G-Jitter Effects on Chaotic Convection in a Rotating Fluid Layer

The effect of gravity modulation and rotation on chaotic convection is investigated. A system of differential equation like Lorenz model has been obtained using the Galerkin-truncated Fourier series approximation. The nonlinear nature of the problem, i.e., chaotic convection, is investigated in a rotating fluid layer in the presence of g-jitter. The NDSolve Mathematica 2017 is employed to obtain the numerical solutions of Lorenz system of equations. It is found that there is a proportional relation between Taylor number and the scaled Rayleigh number R in the presence of modulation. This means that chaotic convection can be delayed (for increasing value of R) or advanced with suitable adjustments of Taylor number and amplitude and frequency of gravity modulation. Further, heat transfer results are obtained in terms of finite amplitude. Finally, we conclude that the transition from steady convection to chaos depends on the values of Taylor number and g-jitter parameter.


Introduction
The study of chaotic convection is of great interest due to its applications in thermal and mechanical engineering and in many other industry applications. It was introduced by Lorenz [1] to illustrate the study of atmospheric three-space model arising from Rayleigh-Benard convection. Some of the applications are production of crystals, oil reservoir modeling, and catalytic packed bed filtration. He developed a simplified mathematical model for atmospheric convection given below: This model is a system of three ordinary differential equations known as the Lorenz equations. These equations are related to the properties of a twodimensional Rayleigh-Benard convection. In particular, the system describes the rate of change of three quantities convection, temperature variation vertically with respect to time. These equations are related to the properties of two-dimensional flow model warmed uniformly from below and cooled from above. In particular, the system describes the rate of change of three quantities of time, x is proportional to the rate of convection, y is the horizontal temperature variation, and z is the vertical temperature variation. The constants Pr, R and β are the system parameters proportional to the Prandtl number, Rayleigh number, and certain physical dimensions of the media. If R < 1 then there is only one equilibrium point at the origin which is represented as no convection point. Further, all orbits converge to the origin, which is a global attractor. When R = 1, then a pitchfork bifurcation occurs, and for R 1 , two additional critical points arise and are known as convection points, and there the system loses its stability. In addition to this model, I would like to add the concept of modulation either to suppress or to enhance nonlinearity. The literature shows that there are different types available; some of them are temperature modulation (Venezian [2]), gravity (Gresho and Sani [3] and Bhadauria and Kiran [4,5]), rotation (Donnelly [6], Kiran and Bhadauria [7]), and magnetic field modulation (Bhadauria and Kiran [8,9]). Their studies are mostly on thermal convection either considering fluid or porous medium. Their ultimate idea behind the research is to find external regulation to the system to control instability and measure the heat mass transfer in the system. But what happens when we consider the external configuration to system Eq. (1). The external configurations are like thermal, gravity, rotation, and magnetic field modulation. In this direction, no data are reported so far. With this, I would like to extend the work of Lorenz along with modulation.
The studies on chaos with respect to the different types of parameters like Rayleigh number and Prandtl number are mostly investigated by the following studies. The transition from steady convection to chaos occurs by a subcritical Hopf bifurcation producing a solitary cycle which may be associated with a homoclinic explosion for low Prandtl number is investigated by Vadasz and Olek [10]. The work of Vadasz [11] suggests an explanation for the appearance of this solitary limit cycle via local analytical results. The effect of magnetic field on chaotic convection in fluid layer is investigated by Mahmud and Hasim [12]. They found that transition from chaotic convection to steady convection occurs by a subcritical Hopf bifurcation producing a homoclinic explosion which may limit the cycle as Hartman number increases. For the moderate values of Prandtl number, the route to chaos occurs by a period of doubling sequence of bifurcations given by Vadasz and Olek [13]. Feki [14] proposed a new simple adaptive controller to control chaotic systems. The constructed linear structure of controller may be used for chaos control as well as for chaotic system synchronization. Yau and Chen [15] found that the Lorenz model could be stabilized, even in the existence of system external distraction. For non-Newtonian fluid case, Sheu et al. [16] have shown that stress relaxation tends to accelerate onset chaos. A weak nonlinear solution to the problem is assumed by Vadasz [17], and it can produce an accurate analytical expression for the transition point as long as the condition of validity and consequent accuracy of the latter solution is fulfilled. Narayana et al. [18] investigated heat mass transfer using truncated Fourier series method. They have also discussed chaotic convection under the effect of binary viscoelastic fluids. The studies related to gravity modulation are given by Kiran et al. [19][20][21][22][23][24][25]. These studies show that the gravity modulation can be used to control heat and mass transfer in the system in terms of frequency and amplitude of modulation.
The above paragraph demonstrated the earlier work on chaotic convection with different configurations and models to control chaos. Recently Vadasz et al. [26] and Kiran et al. [27] have investigated the effect of vertical vibrations and temperature modulation on chaos in a porous media. Their results show that periodic solutions and chaotic solutions alternate as the value of the scaled Rayleigh number changes in the presence of forced vibrations. The root to chaos is also affected by three types of thermal modulations.
The effect of rotation on chaos is investigated by Gupta et al. [28] without any modulation. They found that rotation has delay in chaos and controls nonlinearity. It is also concluded that there are suitable ranges over Ta and R to reduce chaos in the system. Based on the above studies in this chapter, I would like to investigate the study of chaotic convection in the presence of rotation and gravity modulation.

Mathematical model
An infinitely extended horizontal rotating fluid layer about its vertical z-axis is considered. The layer is gravity modulated and the lower plate held at temperature T 0 while the upper plate at T 0 þ ΔT. Here ΔT is the temperature difference in the medium. The mathematical equation of the flow model is given by The thermal boundary conditions are given by where q À > is the velocity of the fluid, Ω À > is the vorticity vector, p À > is the fluid pressure, ρ À > is the density, ν À > is the kinematic viscosity, K T À > is the thermal diffusivity ratio, and α t À > is the thermal expansion coefficient. We consider in our problem the externally imposed gravitational field (given by Gresho and Sani [3]): where δ g , ω g are the amplitude and frequency of gravity modulation.

Basic state
The basic state of the fluid is quiescent and is given by Using the basic state Eq. (10) in the Eqs. (4)-(6), we get the following relations and from Eq.

Perturbed state
On the basic state, we superpose perturbations in the form where the primes denote perturbed quantities. Now substituting Eq. (18) into Eqs. (4)-(7) and using the basic state solutions, we obtain the equations governing the perturbations in the form simplifying the above equation, then we get Similarly we can derive the same for momentum equation of the following form We consider only two-dimensional disturbances and define the stream functions ψ and q by which satisfy the continuity Eq. (19). While introducing the stream function ψ and non-dimensionalizing with the following nondimensional parameters ( after simplifying the above equation, we get Similarly while eliminating the pressure term and using the dimensionless quantities, from the momentum equation (24), we get the following: where is the Rayleigh number. The assumed boundaries are stress free and isothermal; therefore, the boundary conditions are given by The set of partial differential Eqs. (26) and (27) forms a nonlinear coupled system of equations involving stream function and temperature as a function of two variables in x and z. We solve these equations by using the Galerkin method and using Fourier series representation.

Truncated Galerkin expansion
To obtain the solution of nonlinear coupled system of partial differential equations (26) and (27), we represent the stream function and temperature in the form The above are the Galerkin expansion of stream function and temperature. Now substituting these equations in Eqs. (26) and (27) and applying the orthogonal conditions to Eqs. (30) and (31) and finally integrating over the domain [0,1] Â [0,1] yield a set of equations: À2A 1 B 2 aπ cos 2πz cos ax sin πz: Now multiply with cos ax sin πz on both sides, and apply integration from 0 to 1 with respect to x and 0 to 2π a : Now let us consider Eq. (30) and multiply with sin 2πz on both sides of the equation and apply integration from 0 to 1 with respect to x and 0 to 2π a : then by simplifying the above equation, we get Similarly from Eq. (50) where k 2 ¼ π 2 þ a 2 is the total wavenumber and τ ¼ k 2 t is the rescaled time.
Introducing the following dimensionless quantities and rescale the amplitudes in the form of To provide the following set of equations, we consider the following equations and then simplifying the above equation, we get now from the Eq. (50) ∂ ∂τ Similarly from Eq. (28), where the symbol (/) denotes the time derivative dðÞ dτ . Eqs. (56), (59), and (61) are like the Lorenz equations (Lorenz (13), sparrow (14)), although with different coefficients. The final nonlinear differential equations are given by

Stability analyses
To understand the stability of the system, we determine the fixed points of the system and will try to find the nature of these fixed points through eigen equation. The nonlinear dynamics of Lorenz-like system (62)-(65) has been analyzed and solved for σ = 10, γ ¼ À 8 3 corresponding to convection. The basic properties of the system to obtain the eigen function are described next.

Dissipation
The system of Eqs. (62)-(65) is dissipative since If the set of initial solutions is the region of V(0), then after some time t, the endpoints of the trajectories will decrease to a volume: The above expression shows that the volume decreases exponentially with time.

Equilibrium points
System (62)-(65) has the general form, and the equilibrium (fixed or stationary) points are given by: From Eq. (83) we got and similarly we also got the following from Eq. (64): and similarly we also got the following from Eq. (65) for the momentum case: then we get a relation the remaining Y 2,3 , Z 2,3 will be accessed. The fixed points of rescaled system for modulated case are X 1 , Y 1

Stability of equilibrium points
The Jacobian matrix of Eqs. (62)-(65) is as follows: : The characteristic values of the above Jacobian matrix, obtained by solving the zeros of the characteristic polynomial, provide the stability conditions. If all the eigenvalues are negative, then the fixed point is stable (or in the case of complex eigenvalues, they have negative real parts) and unstable, when at least one eigenvalue is positive (or in the case of complex eigenvalues, it has positive real part): : The characteristic equation for the above system at origin is given by |A À λI| ¼ 0 which implies the following The first eigenvalue γ is always negative as γ ¼ À8 3 , but the other three eigenvalues are given by equation The stability of the fixed points corresponding to the convection solution X 2,3 , Y 2,3 , Z 2,3 ð Þ is controlled by the following equation for the eigenvalues λ i , ¼ 1, 2, 3, 4: The loss of stability of the convection fixed points for σ ¼ 10, γ ¼ À 8 3 using Eq. (80) is evaluated to be R c2 ¼ 25:75590 for system parameters T = 0, R c2 for T = 0.1, R c2 ¼ 25:75590 for T = 0.2, R c2 ¼ 29:344020 for T = 0.45, and R c2 ¼ 32:775550 for T = 0.6.

Nusselt number
According to our problem, the horizontally averaged Nusselt number for an oscillatory mode of convection is given by In the absence of the fluid motions, the Nusselt number is equal to 1. And simplifying the above equation, we will get the expressions for heat transfer coefficient:

Result and discussion
In this section we present some numerical simulation of the system of Eqs. (62)-(65) for the time domain 0 ≤ τ ≤ 40. The computational calculations are obtained by using Mathematica 17, fixing the values σ ¼ 10, γ ¼ À8=3, and taking in the initial conditions X(0) = Y(0) = 0.8, Z(0) = 0.9. In the case of T = 0, it is found that at R c 1 ¼ 1, obtained from Eq. (80), the motionless solution loses stability, and the convection solution occurs. Also the eigenvalues from Eq. (80) become equal and complex conjugate when R varies from 24.73684209 to 34.90344691 given by Gupta et al. [28]. The evolution of trajectories over a time domain in the state space for increasing the values of scaled Rayleigh number and modulation terms is given in the figures. The projections of trajectories onto Y-X, Z-Y, Z-Y, and W-Z planes are also drawn (Figure 1). In Figure 2, we observe that the trajectory moves to the steady convection points on a straight line for a Rayleigh number (R = 1:1) just above motionless solutions. It is clear from Figure 3a that the trajectories of the solutions approach the fixed points at R = 12, which means the motionless solution is moving around the fixed points. As the value of R changes around R = 25.75590, there is a sudden change and transition to chaotic solution (in Figure 3b).
In the case of gravity modulation in Figure 4, just keeping the values δ g ¼ 0:05, ω g ¼ 10 in connection with Figure 3, the motionless solution loses stability, and convection solution takes over. Even at the subcritical value of R = 25.75590, transition to chaotic behavior solution occurs, but one can develop fully chaotic nature with suitably adjusting the modulation parameter values δ g ¼ 0:05, ω g ¼ 10.
To see the effect of rotation on chaotic convection for the value of T = 0.45, we get R c 1 ¼ 1:45 from Eq. (80), which concludes that the motionless solution loses stability at this stage and the convection solution takes over. The other second and third eigenvalues become equal and complex conjugate at R = 31.44507647. In this state the convection points lose their stability and move onto the chaotic solution. The corresponding projections of trajectories and evolution of trajectories are presented in Figure 5a and b, planes Y-X, Z-X, Z-Y, and W-Z. At the subcritical value of R = 31.44507647, transition to chaotic behavior solution occurs. Observing Figure 5b it is clearly evident that in the presence of modulation σ ¼ 20, δ g ¼ 0:2,    ω g ¼ 20, the trajectories are manifolds around the fixed points. Which are the interesting results to see that the system is unstable mode with rotation and buoyancy. But with gravity modulation, the system becomes stable mode.
For the value of T = 0.6, we obtain the motionless solution (where the system loss stability) given in Figure 5b. The values of the second and third eigenvalues become equal and complex conjugate when the value of R = 24.73684209; at this point the convection points lose their stability, and chaotic solution must occur. But due to the presence of modulation, the trend is reversed given in Figure 6.
Observing that in the presence of modulation δ g ¼ 0:1, ω g ¼ 2, the system will come to stable mode for large values of R. The effect of frequency of modulation for the values ω g ¼ 2 and ω g ¼ 20 on chaos is presented in Figure 7a and b. It is clear that low-frequency-modulated fluid layer is in stable mode and high-frequencymodulated fluid layer in unstable mode. The reader may have look on the studies of [29][30][31][32][33] for the results corresponding to the modulation effect on chaotic convection. Finally we also derived the heat transfer coefficient (Nu τ ð Þ) given by Eq. (83) and verified the rate of transfer of heat under the effect of gravity modulation. It is clear from Figure 8 that heat transfer in the system is high for low-frequency modulation and for δ g values varies from 0.1 to 0.5. The results corresponding to the gravity modulation may be observed with the studies of [19][20][21][22][23][24][25][26].

Conclusions
In this chapter, we have studied chaotic convection in the presence of rotation and gravity modulation in a rotating fluid layer. It is found that chaotic behavior can be controlled not only by Rayleigh or Taylor numbers but by gravity modulation. The following conclusions are made from the previous analysis: 1. The gravity modulation is to delay the chaotic convection.
2. Taking the suitable ranges of ω g , δ g , and R, the nonlinearity is controlled.
3. The chaos in the system are controlled by gravity modulation either from stable to unstable or unstable to stable depending on the suitable adjustment of the parameter values.
4. The results corresponding to g-jitter may be compared with Vadasz et al. [27], Kiran [31] and Bhadauria and Kiran [33]. 5. It is found that heat transfer is enhanced by amplitude of modulation and reduced by frequency of modulation.