Mathematical Problems in Engineering

Volume 2015, Article ID 905458, 11 pages

http://dx.doi.org/10.1155/2015/905458

## Heat Transfer and Flows of Thermal Convection in a Fluid-Saturated Rotating Porous Medium

Key Laboratory of Gas and Fire Control for Coal Mines, School of Safety Engineering, China University of Mining and Technology, Xuzhou 221116, China

Received 13 December 2014; Accepted 1 March 2015

Academic Editor: Hassan Askari

Copyright © 2015 Jianhong Kang 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

Thermal convection at the steady state for high Rayleigh number in a rotating porous half space is investigated. Taking into account the effect of rotation, Darcy equation is extended to incorporate the Coriolis force term in a rotating reference frame. The velocity and temperature fields of thermal convection are obtained by using the homotopy analysis method. The influences of Taylor number and Rayleigh number on the Nusselt number, velocity profile, and temperature distribution are discussed in detail. It is found that the Nusselt number decreases rapidly with the increase of Taylor number but tends to have an asymptotic value. Besides, the rotation can give rise to downward flow in contrast with the upward thermal convection.

#### 1. Introduction

Thermal convection in porous media has been given a great deal of attention by researchers. Its importance has relevance to a wide range of phenomena in geophysical, astrophysical, and engineering applications [1–3]. Since the pioneering work of Horton and Rogers [4], thermal convection in porous media has been investigated extensively. Lapwood [5] and Katto and Masuoka [6] examined thermal convective instability in a fluid-saturated horizontal porous layer. Wang [7] investigated thermal convective instability in a fluid-saturated rectangular box. Hayat et al. [8] and Yin et al. [9] considered thermal convective stability of a viscoelastic fluid in a porous layer. Malashetty et al. [10] considered thermal convective instability in an anisotropic porous layer. Nield and Kuznetsov [11] studied thermal convective instability in a nanofluid-saturated porous layer. All of these references focused on determining the criterion for the onset of thermal convection in terms of the critical Rayleigh number. Only if the Rayleigh number exceeds the critical value, thermal convection can set in; otherwise, heat transfer is dominated by conduction [12]. At the onset of thermal convection (i.e., the low-Rayleigh-number convection), the deviation of temperature and velocity from the purely conductive basic state is considered to be infinitesimal [13]. Consequently, the nonlinear terms in the governing equations are higher-order infinitesimals compared with other linear terms [14]. For the first-order approximation, the nonlinear terms can be neglected such that the negligence of nonlinearity of the problem is justified [15]. However, with the increase of Rayleigh number, the amplitudes of thermal convection grow rapidly such that the nonlinearity cannot be neglected [16, 17]. Due to the difficulty arising from the nonlinearity, only few works with numerical methods have been done [18, 19]. Till now, the study on high-Rayleigh-number convection in porous media is far less fruitful.

A particularly interesting variation of thermal convection is the case where the sample is rotated about a vertical axis with a uniform angular speed. In this system, the Coriolis and centrifugal forces besides gravity should be taken into account [20]. The Coriolis effect on thermal convection in different rotating systems has been studied extensively in the literature. Chandrasekhar [21] analyzed the thermal convective stability in a rotating horizontal layer of pure fluid, which was extended to the case of porous layer by Vadasz [22] and then the case of viscoelastic fluid by Kang et al. [23]. Malashetty et al. [24, 25] also considered thermal stability in a rotating anisotropic porous layer and in a rotating porous layer by using a thermal nonequilibrium model. Jones et al. [26] studied linear stability of thermal convection in a rapidly rotating sphere and developed the asymptotic theory for the sphere, which was later extended to the case of spherical shells by Dormy et al. [27]. Kang et al. [28] conducted linear stability analysis of viscoelastic fluids in a rotating porous cylindrical annulus. Besides, Busse [29] studied the onset of convection in a narrow cylindrical annulus heated from below and rotating about its vertical axis of symmetry. Ponty et al. [30] studied the onset of thermal convection in a rotating horizontal layer where the rotating axis was inclined at an angle to the vertical. Straughan [31] performed a sharp nonlinear stability threshold in rotating porous convection. These works are based on the assumption that not far from the axis of rotation gravity buoyancy is dominant and the centrifugal buoyancy can be neglected [22, 29]. On the other hand, the centrifugal buoyancy-driven thermal convection was also investigated where gravity buoyancy was assumed to be negligible compared with the centrifugal buoyancy [32, 33]. Assuming a small top aspect ratio of a narrow porous layer, Vadasz [34, 35] originally investigated thermal convection induced by centrifugal force in a rotating porous layer. His work was later extended by many researchers to consider other effects on the centrifugally driven thermal convection in porous media. For example, the anisotropic effects of porous media were studied by Govender [36] and Saravanan and Brindha [37]; the effects of thermal modulation and rotation modulation were considered by Suthar and Bhadauria [38]. Besides, Alloui and Vasseur [39] studied convective heat transfer with centrifugal force field for a horizontal porous annulus of infinite extent. Kang et al. [40] investigated thermal convection driven by centrifugal buoyancy in a rotating porous cylindrical annulus with conical end boundaries. However, to the best of the authors’ knowledge, no paper has investigated the Coriolis effect on the heat transfer rate for the high-Rayleigh-number thermal convection in porous media, which is a coupled nonlinear problem.

In general, it is difficult to solve a nonlinear problem analytically and any analytical solution for nonlinear partial differential equations is always exciting. Several analytical techniques including perturbation, artificial small parameter, the -expansion method, variational iteration, and Adomian decomposition methods have been developed to solve nonlinear problems [41, 42], among which the perturbation method and Adomian decomposition method are most widely applied. The perturbation method transforms a nonlinear problem into a family of infinite linear problems, and the superimposed solutions of linear problems are used to approximate the solution for nonlinear problem [43]. However, perturbation techniques are essentially based on small perturbation quantities, and the so-called “small parameter assumption” of perturbation techniques greatly restricts their applications [44]. Furthermore, the perturbation technique may become invalid for strongly nonlinear problems [45]. In contrast, the Adomian decomposition method is an effective method to solve strongly nonlinear problem and does not depend on any small parameter assumption [46]. The Adomian decomposition method is based on the assumption that the differential operator of nonlinear problems can be divided into linear and nonlinear parts; then the solution for nonlinear problem may be expressed as a polynomial series [47]. The polynomial series obtained by Adomian decomposition method converges quickly, but its convergence radius is relatively small [48]. More recently, Liao [49] developed a new kind of analytic method for nonlinear problems, called the homotopy analysis method (HAM), which is based on the homotopy, a basic concept of topology. Different from perturbation techniques, the homotopy analysis method is valid even for nonlinear problems whose governing equations and/or boundary conditions do not contain any small parameters at all [44]. The homotopy analysis method also provides us with great freedom to select proper base functions to approximate solutions of nonlinear problems [50]. Furthermore, the homotopy analysis method provides us with a family of solution series and a simple way to adjust and control the convergence region and rate of approximation series [48]. Serving as a powerful tool to deal with nonlinear equations, the homotopy analysis method has applied to many nonlinear problems in science and engineering [51, 52]. Abbasbandy [53] used the homotopy analysis method to solve a generalized Hirota-Satsuma coupled KdV equation. Hayat et al. [54] investigated mixed convection flow of a micropolar fluid over a nonlinearly stretching sheet by using the homotopy analysis method. Mehmood et al. [55] presented complete analytic solution to the unsteady heat transfer flow of an incompressible viscous fluid over a permeable plane wall with the homotopy analysis method.

In this work, we study high-Rayleigh-number steady state thermal convection in a rotating porous half space. Our focus is how an external constraint of rotation can affect the flow features and heat transfer in a porous half space. The present analysis combines the effects of high Rayleigh number and rotation together. We solve the coupled nonlinear equations by using the homotopy analysis method (HAM). The other parts of this paper are organized as follows. In Section 2, we give a mathematical formulation of the considered problem. In Section 3, the series solutions for velocity field and temperature distribution are presented using the HAM. Section 4 is devoted to main results and discussion on the convective flow and heat transfer in a rotating porous medium. Finally, this paper closes with a conclusion in Section 5.

#### 2. Mathematical Formulation

Consider a porous half space saturated with a viscous fluid with the density and viscosity , as shown in Figure 1. The porous medium with permeability and porosity is heated from below with constant temperature and cooled from above at infinity with constant temperature . Meanwhile, the porous medium rotates around -axis with uniform angular velocity . Here, a rotating cylindrical coordinate frame of reference is chosen to link together with the porous matrix. Not far (at distances ) from the axis of rotation, one can justify that the gravity buoyancy is dominant and centrifugal buoyancy can be neglected [22].