# Thermal Convective Instabilities and Chaos in a Rotating Hybrid Nanofluid Layer with Cattaneo–Christov Heat Flux Model

Authors: Sèmako Justin Dèdèwanou; Adjimon Vincent Monwanou; Aimé Audran Koukpémèdji; Amoussou Laurent Hinvi; Clément Hodévèwan Miwadinou; Jean Bio Chabi Orou
Journal: Complexity
 (2022)

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## Abstract

The linear and nonlinear dynamics of thermal convection of a rotating hybrid nanofluid layer heated from below with the Cattaneo–Christov heat flux model are studied in this paper. Starting from the flow equations of a hybrid nanofluid and exploiting the free boundary conditions, the analytical expressions of the stationary and oscillatory Rayleigh numbers of the base fluid are determined as a function of the dimensionless parameters of the heat transfer fluid and the thermophysical properties of the hybrid nanofluid. The effects of hybrid nanoparticles and Taylor number on the onset of stationary convection in the base fluid are investigated graphically. Then, a numerical study of the transition from natural convection to chaotic behaviour of the hybrid nanofluid is made using the truncated Galerkin approximation. This approximation allowed us to find a novel six-dimensional nonlinear system depending on the parameters of the base fluid and the thermophysical properties of the hybrid nanofluid that can be reduced to five, four, or three dimensions when we tend some parameters to zero. The different results showed that the addition of hybrid nanoparticles (alumina-copper) to a thermal fluid (water) subjected to the rotation force in the presence or absence of the thermal relaxation time allows control of the chaotic convection in the base fluid. On the other hand, the increase of the rescaled Taylor number and the Cattaneo number widens the domain of chaos in the hybrid nanofluid with the increase of the rescaled Rayleigh number of the base fluid.


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## Body

## 1. Introduction

In 1995, Choi introduced, at the Argonne National Laboratory of the University of Chicago in the U.S., the concept of nanofluid [1]. This new generation of fluids consists of dispersing nanoparticles (assemblies of a few hundred to a few thousand atoms, leading to an object with at least one dimension of thousands of atoms, leading to an object of which at least one of the dimensions is of nanometric size) in a base liquid (water, oil, ethylene glycol, toluene). The use of these nanofluids in some industrial operations that involve heat transfer by convection is a promising alternative solution to improve thermal performance. Thus, the problem of natural convection in a nanofluid layer heated from below has been studied by several researchers [2–6] with the Fourier law. The flow and heat transport of nanomaterial with quadratic radiative heat flux and aggregation kinematics of nanoparticles reported by Mahanthesh [7] revealed that the suspension of the nanoparticles increases the thermal conductivity and, thus, improves the temperature and reduces the heat flux at the plate. The Rayleigh-Bénard convection in nanofluid submerged with dust particles was investigated by Shalini and Mahanthesh [8]. They pointed out that the inclusion of nano and dust particles reduces the Rayleigh number while the rotation postpones the onset of convection and stabilizes the system.Ahuja and Sharma [9] conducted a comprehensive review of the instability of Rayleigh-Bénard convection in nanofluids by summarizing in their work the studies related to the instability of a horizontal nanofluid layer under the impact of various parameters such as rotation magnetic field, Hall currents, and LTNE (local thermal nonequilibrium) effects in porous and nonporous media. The thermal convection in a rotating fluid layer provides a system to study hydrodynamic instabilities, pattern formation, and spatio-temporal chaos in nonlinear dynamical systems with many practical applications in engineering, such as food processing, chemical processes, solidification, centrifugal casting of metals, and rotating machines [10].To compensate for the defect and all the disadvantages of mono nanofluids, it is essential to combine several nanoparticles to prepare a hybrid nanofluid [11]. Natural magnetohydrodynamic convection in a triangular cavity filled with a hybrid (copper-alumina)/water nanofluid with localized heating from below and internal heat has been investigated by Rashad et al. [12]. They concluded that a hybrid nanofluid composed of equal amounts of copper and alumina nanoparticles dispersed in water has no significant improvement on the average Nusselt number compared to the mono nanofluid and that the effect of increasing the hybrid nanoparticles becomes significant in the case where natural convection is very low. Aladin et al. [13] also studied the significant effects of suction and magnetic field on a moving plate containing a hybrid (copper-alumina)/water nanofluid. They have proved that the hybrid nanofluid gives better results than the mono nanofluid.According toMyson and Mahanthesh [14], the hybrid nanofluid delays the convection and will further enhance the heat transfer rate, but the Casson parameter advances the convection while reducing the heat transfer rate. Mackolil and Mahanthesh [15] illustrated the optimization of heat transfer in the thermal Marangoni and nonlinear convective flow of a hybrid nanomaterial with sensitivity analysis. It is shown that the hybrid nanomaterial possesses enhanced thermal fields for nanoparticle volume fractions less than 0.02. The sensitivity computation of nonlinear convective heat transfer in hybrid nanomaterial between two concentric cylinders with irregular heat sources was also made by Thriveni and Mahanthesh [16].Given its advantages and industrial applications, especially in chemical reactions, biological systems, crystal production, petroleum reservoir modeling, and packed-bed catalytic filtration, chaotic convection in a hybrid nanofluid layer should receive considerable attention due to the performance of nanofluids. Jawdat et al. [17], Moaddy et al. [18, 19], and Bhardwaj and Chawla [20] all contributed well to nonlinear dynamical analysis of the thermal convection in a horizontal nanofluid layer heated from below in the presence or absence of a magnetic field. They studied the effect of nanoparticles on chaotic convection in a layer of fluid (water) heated from below and noticed that the stability region can be increased by using nanofluids and that the onset of chaotic convection can be delayed under the influence of nanoparticles. Also, variations in temperature and magnetic field strength cause the system to transition from a steady state to chaos and back to a steady state. The case of hybrid nanofluid was first presented by Dèdèwanou et al. [21] with the Fourier law. They found that the copper nanofluid makes it possible to quickly switch from chaotic to periodic regimes compared to the alumina nanofluid, and the use of hybrid nanoparticles allows further control of the chaos in the base fluid by expanding the convective flow and reducing the chaotic flow.Furthermore, Maxwell and Cattaneo modified Fourier’s law by taking into account the aspect of thermal relaxation time in the propagation of heat [22]. In order to eliminate the heat flow and thus obtain a single equation for the temperature field, Christov [23] proposed a generalization of the material-invariant Maxwell–Cattaneo law, in which the relaxation time of the heat flow is given by the convex Oldroyd upper derivative. This new law was used by Straughan [24] to study thermal convection in an ordinary fluid. He concluded that the thermal relaxation time is significant if the Cattaneo number is large enough, and the convection mechanism changes from stationary to oscillatory convection with narrower cells. Indeed, some researchers used the Cattaneo–Christov model to appreciate the effects of temporal relaxation on the thermal behavior of a nanofluid [25–31]. Alebraheem and Ramzan [26] have studied the heat and mass transfer of Casson nanofluid flow containing gyrotactic microorganisms past a swirling cylinder by considering the Cattaneo-Christov heat flux model. According to their numerical solution of the subject system, which is framed via the bvp4c technique of MATLAB software, the concentration of the fluid is reduced owing to the increase in values of the brownian motion parameter and local Reynolds number, but the diminishing density of microorganisms is perceived for mounting estimates of the bioconvection Péclet number. Multiple slip impacts in the MHD hybrid nanofluid flow with Cattaneo–Christov heat flux and autocatalytic chemical reaction were investigated by Gul et al. [32]. They found that the fluid temperature is diminishing function of the thermal slips parameters but increased for high estimates of the heat source-sink and nanoparticle volume concentration parameters while entropy number augmented for higher thermal relaxation parameter and Reynolds number. Lu et al. [33] have also studied a thin film flow of nanofluid comprising carbon nanotubes influenced by Cattaneo–Christov heat flux and entropy generation. They showed that the velocity and temperature distributions increase as the solid volume fraction escalates. Recently, a three-dimensional flow of gold-silver/engine oil hybrid nanofluid owing to a rotating disk of variable thickness with Cattaneo–Christov heat flux has been addressed by Zhang et al. [34]. They proved that the performance of the hybrid nanofluid is far better than the common nanofluid according to the surface temperature and heat transfer rate. This same remark is made from the results obtained for the model-based comparative study of magnetohydrodynamics unsteady hybrid nanofluid flow between two infinite parallel plates with particle shape effects [35]. Considering hybrid nanofluid Yamada-Ota and Xue flow models in a rotating channel with the modified Fourier law, it is observed that the velocity profile decreases for the higher rotation parameter while it increases for the escalated slip parameter, but the fluid concentration and temperature are on the decline for higher surface catalyzed reaction and thermal relaxation parameters respectively [36]. Ramdan et al. [37] have analyzed the hydrodynamic and heat characteristics of the three-dimensional flow of a steady, laminar, and incompressible convective graphene-copper oxide/water and graphene-silver/water hybrid nanofluids (used as a solar energy absorber) with varied particle shapes in a porous medium. Their study revealed that the rotational parameter has declined the velocity profiles but enhanced the temperature profiles, and the decline effect is significant in the case of graphene-copper oxide/water whereas the enhancement effect of temperature is significant for graphene-silver/water. A comparative analysis of magnetized partially ionized copper, copper oxide-water, and kerosene oil nanofluid flow with Cattaneo–Christov heat flux was made by Abid et al. [38]. They noted the greater effective thermal conductivity for copper-water partially ionized nanofluid as compared to other given partially ionized nanofluids (copper-kerosene oil, copper oxide-water/kerosene oil partially ionized nanofluids). Ramzan et al. [39] developed a mathematical model for the nanofluid flow containing carbon nanotubes with ethylene glycol as a base fluid in a rotating channel with an upper permeable wall by adding the Cattaneo–Christov heat flux’s impact with thermal stratification. The displacement of the lower plate at variable velocity, caused by the rotation of the fluid, produces forced convection with rotation and centripetal impact. Nevertheless, the upper plate is porous. Chu et al. [40] investigated a numerical solution for MHD Maxwell nanofluid with gyrotactic microorganisms, a higher-order chemical reaction in the presence of variable source/sink, and Newtonian heating in a rotating flow on a deformable surface and noted that on incrementing the conjugate heat parameter and thermal relaxation time, the rate of heat transfer augments, but the rate of heat transfer decreases on varying the fluid relaxation time.Despite all the above, the study of nanoparticles requires more attention due to their industrial uses. After inspecting the scientific literature, we noted that no work has yet addressed the chaotic aspect of thermal convection in hybrid nanofluids, taking into account the thermal relaxation time, although this would be very useful in some applications like petroleum reservoir modeling, chemical reactions, thermal transport in biological tissue, and surgical operations. Nevertheless, Layek and Pati [41] studied the effects of thermal lag on the onset of convection, its bifurcations, and the chaos of a horizontal layer of the heated Boussinesq fluid underneath via a five-dimensional nonlinear system. A comparative study of the five-dimensional system obtained for the case of a hybrid nanofluid was made by Dèdèwanou et al. [42]. Therefore, the objective of the present paper is to investigate the effects of hybrid nanoparticles on the occurrence of thermal convection instability and chaos in a rotating fluid layer heated from below with the Cattaneo–Christov heat flux model. A specific objective is to determine the analytical expression for the stationary Rayleigh number that can be used to study the nonlinear dynamics of thermal convection in rotating hybrid nanofluid flow in the presence of thermal relaxation time. This work aims to study the different transition regimes as a function of the thermophysical properties of nanofluids and then to show the effects of hybrid nanoparticles, Taylor number, and Cattaneo number on the chaotic behavior of natural convection in a basic fluid such as water via dynamical systems.In the next section, the thermal convection in an infinite horizontal rotating hybrid nanofluid layer with the hyperbolic Cattaneo–Christov heat flux is outlined. Section3 discusses the theory of conduction, stationary convection, and oscillatory convection, where we generalize and simplify the expression for the Rayleigh number by deriving a number of new analytical results. In order to reduce the set of equations governing the dynamic behavior of thermal convection in the hybrid (alumina-copper)/water nanofluid, discretized models in four and six dimensions are developed in Section 4 using the Galerkin expansion. We have studied the nature of the nonlinear dynamics of the obtained dynamical systems and determined the fixed points by analyzing the stability of the stationary solutions. These analyses have allowed us to justify the influence of the hybrid alumina-copper nanoparticle, the Cattaneo number, and the Taylor number on number on the transition from chaos to periodicity and vice versa in the fluid. In Section 5, we present the different simulations performed, and the results obtained are discussed. The conclusions are drawn in Section 6.


## 2. Mathematical Modeling

### 2.1. Problem Formulation

We consider an infinite rectangular cavity with two horizontal walls maintained at different temperatures. This cavity, heated from below with a thermal relaxation time, is filled with a hybrid nanofluid (water and nanoparticles), subject to gravity acting downwards and to rotation. In order to develop our numerical model, it is necessary to adopt certain assumptions, namely, the flow is assumed to be permanent and incompressible, the mixture is assumed to be homogeneous, single-phase, Newtonian, the nanoparticles are spherical, and the mass transfer between the particles and the fluid is negligible. The Cartesian coordinate system used is such that they-axis follows the horizontal and the vertical z-axis is collinear with gravity. The geometry of our problem is presented in Figure 1.Figure 1
Configuration of the problem.


### 2.2. Governing Equations

In this section, we have studied the equations governing the dynamic and thermal fluxes with boundary conditions and nondimensional parameters characterizing the thermal convection in a rotating hybrid nanofluid layer in the presence of thermal relaxation time. Taking into account the listed assumptions and using the hybrid nanofluid model proposed in Section3, the equations governing the conservation of mass, momentum, energy, and heat flux for a laminar flow of the hybrid nanofluid are written in their dimensional form respectively as follows [10, 42]:(1)∂vi∂xi=0,(2)ρhf∂vi∂t+vj∂vi∂xj=−∂p∂xi−ρgei+μhf∇2vi+2ρhfΩj∂vi∂xjei,(3)ρCphf∂T∂t+vj∂T∂xj=−∂Qj∂xj,(4)∂Qi∂t+vj∂Qi∂xj+Qi=−khf∂T∂xi.In equation (2), ei=0,0,1 is the unit vector and ∇2=∂2/∂x2+∂2/∂y2+∂2/∂z2 is the Laplacian operator. The use of the Boussinesq approximation allows us to define the density as a function of temperature as(5)ρ=ρhnf1−βhfT−T0.The density, thermal expansion coefficient, heat capacity, dynamic viscosity, thermal conductivity of the dynamic viscosity, and thermal conductivity of the hybrid nanofluid, respectively, are defined as follows:(6)ρhf=1−φ21−φ1ρf+φ1ρs1+φ2ρs2,ρβhf=1−φ21−φ1ρβf+φ1ρβs1+φ2ρβs2,ρCphf=1−φ21−φ1ρCpf+φ1ρCps1+φ2ρCps2,ρCphf=1−φ21−φ1ρCpf+φ1ρCps1+φ2ρCps2,μf=μhf1−φ1−2.51−φ2−2.5,khf=kgfks2+ηkgf−ηφ2kgf−ks2ks2+ηkgf+φ2kgf−ks2,to(7)khnf=khfks1+ηkhf−ηφ1khf−ks1ks1+ηkhf+φ1khf−ks1,where(8)η=m−1.The thermophysical properties of nanoparticles and water used in this work are summarized in Table1 (see [13]).Table 1
The thermophysical properties of the water and nanoparticles at 293 K.
ρCpkβ×10−5Water997.141790.61321Alumina3970765400.85Copper89333854011.67


## 2.1. Problem Formulation

We consider an infinite rectangular cavity with two horizontal walls maintained at different temperatures. This cavity, heated from below with a thermal relaxation time, is filled with a hybrid nanofluid (water and nanoparticles), subject to gravity acting downwards and to rotation. In order to develop our numerical model, it is necessary to adopt certain assumptions, namely, the flow is assumed to be permanent and incompressible, the mixture is assumed to be homogeneous, single-phase, Newtonian, the nanoparticles are spherical, and the mass transfer between the particles and the fluid is negligible. The Cartesian coordinate system used is such that they-axis follows the horizontal and the vertical z-axis is collinear with gravity. The geometry of our problem is presented in Figure 1.Figure 1
Configuration of the problem.


## 2.2. Governing Equations

In this section, we have studied the equations governing the dynamic and thermal fluxes with boundary conditions and nondimensional parameters characterizing the thermal convection in a rotating hybrid nanofluid layer in the presence of thermal relaxation time. Taking into account the listed assumptions and using the hybrid nanofluid model proposed in Section3, the equations governing the conservation of mass, momentum, energy, and heat flux for a laminar flow of the hybrid nanofluid are written in their dimensional form respectively as follows [10, 42]:(1)∂vi∂xi=0,(2)ρhf∂vi∂t+vj∂vi∂xj=−∂p∂xi−ρgei+μhf∇2vi+2ρhfΩj∂vi∂xjei,(3)ρCphf∂T∂t+vj∂T∂xj=−∂Qj∂xj,(4)∂Qi∂t+vj∂Qi∂xj+Qi=−khf∂T∂xi.In equation (2), ei=0,0,1 is the unit vector and ∇2=∂2/∂x2+∂2/∂y2+∂2/∂z2 is the Laplacian operator. The use of the Boussinesq approximation allows us to define the density as a function of temperature as(5)ρ=ρhnf1−βhfT−T0.The density, thermal expansion coefficient, heat capacity, dynamic viscosity, thermal conductivity of the dynamic viscosity, and thermal conductivity of the hybrid nanofluid, respectively, are defined as follows:(6)ρhf=1−φ21−φ1ρf+φ1ρs1+φ2ρs2,ρβhf=1−φ21−φ1ρβf+φ1ρβs1+φ2ρβs2,ρCphf=1−φ21−φ1ρCpf+φ1ρCps1+φ2ρCps2,ρCphf=1−φ21−φ1ρCpf+φ1ρCps1+φ2ρCps2,μf=μhf1−φ1−2.51−φ2−2.5,khf=kgfks2+ηkgf−ηφ2kgf−ks2ks2+ηkgf+φ2kgf−ks2,to(7)khnf=khfks1+ηkhf−ηφ1khf−ks1ks1+ηkhf+φ1khf−ks1,where(8)η=m−1.The thermophysical properties of nanoparticles and water used in this work are summarized in Table1 (see [13]).Table 1
The thermophysical properties of the water and nanoparticles at 293 K.
ρCpkβ×10−5Water997.141790.61321Alumina3970765400.85Copper89333854011.67


## 3. Linear Stability Analysis

We consider a classical Rayleigh-Bénard problem of linear stability of convective rolls in a horizontal fluid layer with unconstrained boundary conditions. Thus, the temperature boundary conditions areT=Tc at z = 0 and T=T0 at z = 1 with Tc>T0. As for the velocity, its component along the z axis is zero at the boundaries.


### 3.1. Steady-State Solutions

A time-independent quiescent solution of equations (1)–(4) with temperature and heat flux varying in the z direction only, is obtained by reducing equations (2)–(4) to(9)ρhfvj∂vi∂xj+∂p∂xi+ρgei−μhf∇2vi−2ρhfΩj∂vi∂xjei=0,ρCphfvj∂T∂xj+∂Qj∂xj=0,vj∂Qi∂xj+Qi+khf∂T∂xi=0.Then, the steady state solutions are given(10)vbz=0,Tb=Tc−χxjej,Qbz=χkhfandPbz=P0,with the temperature gradient χ and the profile of P0 defined by(11)χ=∂T∂xiand∂p0∂xi=ρTbgei.


### 3.2. Simplified Set of Equations

In order to simplify the parametric representation of this physical problem and to find the characteristic properties of the system, it is necessary to recast the flow equations. Thus, the following normalized quantities are introduced:(12)x˜i=xiH,v˜i=ρfH.viμf,P˜=H2.Pρf.νf,t∗=μftρf.H2,T˜=βfgkfH2χνfρCpfT,Q˜i=HkfT˜TQi.Considering small perturbations on the basic solutions as follows:(13)v˜i=v˜b+v˜i′,T˜=T˜b+T′,Q˜i=Q˜b+Qi′,P˜=P˜b+P˜′,and neglecting the products of the primed quantities, we obtain the following dimensional equations:(14)∂v˜i′∂x˜i′=0,(15)∂v˜i′∂t˜′=−∂P˜′∂x˜i′+μhfμfρfρhf∇2v˜i′+ρβhfρβfRafT˜ei+Ta∂v˜i′∂x˜jei,(16)ρCphfρCpfPrf∂T˜′∂t˜′−Rafw˜′=−∂Q˜i′∂x˜j′,(17)2PrfCf∂Q˜i′∂t˜′+Q˜i′=−khfkhf∂T˜′∂x˜i′,where the dimensionless parameters are defined by(18)Prf=μfρfαf,Cf=αfτ2H2,Raf=g.H4βfχαfνf.Now we eliminate the pressure from the nondimensional equations (14)–(17) by taking the curl-curl of equation (15), the divergence of equation (15), the inner product of any vector equation with ei;, and denoting the divergence of the heat flux Q=∂Q˜i/∂x˜i, to obtain, after dropping the tilde notation for brevity, the equations(19)∂∂t∇2w=Rafρβhfρβf∂2T∂x2+∂2T∂y2+μhfμfρfρhf∇4w+Ta∂ζ∂z,(20)ρCphfρCpfPrf∂T∂t−Rafw=−Q,(21)2PrfCf∂Q∂t˜′+Q=−khfkhf∇2T.which describe the evolution of the conduction steady state perturbations in a conveniently simplified form with four variables such as the z-component of the velocity field, the vorticity ζ=∂v/∂x−∂u/∂y, the heat flux Q, and the temperature T.An evolutionary equation for the vorticity can be obtained from the equation of motion by taking curl, then the dot product withe3 for the vertical component. Thus, eliminating the pressure and introducing the vorticity in equation (2) allows us to obtain the following:(22)μhfμfρfρhf∇2−∂∂tζ=−Ta∂w∂z.From equations (19) and (22), we obtain the following:(23)μhfμfρfρhf∇2−∂∂t∂∂t∇2−μhfμfρfρhf∇4+Ta∂2∂z2w−Rafρβhfρβfμhfμfρfρhf∇2−∂∂t∂2T∂x2+∂2T∂y2=0.


### 3.3. Normal Modes and Analytical Solution

The linear stability of the conduction solutions is studied by writing the perturbations in separable form and assuming an exponential time dependence(24)w=Wzhx,yeσt,T=Θzhx,yeσt,Q=Φzhx,yeσt,with the plane tiling function satisfying(25)∇2hx,y=−κ2hx,y,where W,Θ,Φ are eigenfunctions. The substitution of equation (24) into the differential equations (20), (21), and (23) leads to(26)D2−κ2γ1D2−κ2−σ2+TaD2W=κ2γ2Rafγ1D2−κ2−σΘ,(27)σPrfΘ=RafW−γ3Φ,(28)2σPrfCfΦ+Φ=−γ4D2−κ2Θ,where D=d/dt,D2=d2/dt2 and(29)γ1=μhfμfρfρhf,γ2=ρβhfρβf,γ3=ρCpfρCphf,γ4=khfkhf.Equations (26)–(28), which represent the starting point for analytical and numerical calculations on thermal convective instability are used to study the occurrence of stationary and oscillatory convection in nanofluids. They are equivalent to those obtained by Straughan [24] and Bissell [43, 44] in the case of an ordinary fluid not subjected to a Coriolis force, i.e. when φ1=φ2=0 and Ta=0. The disappearance of tangent shear stresses at the free surface and the conservation of the mass equation allow us to obtain the boundary conditions of the free surface defined by(30)W=0,D2W=0andΘ=0atz=0,1.In order to obtain an approximate solution of equations (26)–(28), we used the Galerkin weighted residual method by choosing the test function written as(31)W=W0sinπz,Θ=Θ0sinπz,Φ=Φ0sinπz,which fulfill the conditions at the borders mentioned in equation (30).By substituting the test functions defined in equation (31) into equations (26)–(28) and performing some integrations, we obtain the following matrix equation:(32)Jγ1J+σ+π2Taγ1J+σ−κ2γ2Raf0−RafσPrfγ30−γ4J2σPrfCf+1×W0Θ0Φ0=000,where J=D2−κ2. For this matrix equation (32) to admit a nontrivial solution, the Rayleigh number of the base fluid must be in the following form:(33)Raf=γ3γ4J+σPrf2σPrfCf+1Jγ1J+σ+π2Ta/γ1J+σκ2γ22σPrfCf+1.For the following, letσ=jω with j2=−1 and ω the real frequency. Thus, the expression for the Rayleigh number defined in equation (33) becomes(34)Raf=Δ1+jωΔ2,with(35)Δ1=κ2γ21+4ω2Prf2Cf2−1Jγ1γ3γ4J2−ω2Prf+π2Taγ1γ3γ4J2+ω2Prfγ1J2+ω2+2Jω2PrfCfγ3γ4J−2ω2Prf2Cf+2π2ω2PrfCfTa2ω2Prf2Cf−γ3γ4Jγ1J2+ω2,(36)Δ2=κ2γ21+4ω2Prf2Cf2−1J2γ3γ4+γ1J+π2JTaγ1−γ3γ4γ1J2+ω2+2JPrfCf2γ1Jω2Prf2Cf−γ1γ3γ4J2+2π2γ1JPrfCfTa2ω2Prf2Cf−γ3γ4Jγ1J2+ω2.Since the Rayleigh number is a real and positive physical quantity, then for the expression equation (33) to exist, ω must be zero or Δ2=0.


### 3.4. Stationary Convection

According to the stability exchange principle for the stationary case, the stability margin is characterized by the frequency equal to zero. This condition allows to obtain from the expression equation (33), the Rayleigh number of the base fluid of the stationary convection expressed as follows:(37)Rafst=γ3γ4κ2γ2γ1κ2+π23+π2Taγ1.Ifγ1=γ2=γ3=γ4=1 and the cavity is not rotating, i.e. φ1=φ2=0 and Ta=0, equation (37) is equivalent to the classical Rayleigh number of stationary convection in ordinary fluids [42].In the absence of the rotation force, we have(38)Rafst=γ1γ3γ4γ2κ2+π23κ2.The absolute critical Rayleigh number of the hybrid nanofluid in this case is defined as(39)Rafst=27π44γ1γ3γ4γ2,with the corresponding wavenumber(40)κcst=π2.We note that the Rayleigh number obtained for stationary convection in nanofluids is not a function of the Prandtl number or the Cattaneo number of the base fluid. Thus, the same results can be associated with the more usual Fourier law for mono nanofluids or hybrid nanofluids.


### 3.5. Oscillatory Convection

Now, we study the effects of the Cattaneo number, the nanoparticles, and the rotation on the oscillating convection. In this case, we must haveω≠0 and Δ2=0. Therefore, the Rayleigh number of the base fluid for oscillatory convection is given by(41)Rafos=κ2γ21+4ω2Prf2Cf2−1Jγ1γ3γ4J2−ω2Prf+π2Taγ1γ3γ4J2+ω2Prfγ1J2+ω2+2Jω2PrfCfγ3γ4J−2ω2Prf2Cf+2π2ω2PrfCfTa2ω2Prf2Cf−γ3γ4Jγ1J2+ω2.The corresponding oscillatory frequencyω must verify the following equations:(42)4γ1J2Prf3Cf2ω4+J2γ3γ4+γ1Prf+4γ1JPrf3Cf2γ1J3+π2Ta−2γ1γ3γ4PrfCfJ3ω2+γ12Jγ3γ4+γ1Prf+π2JTaγ1−γ3γ4−2γ1γ3γ4PrfCfJ2γ3γ4+γ1Prf=0.For oscillatory instability to be possible, the value ofω2 generated by equation (42) must be positive.


## 3.1. Steady-State Solutions

A time-independent quiescent solution of equations (1)–(4) with temperature and heat flux varying in the z direction only, is obtained by reducing equations (2)–(4) to(9)ρhfvj∂vi∂xj+∂p∂xi+ρgei−μhf∇2vi−2ρhfΩj∂vi∂xjei=0,ρCphfvj∂T∂xj+∂Qj∂xj=0,vj∂Qi∂xj+Qi+khf∂T∂xi=0.Then, the steady state solutions are given(10)vbz=0,Tb=Tc−χxjej,Qbz=χkhfandPbz=P0,with the temperature gradient χ and the profile of P0 defined by(11)χ=∂T∂xiand∂p0∂xi=ρTbgei.


## 3.2. Simplified Set of Equations

In order to simplify the parametric representation of this physical problem and to find the characteristic properties of the system, it is necessary to recast the flow equations. Thus, the following normalized quantities are introduced:(12)x˜i=xiH,v˜i=ρfH.viμf,P˜=H2.Pρf.νf,t∗=μftρf.H2,T˜=βfgkfH2χνfρCpfT,Q˜i=HkfT˜TQi.Considering small perturbations on the basic solutions as follows:(13)v˜i=v˜b+v˜i′,T˜=T˜b+T′,Q˜i=Q˜b+Qi′,P˜=P˜b+P˜′,and neglecting the products of the primed quantities, we obtain the following dimensional equations:(14)∂v˜i′∂x˜i′=0,(15)∂v˜i′∂t˜′=−∂P˜′∂x˜i′+μhfμfρfρhf∇2v˜i′+ρβhfρβfRafT˜ei+Ta∂v˜i′∂x˜jei,(16)ρCphfρCpfPrf∂T˜′∂t˜′−Rafw˜′=−∂Q˜i′∂x˜j′,(17)2PrfCf∂Q˜i′∂t˜′+Q˜i′=−khfkhf∂T˜′∂x˜i′,where the dimensionless parameters are defined by(18)Prf=μfρfαf,Cf=αfτ2H2,Raf=g.H4βfχαfνf.Now we eliminate the pressure from the nondimensional equations (14)–(17) by taking the curl-curl of equation (15), the divergence of equation (15), the inner product of any vector equation with ei;, and denoting the divergence of the heat flux Q=∂Q˜i/∂x˜i, to obtain, after dropping the tilde notation for brevity, the equations(19)∂∂t∇2w=Rafρβhfρβf∂2T∂x2+∂2T∂y2+μhfμfρfρhf∇4w+Ta∂ζ∂z,(20)ρCphfρCpfPrf∂T∂t−Rafw=−Q,(21)2PrfCf∂Q∂t˜′+Q=−khfkhf∇2T.which describe the evolution of the conduction steady state perturbations in a conveniently simplified form with four variables such as the z-component of the velocity field, the vorticity ζ=∂v/∂x−∂u/∂y, the heat flux Q, and the temperature T.An evolutionary equation for the vorticity can be obtained from the equation of motion by taking curl, then the dot product withe3 for the vertical component. Thus, eliminating the pressure and introducing the vorticity in equation (2) allows us to obtain the following:(22)μhfμfρfρhf∇2−∂∂tζ=−Ta∂w∂z.From equations (19) and (22), we obtain the following:(23)μhfμfρfρhf∇2−∂∂t∂∂t∇2−μhfμfρfρhf∇4+Ta∂2∂z2w−Rafρβhfρβfμhfμfρfρhf∇2−∂∂t∂2T∂x2+∂2T∂y2=0.


## 3.3. Normal Modes and Analytical Solution

The linear stability of the conduction solutions is studied by writing the perturbations in separable form and assuming an exponential time dependence(24)w=Wzhx,yeσt,T=Θzhx,yeσt,Q=Φzhx,yeσt,with the plane tiling function satisfying(25)∇2hx,y=−κ2hx,y,where W,Θ,Φ are eigenfunctions. The substitution of equation (24) into the differential equations (20), (21), and (23) leads to(26)D2−κ2γ1D2−κ2−σ2+TaD2W=κ2γ2Rafγ1D2−κ2−σΘ,(27)σPrfΘ=RafW−γ3Φ,(28)2σPrfCfΦ+Φ=−γ4D2−κ2Θ,where D=d/dt,D2=d2/dt2 and(29)γ1=μhfμfρfρhf,γ2=ρβhfρβf,γ3=ρCpfρCphf,γ4=khfkhf.Equations (26)–(28), which represent the starting point for analytical and numerical calculations on thermal convective instability are used to study the occurrence of stationary and oscillatory convection in nanofluids. They are equivalent to those obtained by Straughan [24] and Bissell [43, 44] in the case of an ordinary fluid not subjected to a Coriolis force, i.e. when φ1=φ2=0 and Ta=0. The disappearance of tangent shear stresses at the free surface and the conservation of the mass equation allow us to obtain the boundary conditions of the free surface defined by(30)W=0,D2W=0andΘ=0atz=0,1.In order to obtain an approximate solution of equations (26)–(28), we used the Galerkin weighted residual method by choosing the test function written as(31)W=W0sinπz,Θ=Θ0sinπz,Φ=Φ0sinπz,which fulfill the conditions at the borders mentioned in equation (30).By substituting the test functions defined in equation (31) into equations (26)–(28) and performing some integrations, we obtain the following matrix equation:(32)Jγ1J+σ+π2Taγ1J+σ−κ2γ2Raf0−RafσPrfγ30−γ4J2σPrfCf+1×W0Θ0Φ0=000,where J=D2−κ2. For this matrix equation (32) to admit a nontrivial solution, the Rayleigh number of the base fluid must be in the following form:(33)Raf=γ3γ4J+σPrf2σPrfCf+1Jγ1J+σ+π2Ta/γ1J+σκ2γ22σPrfCf+1.For the following, letσ=jω with j2=−1 and ω the real frequency. Thus, the expression for the Rayleigh number defined in equation (33) becomes(34)Raf=Δ1+jωΔ2,with(35)Δ1=κ2γ21+4ω2Prf2Cf2−1Jγ1γ3γ4J2−ω2Prf+π2Taγ1γ3γ4J2+ω2Prfγ1J2+ω2+2Jω2PrfCfγ3γ4J−2ω2Prf2Cf+2π2ω2PrfCfTa2ω2Prf2Cf−γ3γ4Jγ1J2+ω2,(36)Δ2=κ2γ21+4ω2Prf2Cf2−1J2γ3γ4+γ1J+π2JTaγ1−γ3γ4γ1J2+ω2+2JPrfCf2γ1Jω2Prf2Cf−γ1γ3γ4J2+2π2γ1JPrfCfTa2ω2Prf2Cf−γ3γ4Jγ1J2+ω2.Since the Rayleigh number is a real and positive physical quantity, then for the expression equation (33) to exist, ω must be zero or Δ2=0.


## 3.4. Stationary Convection

According to the stability exchange principle for the stationary case, the stability margin is characterized by the frequency equal to zero. This condition allows to obtain from the expression equation (33), the Rayleigh number of the base fluid of the stationary convection expressed as follows:(37)Rafst=γ3γ4κ2γ2γ1κ2+π23+π2Taγ1.Ifγ1=γ2=γ3=γ4=1 and the cavity is not rotating, i.e. φ1=φ2=0 and Ta=0, equation (37) is equivalent to the classical Rayleigh number of stationary convection in ordinary fluids [42].In the absence of the rotation force, we have(38)Rafst=γ1γ3γ4γ2κ2+π23κ2.The absolute critical Rayleigh number of the hybrid nanofluid in this case is defined as(39)Rafst=27π44γ1γ3γ4γ2,with the corresponding wavenumber(40)κcst=π2.We note that the Rayleigh number obtained for stationary convection in nanofluids is not a function of the Prandtl number or the Cattaneo number of the base fluid. Thus, the same results can be associated with the more usual Fourier law for mono nanofluids or hybrid nanofluids.


## 3.5. Oscillatory Convection

Now, we study the effects of the Cattaneo number, the nanoparticles, and the rotation on the oscillating convection. In this case, we must haveω≠0 and Δ2=0. Therefore, the Rayleigh number of the base fluid for oscillatory convection is given by(41)Rafos=κ2γ21+4ω2Prf2Cf2−1Jγ1γ3γ4J2−ω2Prf+π2Taγ1γ3γ4J2+ω2Prfγ1J2+ω2+2Jω2PrfCfγ3γ4J−2ω2Prf2Cf+2π2ω2PrfCfTa2ω2Prf2Cf−γ3γ4Jγ1J2+ω2.The corresponding oscillatory frequencyω must verify the following equations:(42)4γ1J2Prf3Cf2ω4+J2γ3γ4+γ1Prf+4γ1JPrf3Cf2γ1J3+π2Ta−2γ1γ3γ4PrfCfJ3ω2+γ12Jγ3γ4+γ1Prf+π2JTaγ1−γ3γ4−2γ1γ3γ4PrfCfJ2γ3γ4+γ1Prf=0.For oscillatory instability to be possible, the value ofω2 generated by equation (42) must be positive.


## 4. Dynamical System Analysis

In order to explore how thermal relaxation time, hybrid nanoparticles, and rotation affect the nonlinear stability of the onset of thermal convection in a horizontal layer of ordinary fluid such as water, we reduce the problem to the classical case of two-dimensional convective rolls in a fluid layer with unconstrained horizontal boundaries. Thus, we assume that all physical quantities are independent ofx. We consider the early stages of nonlinear convection when the basic structure of the convective rolls is still determined by the dynamic behavior of the linearized solution. The real components of the fluid velocity are expressed in terms of partial derivatives of the stream function as follows:(43)u=−∂ψy,z,t∂z,w=∂ψy,z,t∂y.Eliminating the pressure from equation (2) and introducing the expressions of the stream function defined in equations (43) into the resulting equation and equations (3) and (4), we obtain with the appropriate dimensionless variables [10, 42], the following new equations:(44)1Prf∂∂t∗−γ5∇22−Ta∂2∂z∗2∂2ψ∗∂y∗2=γ6Raf1Prf∂∂t∗−γ5∇2∂2T∗∂y∗2,(45)1+2Cfddt∗dT∗dt∗−∂ψ∗∂y∗=γ7∂2T∗∂y∗2+∂2T∗∂z∗2,with(46)ddt∗=∂∂t∗−∂ψ∗∂z∗∂∂y∗+∂ψ∗∂y∗∂∂z∗,Raf=g.H3βfTc−T0αfνf,Prf=νfαf,Cf=ταf2H2,γ5=μhfμfρfρhfkfkhfρCphfρCpf,γ6=ρβhfρβfρhfρhf,γ7=khfkfρCpfρCphf.The ratio of the Rayleigh number of the hybrid nanofluid to that of the base fluid gives the effective Rayleigh number of the hybrid nanofluid as a function of the thermophysical properties and Rayleigh number of the heat transfer fluid defined as(47)Raf=ρβhfρβfkfkhfρCphfρCpfμfμhfRaf.Similarly, the Cattaneo number of the hybrid nanofluid is defined as follows:(48)Chf=γ7Cf.


### 4.1. Reduced Set of Equations

The solution of the coupled nonlinear system of partial differential equations (44) and (45) will be obtained by representing the current function and the temperature using the Galerkin expansion in the following form [42]:(49)ψ∗y∗z∗t∗=A11t∗sinκy∗sinπz∗,T∗y∗z∗t∗=B11t∗cosκy∗sinπz∗+B02t∗sinπz∗.This representation is equivalent to a Galerkin expansion of the solution in they and z directions, truncated when i+j=2, where i is the Galerkin summation index in the y direction and j is the Galerkin summation index in the z direction. Substituting equations (49) into equations (44) and (45), multiplying the equations by the orthogonal eigenfunctions corresponding to equations (44), and integrating over the domain and wavelength of the convection cell in the vertical and horizontal directions respectively, i.e., ∫0π/κdy∫01dz., we obtain a set of three ordinary differential equations for the time evolution of the second-order amplitudes expressed by(50)d2A11dt∗2=−2γ5Prfκ2+π2dA11dt∗−γ52Prf2κ2+π22A11−π2Prf2TaA11+γ6κ2PrfRafκ2+π2A11+γ6κ2PrfRafγ5Prf−γ7,d2B02dt∗2=12CfdB02dt∗+πκ2A11B11−4π2γ7B02−πκ22A112+πκ2A11dB11dt∗+B11dA11dt∗+π2κ2A112B11+πκA11dB11dt∗,d2B02dt∗2=12CfdB02dt∗+πκ2A11B11−4π2γ7B02−πκ22A112+πκ2A11dB11dt∗+B11dA11dt∗+π2κ2A112B11+πκA11dB11dt∗.After introducing new variables of amplitudes defined as(51)U=κ/κcA11κ/κc2+2,Y=κRfB11,Z=πRfB02,and the expressions(52)Rf=RafRafc,t∗=κ2+π2t,λ=8κ/κc2+2,Rafc=ϵκ2+π23κ2,δ=12Cfκ2+π2,Tf=π2Taκ2+π23,ϵ=ρβfρβhfαhfαfμhfμf,κc=π2,in equations 50, we obtain the following system:(53)U¨=−2γ5PrfU˙+Prfγ6ϵRf−PrfTf+γ52U−γ6PrfUZ−γ5Prf−γ7Y,Y¨=ϵRfU˙+U2Y−2UZ˙−U˙Z+δϵRfU−Y˙−UZ−γ7Y,Z¨=U˙Y+2UY˙−ϵRfU2+U2Z+δUY−Y˙−λγ7Z.Therefore, we can reduce the amplitude equations of system (53) to a system of first-order nonlinear equations by introducing the amplitudes V=U˙, P=ϵRfU−Y˙−UZ, and S=UY−Z˙. Thus, we obtain the six-dimensional system by describing the nonlinear dynamic behavior of thermal convection in mono or hybrid nanofluids, presented as follows:(54)U˙=V,Y˙=ϵRfU−P−UZ,Z˙=UY−S,P˙=US−δP−γ7Y,S˙=UP−δS−λγ7Z,V˙=−2γ5PrfV+Prfγ6ϵRf−PrfTf+γ52U−γ6PrfUZ−γ5Prf−γ7Y,where the dot (.) denote the time derivative d/dt.Whenφ1=φ2=0, φ1≠0 and φ2=0, φ1=0 and φ2=0, system equation (63) corresponds to the base fluid, alumina-water nanofluid, copper-water nanofluid, respectively.WhenTf=0, system (63) is equivalent to the system obtained by Dèdèwanou et al. [42]. When φ1=φ2=0, Cf=0, system (54) is equivalent to the system obtained by Gupta et al. [10].Whenφ1=φ2=0, Tf=0, system (54) is equivalent to the system obtained by Layek and Pati [41].In the absence of the thermal relaxation time, (54) and (46) are reduced to the following system:(55)U˙=V,Y˙=ϵRfU−γ7Y−UZ,Z˙=UY−λγ7Z,V˙=−2γ5PrfV+Prfγ6ϵRf−PrfTf+γ52U−γ6PrfUZ−γ5Prf−γ7Y.Lorenz [45] has investigated the nonlinear analysis of convection in pure fluid confined in a nonporous cavity by using the Fourier law. His nonlinear dynamic system has been analyzed and solved for Prf=10, so that there are convection cells in the domain and that the boundary conditions are satisfied [45, 46]. Bissell [43] analyzed the oscillatory convection with the Cattaneo–Christov hyperbolic heat-flow model and included the effects owing to Prandtl number, which in some circumstances can be used as a control parameter. He showed that the Cattaneo threshold can be conceived equivalently as a Prandtl threshold, so that system bifurcations could potentially be triggered by varying the Prandtl number. For small values of Cattaneo number, a five-dimensional nonlinear system obtained by Layek and Pati [41] undergoes a subcritical transition to chaos similar to the Lorenz system but undergoes a period-doubling transition to chaos when Prf=5 and Cf=0.001. For increasing values of Prandtl number, he found that the fine-structure of the period-doubling cascade is interrupted and that this is due to the generation of internal noise that fastens the transitional process. With the critical value of the wavenumber corresponding to the convection threshold, the expression of the Rayleigh number of the base fluid defined in equation (53) gives: Rafc=27π4ϵ/4.


### 4.2. Dissipation Effect

The nonlinear dynamical system (54) has the reflection symmetry U,Y,P⟶−U,Y,P and(56)∇⟶.ϑ⟶=∂U˙∂U+∂Y˙∂Y+∂Z˙∂Z+∂P˙∂P+∂S˙∂S+∂V˙∂V,∇⟶.ϑ⟶=−Prf2γ5+γ6+2δ.We note that∇⟶.ϑ⟶<0 whatever the values of Prf, γ5, γ6 and δ. Then system (54) is dissipative and its solutions are bounded in phase space. Therefore, if a set of initial points in phase space occupies the region ϑ0 at t=0, then after some time, t, the end points of the corresponding trajectories will fill a volume(57)ϑt=exp−Prf2γ5+γ6+2δt.


### 4.3. Equilibrium Points and Their Stability

In this section, the nature of the nonlinear dynamics of systems (54) and (55) is determined around the fixed points by analyzing the stability of stationary solutions. The hybrid nanofluid is confined in a nonporous cavity so that there are convection cells in the domain and the boundary conditions are satisfied.


#### 4.3.1. The Case ofCf=0

Considering the general form of system (54) defined by X˙=FXs and the equilibrium (stationary or fixed) points Xs defined by FXs=0, we obtained three fixed equilibrium points of the system, including the first one(58)U1=Y1=Z1=P1=S1=V1=0,is the stationary solution and the other two(59)U2,3=±−λγ5γ6γ7Tf+γ52γ7γ5γ6Tf+γ52−∈Rf,Y2,3=±−λγ7γ5γ6Tf+γ52γ7γ5γ6Tf+γ52−∈Rf,Z2,3=∈Rf−γ7γ5γ6Tf+γ52,V2,3=0,are the convection solutions. The linear stability of the points can be obtained by linearizing the nonlinear dynamical system equation (55). Thus, the resulting Jacobian matrix is as follows:(60)M=0001εrf−Z−γ7−U0YU−λPrf0Prfγ6εrf−PrfTf+γ52γ6Prfγ5Prf−γ7−γ6PrfU2γ5Prf.Using test solutions of the formexpξt, ξ∈ℂ, the stability of the fixed point corresponding to the conduction solution is controlled by the roots of the following characteristic polynomial equation:(61)λγ7+ξξ3+2γ5Prf+γ7ξ2+Prf2Tf+γ52+Prf2γ5γ7−γ6ϵRfξ2+Prf2Tf+γ52−γ5γ6ϵRf=0.This equation (61) generates four eigenvalues. The first one given by ξ=−λγ7 is always negative, and the other three are the solutions of the following equation:(62)ξ3+2γ5Prf+γ7ξ2+Prf2Tf+γ52+Prf2γ5γ7−γ6ϵRfξ+Prf2γ7Tf+γ52−γ5γ6ϵRf=0.From this equation, the fixed point of the stationary solution is stable if and only ifγ5γ6ϵRf<γ7Tf+γ52. Thus, the critical value of the rescaled Rayleigh number of the base fluid at which the fixed point of the stationary solution loses its stability and that of the convection solution takes over is expressed by(63)Rfc1=γ7ϵγ5γ6Tf+γ52.This expression is a function of the thermophysical properties of the hybrid nanofluid and the rescaled Taylor number, so the transition from conduction to stationary convection depends on the volume fraction of the nanoparticles and the effect of rotation as shown in Figure2.Figure 2
Variation of the critical Rayleigh number of the stationary solution as a function of the Taylor number with increasing nanoparticle volume fraction.Data analysis of the curves constructed in Figure2 shows that when the value of Taylor number is less than about 0.33, 0.315, 0.30, and 0.293 for φ1=φ2=0.01,0.02,0.03, and 0.04, respectively, Rfc1 decreases but increases for higher values of Tf. Taking Tf=0 for example, we found Rfc1=1.2 like Gupta [10] for the ordinary fluids (φ=0). But when φ1=φ2=0.01,0.02,0.03,0.04; we have Rfc1≈1.176,1.157,1.142,1.13. Thus, it is then possible to reduce or increase conduction in a heat transfer fluid using hybrid nanoparticles under the effect of rotation. Using the same test solutions of the form expξt, ξ∈ℂ, the stability of the fixed points corresponding to the convection solution is controlled by the roots of the following characteristic polynomial equation:(64)ξ4+γ71+λ+2γ5Prfξ3+λγ5γ6γ7εRfTf+γ52+2γ5γ7Prf1+λ+PrfPrf−γ7γ5Tf+γ52ξ2+2λγ52γ6γ7εRfTf+γ52+λγ7PrfPrf−2γ7γ5Tf+γ52+γ6εRfξ=0.This equation is solved numerically for different values of the parameters to study the stability of the fixed points of the convection solutions.


#### 4.3.2. The Case ofCf≠0

In the presence of the thermal relaxation time, the elimination of a quadratic factor, which is not associated with the beginning of the instability, allowed us to find the characteristic polynomial equation of the fixed point corresponding to the immobile support at the origin whose roots control its stability, which is presented as follows:(65)ξ4+2γ5Prf+δξ3+δ2γ5Prf+δ−Prfγ6ϵRf−PrfTf+γ52ξ2+2γ5γ7δPrf−γ6Prf2γ5Prf−γ7ϵRf−δPrfγ6ϵRf−PrfTf+γ52+δPrf2γ7Tf+γ52−γ5γ6ϵRf=0.Whenξ=0, a stability exchange occurs and stationary convection takes over. The corresponding critical rescaled Rayleigh number of the base fluid from which this phenomenon is observed is equivalent to equation (63).


## 4.1. Reduced Set of Equations

The solution of the coupled nonlinear system of partial differential equations (44) and (45) will be obtained by representing the current function and the temperature using the Galerkin expansion in the following form [42]:(49)ψ∗y∗z∗t∗=A11t∗sinκy∗sinπz∗,T∗y∗z∗t∗=B11t∗cosκy∗sinπz∗+B02t∗sinπz∗.This representation is equivalent to a Galerkin expansion of the solution in they and z directions, truncated when i+j=2, where i is the Galerkin summation index in the y direction and j is the Galerkin summation index in the z direction. Substituting equations (49) into equations (44) and (45), multiplying the equations by the orthogonal eigenfunctions corresponding to equations (44), and integrating over the domain and wavelength of the convection cell in the vertical and horizontal directions respectively, i.e., ∫0π/κdy∫01dz., we obtain a set of three ordinary differential equations for the time evolution of the second-order amplitudes expressed by(50)d2A11dt∗2=−2γ5Prfκ2+π2dA11dt∗−γ52Prf2κ2+π22A11−π2Prf2TaA11+γ6κ2PrfRafκ2+π2A11+γ6κ2PrfRafγ5Prf−γ7,d2B02dt∗2=12CfdB02dt∗+πκ2A11B11−4π2γ7B02−πκ22A112+πκ2A11dB11dt∗+B11dA11dt∗+π2κ2A112B11+πκA11dB11dt∗,d2B02dt∗2=12CfdB02dt∗+πκ2A11B11−4π2γ7B02−πκ22A112+πκ2A11dB11dt∗+B11dA11dt∗+π2κ2A112B11+πκA11dB11dt∗.After introducing new variables of amplitudes defined as(51)U=κ/κcA11κ/κc2+2,Y=κRfB11,Z=πRfB02,and the expressions(52)Rf=RafRafc,t∗=κ2+π2t,λ=8κ/κc2+2,Rafc=ϵκ2+π23κ2,δ=12Cfκ2+π2,Tf=π2Taκ2+π23,ϵ=ρβfρβhfαhfαfμhfμf,κc=π2,in equations 50, we obtain the following system:(53)U¨=−2γ5PrfU˙+Prfγ6ϵRf−PrfTf+γ52U−γ6PrfUZ−γ5Prf−γ7Y,Y¨=ϵRfU˙+U2Y−2UZ˙−U˙Z+δϵRfU−Y˙−UZ−γ7Y,Z¨=U˙Y+2UY˙−ϵRfU2+U2Z+δUY−Y˙−λγ7Z.Therefore, we can reduce the amplitude equations of system (53) to a system of first-order nonlinear equations by introducing the amplitudes V=U˙, P=ϵRfU−Y˙−UZ, and S=UY−Z˙. Thus, we obtain the six-dimensional system by describing the nonlinear dynamic behavior of thermal convection in mono or hybrid nanofluids, presented as follows:(54)U˙=V,Y˙=ϵRfU−P−UZ,Z˙=UY−S,P˙=US−δP−γ7Y,S˙=UP−δS−λγ7Z,V˙=−2γ5PrfV+Prfγ6ϵRf−PrfTf+γ52U−γ6PrfUZ−γ5Prf−γ7Y,where the dot (.) denote the time derivative d/dt.Whenφ1=φ2=0, φ1≠0 and φ2=0, φ1=0 and φ2=0, system equation (63) corresponds to the base fluid, alumina-water nanofluid, copper-water nanofluid, respectively.WhenTf=0, system (63) is equivalent to the system obtained by Dèdèwanou et al. [42]. When φ1=φ2=0, Cf=0, system (54) is equivalent to the system obtained by Gupta et al. [10].Whenφ1=φ2=0, Tf=0, system (54) is equivalent to the system obtained by Layek and Pati [41].In the absence of the thermal relaxation time, (54) and (46) are reduced to the following system:(55)U˙=V,Y˙=ϵRfU−γ7Y−UZ,Z˙=UY−λγ7Z,V˙=−2γ5PrfV+Prfγ6ϵRf−PrfTf+γ52U−γ6PrfUZ−γ5Prf−γ7Y.Lorenz [45] has investigated the nonlinear analysis of convection in pure fluid confined in a nonporous cavity by using the Fourier law. His nonlinear dynamic system has been analyzed and solved for Prf=10, so that there are convection cells in the domain and that the boundary conditions are satisfied [45, 46]. Bissell [43] analyzed the oscillatory convection with the Cattaneo–Christov hyperbolic heat-flow model and included the effects owing to Prandtl number, which in some circumstances can be used as a control parameter. He showed that the Cattaneo threshold can be conceived equivalently as a Prandtl threshold, so that system bifurcations could potentially be triggered by varying the Prandtl number. For small values of Cattaneo number, a five-dimensional nonlinear system obtained by Layek and Pati [41] undergoes a subcritical transition to chaos similar to the Lorenz system but undergoes a period-doubling transition to chaos when Prf=5 and Cf=0.001. For increasing values of Prandtl number, he found that the fine-structure of the period-doubling cascade is interrupted and that this is due to the generation of internal noise that fastens the transitional process. With the critical value of the wavenumber corresponding to the convection threshold, the expression of the Rayleigh number of the base fluid defined in equation (53) gives: Rafc=27π4ϵ/4.


## 4.2. Dissipation Effect

The nonlinear dynamical system (54) has the reflection symmetry U,Y,P⟶−U,Y,P and(56)∇⟶.ϑ⟶=∂U˙∂U+∂Y˙∂Y+∂Z˙∂Z+∂P˙∂P+∂S˙∂S+∂V˙∂V,∇⟶.ϑ⟶=−Prf2γ5+γ6+2δ.We note that∇⟶.ϑ⟶<0 whatever the values of Prf, γ5, γ6 and δ. Then system (54) is dissipative and its solutions are bounded in phase space. Therefore, if a set of initial points in phase space occupies the region ϑ0 at t=0, then after some time, t, the end points of the corresponding trajectories will fill a volume(57)ϑt=exp−Prf2γ5+γ6+2δt.


## 4.3. Equilibrium Points and Their Stability

In this section, the nature of the nonlinear dynamics of systems (54) and (55) is determined around the fixed points by analyzing the stability of stationary solutions. The hybrid nanofluid is confined in a nonporous cavity so that there are convection cells in the domain and the boundary conditions are satisfied.


### 4.3.1. The Case ofCf=0

Considering the general form of system (54) defined by X˙=FXs and the equilibrium (stationary or fixed) points Xs defined by FXs=0, we obtained three fixed equilibrium points of the system, including the first one(58)U1=Y1=Z1=P1=S1=V1=0,is the stationary solution and the other two(59)U2,3=±−λγ5γ6γ7Tf+γ52γ7γ5γ6Tf+γ52−∈Rf,Y2,3=±−λγ7γ5γ6Tf+γ52γ7γ5γ6Tf+γ52−∈Rf,Z2,3=∈Rf−γ7γ5γ6Tf+γ52,V2,3=0,are the convection solutions. The linear stability of the points can be obtained by linearizing the nonlinear dynamical system equation (55). Thus, the resulting Jacobian matrix is as follows:(60)M=0001εrf−Z−γ7−U0YU−λPrf0Prfγ6εrf−PrfTf+γ52γ6Prfγ5Prf−γ7−γ6PrfU2γ5Prf.Using test solutions of the formexpξt, ξ∈ℂ, the stability of the fixed point corresponding to the conduction solution is controlled by the roots of the following characteristic polynomial equation:(61)λγ7+ξξ3+2γ5Prf+γ7ξ2+Prf2Tf+γ52+Prf2γ5γ7−γ6ϵRfξ2+Prf2Tf+γ52−γ5γ6ϵRf=0.This equation (61) generates four eigenvalues. The first one given by ξ=−λγ7 is always negative, and the other three are the solutions of the following equation:(62)ξ3+2γ5Prf+γ7ξ2+Prf2Tf+γ52+Prf2γ5γ7−γ6ϵRfξ+Prf2γ7Tf+γ52−γ5γ6ϵRf=0.From this equation, the fixed point of the stationary solution is stable if and only ifγ5γ6ϵRf<γ7Tf+γ52. Thus, the critical value of the rescaled Rayleigh number of the base fluid at which the fixed point of the stationary solution loses its stability and that of the convection solution takes over is expressed by(63)Rfc1=γ7ϵγ5γ6Tf+γ52.This expression is a function of the thermophysical properties of the hybrid nanofluid and the rescaled Taylor number, so the transition from conduction to stationary convection depends on the volume fraction of the nanoparticles and the effect of rotation as shown in Figure2.Figure 2
Variation of the critical Rayleigh number of the stationary solution as a function of the Taylor number with increasing nanoparticle volume fraction.Data analysis of the curves constructed in Figure2 shows that when the value of Taylor number is less than about 0.33, 0.315, 0.30, and 0.293 for φ1=φ2=0.01,0.02,0.03, and 0.04, respectively, Rfc1 decreases but increases for higher values of Tf. Taking Tf=0 for example, we found Rfc1=1.2 like Gupta [10] for the ordinary fluids (φ=0). But when φ1=φ2=0.01,0.02,0.03,0.04; we have Rfc1≈1.176,1.157,1.142,1.13. Thus, it is then possible to reduce or increase conduction in a heat transfer fluid using hybrid nanoparticles under the effect of rotation. Using the same test solutions of the form expξt, ξ∈ℂ, the stability of the fixed points corresponding to the convection solution is controlled by the roots of the following characteristic polynomial equation:(64)ξ4+γ71+λ+2γ5Prfξ3+λγ5γ6γ7εRfTf+γ52+2γ5γ7Prf1+λ+PrfPrf−γ7γ5Tf+γ52ξ2+2λγ52γ6γ7εRfTf+γ52+λγ7PrfPrf−2γ7γ5Tf+γ52+γ6εRfξ=0.This equation is solved numerically for different values of the parameters to study the stability of the fixed points of the convection solutions.


### 4.3.2. The Case ofCf≠0

In the presence of the thermal relaxation time, the elimination of a quadratic factor, which is not associated with the beginning of the instability, allowed us to find the characteristic polynomial equation of the fixed point corresponding to the immobile support at the origin whose roots control its stability, which is presented as follows:(65)ξ4+2γ5Prf+δξ3+δ2γ5Prf+δ−Prfγ6ϵRf−PrfTf+γ52ξ2+2γ5γ7δPrf−γ6Prf2γ5Prf−γ7ϵRf−δPrfγ6ϵRf−PrfTf+γ52+δPrf2γ7Tf+γ52−γ5γ6ϵRf=0.Whenξ=0, a stability exchange occurs and stationary convection takes over. The corresponding critical rescaled Rayleigh number of the base fluid from which this phenomenon is observed is equivalent to equation (63).


## 4.3.1. The Case ofCf=0

Considering the general form of system (54) defined by X˙=FXs and the equilibrium (stationary or fixed) points Xs defined by FXs=0, we obtained three fixed equilibrium points of the system, including the first one(58)U1=Y1=Z1=P1=S1=V1=0,is the stationary solution and the other two(59)U2,3=±−λγ5γ6γ7Tf+γ52γ7γ5γ6Tf+γ52−∈Rf,Y2,3=±−λγ7γ5γ6Tf+γ52γ7γ5γ6Tf+γ52−∈Rf,Z2,3=∈Rf−γ7γ5γ6Tf+γ52,V2,3=0,are the convection solutions. The linear stability of the points can be obtained by linearizing the nonlinear dynamical system equation (55). Thus, the resulting Jacobian matrix is as follows:(60)M=0001εrf−Z−γ7−U0YU−λPrf0Prfγ6εrf−PrfTf+γ52γ6Prfγ5Prf−γ7−γ6PrfU2γ5Prf.Using test solutions of the formexpξt, ξ∈ℂ, the stability of the fixed point corresponding to the conduction solution is controlled by the roots of the following characteristic polynomial equation:(61)λγ7+ξξ3+2γ5Prf+γ7ξ2+Prf2Tf+γ52+Prf2γ5γ7−γ6ϵRfξ2+Prf2Tf+γ52−γ5γ6ϵRf=0.This equation (61) generates four eigenvalues. The first one given by ξ=−λγ7 is always negative, and the other three are the solutions of the following equation:(62)ξ3+2γ5Prf+γ7ξ2+Prf2Tf+γ52+Prf2γ5γ7−γ6ϵRfξ+Prf2γ7Tf+γ52−γ5γ6ϵRf=0.From this equation, the fixed point of the stationary solution is stable if and only ifγ5γ6ϵRf<γ7Tf+γ52. Thus, the critical value of the rescaled Rayleigh number of the base fluid at which the fixed point of the stationary solution loses its stability and that of the convection solution takes over is expressed by(63)Rfc1=γ7ϵγ5γ6Tf+γ52.This expression is a function of the thermophysical properties of the hybrid nanofluid and the rescaled Taylor number, so the transition from conduction to stationary convection depends on the volume fraction of the nanoparticles and the effect of rotation as shown in Figure2.Figure 2
Variation of the critical Rayleigh number of the stationary solution as a function of the Taylor number with increasing nanoparticle volume fraction.Data analysis of the curves constructed in Figure2 shows that when the value of Taylor number is less than about 0.33, 0.315, 0.30, and 0.293 for φ1=φ2=0.01,0.02,0.03, and 0.04, respectively, Rfc1 decreases but increases for higher values of Tf. Taking Tf=0 for example, we found Rfc1=1.2 like Gupta [10] for the ordinary fluids (φ=0). But when φ1=φ2=0.01,0.02,0.03,0.04; we have Rfc1≈1.176,1.157,1.142,1.13. Thus, it is then possible to reduce or increase conduction in a heat transfer fluid using hybrid nanoparticles under the effect of rotation. Using the same test solutions of the form expξt, ξ∈ℂ, the stability of the fixed points corresponding to the convection solution is controlled by the roots of the following characteristic polynomial equation:(64)ξ4+γ71+λ+2γ5Prfξ3+λγ5γ6γ7εRfTf+γ52+2γ5γ7Prf1+λ+PrfPrf−γ7γ5Tf+γ52ξ2+2λγ52γ6γ7εRfTf+γ52+λγ7PrfPrf−2γ7γ5Tf+γ52+γ6εRfξ=0.This equation is solved numerically for different values of the parameters to study the stability of the fixed points of the convection solutions.


## 4.3.2. The Case ofCf≠0

In the presence of the thermal relaxation time, the elimination of a quadratic factor, which is not associated with the beginning of the instability, allowed us to find the characteristic polynomial equation of the fixed point corresponding to the immobile support at the origin whose roots control its stability, which is presented as follows:(65)ξ4+2γ5Prf+δξ3+δ2γ5Prf+δ−Prfγ6ϵRf−PrfTf+γ52ξ2+2γ5γ7δPrf−γ6Prf2γ5Prf−γ7ϵRf−δPrfγ6ϵRf−PrfTf+γ52+δPrf2γ7Tf+γ52−γ5γ6ϵRf=0.Whenξ=0, a stability exchange occurs and stationary convection takes over. The corresponding critical rescaled Rayleigh number of the base fluid from which this phenomenon is observed is equivalent to equation (63).


## 5. Results and Discussion

We performed numerical simulations to investigate the influence of hybrid nanoparticles and rotation on the dynamic behavior of thermal convection in a base fluid (water) in the presence of thermal relaxation time. Using free boundary conditions, we determined the analytical expressions of Rayleigh numbers of the base fluid for stationary and oscillatory convection as a function of the thermophysical properties of the hybrid nanofluid. We observe that the stationary Rayleigh number of the base fluid does not depend on the Prandtl number and the Cattaneo number. Figure3 shows the variation of the stationary Rayleigh number of the base fluid as a function of wavelength for different values of the volume fraction of the hybrid nanoparticles (alumina-copper) with a fixed value of Taylor number. From these plotted curves, we find that the stationary Rayleigh number increases with the value of the volume fraction of hybrid nanoparticles. Thus, the addition of the hybrid nanoparticles (alumina-copper) to the base fluid (water) subjected to the rotation stabilizes the stationary convection. Figure 4 shows the variation of the stationary Rayleigh number of the base fluid as a function of the wavelength for different values of the Taylor number with a fixed value of the volume fraction of the hybrid nanoparticles (alumina-copper).Figure 3
Variation of the stationary Rayleigh number of the base fluid as a function of the volume fraction of hybrid (aluminum-copper) nanoparticles.Figure 4
Variation of the stationary Rayleigh number of the base fluid as a function of the Taylor number.From these plotted curves, it can be seen that the stationary Rayleigh number increases with an increasing Taylor number. Thus, the rotation stabilizes the stationary convection in the hybrid nanofluid.The fourth-order Runge–Kutta method, the polynomial companion matrix, and the standard eigenvalue solver of the Lapack method are used to numerically solve systems equations (54) and (55). We took the initial conditions U0=Y0=0.8,Z0=0.92195,P0=0.8,S0=0.92195 and V0=0.8. In order to guarantee the results, our different numerical simulations are compared with the results obtained by Dèdèwanou et al. [42] and Gupta [10]. We present in Figures 5–8, the bifurcation diagrams representing the minima and maxima of the posttransient regimes of the solutions of the amplitude Zt as a function of function of Rf when thethermal relaxation time is zero using Prf=10 and λ=8/3. These diagrams show that system equation (55) can have chaotic, periodic, or multiperiodic behavior depending on the parameter values chosen. By comparing the diagrams in Figures 5–7, we notice that, for Tf=0.2, when the volume fraction of the hybrid nanoparticles increases, the domain of the chaotic behavior chaotic behavior decreases with the increase of the values of the rescaled Rayleigh number of the number of the base fluid. On the other hand, comparison of the plots in Figures 6 and 8 shows that, for φ1=φ2=0.02, increasing the values of the Taylor number increases the domain of chaotic behavior with increasing values of the rescaled Rayleigh number of the base fluid.Figure 5
Bifurcation diagram ofZ versus Rf representing maxima and minima of the posttransient solution of Zt for hybrid nanofluid φ1=φ2=0) with Tf=0.2.Figure 6
Bifurcation diagram ofZ versus Rf representing maxima and minima of the posttransient solution of Zt for hybrid nanofluid φ1=φ2=0.02) with Tf=0.5.Figure 7
Bifurcation diagram ofZ versus Rf representing maxima and minima of the posttransient solution of Zt for hybrid nanofluid φ1=φ2=0.04) with Tf=0.5.Figure 8
Bifurcation diagram ofZ versus Rf representing maxima and minima of the posttransient solution of Zt for hybrid nanofluid φ1=φ2=0.02) with Tf=0.2.To confirm this prediction of the bifurcation diagrams, we constructed in Figure9, the chaotic behavior of the system in the base fluid case by choosing Rf=80 and Tf=0.5. As shown in Figure 10, we set the values of the rescaled Taylor and Rayleigh numbers by varying the volume fraction of the hybrid nanoparticles to construct the in-plane phase spaces with their corresponding time evolutions. For φ1=φ2=0,0.02 and 0.04, the system is chaotic in period 2 and period 1, respectively. Therefore, by analyzing these curves shown in Figure 10, it can be deduced that the addition of the hybrid nanoparticles in a heat transfer fluid makes the convection periodic. On the other hand, in the case of Figure 11, we have fixed the values of the volume fraction of hybrid nanoparticles and the rescaled Rayleigh number of the base fluid by varying the value of the rescaled Taylor number. The analysis of these curves shows that the system leaves the chaotic regime to the periodic regime when the value of the rescaled Taylor number decreases. Therefore, increasing the rescaled Taylor number increases the periodicity of the system.Figure 9
Phase portrait and its corresponding time story forφ1=φ2=0), Rf=80 and Tf=0.5.Figure 10
Phase portrait and its corresponding time story forRf=250, Tf=0.5 and φ1=φ2=0,0.02,0.04.Figure 11
Phase portrait forRf=152andφ1=φ2=0.02 with different values of Tf.Furthermore , we present in Figures12–15, the bifurcation diagrams representing the minima and maxima of the posttransient regimes of the solutions of the amplitude Zt as a function of Rf when the thermal relaxation time exists using Prf=5 like Layek and Pati [41]. These diagrams show that system 63 can also have chaotic, periodic, or multiperiodic behavior depending on the parameter values chosen. Comparing the diagrams in Figures 12–14, it can be seen that, for Tf=0.2 and Cf=0.001, increasing the volume fraction of the nanoparticles hybrid nanoparticles decreases the domain of chaotic behavior with the increase of the Rayleigh number values of the base fluid. On the other hand, the comparison of the diagrams in Figures 13 and 15 show that, for φ1=φ2=0.02 and Tf=0.2, increasing the values of the Cattaneo number largely increases the range of chaotic behavior with increasing values of the Rayleigh number of the base fluid. Referring to Figure 14, for φ1=φ2=0.04, Tf=0.2 and Cf=0.001, the system is chaotic for 14<Rf<16.Figure 12
Bifurcation diagram ofZ versus Rf representing maxima and minima of the posttransient solution of Zt for base fluid (φ1=φ2=0) with Tf=0.2 and Cf=0.001.Figure 13
Bifurcation diagram ofZ versus Rf representing maxima and minima of the posttransient solution of Zt for hybrid nanofluid (φ1=φ2=0.02) with Tf=0.2 and Cf=0.001.Figure 14
Bifurcation diagram ofZ versus Rf representing maxima and minima of the posttransient solution of Zt for hybrid nanofluid (φ1=φ2=0.04) with Tf=0.2 and Cf=0.001.Figure 15
Bifurcation diagram ofZ versus Rf representing maxima and minima of the posttransient solution of Zt for hybrid nanofluid (φ1=φ2=0.02) with Tf=0.2 and Cf=0.005.In Figures16 and 17, we have constructed the phase spaces in the X−Z plane for different values of the control parameters of system equation (54). When we set Tf=0.2 and Cf=0.001 (see Figure 16), we notice in the base fluid case (φ1=φ2=0) that the system is in period 4 for Rf=166. For φ1=φ2=0.02, the system is in period 2 and for φ1=φ2=0.04, the system has quasi-chaotic behavior. For φ1=φ2=0.02 and Rf=100 fixed, the system is in period 1 for Tf=0.2 and Cf=0.001. On the other hand, for Tf=0.2 and Cf=0.003, Tf=0.2 and Cf=0.005, Tf=0.3 and Cf=0.005, Tf=0.5 and Cf=0.001, and Tf=1.7 and Cf=0.005, the system is chaotic.Figure 16
Phase portrait and its corresponding time story forTf=0.2 and Cf=0.001.Figure 17
Phase portrait and its corresponding time story forφ1=φ2=0.02 and Rf=100.


## 6. Conclusions

We have studied the occurrence of thermal convective instabilities and chaos in a rotating infinite horizontal hybrid nanofluid layer heated from below with the Cattaneo–Christov heat flux model and subjected to unconstrained boundary conditions. The linear study of the mass, momentum, energy, and heat flow equations governing natural convection allowed us to find the general expression for the stationary Rayleigh number of the base fluid that can be used for the nonlinear dynamic analysis of thermal convection in nanofluids. We noticed that the rotation and the addition of nanoparticles in the base fluid have stabilizing effects on the stationary convection. With the obtained low-dimensional dynamical systems, we notice that the addition of hybrid nanoparticles in the heat transfer fluid subjected to rotation and/or in the presence of the thermal relaxation time reduces the domain of chaos and enlarges the domain of periodicity with the increase of the Rayleigh number of the base fluid. On the other hand, the increase of the Taylor number and Cattaneo number increases the chaotic domain with the increase of the Rayleigh number of the base fluid. The obtained nonlinear system depends on the parameters of the base fluid and the thermophysical properties of the hybrid nanofluid; it will be very useful to predict or control the chaotic behaviour of thermal convection in dynamic and biological systems. Thus, the hybrid nanofluid confers a great advantage for chaos control in many industrial applications like food processing, chemical processes, solidification and centrifugal casting of metals, and rotating machines to achieve the desired results. Obtained results and comparative studies show that the use of hybrid nanoparticles can be useful to control the small thermal relaxation time due to thermal inertia for thermal transport in biological tissues and surgical operations.



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