Advances in Mathematical Physics

Volume 2018, Article ID 1764182, 8 pages

https://doi.org/10.1155/2018/1764182

## Suppression of Chaos in Porous Media Convection under Multifrequency Gravitational Modulation

Laboratory of Mathematics and Applications, Faculty of Sciences and Technologies, University Hassan II of Casablanca, P.O. Box 146, Mohammedia, Morocco

Correspondence should be addressed to Karam Allali; rf.oohay@marakilalla

Received 29 December 2017; Accepted 26 February 2018; Published 3 April 2018

Academic Editor: Giorgio Kaniadakis

Copyright © 2018 Karam Allali. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Suppression of chaos in porous media convection under multifrequency gravitational modulation is investigated in this paper. For this purpose, a two-dimensional rectangular fluid-saturated porous layer heated from below subjected to a vertical gravitational modulation will be considered. The model consists of nonlinear heat equation coupled with a system of equations describing the motion under Darcy law. The time-dependent gravitational modulation is assumed to be with two frequencies and . A spectral method of solution is used in order to reduce the problem to a system of four ordinary differential equations. The system is solved numerically by using the fifth- and a sixth-order Runge-Kutta-Verner method. Oscillating and chaotic convection regimes are observed. It was shown that chaos can be suppressed by appropriate tuning of the frequencies’ ratio .

#### 1. Introduction

Several studies have been devoted to investigating the effect of a periodic gravitational modulation on the convective instability in a fluid layer. Indeed, the modulation of gravity can lead to stabilizing or destabilizing effect on the dynamics and the convective properties of the fluid [1, 2]. In practice, the gravitational modulation can be achieved by oscillating, in the vertical direction, the fluid container that is already subjected to a constant gravitational field. The modulation of gravity, with some specific amplitudes and frequencies, can contribute to the improvement of product performances such as solidification process [3, 4], frontal polymerization [5], or crystal growth [6].

The effect of gravitational modulation on natural convection in a porous layer was studied by Govender [7]. The linear stability analysis method was used to reduce the problem to a Mathieu equation and it was shown that increasing the frequency of vibration stabilizes the fluid convection. Elhajjar et al. [8] studied the influence of small-amplitude and high-frequency vertical vibrations on the Soret-driven convection flow. Both the direct numerical simulations and linear stability analysis were used in order to show that vibrations delay the transition from unicellular to bicellular flow regime. More recently, Vadasz et al. [9, 10] studied the periodic and chaotic natural convection in a porous layer subject to vertical vibrations. It was shown that periodic and chaotic solutions alternate when Rayleigh number increases. All those previous works consider the effect of one frequency gravitational modulation. However, adding another frequency or more can change considerably the onset of convection regime. For instance, it was shown that the gravitational modulation with two incommensurate frequencies produces a convective stabilizing or destabilizing effect depending on the frequencies’ ratio [11, 12]. In the absence of gravitational modulation, it was shown that chaos and thermal convection instabilities can be controlled or even suppressed by tuning of the temperature boundary values [13, 14].

Motivated by the recent works of Vadasz et al. [9, 10], the aim of this present paper is to continue the investigation of the influence of gravitational modulation on convective instability in porous layer heated from below, by assuming that the modulation is with two frequencies. To this end, we will consider that the system containing the fluid and the porous matrix are subjected to two-frequency vertical gravitational modulation. This external excitation causes a time-dependent acceleration given by , where is the gravity acceleration and , where and and and are the amplitudes and the frequencies of the vibration, respectively. We will examine the effect of the two frequencies’ ratio on the convective instability of the fluid flow and whether the chaos regime can be suppressed by tuning of the second frequency. Indeed, in many situations, chaos presents a troublesome phenomenon that may lead to physical system damage, thermal explosion due to irregular temperature oscillations, deadly epilepsy, and cardiac arrhythmias [15–18]. Accordingly, the need for chaos suppression is very important. Among those situations, one will be presented in this paper and we will study the effectiveness of introducing a second frequency gravitational modulation in chaos suppression. The need to add such second frequency gravitational modulation is particularly interesting when the first frequency is constrained to take some specific values or belongs to a narrow interval for which the chaos persists. In this paper, we will study the importance of adding a second frequency gravitational modulation in controlling chaos for both problems dealing with low or high Prandtl number fluids.

The paper is organized as follows: the next section introduces the model, while Section 3 deals with the numerical method used in our simulations. Results and discussions are provided in Section 4. The last section concludes the work.

#### 2. Mathematical Formulation

##### 2.1. The Mathematical Model

We consider a two-dimensional rectangular fluid-saturated porous layer subjected to a vertical gravitational modulation, as shown in Figure 1. We assume that the fluid is incompressible and is heated from bellow. The model of such a process can be described by the energy equation coupled with the hydrodynamics equations under the Boussinesq-Darcy approximation:where denotes the temperature, is the velocity, is the pressure, is the coefficient of thermal diffusivity, is the kinematic viscosity, is the gravity, is a unit vector in the downward vertical direction (in the sense of gravity), is the cold wall temperature, and is the permeability. We consider the following boundary conditions:here is the hot wall temperature; and are the horizontal and the vertical dimensions of the porous medium. The boundary conditions for the temperature correspond to the adiabatic (no-flux) condition at the lateral boundaries and fixed temperature at the lower and upper boundaries. For the velocity, its normal component at the boundary is zero. This means that the fluid does not intersect the boundary. In what follows, we will denote by the characteristic temperature difference.