 23% while the solid Nusselt number arises ~ 7% (Fig. 17). Th interpretation is for the large “Ri”, the natural convection dominates the fluid flow regime, so the fluid temperature difference between the heat source and the other sides decreases in general, but for solid temperature difference is still great due to the non-thermal equilibrium condition. Ultimately, for the cavity geometry, Fig. 18 shows the effect of wave amplitude. As shown, for small A, the contours present uniform vortices as the fluid mass increases. As “A” increases, the average fluid Nusselt number (Fig. 19) decreases ~ 15% because as “A” increases, the fluid flow is hampered. Therefore, the convection regime is deficient and fluid Nusselt number descends.

Fig. 16

Effect of Richardson number variation (Ri = {0.5,0.8,1,10,100} on contours of streamlines (1st row), and isotherms of fluid phase (2nd row), for TiO₂–Cu/water hybrid nanofluid at the moderate values; Ha = 10, Q = 1, H^* = 10, R_d = 0.5, ε = 0.5, γ = 0.5, Φ = π/3, λ = , α = π/3, ϕ_Cu = ϕ_TiO2 = ϕ/2,ϕ = 0.05

Fig. 17

Richardson number profiles of the average Nusselt number of fluid phase (left) and solid phase (right) for TiO₂–Cu/water Hybrid Nanofluid at Ha = 10, Q = 1, Da = 10^–3, Gr_c = 1, H^* = 10, R_d = 0.5, ε = 0.5, γ = 0.5, Φ = π/3, λ = , α = , ϕ_Cu = ϕ _TiO2 = ϕ/2, ϕ = 0.05

Fig. 18

Effect of wave amplitude variation (1st row: A = 0.05, 2nd row: A = 0.15) on the streamline, isotherms of fluid phase, isotherms of solid phase, isoconcentration (from left to right respectively) for TiO₂–Cu/water hybrid nanofluid at the moderate values; Ha = 10, Q = 1, Da = 10^–3, Ri = 1, Gr_c = 1, H^* = 10, R_d = 0.5, ε = 0.5, γ = 0.5, Φ = π/3, λ = , α = , ϕ_Cu = ϕ_TiO2 = ϕ/2, ϕ = 0.05

Fig. 19

Wave amplitude parameter profiles of the fluid average Nusselt number for TiO₂–Cu/water hybrid nanofluid at Ha = 10, Q = 1, Da = 10^–3, Ri = 1, Gr_c = 1, H^* = 10, R_d = 0.5, ε = 0.5, γ = 0.5, Φ = π/3, λ = , α = , ϕ_Cu = ϕ_TiO2 = ϕ/2, ϕ = 0.05

5 Conclusion

The mixed convection phenomenon of hybrid nanofluids within an undulating porous cavity has been investigated within this paper. The contours of the streamlines, isotherms of both fluid and solid phases and isoconcentrations have been inspected as well as the profiles of local and average Nusselt number of the fluid/solid phase under local thermal non-equilibrium case at various key parameters such as the coefficient of heat generation/absorption Q, Hartmann number Ha, porosity parameter ε, an inter-phase heat transfer coefficient H^*, undulation parameter λ, Darcy parameter Da, magnetic field inclination angle Φ, and hybrid nanofluid parameter ϕ. The remarkable points at the thermodynamic model under the study could be concluded as:

• The hybrid nanofluid around the circular heat source is somewhat clotted where there is low temperature and very slow motion,

• This infers that the isolated wavy sides put down the buoyancy force which is the main pillar of natural convection,

• The local thermal non-equilibrium case (LTNE) plays a prominent role as it could create some isolated regions through the entire cavity domain. Therefore, the fluid isotherms are pulled toward the movable surfaces while the solid isotherms concentrated close to the central heat source,

• Consequently, the heat transfer is not accompanied with the mass transfer. Moreover, there is somewhat an isolation between the nanofluid mass and the solid matrix,

• The driven lids mounting at top and bottom surfaces enforces the forced convection dominancy,

• In this arrangement, the circular heat source serves as a thermal barrier and flow resistance.

• The fluid response to heat generation “Q” is much stronger (~ 100%) than the solid response,

• The magnetic field strengthens the thermal equilibrium state which satisfies the balance between the natural convection and the forced convection,

• The thermal effect of the parameters {Ha, R_d, H*, Ri} is reverse when the phase changes (solid/fluid). While the effect of {Q, ε} doesn’t change. This because the last two factors don’t change the convection regime. However, the former parameters are either directly related to the LTNE case or change the dominant regime obviously.

• The average fluid Nusselt number descent due to the porosity is ~ 60% which represents ten times the of that at the solid descent.

• Iso-concentrations contours show a slightly reasonable response to the isotherm’s dynamics when the magnetic field varies.

• The magnetic field has a negative effect on the forced convection.

• The average Nusselt number decreases as the porosity increases.