## Abstract

In this chapter, we present a weakly nonlinear stability analysis of the flow of a nanofluid in a porous medium with stress-free boundary conditions. Some previous studies have investigated cross-diffusion in a nanofluid layer although in most cases these studies mostly deal with linear stability analysis. It is important to study the nonlinear stability in flows subject to cross-diffusion due to the wide range of applications where such flows arise such as in hydrothermal growth, compact heat exchanges, the solidification of binary mixtures, geophysical systems, solar pond, etc. Here we consider flow between parallel plates with an applied magnetic field and zero nanoparticle flux at the boundaries. A truncated Fourier series is introduced reducing the flow equations to a Lorenz-type system of nonlinear evolution equations. The multidomain spectral method is used to solve the equations that describe the growth of the convection amplitudes. The solutions are obtained as sets of trajectories in the phase space. Some interesting spiral trajectories and their sensitivity to the Rayleigh number are given.

### Keywords

- nonlinear instability
- nanofluid flow
- porous medium
- multidomain spectral collocation method

## 1. Introduction

The enhancement of thermal conductivity of a fluid is a matter of supreme interest to engineers due to the important applications of fluids in heat transfer processes. Natural and forced convection plays an important role in heat transfer processes due to continuous molecular movements in fluid. Recent studies show that the suspension of solid nanoparticles in a fluid can substantially improve the fluid’s thermophysical properties, including thermal conductivity.

The term nanofluid describes a liquid containing a suspension of nanometer sized 1–100 nm solid particles [1]. Examples of commonly used nanoparticles include metallic particles such as Al, Cu and Ag, and oxides such as Al_{2}O_{3} and CuO. The base fluid is often a common liquid such as water, ethylene, glycol, or oil. The enhancement of thermochemical properties of a fluid due to the addition of nanoparticles has been observed in experimental studies such as in [2, 3]. Researchers have investigated the influence of seven slip mechanisms, namely, inertia, Brownian diffusion, thermophoresis, diffusiophoresis, magnus effect, fluid drainage, and gravity in nanofluids. It has been shown that, in the absence of turbulence, the most significant among these mechanisms are the Brownian diffusion and thermophoresis.

The classical Rayleigh-Benard convection problem in a heated horizontal layer has been extensively studied in the literature. Among recent studies on nanofluids, Tzou [4] studied the thermal instability and natural convection in nanofluid flow using an eigenfunction expansion method. Narayana et al. [5, 6] studied convection and the stability of a Maxwell fluid in a porous medium. Yadav et al. [7] investigated thermal instability of a rotating nanofluid layer. The studies by Kuznetsov and Nield [8, 9, 10, 11] focused on thermal instability in a porous layer saturated with a nanofluid. They investigated the onset of instability in a horizontal porous layer using a model for the nanofluid that incorporated particle Brownian motion and thermophoresis. Related studies with various assumptions on the geometry and flow structure have been made by [12, 13, 14, 15]. In the last few decades, researchers have also investigated thermal instability in a horizontal nanofluid layer subject to an applied magnetic field [16, 17]. The effects of a magnetic field on convection and the onset of instability have important applications in problems such as in cooling systems, pumps, magnetohydrodynamics and generators. The experimental study by Heris et al. [18] showed that thermal efficiency could be achieved by subjecting the flow to a magnetic field. The studies by Ghasemi et al. [19] and Hamad et al. [20] focused on the flow behavior and heat transfer in an electrically conducting nanofluid under the influence of a magnetic field and subject to Brownian diffusion and thermophoresis. They used a water-based nanofluid containing different types of nanoparticles such as copper, alumina and silver in their numerical simulations. Related studies of interest include [21, 22, 23, 24]. Rana et al. [25] studied thermal convection in a Walters (Model B) fluid in a porous medium. They showed that a magnetic field may introduce oscillatory instability modes and acts to stabilize the system.

In this chapter, we give a weakly nonlinear stability analysis of a nanofluid layer with an applied magnetic field, stress free boundary conditions and under the assumption of zero nanoparticle flux at the boundary. The studies by Kuznetsov and Nield [9] and Nield and Kuznetsov [10, 11] investigated cross-diffusion in a nanofluid layer. However, these studies mostly presented a linear stability analysis. It is important to study the nonlinear regime for a nanofluid flow subject to cross-diffusion due to the wide range of applications where such flows may arise. Typical examples may be found in hydrothermal growth, compact heat exchanges, solidification of binary mixtures, geophysical systems, and so on. Hence, with this in mind, we studied the finite amplitude convection in a nanofluid flows subject to cross-diffusion. By introducing a truncated Fourier series, a Lorenz-type system of seven nonlinear differential equations is obtained. The recent multidomain spectral method is used to solve the nonlinear equations. This method is accurate and very easy to implement compared to older methods such as finite difference methods. An analysis of heat and mass transfer for different parameters such as the Prandtl number, the Dufour and thermophoresis is presented.

## 2. Mathematical formulation

Consider viscous incompressible MHD nanofluid flow in an infinitely extended horizontal porous layer, confined between two boundaries at *z*-axis is vertically upward. The boundaries are perfectly conducting. The temperature at the lower and upper walls is

subject to the boundary conditions

where

where

The basic state is the time independent solution of Eqs. (1)–(5). Solving these equations with boundary conditions, we obtain

## 3. Weakly nonlinear stability analysis

In this section, we restrict the analysis to the case of two-dimensional disturbances. We define the stream function

Eqs. (1)–(5) may now be simplified by introducing the truncated Fourier series

where

subject to

Eqs. (15)–(21) give an approximate description of the full dimensional nonlinear system. An analytical solution of the system of nonlinear ordinary differential Eqs. (15)–(21) is not possible for the general time variable

This is always true if

Eq. (23) shows that the volume decays exponentially with time. Further, it can be noted that the system of Eqs. (15)–(21) are invariant under the transformation

We obtain the possible stationary points of the nonlinear system of equations by setting

The eigenvalues of the above matrix depend on the various parameters. For the specific parameters

with eigenvalues

This stationary point is a saddle point. Nonetheless, because the eigenvalues depend on various parameters, we cannot make general conclusions as to the stability of the system. We note, however, that if we denote the trace of the matrix

and

The trace is always negative, but the sign of determinant depends on the parameter values. If

suggesting a saddle point.

## 4. Method of solution

To study the influence of various physical parameters on the average Nusselt and Sherwood numbers, we solved the nonlinear system of Eqs. (15)–(21) numerically using the multidomain spectral collocation method. This is a novel technique for solving nonlinear initial value problems and parabolic equations with large time domains. It has been suggested in the literature that the method gives better accuracy compared to other methods such as finite difference and Runge-Kutta methods [26]. To apply the multidomain spectral collocation to the nonlinear system of equations, we first divide the interval

is used to transform each subinterval

subject to

The first step in using the multidomain spectral collocation method (MDSCM) concerns the quasilinearization of Eqs. (30)–(36) leading to a system of equations in the form

subject to

where

We approximate the derivatives of the unknown functions

where

is a vector of the unknown functions at the collocation points. Substituting Eq. (41) into Eqs. (38) and reducing the result into matrix form, we obtain

where the matrices

## 5. Heat and mass transfer

The study of heat and mass transfer in a horizontal nanofluid layer heated from below and cooled from above has important engineering applications. We define the rate of heat transfer by the average Nusselt number

Substituting Eqs. (12) and (13) into Eq. (43), we obtain

Similarly, the rate of mass transfer stated in terms of the average Sherwood number is

## 6. Results and discussion

We have studied the weakly nonlinear instability of nanofluid flow in a horizontal layer with stress free boundary conditions. For numerical simulations, the parameter values were chosen from the literature on nanofluid flow such as [4, 7]. In the literature, the critical Rayleigh number is found when the Darcy number is very large. In this study, we investigated the critical Rayleigh number for low Darcy numbers.

The method of solution described in Section 4 was used to solve Eqs. (15)–(21). All computations are carried out up to a value of maximum time

Figures 5–11 show the effect of the Rayleigh number on the trajectories projected onto the

Figures 5–8 show the phase portraits projected onto the ^{4}, the complexity of the trajectories increases leading to certain chaotic forms. Figures 8–11 show the trajectories in the three-dimensional phase space. Here, we observe similar solution sets as in the two-dimensional phase portraits.

Figures 12 and 13 show the streamline, isotherm and isoconcentration contours in the nanofluid flow for different values of the Darcy number and buoyancy ratio. Figure 12 displays the streamlines for various values of the buoyancy ratio term. Two different eddies are observed. The clockwise and anticlockwise flows are shown via negative and positive stream function values, respectively. The anticlockwise rotating flow occupies the largest area of the nanofluid layer.

For low buoyancy ratio parameters, the flow structure is significantly influenced by the buoyancy within the whole enclosure. Increasing the buoyancy ratio causes the boundary layer thickness to become thinner. Also, a high buoyancy ratio changes the flow structure, and this impacts significantly on the concentration field, which builds up a vertical stratification in the enclosure. It is interesting to note that for

We observe that when *N* decreases from 1 to −25, the streamlines become very dense to the left side of nanofluid layer while when *N* increases from 1 to 25, the streamlines are less so. The buoyancy forces that drive the nanofluid motion are mainly due to the temperature gradient.

Three different types of eddies are observed for the isoconcentration contours when *N* decreases from 1 to *N* values, hence the buoyancy ratio has a significant influence on the concentration gradient. As the buoyancy ratio *N* increases from 1 to 25 the isoconcentrations become very dense at the bottom of nanofluid layer.

The effect of the Darcy number on the nanofluid flow in the porous medium is shown in detail in Figure 13. The streamline patterns are similar to those depicted in Figure 12. However, as *Da* increases from 0.05 to 0.07, the rotation of the streamlines changes. Similarly, the isotherm patterns change with increasing Darcy numbers. The value of the center eddies increases with increasing *Da*. Increasing *Da* has the effect of increasing the effective fluid viscosity and reducing the thermal and solutal boundary layers.

## 7. Conclusion

We have investigated the onset of thermal instability in a horizontal porous layer of infinite extent in a cross-diffusive nanofluid flow. The focus of the study has been on stress free boundary conditions with zero nanoparticle flux at the wall. A multidomain spectral collocation method was used to solve the system of nonlinear evolution equations. As the Rayleigh number increases to 10^{4}, the trajectories spiral many times before reaching a fixed point. The nanofluid convection regime is complex for Rayleigh numbers higher than

Additionally, a change in system parameters, such as an increase in the flow Lewis number, improves the rate of heat and mass transfer in the nanofluid saturated porous media. The Dufour parameter has the effect of increasing heat transfer, while increasing the Soret parameter increases the rate of mass transfer.

## Acknowledgments

The authors are grateful to the University of KwaZulu-Natal and the Claude Leon Foundation, South Africa for financial support.

The terms

where

The matrices

where **diag** is an