The Study of Magneto-Convection Heat Transfer in a Partially Open Cavity Based on LBM

Raoudha Chaabane

## Abstract

A free convection heat transfer in a sinusoidally heated enclosure filled with conducting fluid is presented in this chapter by Lattice Boltzmann Method (LBM). The horizontal walls in the enclosures are insulated and there is an opening part on the right wall. The right non-open parts of the vertical wall of the square cavity are maintained at constant cold temperature and the left wall of the cavity is sinusoidally heated. The cavity is get under a uniform in-plane magnetic field. The main aim of this study is to highlight the effectiveness of the LBM mesoscopic approach to predict the effects of pertinent parameters such as the Hartmann number varying from 0 to 150 where Rayleigh number is fixed at moderate value of 10<sup>5</sup> on flow patterns. This in-house numerical code used in this chapter is ascertained and a good agreement with literature is highlighted. The appropriate validation with previous numerical investigations demonstrated that this attitude is a suitable method and a powerful approach for engineering MHD problems. Findings and results show the alterations of Hartmann number that influence the isotherms and the streamlines widely.

Keywords: sinusoidal thermal boundary condition, MHD, partially open cavity, free convection, LBM

## 1. Introduction

Convective flow and heat transfer in an open cavity has been studied due to the extensive range of applications in engineering science and technology that consider various combinations of imposed temperature gradients and enclosure sketches [1–10]. Open cavity with a modified linear or sinusoidal thermal boundary condition is encountered in many practical engineering and industrial applications, such as solar energy collection, cooling of electronic devices, material processing, grain storage, flow and heat transfer in solar ponds, high-performance insulation for buildings, dynamics of lakes, reservoirs and cooling ponds, crystal growing, float glass production, metal casting, food processing, galvanizing, metal coating, and so on.

Besides, pertinent useful numerical research works had been conducted to simulate the MHD free convection under nonuniform thermic boundaries where recent attention has been intensively focused on the cases of mixed boundary conditions [11–14]. However, few results have been reported for free convection caused

simultaneously by both external magnetic field in partially open enclosures subjected to a sinusoidal temperature variation in the left vertical wall although problems of this type are frequently important, and their study is necessary for understanding the performance of complex magneto-convection flow and heat transfer.

MHD forces generated from the interaction of induced electric currents with an applied external magnetic field can alter the flow of an electrically conducting fluid in the presence of magnetic field. An externally imposed magnetic field is an important tool used to control melting flows that grow bulk crystal in semiconductor's applications. A main purpose of electromagnetic control is to stabilize the flow and suppress oscillatory instabilities, which degrades the resulting crystal. In literature, wide ranges of investigations were investigated by researchers in MHD free convection [15–30]. In such complex geometry, the balance is achieved by inertial, viscous, electromagnetic, and buoyancy forces; finding a numerical efficient tool to predict flow and heat pattern inside MHD cavities is a crucial aim for industrial related engineering applications.

The effect of Ra number on free convection MHD in an open cavity was investigated in [29]. Nonetheless several investigations in MHD free convection inside partially open enclosure with sinusoidally wall have been carried out yet.

The mesoscopic approach called Boltzmann method (LBM) joined the microscopic models and the macroscopic dynamics of a fluid. It can recover the Navier-Stokes equation by using the Chapman-Enskog expansion [39]. LB method is not very demanding in terms of memory requirement. In addition, in terms of computational speed, the algorithm is generally simpler and therefore faster than many traditional CFD schemes. It is easy for parallel computation and for implementation of irregular boundary conditions, which is a sought task in many-needed engineering geometry.

conducting fluid at the cold temperature (Tc). The north and south horizontal boundaries are adiabatic. We consider a horizontal uniform magnetic field applied to a two-dimensional Newtonian, laminar, and incompressible conducting fluid. The radiation effects, the viscous dissipation, and Joule heating are neglected in the present study. The thermophysical properties of the conducting fluid are constant, and the density variation in the liquid gallium is approximated by the standard

The Study of Magneto-Convection Heat Transfer in a Partially Open Cavity Based on LBM

Governing equations for MHD free convection are written in terms of the

<sup>∂</sup><sup>y</sup> <sup>¼</sup> <sup>0</sup> (1)

(4)

þ Fx (2)

þ Fy (3)

∂u ∂x þ ∂v

¼ � <sup>∂</sup><sup>p</sup> ∂x þ μ

¼ � <sup>∂</sup><sup>p</sup> ∂y þ μ ∂2 u ∂x<sup>2</sup> þ

∂2 v ∂x<sup>2</sup> þ

T ∂x<sup>2</sup> þ

where ν ¼ μ=ρ is the kinematic viscosity and Fx and Fy are the body forces at horizontal and vertical directions, respectively, and they are defined as follows [30]:

∂2 T ∂y<sup>2</sup>

∂2 u ∂y<sup>2</sup> 

∂2 v ∂y<sup>2</sup> 

Boussinesq (Figure 2).

The standard D2Q9 LBM lattice.

DOI: http://dx.doi.org/10.5772/intechopen.84478

Figure 1.

2.2 Governing equations

Continuity equation

Momentum equations

Energy equation

61

ρ u ∂u ∂x þ v ∂u ∂y

ρ u ∂v ∂x þ v ∂v ∂y

macroscopic variable depending on position x,y as:

u ∂T ∂x þ v ∂T <sup>∂</sup><sup>y</sup> <sup>¼</sup> <sup>α</sup> <sup>∂</sup><sup>2</sup>

To our best knowledge, no previous study on effects of sinusoidally heated boundary on free convection in a partially open MHD enclosure cavity with the LBM had already been studied so far. The main aim of this chapter is to study the effects of linearly heated wall on flow field and temperature distribution in an open MHD cavity filled.

The main aim of the present study is to demonstrate the use of the Lattice Boltzmann Method (LBM) [31–40] for MHD with a simple and clear statement and also solve MHD free convection as a left sinusoidally heated side mixed with a partially open right wall filled with a conducting fluid. Hartmann number varies in a wide range from 0 to 150. First, the results of LBM are validated with previous numerical investigations. Effects of Rayleigh, Hartmann number, and various positions of the open side on flow field and temperature distribution are considered simultaneously.

The proposed configuration with sinusoidal temperatures on the left side wall of a partially open cavity in the presence of a magnetic field has not been focused. A major objective of the present study is to examine the magneto-convection in this configuration filled with a conducting fluid confined between two horizontal walls, which are thermally insulated. The effect of the open side on fluid flow and heat transfer is studied numerically.

## 2. Governing equations and mathematical formulation

#### 2.1 Problem statement

The considered geometries of the present problem are shown in Figure 1. They display a two-dimensional closed, open or partially open east side cavity with the height of H. A constant, linear, or sinusoidal temperature is imposed along the left vertical wall. Then opening side boundaries are correlated with temperature

The Study of Magneto-Convection Heat Transfer in a Partially Open Cavity Based on LBM DOI: http://dx.doi.org/10.5772/intechopen.84478

Figure 1. The standard D2Q9 LBM lattice.

simultaneously by both external magnetic field in partially open enclosures subjected to a sinusoidal temperature variation in the left vertical wall although problems of this type are frequently important, and their study is necessary for understanding the

MHD forces generated from the interaction of induced electric currents with an applied external magnetic field can alter the flow of an electrically conducting fluid in the presence of magnetic field. An externally imposed magnetic field is an important tool used to control melting flows that grow bulk crystal in semiconductor's applications. A main purpose of electromagnetic control is to stabilize the flow and suppress oscillatory instabilities, which degrades the resulting crystal. In literature, wide ranges of investigations were investigated by researchers in MHD free convection [15–30]. In such complex geometry, the balance is achieved by inertial, viscous, electromagnetic, and buoyancy forces; finding a numerical efficient tool to predict flow and heat pattern inside MHD cavities is a crucial aim for industrial related engineering applications. The effect of Ra number on free convection MHD in an open cavity was investigated in [29]. Nonetheless several investigations in MHD free convection inside

performance of complex magneto-convection flow and heat transfer.

Pattern Formation and Stability in Magnetic Colloids

partially open enclosure with sinusoidally wall have been carried out yet.

The main aim of the present study is to demonstrate the use of the Lattice Boltzmann Method (LBM) [31–40] for MHD with a simple and clear statement and also solve MHD free convection as a left sinusoidally heated side mixed with a partially open right wall filled with a conducting fluid. Hartmann number varies in a wide range from 0 to 150. First, the results of LBM are validated with previous numerical investigations. Effects of Rayleigh, Hartmann number, and various positions of the open side on flow field and temperature distribution are considered simultaneously. The proposed configuration with sinusoidal temperatures on the left side wall of a partially open cavity in the presence of a magnetic field has not been focused. A major objective of the present study is to examine the magneto-convection in this configuration filled with a conducting fluid confined between two horizontal walls, which are thermally insulated. The effect of the open side on fluid flow and heat

2. Governing equations and mathematical formulation

The considered geometries of the present problem are shown in Figure 1. They display a two-dimensional closed, open or partially open east side cavity with the height of H. A constant, linear, or sinusoidal temperature is imposed along the left vertical wall. Then opening side boundaries are correlated with temperature

MHD cavity filled.

transfer is studied numerically.

2.1 Problem statement

60

The mesoscopic approach called Boltzmann method (LBM) joined the microscopic models and the macroscopic dynamics of a fluid. It can recover the Navier-Stokes equation by using the Chapman-Enskog expansion [39]. LB method is not very demanding in terms of memory requirement. In addition, in terms of computational speed, the algorithm is generally simpler and therefore faster than many traditional CFD schemes. It is easy for parallel computation and for implementation of irregular boundary conditions, which is a sought task in many-needed engineering geometry. To our best knowledge, no previous study on effects of sinusoidally heated boundary on free convection in a partially open MHD enclosure cavity with the LBM had already been studied so far. The main aim of this chapter is to study the effects of linearly heated wall on flow field and temperature distribution in an open

conducting fluid at the cold temperature (Tc). The north and south horizontal boundaries are adiabatic. We consider a horizontal uniform magnetic field applied to a two-dimensional Newtonian, laminar, and incompressible conducting fluid. The radiation effects, the viscous dissipation, and Joule heating are neglected in the present study. The thermophysical properties of the conducting fluid are constant, and the density variation in the liquid gallium is approximated by the standard Boussinesq (Figure 2).

## 2.2 Governing equations

Governing equations for MHD free convection are written in terms of the macroscopic variable depending on position x,y as:

Continuity equation

$$\frac{\partial u}{\partial \mathbf{x}} + \frac{\partial v}{\partial \mathbf{y}} = \mathbf{0} \tag{1}$$

Momentum equations

$$
\rho \left( u \frac{\partial u}{\partial \mathbf{x}} + v \frac{\partial u}{\partial \mathbf{y}} \right) = -\frac{\partial p}{\partial \mathbf{x}} + \mu \left( \frac{\partial^2 u}{\partial \mathbf{x}^2} + \frac{\partial^2 u}{\partial \mathbf{y}^2} \right) + F\_{\mathbf{x}} \tag{2}
$$

$$
\rho \left( u \frac{\partial v}{\partial \mathbf{x}} + v \frac{\partial v}{\partial \mathbf{y}} \right) = -\frac{\partial p}{\partial \mathbf{y}} + \mu \left( \frac{\partial^2 v}{\partial \mathbf{x}^2} + \frac{\partial^2 v}{\partial \mathbf{y}^2} \right) + F\_\mathbf{y} \tag{3}
$$

Energy equation

$$u\frac{\partial T}{\partial \mathbf{x}} + v\frac{\partial T}{\partial \mathbf{y}} = a\left(\frac{\partial^2 T}{\partial \mathbf{x}^2} + \frac{\partial^2 T}{\partial \mathbf{y}^2}\right) \tag{4}$$

where ν ¼ μ=ρ is the kinematic viscosity and Fx and Fy are the body forces at horizontal and vertical directions, respectively, and they are defined as follows [30]:

Figure 2. Used configurations C1–4 with different boundary conditions.

$$F\_x = R\left(v\sin\chi\cos\chi - u\sin^2\chi\right) \tag{5}$$

$$F\_{\gamma} = R\left(\mu \sin \gamma \cos \gamma - \nu \cos^2 \gamma\right) + \rho \mathbf{g} \beta (T - T\_m) \tag{6}$$

where the Ha number is defined as:

$$\mathbf{H}\mathbf{a} = \mathbf{H}\mathbf{B}\_{\mathbf{x}}\sqrt{\sigma/\mu} \tag{7}$$

where u and ρ are the macroscopic velocity and density, respectively, and w<sup>α</sup> are

w<sup>0</sup> ¼ 4=9 (rest-particle) for ej j <sup>0</sup> ¼ 0, w1�<sup>4</sup> ¼ 1=9 for ej j <sup>1</sup>�<sup>4</sup> ¼ 1, and w5�<sup>9</sup> ¼ 1=36 for

where c = Δx/Δt, Δx and Δt are the lattice space and the lattice time step size,

eα:u c2 s

τ<sup>f</sup> ¼ 0:5 þ 3ν=c

τ<sup>g</sup> ¼ 0:5 þ 3α=c

lattice time step. The buoyancy force term is added as an extra source term to

where α is the diffusion coefficient (thermal diffusion coefficient) and Δt is the

where gy, β, and ΔT are gravitational acceleration, thermal expansion coeffi-

Fx <sup>¼</sup> <sup>3</sup>wk<sup>ρ</sup> R vð Þ� sin <sup>γ</sup> cos <sup>γ</sup> <sup>u</sup> sin <sup>2</sup>

Fy <sup>¼</sup> <sup>3</sup>wk<sup>ρ</sup> gyβð Þþ <sup>T</sup> � Tm R uð Þ� sin <sup>γ</sup> cos <sup>γ</sup> <sup>v</sup> cos <sup>2</sup>

After completing streaming and collision processes where the Boussinesq approximation is considered for free convection, the macroscopic fluid quantities, namely, the macro density, velocity (obtained through moment summations in the

ρð Þ¼ r; t ∑

k

T ¼ ∑ k

k

where <sup>R</sup> <sup>¼</sup> <sup>μ</sup>Ha<sup>2</sup> and <sup>γ</sup> is the direction of the magnetic field.

u rð Þ¼ ; t ∑

Δt τg

<sup>þ</sup> ð Þ <sup>e</sup>α:<sup>u</sup>

2

2

!

2c<sup>4</sup> s

2

gεð Þ� r; t g<sup>α</sup>

� <sup>u</sup>:<sup>u</sup> 2c<sup>2</sup> s

Δt � � (12)

Δt � � (13)

F<sup>α</sup> ¼ 3w<sup>α</sup> gyβΔT (14)

F ¼ Fx þ Fy (15)

γ

f <sup>k</sup>ð Þ r; t (18)

ekf <sup>k</sup>ð Þ r; t =ρð Þ r; t (19)

gkð Þ r; t (20)

<sup>γ</sup> � �� (16)

h i (17)

eqð Þ <sup>r</sup>; <sup>t</sup> � � (10)

(11)

the values of the weighting constant factors for e<sup>α</sup> that must be assigned as

The Study of Magneto-Convection Heat Transfer in a Partially Open Cavity Based on LBM

For scalar function (temperature), another distribution is defined:

gαðr þ eαΔt; t þ ΔtÞ ¼ gαð Þ� r; t

The equilibrium distribution function can be written as:

<sup>α</sup> ¼ wαT 1 þ

where ν is the kinematic viscosity and for the scalar:

geq

cient, and temperature difference, respectively.

velocity space), and temperature are computed.

In the LBM, the total force is

For momentum equation, we have:

j j e5�<sup>9</sup> <sup>¼</sup> ffiffi

Eq. (1) as:

63

2 <sup>p</sup> .

respectively, which are set to unity.

DOI: http://dx.doi.org/10.5772/intechopen.84478

The LBM method [31–40] with standard, two-dimensional, nine velocities (D2Q9) for flow and temperature is used in this chapter (Figure 1a); for completeness, only a brief discussion is given in the following paragraphs. The Bhatnagar-Gross-Krook (BGK) approximation Lattice Boltzmann equation with external forces Ftot can be written as:

$$f\_a(\mathbf{r} + \mathbf{e}\_a \Delta t, t + \Delta t) = f\_a(\mathbf{r}, t) - \frac{\Delta t}{\tau\_f} \left[ f\_a(\mathbf{r}, t) - f\_a^{\ eq}(\mathbf{r}, t) \right] + \Delta t \mathbf{F}\_{tot} \tag{8}$$

where f <sup>α</sup>ð Þ r; t is the particle distribution defined for the finite set of the discrete particle velocity vectors eα. r and t are the coordinates of Eulerian node and time.

where τ<sup>f</sup> is the relaxation time and f <sup>α</sup> eqð Þ <sup>r</sup>; <sup>t</sup> is the local equilibrium distribution function.

The equilibrium distribution can be formulated as:

$$f\_a^{eq} = w\_a \rho \left( \mathbf{1} + \frac{\mathbf{e}\_a \mathbf{u}}{c\_s^2} + \frac{\left(\mathbf{e}\_a \mathbf{u}\right)^2}{2c\_s^4} - \frac{\mathbf{u} \mathbf{u}}{2c\_s^2} \right) \tag{9}$$

The Study of Magneto-Convection Heat Transfer in a Partially Open Cavity Based on LBM DOI: http://dx.doi.org/10.5772/intechopen.84478

where u and ρ are the macroscopic velocity and density, respectively, and w<sup>α</sup> are the values of the weighting constant factors for e<sup>α</sup> that must be assigned as w<sup>0</sup> ¼ 4=9 (rest-particle) for ej j <sup>0</sup> ¼ 0, w1�<sup>4</sup> ¼ 1=9 for ej j <sup>1</sup>�<sup>4</sup> ¼ 1, and w5�<sup>9</sup> ¼ 1=36 for j j e5�<sup>9</sup> <sup>¼</sup> ffiffi 2 <sup>p</sup> .

where c = Δx/Δt, Δx and Δt are the lattice space and the lattice time step size, respectively, which are set to unity.

For scalar function (temperature), another distribution is defined:

$$\mathbf{g}\_a(\mathbf{r} + \mathbf{e}\_a \Delta t, t + \Delta t) = \mathbf{g}\_a(\mathbf{r}, t) - \frac{\Delta t}{\tau\_\S} \left[ \mathbf{g}\_e(\mathbf{r}, t) - \mathbf{g}\_a^{\ eq}(\mathbf{r}, t) \right] \tag{10}$$

The equilibrium distribution function can be written as:

$$\mathbf{g}\_a^{eq} = \boldsymbol{w}\_a T \left( \mathbf{1} + \frac{\mathbf{e}\_a \cdot \mathbf{u}}{c\_s^2} + \frac{(\mathbf{e}\_a \cdot \mathbf{u})^2}{2c\_s^4} - \frac{\mathbf{u} \cdot \mathbf{u}}{2c\_s^2} \right) \tag{11}$$

For momentum equation, we have:

$$
\pi\_f = \mathbf{0}.\mathbf{5} + \left(\mathbf{3}\nu/c^2\Delta t\right) \tag{12}
$$

where ν is the kinematic viscosity and for the scalar:

$$
\pi\_{\mathfrak{g}} = \mathbf{0}.\mathbf{5} + \left(\mathbf{3}a/c^2\Delta t\right) \tag{13}
$$

where α is the diffusion coefficient (thermal diffusion coefficient) and Δt is the lattice time step. The buoyancy force term is added as an extra source term to Eq. (1) as:

$$F\_a = \Im w\_a \mathfrak{g}\_\jmath \beta \Delta T \tag{14}$$

where gy, β, and ΔT are gravitational acceleration, thermal expansion coefficient, and temperature difference, respectively.

In the LBM, the total force is

$$F = F\_{\mathbf{x}} + F\_{\mathbf{y}} \tag{15}$$

$$F\_{\mathbf{x}} = \mathfrak{Z}w\_k \rho \left[ R(\nu \sin \chi \cos \chi) - \mathfrak{u} \sin^2 \chi \right] \tag{16}$$

$$F\_{\mathcal{Y}} = \mathfrak{z}w\_{k}\rho \left[ \mathfrak{g}\_{\mathcal{Y}}\beta(T - T\_{m}) + R(\mathfrak{u}\sin\chi\cos\chi) - \mathfrak{v}\cos^{2}\chi \right] \tag{17}$$

where <sup>R</sup> <sup>¼</sup> <sup>μ</sup>Ha<sup>2</sup> and <sup>γ</sup> is the direction of the magnetic field.

After completing streaming and collision processes where the Boussinesq approximation is considered for free convection, the macroscopic fluid quantities, namely, the macro density, velocity (obtained through moment summations in the velocity space), and temperature are computed.

$$\rho(\mathbf{r},t) = \sum\_{k} f\_{k}(\mathbf{r},t) \tag{18}$$

$$\mathbf{u}(\mathbf{r},t) = \sum\_{k} \mathbf{e}\_{k} f\_{k}(\mathbf{r},t) / \rho(\mathbf{r},t) \tag{19}$$

$$T = \sum\_{k} \mathbf{g}\_{k}(\mathbf{r}, t) \tag{20}$$

Fx <sup>¼</sup> R v sin <sup>γ</sup> cos <sup>γ</sup> � <sup>u</sup> sin <sup>2</sup>

Ha ¼ H Bx

The LBM method [31–40] with standard, two-dimensional, nine velocities (D2Q9) for flow and temperature is used in this chapter (Figure 1a); for completeness, only a brief discussion is given in the following paragraphs. The Bhatnagar-Gross-Krook (BGK) approximation Lattice Boltzmann equation with external

> Δt τf

eα:u c2 s

where f <sup>α</sup>ð Þ r; t is the particle distribution defined for the finite set of the discrete particle velocity vectors eα. r and t are the coordinates of Eulerian node and time.

Fy <sup>¼</sup> R u sin <sup>γ</sup> cos <sup>γ</sup> � <sup>v</sup> cos <sup>2</sup>

where the Ha number is defined as:

Used configurations C1–4 with different boundary conditions.

Pattern Formation and Stability in Magnetic Colloids

f <sup>α</sup>ðr þ eαΔt; t þ ΔtÞ ¼ f <sup>α</sup>ð Þ� r; t

where τ<sup>f</sup> is the relaxation time and f <sup>α</sup>

f eq

The equilibrium distribution can be formulated as:

<sup>α</sup> ¼ wαρ 1 þ

forces Ftot can be written as:

function.

62

Figure 2.

γ � � (5)

σ=μ p (7)

eqð Þ <sup>r</sup>; <sup>t</sup> � � <sup>þ</sup> <sup>Δ</sup>tFtot (8)

eqð Þ <sup>r</sup>; <sup>t</sup> is the local equilibrium distribution

(9)

<sup>γ</sup> � � <sup>þ</sup> <sup>ρ</sup>gβð Þ <sup>T</sup> � Tm (6)

ffiffiffiffiffiffiffiffi

f <sup>α</sup>ð Þ� r; t f <sup>α</sup>

<sup>þ</sup> ð Þ <sup>e</sup>α:<sup>u</sup>

!

2c<sup>4</sup> s

2

� <sup>u</sup>:<sup>u</sup> 2c<sup>2</sup> s

## 3. Results and discussion

In all cases, both the computations for flow and temperature fields are based on the D2Q9 LBM approach.

We have validated our in-house Fortran computer code for the free convection in a square open cavity with insulated horizontal walls filled with air with a uniform right (west) vertical temperature by reference [29]. The obtained numerical results are compared with the numerical ones reported in [29]. It can be seen from Figure 3 that there is a good agreement for the distribution of streamlines and isotherms among the present solution and literature and this for Pr = 0.71, Ha = 0, and Ra = 10<sup>5</sup> . The used configuration is C1 (Figure 2).

In Figure 4, the considered matter is MHD free convection in an open cavity with linearly heated west wall (C2), which is filled with liquid gallium (Pr = 0.025) and Ha = 150. As shown in Figure 4, an increase in Rayleigh number makes the thermal boundary layer to become narrower for the constant parameters Pr = 0.025 and Ha = 150.

Figure 4.

Figure 5.

65

Steady-state isotherms (b) at Pr = 0.025 and Ha = 150 for different Rayleigh numbers.

The Study of Magneto-Convection Heat Transfer in a Partially Open Cavity Based on LBM

DOI: http://dx.doi.org/10.5772/intechopen.84478

Steady-state isotherms (a) and isotherms (b) at Pr = 0.025, Ra = 106, and Ha = 150.

Figure 5 displays steady-state contour maps for the isotherms and the streamline contours at Pr = 0.025 for a high Rayleigh number Ra = 10<sup>6</sup> and high Hartmann

#### Figure 3.

Comparison of the steady state isotherms (a) and streamlines (b) at Pr = 0.71 for Ha = 0 and Ra = 105 between [29] and the present work.

The Study of Magneto-Convection Heat Transfer in a Partially Open Cavity Based on LBM DOI: http://dx.doi.org/10.5772/intechopen.84478

#### Figure 4.

3. Results and discussion

Pattern Formation and Stability in Magnetic Colloids

the D2Q9 LBM approach.

Ra = 10<sup>5</sup>

and Ha = 150.

Figure 3.

64

[29] and the present work.

In all cases, both the computations for flow and temperature fields are based on

We have validated our in-house Fortran computer code for the free convection in a square open cavity with insulated horizontal walls filled with air with a uniform right (west) vertical temperature by reference [29]. The obtained numerical results are compared with the numerical ones reported in [29]. It can be seen from Figure 3 that there is a good agreement for the distribution of streamlines and isotherms among the present solution and literature and this for Pr = 0.71, Ha = 0, and

In Figure 4, the considered matter is MHD free convection in an open cavity with linearly heated west wall (C2), which is filled with liquid gallium (Pr = 0.025) and Ha = 150. As shown in Figure 4, an increase in Rayleigh number makes the thermal boundary layer to become narrower for the constant parameters Pr = 0.025

Figure 5 displays steady-state contour maps for the isotherms and the streamline contours at Pr = 0.025 for a high Rayleigh number Ra = 10<sup>6</sup> and high Hartmann

Comparison of the steady state isotherms (a) and streamlines (b) at Pr = 0.71 for Ha = 0 and Ra = 105 between

. The used configuration is C1 (Figure 2).

Steady-state isotherms (b) at Pr = 0.025 and Ha = 150 for different Rayleigh numbers.

Figure 5. Steady-state isotherms (a) and isotherms (b) at Pr = 0.025, Ra = 106, and Ha = 150.

number Ha = 150. Configuration C2 is considered for this case. The effect of the open right side wall is depicted in Figure 5a via streamline display.

filled with liquid gallium, and the simulation is done for Ha = 50 and a moderate

The Study of Magneto-Convection Heat Transfer in a Partially Open Cavity Based on LBM

Many researchers were considering a sinusoidal heating wall in their simulation

Steady-state isotherms (a) and streamlines (b) of sinusoidally heated right side wall partially open MHD cavity

.

.

[11–14]. For brevity, we resume their important findings.

Rayleigh number of Ra = 10<sup>5</sup>

DOI: http://dx.doi.org/10.5772/intechopen.84478

Figure 7.

67

for Ha = 150, Pr = 0.025 and Ra = 106

In this chapter, we aim to test the ability of the LBM to deal with a future complex configuration (C4), which is a MHD partially open cavity with sinusoidal input excitation on the west wall. The horizontal walls are adiabatic. The cavity is

Steady-state isotherms (a) and streamlines (b) of sinusoidally heated side walls in a partially open MHD cavity for Ha = 150, Pr = 0.025 and Ra = 105 .

The Study of Magneto-Convection Heat Transfer in a Partially Open Cavity Based on LBM DOI: http://dx.doi.org/10.5772/intechopen.84478

filled with liquid gallium, and the simulation is done for Ha = 50 and a moderate Rayleigh number of Ra = 10<sup>5</sup> .

Many researchers were considering a sinusoidal heating wall in their simulation [11–14]. For brevity, we resume their important findings.

Steady-state isotherms (a) and streamlines (b) of sinusoidally heated right side wall partially open MHD cavity for Ha = 150, Pr = 0.025 and Ra = 106 .

number Ha = 150. Configuration C2 is considered for this case. The effect of the

In this chapter, we aim to test the ability of the LBM to deal with a future complex configuration (C4), which is a MHD partially open cavity with sinusoidal input excitation on the west wall. The horizontal walls are adiabatic. The cavity is

Steady-state isotherms (a) and streamlines (b) of sinusoidally heated side walls in a partially open MHD cavity

.

open right side wall is depicted in Figure 5a via streamline display.

Pattern Formation and Stability in Magnetic Colloids

Figure 6.

66

for Ha = 150, Pr = 0.025 and Ra = 105

In literature, we find that the heat transfer rate was increased as the amplitude ratio of the sinusoidal excitation increases with Rayleigh and Hartmann numbers.

present chapter is therefore to predict dynamic and thermic heat transfer in a crucial engineering application. The cavity is investigated at the high Ra number of 105

been made of free convection in a square enclosure with spatially varying sinusoidal temperature distributions on the vertical left sidewall, whereas the horizontal walls are thermally insulated. The right wall is a partially open one. Lattice Boltzmann method is considered for flow and heat transfer simulation of this problem inside the cavity. After validation, the present in-house Fortran code is extended to deal with the present complex geometry in order to highlight the workability and the ability of LBM to deal with such mixed boundary condition sketch. This investigation demonstrated ability of LBM for simulation of different boundary conditions at various elements affecting the stream in a partially open cavity with sinusoidal heating vertical wall. An analysis of the opening mass flow is highlighted in the dynamic and

Ha number (Ha = 50), and Pr number of 0.025. The present chapter extends the study to deal with free convection in MHD open cavity with sinusoidal heated west

The Study of Magneto-Convection Heat Transfer in a Partially Open Cavity Based on LBM

wall which is filled with liquid gallium with Ha = 50 for Ra = 105

thermal behavior of the streamlines and the isotherms.

cp specific heat at constant pressure

eq equilibrium density distribution functions

eq equilibrium internal energy distribution functions

Nomenclature

f

g

C lattice speed

F external forces

g gravity

ci discrete particle speeds

DOI: http://dx.doi.org/10.5772/intechopen.84478

G buoyancy per unit mass H enclosure height Ma Mach number Nu Nusselt number Pr Prandtl number Ra Rayleigh number T temperature

x,y Cartesian coordinates

Ha Hartmann number

τ relaxation time ν kinematic viscosity Δt time increment α thermal diffusivity

av average H hot C cold b bottom

Greek letters

Subscripts

69

X horizontal length of the cavity Y vertical length of the cavity

ω<sup>i</sup> weighted factor indirection i β thermal expansion coefficient , high

. Numerical study has

For a uniform heating wall, the heat transfer rate remains low, which makes that the nonuniform heating of both walls is advised for enhancing and improving heat transfer. It is proven that the heat transfer rate is increased first and then decreased on increasing the phase deviation from 0 to pi [11–14].

In addition, with a phase deviation from 0 to 3pi/4, heat transfer rate is enhanced for all Rayleigh numbers, and the average Nusselt number reaches its highest value at 3pi/4.

A further finding proves that when both walls are in the same temperature distribution in the absence of phase deviation, the heat transfer rate is low for all values of Hartmann and Rayleigh numbers. The right wall is widely influenced by the variation of the amplitude ratio and the phase deviation of the sinusoidal temperature distribution. However, those physical variations have very little effect on the left wall. Besides, an increase in Hartmann number decreases the heat transfer [11–14].

In the present chapter, we seek to deal with a new configuration C4. For this goal, we deal first with the configuration (C3) of Figure 2. The sidewalls of the cavity have spatially varying sinusoidal temperature distributions. The horizontal walls are adiabatic. Simulation is established for MHD cavity for Ha = 150, Pr = 0.025, and Ra = 10<sup>6</sup> . The heat transfer rate is highlighted within the evolution of isotherms and streamlines inside the cavity in Figure 6.

Figure 7 shows the dynamic and thermic behavior of configuration C4. Numerical results in terms of flow and thermic structure show that the flow within the cavity takes place owing to the thermic buoyancy effects caused by the sinusoidally heated right wall. In C3, the flow is characterized by a symmetric multicellular behavior in which the recirculating eddies or cells of relatively high velocity are formed within the enclosure and this in the presence of a high Hartmann Ha = 150 and a high Rayleigh number Ra = 10<sup>5</sup> .

In C4, the presence of the partially open sidewall changes the flow and heat transfer. We note the presence of a one dominant cell in the core region of the cavity and a little cell that occurs at upper left side.

As forecasted, because of the partially open side effects, the temperature field was sketched by a noticeable drop in its behavior near the open side of the cavity and these both in flow and heat behaviors. Besides, we highlight that the temperature contour maps lose the sinusoidal behavior as they move to the partially open side.

We infer that a partially open sidewall has the tendency to control efficiently the movement of the fluid in such given configuration (C4).

We subject in the convergence criterion that the relative change in two successive iterates of the solution (temperate and velocities) at each computational point be below a prescribed small value of 10�<sup>6</sup> .

For all simulations, the criterion convergence is considered to be reached, for velocity and temperature when the following convergence is satisfied:

$$\left|\left|\xi^{\sigma+1} - \xi^{\sigma}\right| \leq 10^{-6} \tag{21}$$

where ξ is velocity or temperature and σ is the iteration number.

## 4. Conclusions

To the author's knowledge, studies have thus far addressed a mesoscopic approach in an MHD open cavity with sinusoidally heated wall (Figure 2). The objective of the The Study of Magneto-Convection Heat Transfer in a Partially Open Cavity Based on LBM DOI: http://dx.doi.org/10.5772/intechopen.84478

present chapter is therefore to predict dynamic and thermic heat transfer in a crucial engineering application. The cavity is investigated at the high Ra number of 105 , high Ha number (Ha = 50), and Pr number of 0.025. The present chapter extends the study to deal with free convection in MHD open cavity with sinusoidal heated west wall which is filled with liquid gallium with Ha = 50 for Ra = 105 . Numerical study has been made of free convection in a square enclosure with spatially varying sinusoidal temperature distributions on the vertical left sidewall, whereas the horizontal walls are thermally insulated. The right wall is a partially open one. Lattice Boltzmann method is considered for flow and heat transfer simulation of this problem inside the cavity. After validation, the present in-house Fortran code is extended to deal with the present complex geometry in order to highlight the workability and the ability of LBM to deal with such mixed boundary condition sketch. This investigation demonstrated ability of LBM for simulation of different boundary conditions at various elements affecting the stream in a partially open cavity with sinusoidal heating vertical wall. An analysis of the opening mass flow is highlighted in the dynamic and thermal behavior of the streamlines and the isotherms.

## Nomenclature

In literature, we find that the heat transfer rate was increased as the amplitude ratio of the sinusoidal excitation increases with Rayleigh and Hartmann numbers. For a uniform heating wall, the heat transfer rate remains low, which makes that the nonuniform heating of both walls is advised for enhancing and improving heat transfer. It is proven that the heat transfer rate is increased first and then decreased

In addition, with a phase deviation from 0 to 3pi/4, heat transfer rate is enhanced for all Rayleigh numbers, and the average Nusselt number reaches its

A further finding proves that when both walls are in the same temperature distribution in the absence of phase deviation, the heat transfer rate is low for all values of Hartmann and Rayleigh numbers. The right wall is widely influenced by the variation of the amplitude ratio and the phase deviation of the sinusoidal temperature distribution. However, those physical variations have very little effect on the left wall. Besides, an increase in Hartmann number decreases the heat

In the present chapter, we seek to deal with a new configuration C4. For this goal, we deal first with the configuration (C3) of Figure 2. The sidewalls of the cavity have spatially varying sinusoidal temperature distributions. The horizontal walls are adiabatic. Simulation is established for MHD cavity for Ha = 150,

Figure 7 shows the dynamic and thermic behavior of configuration C4. Numerical results in terms of flow and thermic structure show that the flow within the cavity takes place owing to the thermic buoyancy effects caused by the sinusoidally heated right wall. In C3, the flow is characterized by a symmetric multicellular behavior in which the recirculating eddies or cells of relatively high velocity are formed within the enclosure and this in the presence of a high Hartmann Ha = 150

As forecasted, because of the partially open side effects, the temperature field was sketched by a noticeable drop in its behavior near the open side of the cavity and these both in flow and heat behaviors. Besides, we highlight that the temperature contour maps lose the sinusoidal behavior as they move to the partially open side. We infer that a partially open sidewall has the tendency to control efficiently the

We subject in the convergence criterion that the relative change in two successive iterates of the solution (temperate and velocities) at each computational point

. For all simulations, the criterion convergence is considered to be reached, for

To the author's knowledge, studies have thus far addressed a mesoscopic approach in an MHD open cavity with sinusoidally heated wall (Figure 2). The objective of the

velocity and temperature when the following convergence is satisfied:

where ξ is velocity or temperature and σ is the iteration number.

<sup>ξ</sup><sup>σ</sup>þ<sup>1</sup> � <sup>ξ</sup><sup>σ</sup> 

. In C4, the presence of the partially open sidewall changes the flow and heat transfer. We note the presence of a one dominant cell in the core region of the

. The heat transfer rate is highlighted within the evolution

<sup>≤</sup>10�<sup>6</sup> (21)

on increasing the phase deviation from 0 to pi [11–14].

Pattern Formation and Stability in Magnetic Colloids

of isotherms and streamlines inside the cavity in Figure 6.

highest value at 3pi/4.

transfer [11–14].

Pr = 0.025, and Ra = 10<sup>6</sup>

and a high Rayleigh number Ra = 10<sup>5</sup>

be below a prescribed small value of 10�<sup>6</sup>

4. Conclusions

68

cavity and a little cell that occurs at upper left side.

movement of the fluid in such given configuration (C4).


## Greek letters



### Subscripts


References

[1] Lauriat G, Desrayaud G. Effect of surface radiation on conjugate natural convection in partially open enclosures. International Journal of Thermal Sciences. 2006;45(4):335-346

DOI: http://dx.doi.org/10.5772/intechopen.84478

method. International Journal of Thermal Sciences. 2010;49:1944-1953

[11] Bilgen E, Yedder RB. Natural convection in enclosure with heating and cooling by sinusoidal temperature profiles on one side. International Journal of Heat and Mass Transfer.

[12] Sarris IE, Lekakis I, Vlachos NS. Natural convection in a 2D enclosure with sinusoidal upper wall temperature. Numerical Heat Transfer A. 2002;42:

[13] Varol Y, Oztop HF, Pop I. Numerical analysis of natural convection for a porous rectangular enclosure with sinusoidally varying temperature profile on the bottom wall. International Communications in Heat and Mass Transfer. 2008;35:56-64

[14] Saeid NH, Yaacob Y. Natural convection in a square cavity with spatial side wall temperature variation. Numerical Heat Transfer A. 2006;49:

[15] Fidaros D, Grecos A, Vlachos N. Development of numerical tool for 3D MHD natural convection. Annex xx.

[16] Sathiyamoorthy M, Chamkha A. Effect of magnetic field on natural convection flow in a liquid gallium filled square cavity for linearly heated side wall(s). International Journal of Thermal Sciences. 2010;49:1856-1865

[17] Alchaar S, Vasseur P, Bilgen E. Natural convection heat transfer in a rectangular enclosure with a transverse

2007;50:139-150

513-530

The Study of Magneto-Convection Heat Transfer in a Partially Open Cavity Based on LBM

683-697

pp. 73-74

[10] Polat O, Bilgen E. Laminar natural convection in inclined open shallow cavities. International Journal of Thermal Sciences. 2002;41:360-368

International Journal of Heat and Mass Transfer. 2005;48(8):1470-1479

[3] Chan YL, Tien CL. A numerical study of two-dimensional laminar natural convection in shallow open cavities. International Journal of Heat Mass Transfer. 1985;28(3):603-612

[4] Xia JL, Zhou ZW. Natural convection in an externally heated partially open cavity with a heated protrusion. FEDvol. 143/HTD, Vol. 232. Measurement and Modeling of Environmental Flows—ASME. 1992;232:201-208

[5] Angirasa D, Eggels JGM, Nieuwstadt FTM. Numerical simulation of transient natural convection from an isothermal cavity open on a side. Numerical Heat Transfer, Part A: Applications. 1995;

[6] Mohamad AA. Natural convection in open cavities and slots. Numerical Heat Transfer, Part A: Applications. 1995;27:

[7] Kennedy P, Zheng R. Flow Analysis of Injection Molds. Munich: Hanser;

[8] Mohamad AA, El-Ganaoui M, Bennacer R. Lattice Boltzmann simulation of natural convection in an open ended cavity. International Journal of Thermal Sciences. 2009;48:1870-1875

[9] Mohamad AA, Bennacer R, El-Ganaoui M. Double dispersion natural convection in an open end cavity simulation via Lattice Boltzmann

28(6):755-768

705-716

2013

71

[2] Bilgen E, Oztop H. Natural convection heat transfer in partially open inclined square cavities.

## Author details

Raoudha Chaabane1,2

1 Laboratory of Thermal and Energetic Systems Studies (LESTE), National School of Engineering of Monastir, University of Monastir, Tunisia

2 Preparatory Institute of Engineering Studies of Monastir (IPEIM), University of Monastir, Tunisia

\*Address all correspondence to: raoudhach@gmail.com

© 2019 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

The Study of Magneto-Convection Heat Transfer in a Partially Open Cavity Based on LBM DOI: http://dx.doi.org/10.5772/intechopen.84478

## References

[1] Lauriat G, Desrayaud G. Effect of surface radiation on conjugate natural convection in partially open enclosures. International Journal of Thermal Sciences. 2006;45(4):335-346

[2] Bilgen E, Oztop H. Natural convection heat transfer in partially open inclined square cavities. International Journal of Heat and Mass Transfer. 2005;48(8):1470-1479

[3] Chan YL, Tien CL. A numerical study of two-dimensional laminar natural convection in shallow open cavities. International Journal of Heat Mass Transfer. 1985;28(3):603-612

[4] Xia JL, Zhou ZW. Natural convection in an externally heated partially open cavity with a heated protrusion. FEDvol. 143/HTD, Vol. 232. Measurement and Modeling of Environmental Flows—ASME. 1992;232:201-208

[5] Angirasa D, Eggels JGM, Nieuwstadt FTM. Numerical simulation of transient natural convection from an isothermal cavity open on a side. Numerical Heat Transfer, Part A: Applications. 1995; 28(6):755-768

[6] Mohamad AA. Natural convection in open cavities and slots. Numerical Heat Transfer, Part A: Applications. 1995;27: 705-716

[7] Kennedy P, Zheng R. Flow Analysis of Injection Molds. Munich: Hanser; 2013

[8] Mohamad AA, El-Ganaoui M, Bennacer R. Lattice Boltzmann simulation of natural convection in an open ended cavity. International Journal of Thermal Sciences. 2009;48:1870-1875

[9] Mohamad AA, Bennacer R, El-Ganaoui M. Double dispersion natural convection in an open end cavity simulation via Lattice Boltzmann

method. International Journal of Thermal Sciences. 2010;49:1944-1953

[10] Polat O, Bilgen E. Laminar natural convection in inclined open shallow cavities. International Journal of Thermal Sciences. 2002;41:360-368

[11] Bilgen E, Yedder RB. Natural convection in enclosure with heating and cooling by sinusoidal temperature profiles on one side. International Journal of Heat and Mass Transfer. 2007;50:139-150

[12] Sarris IE, Lekakis I, Vlachos NS. Natural convection in a 2D enclosure with sinusoidal upper wall temperature. Numerical Heat Transfer A. 2002;42: 513-530

[13] Varol Y, Oztop HF, Pop I. Numerical analysis of natural convection for a porous rectangular enclosure with sinusoidally varying temperature profile on the bottom wall. International Communications in Heat and Mass Transfer. 2008;35:56-64

[14] Saeid NH, Yaacob Y. Natural convection in a square cavity with spatial side wall temperature variation. Numerical Heat Transfer A. 2006;49: 683-697

[15] Fidaros D, Grecos A, Vlachos N. Development of numerical tool for 3D MHD natural convection. Annex xx. pp. 73-74

[16] Sathiyamoorthy M, Chamkha A. Effect of magnetic field on natural convection flow in a liquid gallium filled square cavity for linearly heated side wall(s). International Journal of Thermal Sciences. 2010;49:1856-1865

[17] Alchaar S, Vasseur P, Bilgen E. Natural convection heat transfer in a rectangular enclosure with a transverse

Author details

Monastir, Tunisia

70

Raoudha Chaabane1,2

1 Laboratory of Thermal and Energetic Systems Studies (LESTE), National School

2 Preparatory Institute of Engineering Studies of Monastir (IPEIM), University of

© 2019 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium,

of Engineering of Monastir, University of Monastir, Tunisia

Pattern Formation and Stability in Magnetic Colloids

\*Address all correspondence to: raoudhach@gmail.com

provided the original work is properly cited.

magnetic field. Journal of Heat Transfer. 1995;117:668-673

[18] Garandet J, Alboussiere T, Moreau R. Buoyancy drive convection in a rectangular enclosure with a transverse magnetic field. International Journal of Heat and Mass Transfer. 1992;35: 741-748

[19] Rudraiah N, Barron R, Venkatachalappa M, Subbaraya C. Effect of a magnetic field on free convection in a rectangular enclosure. International Journal of Engineering Science. 1995;33:1075-1084

[20] Cowley M. Natural convection in rectangular enclosures of arbitrary orientation with magnetic field vertical. Magnetohydrodynamics. 1996;32: 390-398

[21] Teamah M. Hydro-magnetic double-diffusive natural convection in a rectangular enclosure with imposing an inner heat source or sink. Alexandria Engineering Journal. 2006;45(4): 401-415

[22] Ozoe H, Okada K. The effect of the direction of the external magnetic field on the three-dimensional natural convection in a cubic enclosure. International Journal of Heat and Mass Transfer. 1989;32:1939-1953

[23] Ece M, Buyuk E. Natural convection flow under a magnetic field in an inclined rectangular enclosure heated and cooled on adjacent walls. Fluid Dynamics Research. 2006;38(5): 546-590

[24] Al-Najem N, Khanafer K, El-Refaee M. Numerical study of laminar natural convection in tilted enclosure with transverse magnetic field. International Journal of Numerical Methods for Heat and Fluid Flow. 1998;8:651-672

[25] Jalil J, Al-Taey K. MHD turbulent natural convection in a liquid metal filed square enclosure. Emirates Journal for Engineering Research. 2007;12(2):31-40

anisotropically scattering participating enclosure using the Lattice Boltzmann method and the control volume finite element method. Journal of Computer Physics Communications. 2011;182(7):

DOI: http://dx.doi.org/10.5772/intechopen.84478

The Study of Magneto-Convection Heat Transfer in a Partially Open Cavity Based on LBM

[34] Lamsaadi M, Naimi M, Hasnaoui M, Mamou M. Natural convection in a vertical rectangular cavity filled with a non-Newtonian power law fluid and subjected to a horizontal temperature gradient. Numerical Heat Transfer Part A: Application. 2006;49:969-990

[35] Chaabane R, Askri F, Ben Nasrallah S. Application of the Lattice Boltzmann method to transient conduction and radiation heat transfer in cylindrical media. Journal of Quantitative Spectroscopy and Radiative Transfer.

2011;112(12):2013-2027

[36] Chaabane R, Askri F, Jemni A, Nasrallah SB. Numerical study of transient convection with volumetric radiation using an hybrid lattice Boltzmann BGK-control volume finite element method. Journal of Heat Transfer. 2017;139(9):092701-072017

[37] Chaabane R, Askri F, Jemni A, Nasrallah SB. Analysis of Rayleigh-Bénard convection with thermal volumetric radiation using Lattice Boltzmann formulation. Journal of Thermal Science and Technology. 2017;

[38] Mohamad AA. Applied Lattice Boltzmann Method for Transport Phenomena, Momentum, Heat and Mass Transfer. Calgary: Sure; 2007

[39] Succi S. The Lattice Boltzmann Equation for Fluid Dynamics and Beyond. Oxford, London: Clarendon

[40] Series RW, Hurle DTJ. The use of magnetic fields in semiconductor crystal growth. Journal of Crystal Growth.

12(2)

Press; 2001

73

1991;133:305-328

1402-1413

[26] Gelfgat A, Bar-Yoseph P. The effect of an external magnetic field on oscillatory instability of convective flows in a rectangular cavity. Physics of Fluids. 2001;13(8):2269-2278

[27] Aleksandrova S, Molokov S. Threedimensional buoyant convection in a rectangular cavity with differentially heated walls in a strong magnetic field. Fluid Dynamics Research. 2004;35: 37-66

[28] Kahveci K, Oztuna S. MHD natural convection flow and heat transfer in a laterally heated partitioned enclosure. European Journal of Mechanics-B/ Fluids. 2009;28:744-752

[29] Kefayati GHR, Gorji M, Ganji DD, Sajjadi H. Investigation of Prandtl number effect on natural convection MHD in an open cavity by Lattice Boltzmann method. Engineering Computations. 2013;30:97-116

[30] Martinez D, Chen S, Matthaeus W. Lattice Boltzmann magneto hydrodynamics. Physics of Plasmas. 1994;(6):1850-1867

[31] Chaabane R, Askri F, Nasrallah SB. Parametric study of simultaneous transient conduction and radiation in a two-dimensional participating medium. Communications in Nonlinear Science and Numerical Simulation. 2011;16(10): 4006-4020

[32] Saha LK, Hossain MA, Gorla RSR. Effect of Hall current on the MHD laminar natural convection flow from a vertical permeable flat plate with uniform surface temperature. International Journal of Thermal Science. 2007;46:790-801

[33] Chaabane R, Askri F, Nasrallah SB. Analysis of two-dimensional transient conduction-radiation problems in an

The Study of Magneto-Convection Heat Transfer in a Partially Open Cavity Based on LBM DOI: http://dx.doi.org/10.5772/intechopen.84478

anisotropically scattering participating enclosure using the Lattice Boltzmann method and the control volume finite element method. Journal of Computer Physics Communications. 2011;182(7): 1402-1413

magnetic field. Journal of Heat Transfer.

Pattern Formation and Stability in Magnetic Colloids

square enclosure. Emirates Journal for Engineering Research. 2007;12(2):31-40

[26] Gelfgat A, Bar-Yoseph P. The effect

[27] Aleksandrova S, Molokov S. Threedimensional buoyant convection in a rectangular cavity with differentially heated walls in a strong magnetic field. Fluid Dynamics Research. 2004;35:

[28] Kahveci K, Oztuna S. MHD natural convection flow and heat transfer in a laterally heated partitioned enclosure. European Journal of Mechanics-B/

[29] Kefayati GHR, Gorji M, Ganji DD, Sajjadi H. Investigation of Prandtl number effect on natural convection MHD in an open cavity by Lattice Boltzmann method. Engineering Computations. 2013;30:97-116

[30] Martinez D, Chen S, Matthaeus W.

[31] Chaabane R, Askri F, Nasrallah SB. Parametric study of simultaneous transient conduction and radiation in a two-dimensional participating medium. Communications in Nonlinear Science and Numerical Simulation. 2011;16(10):

[32] Saha LK, Hossain MA, Gorla RSR. Effect of Hall current on the MHD laminar natural convection flow from a vertical permeable flat plate with uniform surface temperature. International Journal of Thermal Science. 2007;46:790-801

[33] Chaabane R, Askri F, Nasrallah SB. Analysis of two-dimensional transient conduction-radiation problems in an

hydrodynamics. Physics of Plasmas.

Lattice Boltzmann magneto

1994;(6):1850-1867

4006-4020

of an external magnetic field on oscillatory instability of convective flows in a rectangular cavity. Physics of

Fluids. 2001;13(8):2269-2278

Fluids. 2009;28:744-752

37-66

[18] Garandet J, Alboussiere T, Moreau R. Buoyancy drive convection in a rectangular enclosure with a transverse magnetic field. International Journal of Heat and Mass Transfer. 1992;35:

1995;117:668-673

[19] Rudraiah N, Barron R,

Science. 1995;33:1075-1084

Venkatachalappa M, Subbaraya C. Effect of a magnetic field on free convection in a rectangular enclosure. International Journal of Engineering

[20] Cowley M. Natural convection in rectangular enclosures of arbitrary orientation with magnetic field vertical. Magnetohydrodynamics. 1996;32:

[21] Teamah M. Hydro-magnetic

double-diffusive natural convection in a rectangular enclosure with imposing an inner heat source or sink. Alexandria Engineering Journal. 2006;45(4):

[22] Ozoe H, Okada K. The effect of the direction of the external magnetic field on the three-dimensional natural convection in a cubic enclosure. International Journal of Heat and Mass

[23] Ece M, Buyuk E. Natural convection flow under a magnetic field in an inclined rectangular enclosure heated and cooled on adjacent walls. Fluid Dynamics Research. 2006;38(5):

[24] Al-Najem N, Khanafer K, El-Refaee M. Numerical study of laminar natural convection in tilted enclosure with transverse magnetic field. International Journal of Numerical Methods for Heat

and Fluid Flow. 1998;8:651-672

[25] Jalil J, Al-Taey K. MHD turbulent natural convection in a liquid metal filed

Transfer. 1989;32:1939-1953

741-748

390-398

401-415

546-590

72

[34] Lamsaadi M, Naimi M, Hasnaoui M, Mamou M. Natural convection in a vertical rectangular cavity filled with a non-Newtonian power law fluid and subjected to a horizontal temperature gradient. Numerical Heat Transfer Part A: Application. 2006;49:969-990

[35] Chaabane R, Askri F, Ben Nasrallah S. Application of the Lattice Boltzmann method to transient conduction and radiation heat transfer in cylindrical media. Journal of Quantitative Spectroscopy and Radiative Transfer. 2011;112(12):2013-2027

[36] Chaabane R, Askri F, Jemni A, Nasrallah SB. Numerical study of transient convection with volumetric radiation using an hybrid lattice Boltzmann BGK-control volume finite element method. Journal of Heat Transfer. 2017;139(9):092701-072017

[37] Chaabane R, Askri F, Jemni A, Nasrallah SB. Analysis of Rayleigh-Bénard convection with thermal volumetric radiation using Lattice Boltzmann formulation. Journal of Thermal Science and Technology. 2017; 12(2)

[38] Mohamad AA. Applied Lattice Boltzmann Method for Transport Phenomena, Momentum, Heat and Mass Transfer. Calgary: Sure; 2007

[39] Succi S. The Lattice Boltzmann Equation for Fluid Dynamics and Beyond. Oxford, London: Clarendon Press; 2001

[40] Series RW, Hurle DTJ. The use of magnetic fields in semiconductor crystal growth. Journal of Crystal Growth. 1991;133:305-328

Chapter 5

Abstract

Generation

Convection Flow of MHD Couple

Entropy generation of fully developed steady, viscous, incompressible couple

stress fluid in a vertical micro-porous-channel in the presence of horizontal magnetic field is analysed in this work. The governing equations for the flow are derived, and nondimensionalised and the resulting nonlinear ordinary differential equations are solved via a rapidly convergent technique developed by Zhou. The solution of the velocity and temperature profiles are utilised to obtain the flow irreversibility and Bejan number. The effects of couple stresses, fluid wall interaction parameter (FSIP), effective temperature ratio (ETR), rarefaction and magnetic parameter on the velocity profile, temperature profile, entropy generation and Bejan number are presented and discussed graphically.

Keywords: microchannel, entropy generation, MHD, natural convection,

In the last decades, the study of microchannel flows has become an important subject for researchers because of the reduction in the size of such devices which increases the dissipated heat per unit area. The effective performance of these devices is dependent on the temperature; as a result a comprehensive knowledge of such flow behaviours is required for accurate prediction of performance during the design process. These microfluidics have characteristic lengths of 1 � 100 μm and are categorised by the dimensionless quantity called the Knudsen number ð Þ Kn . Researchers have shown that microchannel flows are influenced by several

parameters of which velocity slip and temperature jump occurring at the solid-fluid interface in small-scale systems are the most important [2–4]. Velocity slip and temperature jump become more significant at higher Knudsen number. The latter boundary conditions are assumed at Knudsen number greater than 0.01 since below this value the classical Navier-Stokes equations is no longer valid.

The influence of velocity slip and temperature jump on microchannel flows has

been extensively studied. Khadrawi and Al-Shyyab [5] obtained a close form

Stress Fluid in Vertical

Hilary I. Okagbue and Sheila A. Bishop

differential transform method (DTM)

1. Introduction

75

Microchannel with Entropy

Abiodun A. Opanuga, Olasunmbo O. Agboola,

## Chapter 5
