Advances in Mathematical Physics

Volume 2017, Article ID 2789024, 12 pages

https://doi.org/10.1155/2017/2789024

## Effect of Internal Heat Source on the Onset of Double-Diffusive Convection in a Rotating Nanofluid Layer with Feedback Control Strategy

^{1}Institute for Mathematical Research, Universiti Putra Malaysia, 43400 Serdang, Selangor, Malaysia^{2}Department of Mathematics, Faculty of Science, Universiti Putra Malaysia, 43400 Serdang, Selangor, Malaysia^{3}School of Mathematical Sciences, Faculty of Science and Technology, National University of Malaysia, 43600 Bangi, Selangor, Malaysia^{4}Department of Chemistry, Faculty of Science, Universiti Putra Malaysia, 43400 Serdang, Selangor, Malaysia

Correspondence should be addressed to N. F. M. Mokhtar; moc.liamg@rathkom.hallizdafron

Received 15 February 2017; Revised 1 May 2017; Accepted 5 June 2017; Published 13 July 2017

Academic Editor: Xavier Leoncini

Copyright © 2017 I. K. Khalid et al. 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

A linear stability analysis has been carried out to examine the effect of internal heat source on the onset of Rayleigh–Bénard convection in a rotating nanofluid layer with double diffusive coefficients, namely, Soret and Dufour, in the presence of feedback control. The system is heated from below and the model used for the nanofluid layer incorporates the effects of thermophoresis and Brownian motion. Three types of bounding systems of the model have been considered which are as follows: both the lower and upper bounding surfaces are free, the lower is rigid and the upper is free, and both of them are rigid. The eigenvalue equations of the perturbed state were obtained from a normal mode analysis and solved using the Galerkin method. It is found that the effect of internal heat source and Soret parameter destabilizes the nanofluid layer system while increasing the Coriolis force, feedback control, and Dufour parameter helps to postpone the onset of convection. Elevating the modified density ratio hastens the instability in the system and there is no significant effect of modified particle density in a nanofluid system.

#### 1. Introduction

The importance of understanding convective heat transfer in nanofluids has been a topic of interest for the last few years. A nanofluid, that is, a colloidal mixture of nanosized particles (1–100 nm), and a base fluid (nanoparticle fluid suspension) was first introduced by Choi [1]. Nanofluids possess their stability due to the small size of their particles, low weight, and less chance of sedimentation. Subsequently, there have been tremendous attempts to observe the enhancement of thermal conductivity in a nanofluid. The earliest investigation was reported by Masuda et al. [2], followed by Xuan and Li [3], Eastman et al. [4], and Das et al. [5]. Buongiorno [6] discussed the seven slip mechanisms that can produce a relative velocity between the nanoparticles and the base fluid. These are inertia, Brownian diffusion, thermophoresis, diffusiophoresis, Magnus effect, fluid drainage, and gravity settling. He concluded that the effects of Brownian diffusion and thermophoresis are important for convective transport in nanofluids and his model is the basis for the present study. Tzou [7, 8] employed Buongiorno’s model to investigate the thermal instability problem and found that nanofluids are less stable than regular fluids. Alloui et al. [9] studied the natural convection of nanofluids in a shallow cavity heated from below. Nield and Kuznetsov [10] studied the parameters involved in the onset of convection in a horizontal nanofluid layer of finite depth. A linear analysis of the Rayleigh–Bénard instability for a nanofluid is performed by Yadav et al. [11]. Haddad et al. [12] reported that the thermophoresis and Brownian motion are significant in the thermal enhancement of the natural convection in a nanofluid layer. Recently, Gupta et al. [13] investigated the thermosolutal convection in a horizontal nanofluid layer heated from below.

Soret diffusion, or also known as thermal diffusion, and Dufour diffusion, also known as thermodiffusion [14], are important in both Newtonian and non-Newtonian convective heat and mass transfer, and they are often encountered in chemical process engineering and in high-speed aerodynamics. Such effects are significant for gases of intermediate molecular weight in the coupled heat and mass transfer in binary systems. Hurle and Jakeman [15] demonstrated the Soret-driven thermosolutal convection both theoretically and experimentally using a water-methanol mixture. Then, Platten and Chavepeyer [16] continued using a water-ethanol mixture while Caldwell [17] extended the investigation using a salt solution. The linear stability of experimental Soret convection in a water-ethanol mixture under various boundary conditions had been investigated by Knobloch and Moore [18] with an emphasis on the Biot number. The thermocapillary instability in a binary fluid on the onset of convection with Soret effect and other physical influences also had been studied [19–21]. Later, Nield and Kuznetsov [22] extended [10] and used the linear instability theory to study the onset of double-diffusive convection in a horizontal layer of a nanofluid. Kuznetsov and Nield [23], Yadav et al. [24, 25], and Agarwal et al. [26] investigated double-diffusive convection in a porous medium permeated by a nanofluid layer. The authors in [27–29] added other effects to the system of double-diffusive convection in a nanofluid layer.

The thermal instability induced by uniform internal heat generation arising in a horizontal fluid layer has attracted the attention of many researchers [30–38]. It is found that an internal heat source decreases the stability of the fluid layer. Yadav et al. [39, 40] and Nield and Kuznetsov [41] included the effect of an internal heat source on the thermal instability in a nanofluid layer and found that the heat source advances the onset of convection. Recently, Wakif et al. [42] studied the combined effects of rotation and internal heating on a radiating nanofluid layer using the power series method (PSM) solely at both the upper and lower rigid boundaries.

The Coriolis force which is due to rotation in a system has an important effect on convective instability. Chandrasekhar [43] discussed Rayleigh–Bénard convection in a regular fluid with a linear temperature profile, both with and without the effect of rotation. In 1966, Vidal and Acrivos [44] included the effect of Coriolis force on the thermocapillary type of convection. Later, McConaghy and Finlayson [45], Takashima and Namikawa [46], Friedrich and Rudraiah [47], and Douiebe et al. [48] extended the previous analysis with other aspects of the problem. Meanwhile, several authors [49–52] have investigated the convective instability in a rotating fluid layer induced by buoyancy and thermocapillary, which has received profound attention from the engineering industry. Yadav et al. [53–56] studied the effect of rotation in a nanofluid layer and found that an increase in the Taylor number delays the onset of convection.

The use of feedback control in stabilizing the thermal convection was by Wang et al. [57] and they managed to inhibit the chaotic behaviour in the fluid layer by applying proportional control in a thermal convection loop. Later, Tang [58] and Tang and Bau [59, 60] showed that, with the use of a feedback controller, the critical Rayleigh number for the onset of convection can be significantly increased. Tang and Bau [61–64] and Howle [65–67] demonstrated experimentally that feedback control can be used to stable the system through the use of the control strategy. It is interesting to note that similar control strategies can be used to modify the flow patterns of shear flows [68] and surface-tension driven flows [69]. Further, many researchers have attempted to include other effects with the feedback control [70–75] on convection due to buoyancy and surface-tension.

In this paper, we intend to scrutinize the effect of internal heat source on the thermal instability of double-diffusive convection in a rotating nanofluid layer with feedback control. As in industrial heat transfer, heat needs to be efficiently controlled whether being added, removed, or transferred from one process stream to another. It is reported that between 20% and 50% of industrial energy input is lost as waste heat [76], which has attracted numerous studies on energy efficiency, recovering waste heat, and so on. This motivated us to contribute mathematically by providing a mathematical model that can simulate the influence of an internal heat source and feedback control in a nanofluid layer with other crucial parameters. We assume the nanofluid layer is heated from below and the lower-upper boundary conditions are considered to be free-free, rigid-free, and rigid-rigid. For systems that have thermal gradient and energy flux due to a mass concentration, Soret and Dufour coefficients are vital. To understand deeply the reaction of these two coefficients in a rotating nanofluid layer with internal heat source and feedback controller, we considered the interdiffusion parameter in the system. A linear stability analysis is performed, and the eigenvalue is obtained by employing the Galerkin technique. Numerical computations of the various relevant parameters are presented graphically.

#### 2. Mathematical Formulation

Cartesian coordinates are used, where the -axis points vertically upward. From now on, we denote the dimensional (nondimensional) variables with (without) asterisks. Consider a horizontal layer of a rotating incompressible nanofluid of thickness* L* confined between the planes and subjected to a uniform internal heat source is heated from below as shown in Figure 1. The nanofluid layer rotates about the vertical axis at a constant angular velocity, . The stability of a horizontal rotating nanofluid layer in the presence of internal heat source is examined. The temperature, solute concentration, and nanoparticle volume fraction at the lower and upper walls are denoted by , , and at and , , and at , respectively. Following Nield and Kuznetsov [22], the governing equations that describe the Boussinesq flow under this model with the presence of the effect of rotation and internal heat source arewhere is the velocity, is the density, is the nanofluid density at the reference temperature , is the nanoparticle mass density, is time, is the pressure, is the viscosity,** g** is the gravitational force, is the nanoparticle volume fraction, is the uniform internal heat source, is the thermal volumetric coefficient, is the temperature, is the solutal volumetric coefficient, is the solute concentration, is the specific heat, is the specific heat of the nanoparticles, is the nanofluid’s thermal conductivity, is the Dufour diffusivity, is the Brownian diffusion coefficient, is the thermophoretic diffusion coefficient, is the solutal diffusivity, and is the Soret diffusivity.