# Regular and Chaotic Behavior on the Convection in a Rotating Ferrofluid in Porous Medium under Helical Force

Authors: M.L. Hounvènou; Vincent Adjimon Monwanou
Journal: Complexity
 (2022)

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## Abstract

This article deals with the control of chaos on the convective motion in a ferrofluid filled in a rotating porous medium and under the helical force effect. We performed a truncated expansion of Galerkin and found the Lorenz-type model which described the system. The dynamical system is characterized using appropriate and subsequent criteria. We noticed that the system presents regular and chaotic behaviors according to the parameters at present. A considerable reduction of the chaotic domain is noticed with the increase of the intensity of the helical force and of the Darcy number, thus making it possible to delay the heat transfer in the system. The increase in the viscosity ratio and the Taylor number widen the chaotic domain leading to an acceleration of the heat transfer in the system.


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## Body

## 1. Introduction

Since its discovery in the 20th century, chaos has been one of the most interesting for dynamic systems in areas such as physics, mathematics, chemistry, biochemistry, economics and finance, epidemiology, and engineering. [1–6]. Depending on the field of study, chaos is sometimes useful or undesirable to the point where many researchers are interested in its prediction and/or control. For example, in communications engineering, chaos allows to encode information or to encrypt images (synchronization and/or antisynchronization of chaotic systems) [7]. In the construction engineering of large infrastructures (the case of bridges for example), chaos is to be avoided. Naturally and this in almost all domains, the dynamic systems encountered are nonlinear. But the most important factors favoring chaos are nonlinearity and sensitivity to initial conditions, so nonlinear dynamic systems have a strong chance to develop chaotic behaviors followed by strong instabilities. Due to the complexity of these nonlinear dynamic systems, analytically process and finding exact solutions are often difficult. For this reason, several approximate analytical methods and numerical simulations are used.The theory of ferrofluids has attracted attention of many researchers since their appearance, because their applications are numerous in various fields. Ferroﬂuids are also called magnetic ﬂuids. They are composed of colloidal suspensions of nanosized ferromagnetic particles stably dispersed in organic or nonorganic carrier ﬂuids which can be water, kerosene, and hydrocarbon. Due to their use in a variety of engineering applications, commercial production has been of a large quantity. Also, the heat transfer with magnetic ﬂuids as the support has been one of the leading area of scientiﬁc study due to its technological applications.Chaotic convection in fluid layer is very interesting due to its relevance in a wide range of industrial applications. In a three-dimensional phase space, chaos was obtained for the Lorenz system [8] arising from the truncation of the classical Rayleigh–Bénard convection model. In a fluid layer, chaotic behavior can be actually advantageous in various industrial applications such as the production of crystals, oil reservoir modeling, and catalytic packed bed filtration.The study of chaos can generally be carried out in various fields of physics. For example, a complete analytical study on the bifurcations and chaotic phenomena raised in certain second-order, nonautonomous, dissipative chaotic systems is reported by the author of [9].Singh et al. [10] studied feedback control of chaotic convection in a porous medium under gravity modulation. They obtained a nonautonomous system with three differential equations by employing the truncated Galerkin expansion method in the modulated momentum and energy equations. The control with the Rayleigh number demonstrates either periodic or chaotic behavior of the system. The influence of the modulated amplitude is also found to advance the chaotic nature of the system whereas the feedback control and frequency of modulation parameters delay the chaotic behavior.Control of chaos in a thermal convection loop by state space linearization has been investigated by Rana et al. [11]. A laminar flow has been obtained from a chaotic flow of the thermal convection loop by applying a technique state space linearization converting the nonlinear system into the linear system through a coordinate transformation, based on the concept of Lie brackets.Ahmed et al. [12] studied natural convection flows of an incompressible Newtonian fluid filled in a circular cylinder and described the heat transfer process by a generalized fractional constitutive equation for the thermal flux-temperature gradient. They obtained the analytical solutions to the fluid temperature, thermal flux, fluid velocity, and volume flow rate using Laplace transform and finite Hankel transform.Izadi et al. [13] treated computational analysis of thermal gravitational convection within a porous chamber under the impact of tilted periodic magnetic force and used Galerkin finite element method to resolve governing equations found from the study. They then have examined the impacts of the Darcy number, Hartmann number, Rayleigh number, periodicity of the magnetic field, magnetic field inclination angle, thermal conductivity ratio, and medium porosity on flow and thermal patterns. The study pointed out that parameters of the periodic magnetic field have the nonmonotonic influence of the heat transfer performance.Convective chaos in fluids has received particular attention in recent years. Many works of chaotic convection have been done before with various additional effects in a different type of fluid due to its importance. Indeed, it makes it possible to highlight heat exchanges for large applications in the laboratory and in engineering. The case of magnetic fluid, due to its technological applications, has attracted the attention of several researchers. The chaos control on convection in fluids was carried out with the use of different types of fluid from simple fluid to magnetic one and in porous and nonporous media [14–23].Laroze et al. [24] studied before theoretically and numerically the thermal convection in a magnetic suspension and modeled magnetic properties as those of electrically nonconducting superparamagnets by performing a truncated Galerkin expansion. Based on bifurcation diagrams, and Lyapunov exponents, they found that the system exhibits multiple transitions according to the parameters at present. The convection dynamics was studied in viscoelastic fluid in a porous medium [16] where the authors applied truncated expansion of Galerkin and obtained a generalized four-dimensional Lorenz system. Using the appropriate bifurcation curves and their corresponding Lyapunov exponents, they characterized the nonlinear dynamics of the system and found within a range of moderate and high Rayleigh numbers the proportional controller gain to enhance the stabilization and destabilization effects on the thermal convection. The system also exhibits remarkable topological structures.Chaotic convection was carried out in a viscoelastic fluid with various technics and considerations as well as the stability analysis and has given very varied and very interesting results [25–37].In a convective system, the pseudo-vector nature force responsible for helical turbulence is helical force. Hounvenou and Monwanou [19], in a recent work, studied the action of this force on chaotic convection in a magnetic fluid and come to the conclusion that this force reduces the chaotic domain.Summarizing the literature, it comes that only the effect of rotation has been studied on the chaotic convection of a magnetic fluid in one case, then only the effect of porosity in another case. The combined effect of the three effects on the chaotic convection of a magnetic fluid has not been studied because of the difficulties that this could cause. Overcoming all these difficulties would therefore be a major scientific challenge. So, in our work, we have decided to study the effect of this pseudo-vector type force on the chaotic convection in a rotating porous medium filled by a magnetic fluid. This study can reveal useful behaviors for its application in the field of technology and engineering.We derive a set of nonlinear differential equations for the magnitudes of flow rate, temperature, magnetic particle concentration, and magnetic potential using a truncated Galerkin expansion. The dynamic behavior of the system has been characterized by plotting bifurcation diagrams, Lyapunov exponents, and phase diagrams which show chaotic and periodic behaviors.The paper is organized as follows: in section2, the basic equations for the convection in a rotating porous medium filled by a magnetic fluid and subject to the helical force are presented. Section 3 deals with the deriving of Lorenz-type equations using a truncated expansion of Galerkin. In section 4, the analysis of the stability of the solutions is made. Numerical simulations are performed, and the results are explained and discussed concerning the chaotic behavior in section 5. Finally, a conclusion is presented in the last section.


## 2. Basic Equations

We consider a porous rotating horizontal layer of thicknessd of incompressible binary magnetic fluid subject to the effect of helical force. The configuration of the problem is illustrated in Figure 1.Figure 1
Problem configuration.With the Boussinesq approximation, the equations describing the evolution of the binary ferrofluid under effect of helical force within the rotating porous medium in dimensionless form are written [38, 39] as(1)∇⋅u=0,(2)ε−1P−1∂t+ε−1u.∇u=−∇Peff+Λ∇2u+RaΣ+sf+Ta1/2u∧ez−Da−1u,(3)dtθ−M4∂zϕ=1−M4w+∇2θ+F∇2C−M2∂zϕ,(4)dtC=L∇2C+θ−M2∂zϕ,(5)∂z2+M3∇⊥2ϕ=∂zθ−ψmC,(6)∇ϕext2=0,where u,θ,ϕ are, respectively, the dimensionless perturbations of velocity, temperature, magnetic particle concentration, and magnetic potential.dtf=∂tf+u⋅∇f is the particular derivative. Peff is the effective pressure. Also, Σ=Π1θ,C,ϕz^−M1θ−ψmC∇∂zϕ with Π1θ,C,ϕ=1+M1θ−ψ+M1ψmC and ∇⊥2=∂x2+∂y2.The following dimensionless numbers appear in the equations (1)–(5):Rayleigh number:Ra=αTρogβd4/κμPrandtl number:P=μ/ρoκ,Darcy number:Da=kp/d2,Brinkman number:Λ=μe/μ,Intensitys of the helical force f: s=aϖε−1d2/μ.Magnetic numbers are as follows:(7)M1=βχT2Ho2ρogαT1+χ,M2=D˜χcχTHo2ρoD¯T1+χ,M3=1+χo+χHHo21+χ,M4=χTHo2ToCH1+χ,M5=αHχTHo2αT1+χ,and ψm=−χcDs/χTDc.Lewis number:L=Dc/κ,Separation report:ψ=ρoD¯Tαc/αTD˜γH, Dufour number: F=D¯T2/D˜κ.And, Taylor number:Ta=2ρoε−1ϖd2/μ2 with Dc=D˜γH/ρo2 and Ds=D¯T/ρo.The magnetic numbers used have the following orders of magnitude:M1=10−4−10, M3≃1.1, M4≃M5≃10−6[24].The values of the numbersM4, M5, and F are very small. This is why they will be neglected in the following calculations.We express the velocity field as (the analysis is limited to a flow in dimension 2):u=−∂zΨ,0,∂xΨ. Eliminating the pressure and the vorticity in equations (2)–(5), we obtain(8)ε−1P−1∂t−Λ∇+2+Da−12∇+2−s2∂x2−∂z2∂z2+Ta∂z2+Ta1/2s∂z3−∂x2−∂z2∂z∂zΨ=Raε−1P−1∂t−Λ∇+2+Da−1∂x2Ξ,(9)dtθ=∂xΨ+∇+2θ,(10)dtC=L∇+2C+θ−M2∂zϕ,(11)∂z2+M3∂x2ϕ=∂zθ−ψmC,(12)dtf=∂tf+∂xΨ∂zf−∂zΨ∂xf,Ξ=1+M1θ−ψ+M1ψmC=M1∂zϕ and. ∇+2f=∂x2f+∂z2f.We impose ideal boundary conditionsθ=C=Ψ=∂z2Ψ=∂zϕ=0 at z=0,1 for temperature, magnetic particle concentration, and scalar magnetic potential stream function in order to solve equations (8)–(11) using the Galerkin method.


## 3. Galerkin Truncated Extension

We will restrict ourselves to the fundamental mode for the numerical simulations in the lateral direction since we consider a large container. In thez-direction across the layer, a multimode description will be used where necessary. Higher harmonics describe deviations of the variables from the linear regime [24]. According to the boundary conditions, we can represent the stream function, the temperature, and the concentration of the magnetic particles and the magnetic potential on the form below in order to obtain the solution of (8)–(11):(13)kΨt,z,x=a1tsinπzsinkx,(14)θt,z,x=a2tsinπzcoskx+a3tsin2πz,(15)Ct,z,x=a4tsinπzcoskx+a5tsin2πz,(16)ϕt,z,x=a6tcosπzcoskx+a7tcos2πz,with(17)a6t=−πa2tπ2+k2M3anda7t=−a3t2π.As in the Lorenz model, we only consider the temperature, the concentration, and consequently the scalar magnetic potential, the effect of the second harmonics. We have neglected the second harmonic of the current function due to the assumption of small convective motions [8].Calculating the terms of the equations (8)–(11) with the Galerkin functions defined from (12)–(15), then multiplying the equations by the orthogonal eigenfunctions, and integrating them in space over the wavelength of a convection cell, ∫−π/kπ/k∫01∫01dxdz, a set of ordinary differential equations is obtained for the temporal evolution of the amplitudes:(18)a¨1t=−2εPΛq2+Da−1a˙1+rεPq4−ε2P2Λ2q4+Da−2+ΛDa−11+q2−π2ε2P2q2Ta+k2−π2s2a1t+πrεPq4a1ta3t−πRak2Pq2ψ+M1ψma1ta5t+rPq6−rε2P2Λq2+Da−1q4+σRak2Pψ+M1ψma1t+ε2P2k2Raq2ψ+M1ψmΛq2+Da−1−ε−1P−1q2La4t,a˙2t=−a1t−q2a2t−πa1ta3t,a˙3t=−π2a1ta2t−4π2a3t,a˙4t=−πa1ta5t−Lq2a4t+σq2a2t,a5t=π2a1ta4t−4π2La5t−4π2L1−M2a3t.q2=k2+π2,r=Ra/Ras and(19)σ=LM2π2−Lπ2+k2M3π2+k2M3.r is the reduced number of Rayleigh [24]; Ras is the stationary Rayleigh number in the case of linear stability when ψ=0 and ψm=0 (Finlayson [40]).(20)Ras=q6π2+k2M3k2π2+k21+M1M3.


## 4. Stability Analysis of the System

We use [19] and we setτ=q2t and new variables Xτ=π/q22a1τ, Yτ=πr/2−a2τ, Zτ=−πra3τ, Rτ=−πr/2a4τ, and Sτ=−πra5τ.We get(21)X˙τ=Wτ,(22)Y˙τ=rXτ−Yτ−XτZτ,(23)Z˙τ=XτYτ−ηZτ,(24)R˙τ=−LRτ+σYτ−XτSτ,(25)S˙τ=−ηLSτ+XτRτ−ηL1−M2Zτ,(26)W˙τ=−2εPΛ+23π2Da−1Wτ−εPXτZτ+εPεPΛ+23π2Da−1−1Yτ+εPr−εP49π2+23π2ΛDa−1+Λ2+427π4Ta−427π2s2+49π2Da−2Xτ.withη=4π2/q2 and f˙τ=dfτ/dτ.We recover existing results in literature from the above formula. WhenC⟶0, M1=0 or M3=0Q13=0, and q2⟶3π2/2, we retrieve Lorenz's system [8]. When C=0,Ta=0, and s=0, the Lorenz system obtained by Laroze [24] is recovered.Ifs⟶0, Da−1≈0,Λ⟶1, and ε⟶1, we recover the system studied in [28].


### 4.1. Dissipation Effect

(27)∇V=∂X˙∂X+∂Y˙∂Y+∂Z˙∂Z+∂R˙∂R+∂S˙∂S+∂W˙∂W=−η1+L−1+L+2εPΛ+23π2Da−1<0.The system (21)–(26) is then dissipative. Therefore, the endpoints of the trajectories of a set of initial points in phase space, which occupies the region V0 at time t=0, will decrease in volume after some time t such that(28)Vt=V0exp−η1+L−1+L+2εPΛ+23π2Da−1t.The volume then decreases exponentially over time.


### 4.2. Study of the Equilibrium Points

The system (21)–(26) has the form X˙=fX, and the fixed points XS are given by fXS. The trivial equilibrium point E1 is(29)X1,Y1,Z1,R1,S1,W1=0,0,0,0,0,0.


### 4.3. Analysis of the System Stability at the Trivial Equilibrium Points

The matrix associated with the dynamic system studied is as follows due to the stability conditions:(30)JE1=000001r−1000000−η0000σ0−L0000−ηmL0−ηL0Yζ000−βwith(31)Υ=εPr−εP49π2+23π2ΛDa−1+Λ2+427π4Ta−427π2s2+49π2Da−2,ζ=εPεPΛ+23π2Da−1−1,β=−2εPΛ+23π2Da−1.The trivial fixed point polynomial characteristic equation is(32)Pλ≡b0λ6+b1λ5+b2λ4+b3λ3+b4λ2+b5λ+b6,with(33)b0=1,b1=η+L+ηL+1+β,b2=η1+L2+1+ηL1+η+β1+L−Y,b3=1+ηηLβ+βηL+ηL1+η+βL+βη−1+η1+L+ηY,b4=η2L2+βηL1+η1+L−1+η1+L+ηY,b5=βη2L2−ηL+ηL21+ηY,b6=−η2L2Y.To determine the eigenvaluesλi, we use the Routh–Hurwitz criterion [19, 41, 42].Whateverr>0, all the coefficients of the polynomial Pλ are positive. For <1.1, the Hurwitz determinants verify Δn−1>0,Δn−3>0 for Λ=1,Da−1≈0,s=0, and Ta=0. The trivial fixed point is then stable for r<1.1 when Λ=1,Da−1≈0,s=0, and Ta=0, and unstable otherwise. The fixed point E1 is linearly unstable when the critical value of r is rc=1.1.


### 4.4. Regular and Chaotic Behavior Analysis

Numerical simulations are carried out in order to examine the effects of viscosity ratioΛ, the number of Darcy Da characteristic of the porosity of the medium, the helical force s, and the Taylor number Ta on the chaotic convection of the magnetic fluid. The fourth-order Runge-Kutta method is employed to solve numerically (21)–(26). We set the values P=10, η=8/3, L=1, M2=1, and σ=−1/3. The initial conditions are X0=Y0=R0=W0=0.8 and Z0=S0=0.92195.We controlled the system formed by equations (21)–(26) with the reduced Rayleigh number r and the parameters above in presence.In order to appreciate the effects of the parameters in presence on the system, we realize the bifurcation diagrams and their corresponding Lyapunov exponents, the phase portraits, and time stories.The system has the possibility of having regular and chaotic behaviors according to the range of values to reduce Rayleigh numberr (Figures 2–19).Figure 2
Diagrams of bifurcations ofZ as a function of r representing the maxima and minima of the posttransient solution of Zτ showing the effect of the viscosity ratio Λ on the chaos of the system for Da=1, s=0, and Ta=0 with P=10, L=M2=1,η=8/3, k=π/2, ε=1, and σ=−1/3.Figure 3
Phase portraits and time diagrams of the system for a fixed reduced Rayleigh numberr=765 and different values of Λ with the parameter values of Figure 2.Figure 4
Phase portraits and time diagrams for different values of the reduced Rayleigh numberr when Λ=2 with the parameter values of Figure 2.Figure 5
Diagrams of bifurcations ofZ as a function of r showing the effect of the helical force on the chaos domain of the system when Λ=1Da=0.09, Ta=0, P=10, L=M2=1,η=8/3, k=π/2, ε=1, andσ=−1/3.Figure 6
Phase portraits and time stories of the system for a fixed reduced Rayleigh numberr=670 and different values of s with the parameter values of Figure 5.Figure 7
Phase portraits and time stories for different values of the reduced number of Rayleighr when s=2 with the parameter values of Figure 5.Figure 8
Diagrams of bifurcations ofZ as a function of r showing the effect of the Darcy number Da on the chaos of the system when s=0 with the parameter values of Figure 5.Figure 9
Phase portraits and time stories of the system for a fixed reduced number of Rayleighr=362 and different values of Da when s=0 with the parameter values of Figure 5.Figure 10
Phase portraits and time stories for different values ofr when Da=0.1, Λ=1, s=0, Ta=0, P=10, k=π/2, L=M2=1,η=8/3, ε=1, and σ=−1/3.Figure 11
Diagrams of bifurcations ofZ as a function of r showing the effect of the Taylor number Ta when Λ=1, Da=1, s=0, P=10, k=π/2, L=M2=1,η=8/3, ε=1, and σ=−1/3.Figure 12
Phase portraits and time stories for a fixed reduced Rayleigh numberr=464 and various values of Ta for Λ=1,s=0, Da=1,P=10, k=π/2, L=M2=1,η=8/3, ε=1, and σ=−1/3.Figure 13
Phase portraits and time stories for various values ofr when Ta=200, Λ=1, s=0, Da=1, P=10, L=M2=1,η=8/3, k=π/2, ε=1, and σ=−1/3.Figure 14
Diagrams of bifurcations ofZ as a function of r representing the maxima and minima of the posttransient solution of Zτ for Da=0.01Λ=1, s=0, Ta=0, k=π/2, L=M2=1,η=8/3,,P=10, ε=1, and σ=−1/3.Figure 15
Phase portraits and time stories for various values ofr when Da=0.01, Λ=1, s=0, Ta=0,k=π/2, L=M2=1,η=8/3,P=10, ε=1, and σ=−1/3.Figure 16
Diagrams of bifurcation and Lyapunov exponents for different values ofΛ with Da=1, s=0, Ta=0, P=10, k=π/2, L=M2=1,η=8/3, ε=1, and σ=−1/3.Figure 17
Diagrams of bifurcation and Lyapunov exponents for various values ofs with Λ=1, Da=0.09, Ta=0, P=10, k=π/2, L=M2=1,η=8/3, ε=1, and σ=−1/3.Figure 18
Bifurcation diagrams and Lyapunov exponents for various values ofDa with Λ=1, s=0, Ta=0, P=10, k=π/2, L=M2=1,η=8/3, ε=1, and σ=−1/3.Figure 19
Bifurcation diagrams and its corresponding Lyapunov exponents for various values ofTa with Λ=1, Da=0, s=0, P=10, k=π/2, L=M2=1,η=8/3, ε=1, and σ=−1/3.Figures2 and 3 show that at a certain increase in the viscosity ratio, with the fixed reduced Rayleigh number, the system is completely chaotic after a linear, an oscillatory, and two small periodic sequences, i.e., the heat transfer is progressively accelerated. The increase in the viscosity ratio makes the system completely chaotic.Figure4 shows that the system is linearly stable for r=1.56, oscillatory for 46.95≤r≤58.125, chaotic for 58.237764359≤r≤549.828, 556.575≤r≤559.375, and r≥668,754, and periodic for 551.978≤r≤555.979 and 661.505≤r≤668.251.Figures5 and 6 show a reduction in the chaotic domain with the helical force. For a certain fixed value of r (Figure 6), for example, for r=670 when the other parameters are kept fixed, the system is chaotic for s=0 and s=2 then periodic thereafter. We then conclude that for a value of the reduced number of Rayleigh r fixed, the system changes state, and the chaos slowly disappears when s increases, leaving the system in a periodic state of decreasing period. It can therefore be deduced that the increase in the helical force leads to a reduction of the chaotic domain of the system, corresponding to a delay of the heat transfer.The system is linearly stable forr=1.447, oscillatory for 25.899≤r≤35.6994, chaotic for 37.83909≤r≤384.086 and 429.00251≤r≤680.86, and then periodic for 386.237≤r≤424,946 and r≥683,6 (Figure 7).Figures8 and 9 show a reduction in the chaotic domain with the Darcy number leaving the system in a periodic state. Considering Figure 9, we notice that for a certain fixed value of r, for example, for r=362, when the other parameters are kept fixed, the system is chaotic for Da=0.09 then periodic: of period 2 for Da=0.5 and of period 1 for Da=1. It can be therefore concluded that the Darcy number Da reduces the chaotic domain of the system; i.e., the heat transfer is gradually retarded.The system is linearly stable forr=1.45, oscillatory for 25.59≤r≤35.6994, chaotic for 36.9094≤r≤362.581 and 403.441≤r≤637.849, and then periodic for 364.731≤r≤399.14 and r≥640 (Figure 10).We notice that for a low value of the Darcy numberDa, the system presents a surprising and advantageous behavior for the convection (Figure 14). The system is linearly stable for r=7.8, oscillatory for 374.142≤r≤376.961, chaotic for 377.00892639168≤r≤468.873 and 494.608≤r≤530.147, and then periodic for 470.098≤r≤492.157 and r≥531.377 (Figure 15).Figures11 and 12 show that the Taylor number Ta makes the system progressively chaotic and the heat transfer gradually increases. Considering the Figure 11, for r=464, the system is periodic: of period 1 for Ta=0, of period 2 for Ta=200 then of period 4 for Ta=500 and chaotic for Ta=1000. The Taylor number then enlarges the chaotic domain of the system.The system with Figure13 is linearly stable for r=1.449, oscillatory for 22.15≤r≤27.0973, chaotic for 28.55984405≤r≤231.398 and 252.903≤r≤328.823, and then periodic for 237.85≤r≤250. 753 and r≥330


## 4.1. Dissipation Effect

(27)∇V=∂X˙∂X+∂Y˙∂Y+∂Z˙∂Z+∂R˙∂R+∂S˙∂S+∂W˙∂W=−η1+L−1+L+2εPΛ+23π2Da−1<0.The system (21)–(26) is then dissipative. Therefore, the endpoints of the trajectories of a set of initial points in phase space, which occupies the region V0 at time t=0, will decrease in volume after some time t such that(28)Vt=V0exp−η1+L−1+L+2εPΛ+23π2Da−1t.The volume then decreases exponentially over time.


## 4.2. Study of the Equilibrium Points

The system (21)–(26) has the form X˙=fX, and the fixed points XS are given by fXS. The trivial equilibrium point E1 is(29)X1,Y1,Z1,R1,S1,W1=0,0,0,0,0,0.


## 4.3. Analysis of the System Stability at the Trivial Equilibrium Points

The matrix associated with the dynamic system studied is as follows due to the stability conditions:(30)JE1=000001r−1000000−η0000σ0−L0000−ηmL0−ηL0Yζ000−βwith(31)Υ=εPr−εP49π2+23π2ΛDa−1+Λ2+427π4Ta−427π2s2+49π2Da−2,ζ=εPεPΛ+23π2Da−1−1,β=−2εPΛ+23π2Da−1.The trivial fixed point polynomial characteristic equation is(32)Pλ≡b0λ6+b1λ5+b2λ4+b3λ3+b4λ2+b5λ+b6,with(33)b0=1,b1=η+L+ηL+1+β,b2=η1+L2+1+ηL1+η+β1+L−Y,b3=1+ηηLβ+βηL+ηL1+η+βL+βη−1+η1+L+ηY,b4=η2L2+βηL1+η1+L−1+η1+L+ηY,b5=βη2L2−ηL+ηL21+ηY,b6=−η2L2Y.To determine the eigenvaluesλi, we use the Routh–Hurwitz criterion [19, 41, 42].Whateverr>0, all the coefficients of the polynomial Pλ are positive. For <1.1, the Hurwitz determinants verify Δn−1>0,Δn−3>0 for Λ=1,Da−1≈0,s=0, and Ta=0. The trivial fixed point is then stable for r<1.1 when Λ=1,Da−1≈0,s=0, and Ta=0, and unstable otherwise. The fixed point E1 is linearly unstable when the critical value of r is rc=1.1.


## 4.4. Regular and Chaotic Behavior Analysis

Numerical simulations are carried out in order to examine the effects of viscosity ratioΛ, the number of Darcy Da characteristic of the porosity of the medium, the helical force s, and the Taylor number Ta on the chaotic convection of the magnetic fluid. The fourth-order Runge-Kutta method is employed to solve numerically (21)–(26). We set the values P=10, η=8/3, L=1, M2=1, and σ=−1/3. The initial conditions are X0=Y0=R0=W0=0.8 and Z0=S0=0.92195.We controlled the system formed by equations (21)–(26) with the reduced Rayleigh number r and the parameters above in presence.In order to appreciate the effects of the parameters in presence on the system, we realize the bifurcation diagrams and their corresponding Lyapunov exponents, the phase portraits, and time stories.The system has the possibility of having regular and chaotic behaviors according to the range of values to reduce Rayleigh numberr (Figures 2–19).Figure 2
Diagrams of bifurcations ofZ as a function of r representing the maxima and minima of the posttransient solution of Zτ showing the effect of the viscosity ratio Λ on the chaos of the system for Da=1, s=0, and Ta=0 with P=10, L=M2=1,η=8/3, k=π/2, ε=1, and σ=−1/3.Figure 3
Phase portraits and time diagrams of the system for a fixed reduced Rayleigh numberr=765 and different values of Λ with the parameter values of Figure 2.Figure 4
Phase portraits and time diagrams for different values of the reduced Rayleigh numberr when Λ=2 with the parameter values of Figure 2.Figure 5
Diagrams of bifurcations ofZ as a function of r showing the effect of the helical force on the chaos domain of the system when Λ=1Da=0.09, Ta=0, P=10, L=M2=1,η=8/3, k=π/2, ε=1, andσ=−1/3.Figure 6
Phase portraits and time stories of the system for a fixed reduced Rayleigh numberr=670 and different values of s with the parameter values of Figure 5.Figure 7
Phase portraits and time stories for different values of the reduced number of Rayleighr when s=2 with the parameter values of Figure 5.Figure 8
Diagrams of bifurcations ofZ as a function of r showing the effect of the Darcy number Da on the chaos of the system when s=0 with the parameter values of Figure 5.Figure 9
Phase portraits and time stories of the system for a fixed reduced number of Rayleighr=362 and different values of Da when s=0 with the parameter values of Figure 5.Figure 10
Phase portraits and time stories for different values ofr when Da=0.1, Λ=1, s=0, Ta=0, P=10, k=π/2, L=M2=1,η=8/3, ε=1, and σ=−1/3.Figure 11
Diagrams of bifurcations ofZ as a function of r showing the effect of the Taylor number Ta when Λ=1, Da=1, s=0, P=10, k=π/2, L=M2=1,η=8/3, ε=1, and σ=−1/3.Figure 12
Phase portraits and time stories for a fixed reduced Rayleigh numberr=464 and various values of Ta for Λ=1,s=0, Da=1,P=10, k=π/2, L=M2=1,η=8/3, ε=1, and σ=−1/3.Figure 13
Phase portraits and time stories for various values ofr when Ta=200, Λ=1, s=0, Da=1, P=10, L=M2=1,η=8/3, k=π/2, ε=1, and σ=−1/3.Figure 14
Diagrams of bifurcations ofZ as a function of r representing the maxima and minima of the posttransient solution of Zτ for Da=0.01Λ=1, s=0, Ta=0, k=π/2, L=M2=1,η=8/3,,P=10, ε=1, and σ=−1/3.Figure 15
Phase portraits and time stories for various values ofr when Da=0.01, Λ=1, s=0, Ta=0,k=π/2, L=M2=1,η=8/3,P=10, ε=1, and σ=−1/3.Figure 16
Diagrams of bifurcation and Lyapunov exponents for different values ofΛ with Da=1, s=0, Ta=0, P=10, k=π/2, L=M2=1,η=8/3, ε=1, and σ=−1/3.Figure 17
Diagrams of bifurcation and Lyapunov exponents for various values ofs with Λ=1, Da=0.09, Ta=0, P=10, k=π/2, L=M2=1,η=8/3, ε=1, and σ=−1/3.Figure 18
Bifurcation diagrams and Lyapunov exponents for various values ofDa with Λ=1, s=0, Ta=0, P=10, k=π/2, L=M2=1,η=8/3, ε=1, and σ=−1/3.Figure 19
Bifurcation diagrams and its corresponding Lyapunov exponents for various values ofTa with Λ=1, Da=0, s=0, P=10, k=π/2, L=M2=1,η=8/3, ε=1, and σ=−1/3.Figures2 and 3 show that at a certain increase in the viscosity ratio, with the fixed reduced Rayleigh number, the system is completely chaotic after a linear, an oscillatory, and two small periodic sequences, i.e., the heat transfer is progressively accelerated. The increase in the viscosity ratio makes the system completely chaotic.Figure4 shows that the system is linearly stable for r=1.56, oscillatory for 46.95≤r≤58.125, chaotic for 58.237764359≤r≤549.828, 556.575≤r≤559.375, and r≥668,754, and periodic for 551.978≤r≤555.979 and 661.505≤r≤668.251.Figures5 and 6 show a reduction in the chaotic domain with the helical force. For a certain fixed value of r (Figure 6), for example, for r=670 when the other parameters are kept fixed, the system is chaotic for s=0 and s=2 then periodic thereafter. We then conclude that for a value of the reduced number of Rayleigh r fixed, the system changes state, and the chaos slowly disappears when s increases, leaving the system in a periodic state of decreasing period. It can therefore be deduced that the increase in the helical force leads to a reduction of the chaotic domain of the system, corresponding to a delay of the heat transfer.The system is linearly stable forr=1.447, oscillatory for 25.899≤r≤35.6994, chaotic for 37.83909≤r≤384.086 and 429.00251≤r≤680.86, and then periodic for 386.237≤r≤424,946 and r≥683,6 (Figure 7).Figures8 and 9 show a reduction in the chaotic domain with the Darcy number leaving the system in a periodic state. Considering Figure 9, we notice that for a certain fixed value of r, for example, for r=362, when the other parameters are kept fixed, the system is chaotic for Da=0.09 then periodic: of period 2 for Da=0.5 and of period 1 for Da=1. It can be therefore concluded that the Darcy number Da reduces the chaotic domain of the system; i.e., the heat transfer is gradually retarded.The system is linearly stable forr=1.45, oscillatory for 25.59≤r≤35.6994, chaotic for 36.9094≤r≤362.581 and 403.441≤r≤637.849, and then periodic for 364.731≤r≤399.14 and r≥640 (Figure 10).We notice that for a low value of the Darcy numberDa, the system presents a surprising and advantageous behavior for the convection (Figure 14). The system is linearly stable for r=7.8, oscillatory for 374.142≤r≤376.961, chaotic for 377.00892639168≤r≤468.873 and 494.608≤r≤530.147, and then periodic for 470.098≤r≤492.157 and r≥531.377 (Figure 15).Figures11 and 12 show that the Taylor number Ta makes the system progressively chaotic and the heat transfer gradually increases. Considering the Figure 11, for r=464, the system is periodic: of period 1 for Ta=0, of period 2 for Ta=200 then of period 4 for Ta=500 and chaotic for Ta=1000. The Taylor number then enlarges the chaotic domain of the system.The system with Figure13 is linearly stable for r=1.449, oscillatory for 22.15≤r≤27.0973, chaotic for 28.55984405≤r≤231.398 and 252.903≤r≤328.823, and then periodic for 237.85≤r≤250. 753 and r≥330


## 5. Conclusions

In this work, we studied the effects of the viscosity ratioΛ, the helical force intensity s, the Darcy number Da, and the Taylor number Ta on the regular and chaotic behavior of the convection in a rotating magnetic fluid in a porous medium under the helical force effect. The classical nonlinear stability theory is used. Lorenz-type model is obtained by derived from the magnetohydrodynamics equations using the lowest Fourier modes a set of first-order ordinary differential equations.In the absence of a magnetic field and helical in a nonrotating and a nonporous medium, the classical Lorenz model is recovered. The stability of the trivial equilibrium point has been studied with the Routh–Hurwitz criterion. The parameter regions where stationary states can occur have been identified as well as those of regular and chaotic behaviors by plotting successively bifurcation diagrams, Lyapunov exponent, and phase portraits. Throughout this study, the following results have been obtained:(i)
Increasing the ratio of viscositiesΛ increases the chaotic domain, so serves to advance the heat transfer in the magnetic fluid.(ii)
Increasing the intensitys of the helical force gradually contributes to reduce the chaotic domain of the system leaving it in a periodic state. This favors the heat transfer to decrease in the fluid.(iii)
Increasing the Darcy numberDa considerably reduces the domain of chaos leaving the system in a periodic state, hence serves to delay the heat transfer in the magnetic fluid.(iv)
Increasing the Taylor numberTa increases the chaotic domain, so serves to accelerate the heat transfer in the system.We then conclude that the ratio of viscosities and the rotation enlarge the domain where chaos occurs and accelerate the heat transfer whereas the helical force and the porosity of the medium reduce the chaotic domain and delay the heat transfer. The ratio of viscosities, the porosity, the rotation, and the helical force, therefore, make it possible to control the convective chaos in the magnetic fluid.



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