**1. Introduction**

The flow due to rotating disks is of great interest in many practical and engineering aspects. Rotating disk flows of electrically conducting fluids have practical applications in many areas, such as rotating machinery, lubrication, oceanography, computer storage devices, viscometer and crystal growth processes etc. Also, the study is interesting from the mathematical point of view. During the last two decades, research on renewable energy sources, as for example, solar, wind energy or energy from hydro-power and the preparation of oxygenated additives to blend diesel fuel, has been intensified.

Pioneering study of fluid flow due to an infinite rotating disk was carried by authors [1–3]. Chemical reactions usually accompany a large amount of exothermic and endothermic reactions. These characteristics can be easily seen in a lot of industrial processes, it has been realized that it is not always permissible to neglect the convection effects in porous constructed chemical reactors [4]. The reaction produced in a porous medium was extraordinarily in common, such as the topic of

PEM fuel cells modules and the polluted underground water because of discharging the toxic substance, etc.

**2. Mathematical formulation**

*DOI: http://dx.doi.org/10.5772/intechopen.82186*

Consider the steady, axially symmetric, incompressible flow of an electrically conducting fluid with heat and mass transfer flow due to a rotating porous disk in the presence of radiation has been considered. Assume that the fluid is infinite in extent in the positive z-direction. Let ð Þ *r; φ; z* be the set of cylindrical polar coordinates and let the disk rotate with constant angular velocity Ω and is placed at *z* ¼ 0. The components of the flow velocity are ð Þ *u; v; w* in the directions of increasing ð Þ *r; φ; z* , respectively. *p*, *T* and *C* are the pressure, the temperature and the concentration distribution, respectively. The surface of the rotating disk is maintained at a uniform temperature *Tw* and uniform concentration *Cw*. Far away from the surface, the free stream is kept at a constant temperature *T*∞, at a constant concentration *C*<sup>∞</sup>

*Thermal Radiation and Thermal Diffusion for Soret and Dufour's Effects on MHD Flow over…*

and at a constant pressure *p*∞. The fluid is assumed to be gray, emitting and absorbing heat, but not scattering medium and is assumed to be Newtonian. The

*divB* ¼ 0*, CurlB* ¼ *μmJ* and *divE* ¼ 0

The magnetic body force *<sup>J</sup>* � *<sup>B</sup>* takes the form *<sup>σ</sup> <sup>V</sup>* � *<sup>B</sup>* � *<sup>B</sup>*, therefore,

where *J* is the electric current density, *B* ¼ *B* þ *b* is the total magnetic field, *μ<sup>m</sup>* is the magnetic permeability and *b* is the induced magnetic field. The external uniform magnetic field *B* is imposed in the direction normal to the surface of the disk which is assumed unchanged by taking small magnetic Reynolds number, so that the flow induction distortion of the applied magnetic field can be neglected as in the case with most of conducting fluids. In addition, a uniform suction is applied at the

*V=r*, where *σ* is the electrical conductivity of fluid and *V* is

physical model and geometrical coordinates are shown in **Figure 1**.

The MHD body forces *J* � *B* the Maxwell's equations:

surface of the disk for the entire range.

*<sup>σ</sup> <sup>V</sup>* � *<sup>B</sup>* � *<sup>B</sup>* ¼ �*σB*<sup>2</sup>

**Figure 1.**

**103**

*Schematic diagram of the problem.*

Fourier's law, for instance, described the relation between energy flux and temperature gradient. In other aspects, Fick's law was determined by the correlation of mass flux and concentration gradient. Moreover, it was found that energy flux can also be generated by composition gradients, pressure gradients, or body forces. The energy flux caused by a composition gradient was discovered in 1873 by Dufour and was correspondingly referred to the Dufour effect.

It was also called the diffusion-thermo effect. On the other hand, mass flux can also be created by a temperature gradient, as was established by Soret. This is the thermal-diffusion effect. In general, the thermal-diffusion and the diffusion-thermo effects were of a smaller order of magnitude than the effects described by Fourier's or Fick's law and were often neglected in heat and mass transfer processes. There were still some exceptional conditions. The thermal-diffusion effect has been utilized for isotope separation and in mixtures between gases with very light molecular weight ð Þ *H*2*; He* and of medium molecular weight ð Þ *N*2*; air* , the diffusion-thermo effect was found to be of a magnitude such that it may not be neglected in certain conditions [5].

The first traceable interest in magnetohydrodynamics (MHD) flow was in 1907, when Northrop built an MHD pump prototype [6, 7]. Since then, analysis of the effects of both rotation and magnetic fields on fluid flows has been an active area of research. While technology expanded in many directions, the subject of MHD has developed in the use of magnetic fields and the range of fluid and thermal processes by [8–13]. This study considers the effect of slip as a result of rarefied effect, a type of flow commonly encountered in many engineering tasks such as high altitude flight, micro-machines, vacuum technology, aerosol reactors, etc. In this study, the slip and no-slip regimes that lie in the range 0.1 > *Kn* > 0 are considered. A completely different extension of von Karman's one-disk problem is the analysis of Sparrow et al. [14]. They considered the flow of a Newtonian fluid due to the rotation of a porous-surfaced disk and for that purpose replaced the conventional no-slip boundary conditions at the disk surface with a set of linear slip flow conditions. A substantial reduction in torque then occurred as a result of surface slip. Recently Frusteri and Osalusi [15] studied the effects of variable properties on MHD and slip flow over a porous rotating disk.

In all these studies Soret and Dufour effects were assumed to be negligible. Such effects are significant when density differences exist in the flow regime. For example, when species are introduced at a surface in fluid domain, with different (lower) density than the surrounding fluid, both Soret (thermo-diffusion) and Dufour (diffusion-thermal) effects can be influential. An analytical study of convection along a horizontal cylinder for a Helium-air system was reported subsequently by Sparrow et al. [16]. In view of the importance of above mentioned effects, Maleque [17] studied Soret effect on convective heat and mass transfer past a rotating porous disk and he neglected the Dufour effect. Ahmed [18] investigated the Dufour and Soret effects on free convective heat and mass transfer over a stretching surface considering suction or injection. Recently, numerical study of free convection magnetohydrodynamic heat and mass transfer due to a stretching surface under saturated porous medium with Soret and Dufour effects was also discussed by Anwar Beg et al. [19].

In these two papers [15, 20] they have studied the effect of the magnetic field on the equations of motion that I found them used in the cylindrical coordinate, although the magnetic field parameters are in the Cartesian coordinate, which is wrong.

*Thermal Radiation and Thermal Diffusion for Soret and Dufour's Effects on MHD Flow over… DOI: http://dx.doi.org/10.5772/intechopen.82186*

### **2. Mathematical formulation**

PEM fuel cells modules and the polluted underground water because of discharging

It was also called the diffusion-thermo effect. On the other hand, mass flux can also be created by a temperature gradient, as was established by Soret. This is the thermal-diffusion effect. In general, the thermal-diffusion and the diffusion-thermo effects were of a smaller order of magnitude than the effects described by Fourier's or Fick's law and were often neglected in heat and mass transfer processes. There were still some exceptional conditions. The thermal-diffusion effect has been utilized for isotope separation and in mixtures between gases with very light molecular weight ð Þ *H*2*; He* and of medium molecular weight ð Þ *N*2*; air* , the diffusion-thermo effect was found to be of a magnitude such that it may not be neglected in certain

The first traceable interest in magnetohydrodynamics (MHD) flow was in 1907, when Northrop built an MHD pump prototype [6, 7]. Since then, analysis of the effects of both rotation and magnetic fields on fluid flows has been an active area of research. While technology expanded in many directions, the subject of MHD has developed in the use of magnetic fields and the range of fluid and thermal processes by [8–13]. This study considers the effect of slip as a result of rarefied effect, a type of flow commonly encountered in many engineering tasks such as high altitude flight, micro-machines, vacuum technology, aerosol reactors, etc. In this study, the slip and no-slip regimes that lie in the range 0.1 > *Kn* > 0 are considered. A completely different extension of von Karman's one-disk problem is the analysis of Sparrow et al. [14]. They considered the flow of a Newtonian fluid due to the rotation of a porous-surfaced disk and for that purpose replaced the conventional no-slip boundary conditions at the disk surface with a set of linear slip flow conditions. A substantial reduction in torque then occurred as a result of surface slip. Recently Frusteri and Osalusi [15] studied the effects of variable properties on MHD

In all these studies Soret and Dufour effects were assumed to be negligible. Such effects are significant when density differences exist in the flow regime. For example, when species are introduced at a surface in fluid domain, with different (lower) density than the surrounding fluid, both Soret (thermo-diffusion) and Dufour (diffusion-thermal) effects can be influential. An analytical study of convection along a horizontal cylinder for a Helium-air system was reported subsequently by Sparrow et al. [16]. In view of the importance of above mentioned effects, Maleque [17] studied Soret effect on convective heat and mass transfer past a rotating porous disk and he neglected the Dufour effect. Ahmed [18] investigated the Dufour and Soret effects on free convective heat and mass transfer over a stretching surface considering suction or injection. Recently, numerical study of free convection magnetohydrodynamic heat and mass transfer due to a stretching surface under saturated porous medium with Soret and Dufour effects was also discussed by Anwar

In these two papers [15, 20] they have studied the effect of the magnetic field on

the equations of motion that I found them used in the cylindrical coordinate, although the magnetic field parameters are in the Cartesian coordinate, which is

Fourier's law, for instance, described the relation between energy flux and temperature gradient. In other aspects, Fick's law was determined by the correlation of mass flux and concentration gradient. Moreover, it was found that energy flux can also be generated by composition gradients, pressure gradients, or body forces. The energy flux caused by a composition gradient was discovered in 1873 by Dufour

and was correspondingly referred to the Dufour effect.

and slip flow over a porous rotating disk.

the toxic substance, etc.

*Nanofluid Flow in Porous Media*

conditions [5].

Beg et al. [19].

wrong.

**102**

Consider the steady, axially symmetric, incompressible flow of an electrically conducting fluid with heat and mass transfer flow due to a rotating porous disk in the presence of radiation has been considered. Assume that the fluid is infinite in extent in the positive z-direction. Let ð Þ *r; φ; z* be the set of cylindrical polar coordinates and let the disk rotate with constant angular velocity Ω and is placed at *z* ¼ 0. The components of the flow velocity are ð Þ *u; v; w* in the directions of increasing ð Þ *r; φ; z* , respectively. *p*, *T* and *C* are the pressure, the temperature and the concentration distribution, respectively. The surface of the rotating disk is maintained at a uniform temperature *Tw* and uniform concentration *Cw*. Far away from the surface, the free stream is kept at a constant temperature *T*∞, at a constant concentration *C*<sup>∞</sup> and at a constant pressure *p*∞. The fluid is assumed to be gray, emitting and absorbing heat, but not scattering medium and is assumed to be Newtonian. The physical model and geometrical coordinates are shown in **Figure 1**.

The MHD body forces *J* � *B* the Maxwell's equations:

$$\operatorname{div} \overline{B} = 0, \operatorname{Curl} \overline{B} = \mu\_m \overline{J} \text{ and } \operatorname{div} \overline{E} = 0.$$

where *J* is the electric current density, *B* ¼ *B* þ *b* is the total magnetic field, *μ<sup>m</sup>* is the magnetic permeability and *b* is the induced magnetic field. The external uniform magnetic field *B* is imposed in the direction normal to the surface of the disk which is assumed unchanged by taking small magnetic Reynolds number, so that the flow induction distortion of the applied magnetic field can be neglected as in the case with most of conducting fluids. In addition, a uniform suction is applied at the surface of the disk for the entire range.

The magnetic body force *<sup>J</sup>* � *<sup>B</sup>* takes the form *<sup>σ</sup> <sup>V</sup>* � *<sup>B</sup>* � *<sup>B</sup>*, therefore, *<sup>σ</sup> <sup>V</sup>* � *<sup>B</sup>* � *<sup>B</sup>* ¼ �*σB*<sup>2</sup> *V=r*, where *σ* is the electrical conductivity of fluid and *V* is

**Figure 1.** *Schematic diagram of the problem.*

velocity vector *V* ¼ ð Þ *u; v; w* and *B* ¼ ð Þ 0*;* 0*; B* . The Lorentz force (MHD body force) has two components:

$$F\_r = -\sigma B^2 \boldsymbol{\nu}/r,\\ F\_\theta = -\sigma B^2 \boldsymbol{\nu}/r.$$

Under these assumptions, the governing equations for the continuity, momentum, energy and concentration in laminar incompressible flow can be written as follows:

$$\frac{\partial u}{\partial r} + \frac{u}{r} + \frac{\partial w}{\partial z} = \mathbf{0} \ , \tag{1}$$

where *k* is the thermal conductivity, *ν* ¼ *μ=ρ* is the kinematic viscosity of the

*Thermal Radiation and Thermal Diffusion for Soret and Dufour's Effects on MHD Flow over…*

medium, *g* is the gravitational acceleration, *β<sup>T</sup>* and *β<sup>C</sup>* are the expansion coefficients of temperature and concentration, *ρ, μ* and *cp* are the density, dynamic viscosity and the specific heat at constant pressure, respectively, *Q* is the volumetric heat

*kT, cs, Tm* and *qr* are the thermal-diffusion rate, concentration susceptibility, the mean fluid temperature and the radiative heat flux. Using the Rosseland approxi-

> *qr* ¼ � <sup>4</sup>*σ*<sup>∗</sup> 3*k*<sup>∗</sup>

Where the *σ*<sup>∗</sup> represents the Stefan-Boltzman constant and *k*<sup>∗</sup> is the Rosseland

Assuming that the temperature difference within the flow is sufficiently small

For the flow under study, it is relevant to assume that the applied magnetic field

<sup>∞</sup>*<sup>T</sup>* � <sup>3</sup>*T*<sup>4</sup>

*∂T*<sup>4</sup> *∂ z*

generation/absorption rate, *D* is the molecular diffusion coefficient,

mation (Rashed [21]), the radiative heat flux *qr* could be expressed by

such that *T*<sup>4</sup> could be approached as the linear function of temperature

<sup>Ω</sup>*<sup>L</sup> , <sup>V</sup>* <sup>¼</sup> *<sup>v</sup>*

*Tw* � *T*<sup>∞</sup>

*∂U ∂R* þ *U R* þ *∂W*

*∂*<sup>2</sup>*V ∂R*<sup>2</sup> þ 1 *R ∂V <sup>∂</sup><sup>R</sup>* � *<sup>V</sup> R*2 þ *∂*<sup>2</sup>*V ∂Z*<sup>2</sup>

*<sup>∂</sup><sup>Z</sup>* ¼ � *<sup>∂</sup><sup>P</sup> ∂Z* þ *ν*

> 1 *R ∂T ∂R* þ *∂*<sup>2</sup> *T ∂Z*<sup>2</sup>

� �

� �

*∂*<sup>2</sup> *T <sup>∂</sup>R*<sup>2</sup> <sup>þ</sup>

> *∂*<sup>2</sup>*C ∂R*<sup>2</sup> þ 1 *R ∂C ∂R* þ *∂*<sup>2</sup>*C ∂Z*<sup>2</sup>

*<sup>L</sup>*<sup>2</sup> *, <sup>T</sup>* <sup>¼</sup> *<sup>T</sup>* � *<sup>T</sup>*<sup>∞</sup>

*<sup>T</sup>*<sup>4</sup> ffi <sup>4</sup>*T*<sup>3</sup>

ffiffi

To obtain the non-dimensional form of the above equations, the following

<sup>Ω</sup>*<sup>L</sup> , <sup>λ</sup>* <sup>¼</sup> *<sup>λ</sup>*

Substituting Eqs. (9)–(11) in Eqs. (1)–(6), we obtain the following dimension-

*∂*<sup>2</sup> *U ∂R*<sup>2</sup> þ 1 *R ∂U <sup>∂</sup><sup>R</sup>* � *<sup>U</sup> R*2 þ *∂*<sup>2</sup> *U ∂Z*<sup>2</sup>

*<sup>L</sup> C,*

� �

*∂*<sup>2</sup> *W ∂R*<sup>2</sup> þ 1 *R*

þ

*<sup>T</sup>* <sup>þ</sup> *<sup>g</sup> <sup>β</sup>c*ð Þ *Cw* � *<sup>C</sup>*<sup>∞</sup> Ω2

*, <sup>C</sup>* <sup>¼</sup> *<sup>C</sup>* � *<sup>C</sup>*<sup>∞</sup> *Cw* � *C*<sup>∞</sup>

� �

*<sup>L</sup> ,W* <sup>¼</sup> *<sup>w</sup>*

<sup>1</sup> is the permeability of the porous

<sup>∞</sup> (10)

*<sup>r</sup>* <sup>p</sup> , where *<sup>B</sup>*<sup>0</sup> is constant magnetic flux density.

*, k*<sup>∗</sup> <sup>1</sup> <sup>¼</sup> *<sup>k</sup>*<sup>∗</sup> *L*2

<sup>Ω</sup>*<sup>L</sup> , <sup>P</sup>* <sup>¼</sup> *<sup>p</sup>* � *<sup>p</sup>*<sup>∞</sup> *ρ*Ω<sup>2</sup> *L*2 *,*

*<sup>∂</sup><sup>Z</sup>* <sup>¼</sup> <sup>0</sup> *,* (12)

� *<sup>σ</sup> <sup>B</sup>*<sup>2</sup> 0 *ρ*Ω

*∂W ∂R* þ *∂*<sup>2</sup> *W ∂Z*<sup>2</sup> � �*,* (15)

> 16*σ*<sup>∗</sup>*T*<sup>3</sup> ∞ 3*ρcp*Ω*k*<sup>∗</sup> *L*<sup>2</sup>

<sup>þ</sup> *<sup>σ</sup> <sup>B</sup>*<sup>2</sup>

� *<sup>σ</sup> <sup>B</sup>*<sup>2</sup> 0 *ρ*Ω

*<sup>V</sup>* � *<sup>ν</sup> k*1

*∂*<sup>2</sup> *T ∂Z*<sup>2</sup>

> *U*2 *R*

<sup>þ</sup> *<sup>Q</sup> <sup>ρ</sup>cp* <sup>Ω</sup> *T,*

(16)

<sup>0</sup>Ω*L ρcp*ð Þ *Tw* � *T*<sup>∞</sup> *<sup>U</sup>* � *<sup>ν</sup> k*1 *U*

(13)

*V,* (14)

(9)

(11)

ambient fluid, *σ* is the electrical conductivity, *K*<sup>∗</sup>

*DOI: http://dx.doi.org/10.5772/intechopen.82186*

mean absorption coefficient.

*B r*ð Þ has the form cobble [22] *B* ¼ *B*<sup>0</sup>

*<sup>L</sup> , <sup>Z</sup>* <sup>¼</sup> *<sup>z</sup>*

<sup>Ω</sup>*L*<sup>2</sup> *, <sup>k</sup>*<sup>1</sup> <sup>¼</sup> *<sup>k</sup>*<sup>1</sup>

*<sup>R</sup>* <sup>¼</sup> *<sup>r</sup>*

*<sup>ν</sup>* <sup>¼</sup> *<sup>ν</sup>*

less equations:

*<sup>U</sup> <sup>∂</sup><sup>U</sup> <sup>∂</sup><sup>R</sup>* � *<sup>V</sup>*<sup>2</sup> *R*

*<sup>U</sup> <sup>∂</sup><sup>V</sup>*

*<sup>U</sup> <sup>∂</sup><sup>T</sup> ∂R*

**105**

*<sup>∂</sup><sup>R</sup>* � *<sup>U</sup> <sup>V</sup> R*

<sup>þ</sup> *<sup>W</sup> <sup>∂</sup><sup>T</sup>*

*cscp*Ω*L*<sup>2</sup>

þ

dimensionless variables are introduced.

<sup>þ</sup> *<sup>W</sup> <sup>∂</sup><sup>U</sup>*

<sup>þ</sup> *<sup>W</sup> <sup>∂</sup><sup>V</sup>*

<sup>þ</sup> *<sup>g</sup> <sup>β</sup>T*ð Þ *Tw* � *<sup>T</sup>*<sup>∞</sup> Ω2 *L*

> *<sup>U</sup> <sup>∂</sup><sup>W</sup> ∂R*

*<sup>∂</sup><sup>Z</sup>* <sup>¼</sup> *<sup>k</sup>*

*D kT*ð Þ *Cw* � *C*<sup>∞</sup>

*ρcp*Ω*L*<sup>2</sup>

ð Þ *Tw* � *T*<sup>∞</sup>

*<sup>∂</sup><sup>Z</sup>* ¼ � *<sup>∂</sup><sup>P</sup> ∂R* þ *ν*

*<sup>∂</sup><sup>Z</sup>* <sup>¼</sup> *<sup>ν</sup>*

<sup>þ</sup> *<sup>W</sup> <sup>∂</sup><sup>W</sup>*

*<sup>L</sup> , <sup>U</sup>* <sup>¼</sup> *<sup>u</sup>*

$$\begin{split} u\frac{\partial u}{\partial r} - \frac{v^2}{r} + w\frac{\partial u}{\partial z} &= -\frac{1}{\rho}\frac{\partial p}{\partial r} + \nu \left( \frac{\partial^2 u}{\partial r^2} + \frac{1}{r}\frac{\partial u}{\partial r} - \frac{u}{r^2} + \frac{\partial^2 u}{\partial z^2} \right) \\ &- \frac{\sigma B^2}{\rho r}u - \frac{\nu}{K\_1^\*}u + \mathbf{g}\,\beta\_T (T - T\_\infty) + \mathbf{g}\,\beta\_C (\mathbf{C} - \mathbf{C}\_\infty), \end{split} \tag{2}$$

$$u\frac{\partial v}{\partial r} + \frac{uv}{r} + w\frac{\partial v}{\partial z} = \nu \left( \frac{\partial^2 v}{\partial r^2} + \frac{1}{r}\frac{\partial v}{\partial r} - \frac{v}{r^2} + \frac{\partial^2 v}{\partial z^2} \right) - \frac{\sigma B^2}{\rho r}v - \frac{\nu}{K\_1^\*}v,\tag{3}$$

$$u\frac{\partial w}{\partial r} + w\frac{\partial w}{\partial z} = -\frac{1}{\rho}\frac{\partial p}{\partial z} + \nu \left(\frac{\partial^2 w}{\partial r^2} + \frac{1}{r}\frac{\partial w}{\partial r} + \frac{\partial^2 w}{\partial z^2}\right),\tag{4}$$

$$\begin{split} u\frac{\partial T}{\partial r} + w\frac{\partial T}{\partial z} &= \frac{k}{\rho c\_p} \left( \frac{\partial^2 T}{\partial r^2} + \frac{1}{r}\frac{\partial T}{\partial r} + \frac{\partial^2 T}{\partial z^2} \right) - \frac{1}{\rho c\_p} \frac{\partial q\_r}{\partial z} + \frac{Dk\_T}{c\_r c\_p} \left( \frac{\partial^2 C}{\partial r^2} + \frac{1}{r}\frac{\partial C}{\partial r} + \frac{\partial^2 C}{\partial z^2} \right) \\ &+ \frac{\sigma B^2}{\rho c\_p r} u^2 + \frac{Q}{\rho c\_p} (T - T\_\infty), \end{split} \tag{5}$$

$$u\frac{\partial C}{\partial r} + w\frac{\partial C}{\partial z} = D\left(\frac{\partial^2 C}{\partial r^2} + \frac{1}{r}\frac{\partial C}{\partial r} + \frac{\partial^2 C}{\partial z^2}\right) + \frac{Dk\_T}{T\_m}\left(\frac{\partial^2 T}{\partial r^2} + \frac{1}{r}\frac{\partial T}{\partial r} + \frac{\partial^2 T}{\partial z^2}\right) - k\_1(C - C\_\infty) \tag{6}$$

When the mean free path of the fluid particle is comparable to the characteristic dimensions of the flow field domain, Naiver-Stokes equations break down since the assumption of continuum media fails. In the range of 0*:*1 < *kn* < 10 Knudsen Number, the higher order continuum equation, for example, Burnett equation should be used. For the range of 0*:*001 ≤*kn* ≤0*:*1, no slip boundary conditions cannot be used and should be replaced with the following expression (Gad-el-Hak [20]):

$$U\_t = \frac{2 - \xi}{\xi} \lambda \frac{\partial U\_t}{\partial n},\tag{7}$$

Where *Ut* is the tangential velocity, *n* is the normal direction to the wall, *ξ* is the tangent momentum accommodation coefficient and *λ* is the mean free path. For *kn* < 0*:*001, the no-slip boundary condition is valid, therefore, the velocity at the surface is equal to zero. In this study the slip and the no-slip regimes of the Knudsen number that lies in the range 0*:*1>*kn*>0 is considered. By using Eq. (7), the appropriate boundary conditions for the flow induced by an infinite disk ð Þ *z* ¼ 0 which rotates with constant angular velocity Ω subjected to uniform suction *w*<sup>0</sup> through the disk are given by

$$\begin{aligned} z = 0: \quad & u = \frac{2 - \xi}{\xi} \lambda \frac{\partial u}{\partial z}, \quad v = \Omega r + \frac{2 - \xi}{\xi} \lambda \frac{\partial v}{\partial z}, \; w = w\_0, \; T = T\_w, \; C = C\_w, \\\ z \to \infty: \quad & u \to 0, v \to 0, \quad T \to T\_\infty, \quad C \to C\_\infty, \; p \to p\_\infty. \end{aligned} \tag{8}$$

*Thermal Radiation and Thermal Diffusion for Soret and Dufour's Effects on MHD Flow over… DOI: http://dx.doi.org/10.5772/intechopen.82186*

where *k* is the thermal conductivity, *ν* ¼ *μ=ρ* is the kinematic viscosity of the ambient fluid, *σ* is the electrical conductivity, *K*<sup>∗</sup> <sup>1</sup> is the permeability of the porous medium, *g* is the gravitational acceleration, *β<sup>T</sup>* and *β<sup>C</sup>* are the expansion coefficients of temperature and concentration, *ρ, μ* and *cp* are the density, dynamic viscosity and the specific heat at constant pressure, respectively, *Q* is the volumetric heat generation/absorption rate, *D* is the molecular diffusion coefficient, *kT, cs, Tm* and *qr* are the thermal-diffusion rate, concentration susceptibility, the mean fluid temperature and the radiative heat flux. Using the Rosseland approximation (Rashed [21]), the radiative heat flux *qr* could be expressed by

$$q\_r = -\frac{4\sigma^\*}{3k^\*} \frac{\partial T^4}{\partial z} \tag{9}$$

Where the *σ*<sup>∗</sup> represents the Stefan-Boltzman constant and *k*<sup>∗</sup> is the Rosseland mean absorption coefficient.

Assuming that the temperature difference within the flow is sufficiently small such that *T*<sup>4</sup> could be approached as the linear function of temperature

$$T^4 \cong 4T\_\infty^3 T - 3T\_\infty^4 \tag{10}$$

For the flow under study, it is relevant to assume that the applied magnetic field *B r*ð Þ has the form cobble [22] *B* ¼ *B*<sup>0</sup> ffiffi *<sup>r</sup>* <sup>p</sup> , where *<sup>B</sup>*<sup>0</sup> is constant magnetic flux density.

To obtain the non-dimensional form of the above equations, the following dimensionless variables are introduced.

$$\begin{aligned} \overline{R} &= \frac{r}{L}, \ \overline{Z} = \frac{z}{L}, \ \overline{U} = \frac{u}{\Omega L}, \ \overline{V} = \frac{v}{\Omega L}, \ \overline{\lambda} = \frac{\lambda}{L}, \ \overline{W} = \frac{w}{\Omega L}, \ \overline{P} = \frac{p - p\_{\infty}}{\rho \Omega^2 L^2}, \\\overline{\nu} &= \frac{\nu}{\Omega L^2}, \ \overline{k\_1} = \frac{k\_1}{L^2}, \ \overline{T} = \frac{T - T\_{\infty}}{T\_w - T\_{\infty}}, \ \overline{C} = \frac{C - C\_{\infty}}{C\_w - C\_{\infty}}, \ k\_1^\* = \frac{\overline{k\_\*}}{L^2} \end{aligned} \tag{11}$$

Substituting Eqs. (9)–(11) in Eqs. (1)–(6), we obtain the following dimensionless equations:

$$\frac{\partial \overline{U}}{\partial \overline{R}} + \frac{\overline{U}}{\overline{R}} + \frac{\partial \overline{W}}{\partial \overline{Z}} = \mathbf{0} \,, \tag{12}$$

$$\begin{split} \overline{U}\frac{\partial\overline{U}}{\partial\overline{R}} - \frac{\overline{V}^{2}}{\overline{R}} + \overline{W}\frac{\partial\overline{U}}{\partial\overline{Z}} &= -\frac{\partial\overline{P}}{\partial\overline{R}} + \overline{\nu}\left(\frac{\partial^{2}\overline{U}}{\partial\overline{R}^{2}} + \frac{1}{\overline{R}}\frac{\partial\overline{U}}{\partial\overline{R}} - \frac{\overline{U}}{\overline{R}^{2}} + \frac{\partial^{2}\overline{U}}{\partial\overline{Z}^{2}}\right) - \frac{\sigma B\_{0}^{2}}{\rho\Omega}\overline{U} - \frac{\overline{\nu}}{\overline{k}\_{1}}\overline{U} \\ &+ \frac{\mathbf{g}\beta\_{T}(T\_{w} - T\_{\infty})}{\Omega^{2}L}\overline{T} + \frac{\mathbf{g}\beta\_{c}(C\_{w} - C\_{\infty})}{\Omega^{2}L}\overline{C}, \end{split} \tag{13}$$

$$
\overline{U}\frac{\partial\overline{V}}{\partial\overline{R}} - \frac{\overline{U}\overline{V}}{\overline{R}} + \overline{W}\frac{\partial\overline{V}}{\partial\overline{Z}} = \overline{\nu}\left(\frac{\partial^2\overline{V}}{\partial\overline{R}^2} + \frac{1}{\overline{R}}\frac{\partial\overline{V}}{\partial\overline{R}} - \frac{\overline{V}}{\overline{R}^2} + \frac{\partial^2\overline{V}}{\partial\overline{Z}^2}\right) - \frac{\sigma B\_0^2}{\rho\Omega}\overline{V} - \frac{\overline{\nu}}{\overline{k}\_1}\overline{V},\tag{14}
$$

$$
\overline{U}\frac{\partial\overline{W}}{\partial\overline{R}} + \overline{W}\frac{\partial\overline{W}}{\partial\overline{Z}} = -\frac{\partial\overline{P}}{\partial\overline{Z}} + \overline{\nu}\left(\frac{\partial^2\overline{W}}{\partial\overline{R^2}} + \frac{1}{\overline{R}}\frac{\partial\overline{W}}{\partial\overline{R}} + \frac{\partial^2\overline{W}}{\partial\overline{Z^2}}\right),\tag{15}
$$

$$\begin{split} \overline{U}\frac{\partial\overline{T}}{\partial\overline{R}} + \overline{W}\frac{\partial\overline{T}}{\partial\overline{Z}} &= \frac{k}{\rho c\_{p}\Omega L^{2}} \left( \frac{\partial^{2}\overline{T}}{\partial\overline{R}^{2}} + \frac{1}{R}\frac{\partial\overline{T}}{\partial\overline{R}} + \frac{\partial^{2}\overline{T}}{\partial\overline{Z}^{2}} \right) + \frac{16\sigma^{\*}T\_{\infty}^{3}}{3\rho c\_{p}\Omega k^{\*}L^{2}} \frac{\partial^{2}\overline{T}}{\partial\overline{Z}^{2}} \\ &+ \frac{Dk\_{T}(\mathbf{C}\_{w} - \mathbf{C}\_{\infty})}{c\_{c}c\_{p}\Omega L^{2}(T\_{w} - T\_{\infty})} \left( \frac{\partial^{2}\overline{C}}{\partial\overline{R}^{2}} + \frac{1}{\overline{R}}\frac{\partial\overline{C}}{\partial\overline{R}} + \frac{\partial^{2}\overline{C}}{\partial\overline{Z}^{2}} \right) + \frac{\sigma B\_{0}^{2}\Omega L}{\rho c\_{p}(T\_{w} - T\_{\infty})} \frac{\overline{U}^{2}}{\overline{R}} + \frac{Q}{\rho c\_{p}\Omega} \overline{T}, \end{split} \tag{16}$$

velocity vector *V* ¼ ð Þ *u; v; w* and *B* ¼ ð Þ 0*;* 0*; B* . The Lorentz force (MHD body force)

*<sup>u</sup>=r, F<sup>θ</sup>* ¼ �*σB*<sup>2</sup>

*∂*<sup>2</sup> *u ∂ r*<sup>2</sup> þ

*u* þ *g β<sup>T</sup>* ð Þþ *T* � *T*<sup>∞</sup> *g βC*ð Þ *C* � *C*<sup>∞</sup> *,*

1 *r ∂u ∂ r* � *u r*<sup>2</sup> þ

*∂*<sup>2</sup> *v ∂ z*<sup>2</sup>

> 1 *r ∂w ∂ r* þ *∂*<sup>2</sup> *w ∂ z*<sup>2</sup>

*∂qr ∂ z* þ *D kT cscp*

*∂*2 *T ∂ r*<sup>2</sup> þ 1 *r ∂T ∂ r* þ *∂*2 *T ∂ z*<sup>2</sup>

Under these assumptions, the governing equations for the continuity, momentum, energy and concentration in laminar incompressible flow can be written as follows:

*v=r:*

*<sup>∂</sup> <sup>z</sup>* <sup>¼</sup> <sup>0</sup> *,* (1)

� *<sup>σ</sup> <sup>B</sup>*<sup>2</sup> *ρr*

> *∂*2 *C ∂ r*<sup>2</sup> þ

*<sup>∂</sup><sup>n</sup> ,* (7)

*, w* ¼ *w*0*, T* ¼ *Tw, C* ¼ *Cw,*

*∂*<sup>2</sup> *u ∂ z*<sup>2</sup>

*<sup>v</sup>* � *<sup>ν</sup> K*∗ 1

> 1 *r ∂C ∂r* þ *∂*2 *C ∂ z*<sup>2</sup>

(2)

(5)

(6)

(8)

� *k*1ð Þ *C* � *C*<sup>∞</sup>

*v,* (3)

*,* (4)

*Fr* ¼ �*σB*<sup>2</sup>

*∂u ∂ r* þ *u r* þ *∂w*

> *∂*<sup>2</sup> *v ∂ r*<sup>2</sup> þ

1 *r ∂ v ∂ r* � *v r*<sup>2</sup> þ

þ *DkT Tm*

and should be replaced with the following expression (Gad-el-Hak [20]):

*Ut* <sup>¼</sup> <sup>2</sup> � *<sup>ξ</sup> <sup>ξ</sup> <sup>λ</sup>*

Where *Ut* is the tangential velocity, *n* is the normal direction to the wall, *ξ* is the tangent momentum accommodation coefficient and *λ* is the mean free path. For *kn* < 0*:*001, the no-slip boundary condition is valid, therefore, the velocity at the surface is equal to zero. In this study the slip and the no-slip regimes of the Knudsen number that lies in the range 0*:*1>*kn*>0 is considered. By using Eq. (7), the appropriate boundary conditions for the flow induced by an infinite disk ð Þ *z* ¼ 0 which rotates with constant angular velocity Ω subjected to uniform suction *w*<sup>0</sup> through the disk are given by

> 2 � *ξ <sup>ξ</sup> <sup>λ</sup> ∂v ∂z*

When the mean free path of the fluid particle is comparable to the characteristic dimensions of the flow field domain, Naiver-Stokes equations break down since the assumption of continuum media fails. In the range of 0*:*1 < *kn* < 10 Knudsen Number, the higher order continuum equation, for example, Burnett equation should be used. For the range of 0*:*001 ≤*kn* ≤0*:*1, no slip boundary conditions cannot be used

*∂Ut*

*∂*<sup>2</sup> *w ∂ r*<sup>2</sup> þ

� 1 *ρcp*

has two components:

*Nanofluid Flow in Porous Media*

*u ∂u ∂ r* � *v*2 *r* þ *w ∂u <sup>∂</sup> <sup>z</sup>* ¼ � <sup>1</sup> *ρ ∂p ∂ r* þ *ν*

*u ∂ v ∂ r* þ *uv r* þ *w ∂ v <sup>∂</sup> <sup>z</sup>* <sup>¼</sup> *<sup>ν</sup>*

<sup>þ</sup> *<sup>σ</sup>B*<sup>2</sup> *ρcpr*

*<sup>z</sup>* <sup>¼</sup> <sup>0</sup> : *<sup>u</sup>* <sup>¼</sup> <sup>2</sup> � *<sup>ξ</sup>*

**104**

*<sup>ξ</sup> <sup>λ</sup>*

*∂u ∂z*

*, v* ¼ Ω*r* þ

*z* ! ∞ : *u* ! 0*, v* ! 0*, T* ! *T*∞*, C* ! *C*∞*, p* ! *p*∞*:*

*u ∂T ∂ r* þ *w ∂T <sup>∂</sup> <sup>z</sup>* <sup>¼</sup> *<sup>k</sup> ρcp*

*u ∂C ∂ r* þ *w ∂C <sup>∂</sup> <sup>z</sup>* <sup>¼</sup> *<sup>D</sup> <sup>∂</sup>*<sup>2</sup>

� *<sup>σ</sup> <sup>B</sup>*<sup>2</sup> *ρr*

> *∂*2 *T ∂ r*<sup>2</sup> þ

*C ∂ r*<sup>2</sup> þ 1 *r ∂T ∂r* þ *∂*2 *T ∂ z*<sup>2</sup>

ð Þ *T* � *T*<sup>∞</sup> *,*

1 *r ∂C ∂ r* þ *∂*2 *C ∂ z*<sup>2</sup>

*u ∂w ∂ r* þ *w ∂w <sup>∂</sup> <sup>z</sup>* ¼ � <sup>1</sup> *ρ ∂p ∂z* þ *ν*

*<sup>u</sup>*<sup>2</sup> <sup>þ</sup> *<sup>Q</sup> ρcp* *<sup>u</sup>* � *<sup>ν</sup> K*∗ 1

$$\begin{split} \overline{U}\frac{\partial\overline{C}}{\partial\overline{R}} + \overline{W}\frac{\partial\overline{C}}{\partial\overline{Z}} &= \frac{D}{\Omega L^{2}} \left( \frac{\partial^{2}\overline{C}}{\partial\overline{R}^{2}} + \frac{1}{\overline{R}}\frac{\partial\overline{C}}{\partial\overline{R}} + \frac{\partial^{2}\overline{C}}{\partial\overline{Z}^{2}} \right) \\ &+ \frac{Dk\_{T}(T\_{w} - T\_{\infty})}{T\_{m}\mathfrak{Q}L^{2}(\mathbb{C}\_{w} - \mathbb{C}\_{\infty})} \left( \frac{\partial^{2}\overline{T}}{\partial\overline{R}^{2}} + \frac{1}{\overline{R}}\frac{\partial\overline{T}}{\partial\overline{R}} + \frac{\partial^{2}\overline{T}}{\partial\overline{Z}^{2}} \right) - \frac{\overline{k}}{\Omega L^{2}}\overline{C}. \end{split} \tag{17}$$

The boundary conditions (8) are reduced to

$$\begin{aligned} \overline{Z} &= 0: \qquad \overline{U} = \frac{2 - \xi}{\xi} \overline{\lambda} \frac{\partial \overline{U}}{\partial \overline{Z}}, & \overline{V} = \overline{R} + \frac{2 - \xi}{\xi} \overline{\lambda} \frac{\partial \overline{V}}{\partial \overline{Z}}, & \overline{W} = \frac{w\_0}{\Omega L}, & \overline{T} = 1, & \overline{C} = 1, \\\overline{Z} &\to \infty: \quad \overline{U} \to 0, \overline{V} \to 0, & \overline{T} \to 0, & \overline{C} \to 0, & \overline{P} \to 0. \end{aligned} \tag{18}$$

The governing equations are obtained by introducing a dimensionless normal distance from the disk, *<sup>η</sup>* <sup>¼</sup> *<sup>Z</sup><sup>=</sup>* ffiffi *<sup>ν</sup>* <sup>p</sup> along with the von-Karman transformations,

$$\begin{aligned} \overline{U} &= \overline{R}F(\eta), \quad \overline{V} = \overline{R}G(\eta), \quad \overline{W} = \sqrt{\overline{\nu}}H(\eta). \\ \overline{T} &= \theta(\eta), \quad \overline{C} = \rho(\eta), \quad \overline{P} = \overline{\nu}\overline{P}(\eta) \end{aligned} \tag{19}$$

Where *F, G, H, θ, φ* and *P* are non-dimensionless functions in terms of vertical co-ordinate *η*. Substituting these transformations into Eqs. (12)–(17) gives the nonlinear ordinary differential equations, expressed as

$$2F + H^{'} = \mathbf{0},\tag{20}$$

*τφ<sup>z</sup>* ¼ *μ*

*τzr* ¼ *μ*

*Cf* <sup>1</sup> <sup>¼</sup> *τφz=ρ*Ω<sup>2</sup>

*Cf* <sup>2</sup> <sup>¼</sup> *<sup>τ</sup>zr=ρ*Ω<sup>2</sup>

*∂ z* � �

*Nu* <sup>¼</sup> *L qw*

*Sh* <sup>¼</sup> *LMw*

*=ν* is the rotational Reynolds number.

results with those of Frusteri and Osalusi [15], Osalusi and Sibanda [23] and

for various values of *Ws*. The comparisons in all above cases are found to be

**Figures 2**, **3** and **11a–d** display the velocity (radial, axial and tangential), temperature, concentration and pressure profiles under the magnetic field parameter, porosity parameter and Schmidt number. The (radial, axial and tangential) components of the velocity and pressure profile decrease with increase of magnetic field due to the inhibiting influence of the Lorentz force and increasing of all porosity parameter and Schmidt number, while the temperature and the concentration profiles increase with increasing of all magnetic field parameter, porosity parameter and Schmidt number. In **Figures 4** and **5a–d**, it is clear that the (radial and axial)

Maleque and Sattar [24]. **Table 1** shows the values of *F<sup>=</sup>*

*z*¼0

Hence the Nusselt number, *Nu* and Sherwood number, *Sh* are obtained as

*k T*ð Þ *<sup>w</sup>* � *<sup>T</sup>*<sup>∞</sup> ð Þ*<sup>ν</sup>* �1*=*<sup>2</sup> ¼ �*θ<sup>=</sup>*

*D C*ð Þ *<sup>w</sup>* � *<sup>C</sup>*<sup>∞</sup> ð Þ*<sup>ν</sup>* �1*=*<sup>2</sup> ¼ �*φ<sup>=</sup>*

The system of non-linear ordinary differential Eqs. (20)–(25) together with the boundary conditions (26) are locally similar and solved numerically by using the Keller-box method. In order to gain physical insight, the velocity (radial, axial and tangential), temperature, concentration and pressure profiles have been discussed by assigning numerical values to the parameter, encountered in the problem which the numerical results are tabulated and displayed with the graphical illustrations. In order to verify the accuracy of our present method, we have compared our

*qw* ¼ �*<sup>k</sup> <sup>∂</sup><sup>T</sup>*

Eq. (27):

And

**107**

Where *Re* <sup>¼</sup> *<sup>R</sup>*<sup>2</sup>

**4. Numerical results and discussion**

excellent and agreed, so it is good.

Now the heat flux *qw*

*DOI: http://dx.doi.org/10.5772/intechopen.82186*

Eqs. (28) and (29), respectively:

*∂ v ∂ z* þ 1 *r ∂w ∂φ*

*Thermal Radiation and Thermal Diffusion for Soret and Dufour's Effects on MHD Flow over…*

*∂u ∂ z* þ *∂w ∂ r* � �

The tangential and radial skin-friction coefficient are, respectively, given by,

*<sup>L</sup>*<sup>2</sup> <sup>¼</sup> ffiffiffiffiffi *Re* <sup>p</sup> *<sup>G</sup><sup>=</sup>*

*<sup>L</sup>*<sup>2</sup> <sup>¼</sup> ffiffiffiffiffi *Re* <sup>p</sup> *<sup>F</sup><sup>=</sup>*

� �

*z*¼0 *,*

*z*¼0 *,*

� � and the mass flux ð Þ *Jw* at the surface are given by.

*,* and *Mw* ¼ �*<sup>D</sup> <sup>∂</sup><sup>C</sup>*

ð Þ 0 *,*

*∂ z* � �

*z*¼0 *:*

ð Þ <sup>0</sup> *,* � *<sup>G</sup><sup>=</sup>*

ð Þ <sup>0</sup> and � *<sup>θ</sup><sup>=</sup>*

ð Þ 0

ð Þ <sup>0</sup> *,* (27)

ð Þ 0 *,* (28)

ð Þ 0 *:* (29)

$$\mathbf{F}^{\prime\prime} - \mathbf{H}\mathbf{F}^{\prime} - \mathbf{F}^2 + \mathbf{G}^2 - (\mathbf{M} + \mathbf{S})\mathbf{F} + a\theta + N\rho = \mathbf{0},\tag{21}$$

$$\rm{G}^{\prime\prime} - \rm{HG}^{\prime} - 2FG - (M+S)G = 0,\tag{22}$$

$$-\mathbf{H}^{\prime \prime} - \mathbf{H} \mathbf{H}^{\prime} - \mathbf{P}^{\prime} = \mathbf{0},\tag{23}$$

$$\frac{1}{P\_r} \left( 1 + \frac{4}{3R\_d} \right) \theta^{\prime \prime} - H \theta^{\prime} + D\_u \rho^{\prime \prime} + f \mathbf{F}^2 + \delta \theta = \mathbf{0},\tag{24}$$

$$\frac{1}{S\_{\varepsilon}}\boldsymbol{\varrho}^{\prime/} - \boldsymbol{H}\boldsymbol{\varrho}^{\prime} + \mathbb{S}\_{0}\boldsymbol{\theta}^{\prime/} - \boldsymbol{\beta}\boldsymbol{\varrho} = \mathbf{0}.\tag{25}$$

With the appropriate boundary conditions:

$$\begin{aligned} \eta = \mathbf{0}; F(\mathbf{0}) = \gamma F'(\mathbf{0}), &\; G(\mathbf{0}) = \mathbf{1} + \gamma G'(\mathbf{0}), H(\mathbf{0}) = W\_{\infty} &\; \boldsymbol{\theta}(\mathbf{0}) = \mathbf{1}, &\; \boldsymbol{\rho}(\mathbf{0}) = \mathbf{1}, \\ \eta \to \infty; F(\infty) = \mathbf{0}, G(\infty) = \mathbf{0}, &\; \boldsymbol{\theta}(\infty) = \mathbf{0}, &\; \boldsymbol{\rho}(\infty) = \mathbf{0}, \overline{P}(\infty) = \mathbf{0}. \end{aligned} \tag{26}$$

Where *<sup>γ</sup>* <sup>¼</sup> ð Þ <sup>2</sup> � *<sup>ξ</sup> <sup>λ</sup>* ffiffiffiffi <sup>Ω</sup> � � <sup>p</sup> *<sup>=</sup><sup>ξ</sup>* ffiffi *<sup>ν</sup>* <sup>p</sup> the slip is factor and *Ws* <sup>¼</sup> *<sup>w</sup>*0*<sup>=</sup>* ffiffiffiffiffiffiffi *<sup>ν</sup>*<sup>Ω</sup> <sup>p</sup> represents a uniform suction ð Þ *Ws* < 0 at the disk surface. The boundary conditions given by Eq. (26) imply that the radial ð Þ *F* , the tangential ð Þ *G* components of velocity, temperature and concentration vanish sufficiently far away from the disk, whereas the axial velocity component ð Þ *H* is anticipated to approach a yet unknown asymptotic limit for sufficiently large *η*-values.

#### **3. Skin-friction coefficient, Nusselt number and Sherwood number**

The parameters of engineering interest for the present problem are the local skin-friction coefficients and the local rates of heat and mass transfer to the surface are calculated. The radial shear stress and tangential shear stress are given by:

*Thermal Radiation and Thermal Diffusion for Soret and Dufour's Effects on MHD Flow over… DOI: http://dx.doi.org/10.5772/intechopen.82186*

$$\begin{aligned} \boldsymbol{\tau}\_{\mathrm{qz}} &= \mu \left( \frac{\partial \boldsymbol{v}}{\partial z} + \frac{1}{r} \frac{\partial \boldsymbol{w}}{\partial \boldsymbol{\rho}} \right)\_{z=0}, \\ \boldsymbol{\tau}\_{\mathrm{zr}} &= \mu \left( \frac{\partial \boldsymbol{u}}{\partial z} + \frac{\partial \boldsymbol{w}}{\partial r} \right)\_{z=0}. \end{aligned}$$

The tangential and radial skin-friction coefficient are, respectively, given by, Eq. (27):

$$\begin{aligned} \mathbf{C}\_{\mathbf{f}\_1} &= \boldsymbol{\tau}\_{q\mathbf{z}} / \rho \, \boldsymbol{\Omega}^2 L^2 = \sqrt{R\_\epsilon} \, \mathbf{G}^{\!/} (\mathbf{0}),\\ \mathbf{C}\_{\mathbf{f}\_2} &= \boldsymbol{\tau}\_{xr} / \rho \, \boldsymbol{\Omega}^2 L^2 = \sqrt{R\_\epsilon} F^{\!/} (\mathbf{0}), \end{aligned} \tag{27}$$

Now the heat flux *qw* � � and the mass flux ð Þ *Jw* at the surface are given by.

$$q\_w = -k \left(\frac{\partial T}{\partial z}\right)\_{z=0}, \text{and } M\_w = -D \left(\frac{\partial C}{\partial z}\right)\_{z=0}.$$

Hence the Nusselt number, *Nu* and Sherwood number, *Sh* are obtained as Eqs. (28) and (29), respectively:

$$N\_u = \frac{Lq\_w}{k \left(T\_w - T\_\infty\right) \left(\overline{\nu}\right)^{-1/2}} = -\theta^\prime(\mathbf{0}),\tag{28}$$

And

*<sup>U</sup> <sup>∂</sup><sup>C</sup>*

*Nanofluid Flow in Porous Media*

*<sup>∂</sup><sup>R</sup>* <sup>þ</sup> *<sup>W</sup> <sup>∂</sup><sup>C</sup>*

þ

*<sup>Z</sup>* <sup>¼</sup> <sup>0</sup> : *<sup>U</sup>* <sup>¼</sup> <sup>2</sup> � *<sup>ξ</sup>*

distance from the disk, *<sup>η</sup>* <sup>¼</sup> *<sup>Z</sup><sup>=</sup>* ffiffi

1 *Pr*

*<sup>η</sup>* <sup>¼</sup> <sup>0</sup>*; F*ð Þ¼ <sup>0</sup> *<sup>γ</sup>F<sup>=</sup>*

**106**

Where *<sup>γ</sup>* <sup>¼</sup> ð Þ <sup>2</sup> � *<sup>ξ</sup> <sup>λ</sup>* ffiffiffiffi

1 þ 4 3*Rd* � �

> 1 *Sc*

ð Þ <sup>0</sup> *, G*ð Þ¼ <sup>0</sup> <sup>1</sup> <sup>þ</sup> *<sup>γ</sup>G<sup>=</sup>*

With the appropriate boundary conditions:

<sup>Ω</sup> � � <sup>p</sup> *<sup>=</sup><sup>ξ</sup>* ffiffi

totic limit for sufficiently large *η*-values.

*Tm*Ω*L*<sup>2</sup>

*<sup>∂</sup><sup>Z</sup>* <sup>¼</sup> *<sup>D</sup>* Ω*L*<sup>2</sup>

*D kT*ð Þ *Tw* � *T*<sup>∞</sup>

The boundary conditions (8) are reduced to

*∂U*

nonlinear ordinary differential equations, expressed as

*<sup>ξ</sup> <sup>λ</sup>*

ð Þ *Cw* � *C*<sup>∞</sup>

*∂*<sup>2</sup>*C <sup>∂</sup>R*<sup>2</sup> <sup>þ</sup>

*<sup>∂</sup><sup>Z</sup> , <sup>V</sup>* <sup>¼</sup> *<sup>R</sup>* <sup>þ</sup>

*Z* ! ∞ : *U* ! 0*,V* ! 0*, T* ! 0*, C* ! 0*, P* ! 0*:*

1 *R ∂C <sup>∂</sup><sup>R</sup>* <sup>þ</sup>

*∂*<sup>2</sup> *T <sup>∂</sup>R*<sup>2</sup> <sup>þ</sup>

� �

1 *R ∂T <sup>∂</sup><sup>R</sup>* <sup>þ</sup>

2 � *ξ <sup>ξ</sup> <sup>λ</sup>*

The governing equations are obtained by introducing a dimensionless normal

Where *F, G, H, θ, φ* and *P* are non-dimensionless functions in terms of vertical co-ordinate *η*. Substituting these transformations into Eqs. (12)–(17) gives the

*<sup>η</sup>* ! <sup>∞</sup>*; F*ð Þ¼ <sup>∞</sup> <sup>0</sup>*, G*ð Þ¼ <sup>∞</sup> <sup>0</sup>*, <sup>θ</sup>* ð Þ¼ <sup>∞</sup> <sup>0</sup>*, <sup>φ</sup>*ð Þ¼ <sup>∞</sup> <sup>0</sup>*, <sup>P</sup>*ð Þ¼ <sup>∞</sup> <sup>0</sup>*:* (26)

uniform suction ð Þ *Ws* < 0 at the disk surface. The boundary conditions given by Eq. (26) imply that the radial ð Þ *F* , the tangential ð Þ *G* components of velocity, temperature and concentration vanish sufficiently far away from the disk, whereas the axial velocity component ð Þ *H* is anticipated to approach a yet unknown asymp-

**3. Skin-friction coefficient, Nusselt number and Sherwood number**

The parameters of engineering interest for the present problem are the local skin-friction coefficients and the local rates of heat and mass transfer to the surface are calculated. The radial shear stress and tangential shear stress are given by:

*<sup>F</sup>==* � *HF<sup>=</sup>* � *<sup>F</sup>*<sup>2</sup> <sup>þ</sup> *<sup>G</sup>*<sup>2</sup> � ð Þ *<sup>M</sup>* <sup>þ</sup> *<sup>S</sup> <sup>F</sup>* <sup>þ</sup> *αθ* <sup>þ</sup> *<sup>N</sup><sup>φ</sup>* <sup>¼</sup> <sup>0</sup>*,* (21)

*<sup>G</sup>==* � *HG<sup>=</sup>* � <sup>2</sup>*FG* � ð Þ *<sup>M</sup>* <sup>þ</sup> *<sup>S</sup> <sup>G</sup>* <sup>¼</sup> <sup>0</sup>*,* (22)

*<sup>H</sup>==* � *HH<sup>=</sup>* � *<sup>P</sup><sup>=</sup>* <sup>¼</sup> <sup>0</sup>*,* (23)

*<sup>θ</sup>==* � *<sup>H</sup>θ<sup>=</sup>* <sup>þ</sup> *Duφ==* <sup>þ</sup> *JF*<sup>2</sup> <sup>þ</sup> *δ θ* <sup>¼</sup> <sup>0</sup>*,* (24)

*<sup>φ</sup>==* � *<sup>H</sup>φ<sup>=</sup>* <sup>þ</sup> *<sup>S</sup>*0*θ==* � *β φ* <sup>¼</sup> <sup>0</sup>*:* (25)

ð Þ 0 *, H*ð Þ¼ 0 *Ws, θ* ð Þ¼ 0 1*, φ*ð Þ¼ 0 1*,*

*<sup>ν</sup>* <sup>p</sup> the slip is factor and *Ws* <sup>¼</sup> *<sup>w</sup>*0*<sup>=</sup>* ffiffiffiffiffiffiffi

*<sup>U</sup>* <sup>¼</sup> *RF*ð Þ*<sup>η</sup> , <sup>V</sup>* <sup>¼</sup> *RG*ð Þ*<sup>η</sup> , <sup>W</sup>* <sup>¼</sup> ffiffi

*T* ¼ *θ η*ð Þ*, C* ¼ *φ η*ð Þ*, P* ¼ *νP*ð Þ*η*

� �

*∂*<sup>2</sup>*C ∂Z*<sup>2</sup>

*∂V*

*∂*<sup>2</sup> *T ∂Z*<sup>2</sup>

*<sup>∂</sup><sup>Z</sup> , <sup>W</sup>* <sup>¼</sup> *<sup>w</sup>*<sup>0</sup>

*<sup>ν</sup>* <sup>p</sup> along with the von-Karman transformations,

*<sup>ν</sup>* <sup>p</sup> *<sup>H</sup>*ð Þ*<sup>η</sup> :*

<sup>2</sup>*<sup>F</sup>* <sup>þ</sup> *<sup>H</sup><sup>=</sup>* <sup>¼</sup> <sup>0</sup>*,* (20)

� *<sup>k</sup>* <sup>Ω</sup>*L*<sup>2</sup> *<sup>C</sup>:*

<sup>Ω</sup>*<sup>L</sup> , <sup>T</sup>* <sup>¼</sup> <sup>1</sup>*, <sup>C</sup>* <sup>¼</sup> <sup>1</sup>*,*

(17)

(18)

(19)

*<sup>ν</sup>*<sup>Ω</sup> <sup>p</sup> represents a

$$\mathbf{S}\_{h} = \frac{LM\_{w}}{D\left(\mathbf{C}\_{w} - \mathbf{C}\_{\infty}\right) \left(\overline{\nu}\right)^{-1/2}} = -\rho^{/}\left(\mathbf{0}\right). \tag{29}$$

Where *Re* <sup>¼</sup> *<sup>R</sup>*<sup>2</sup> *=ν* is the rotational Reynolds number.

#### **4. Numerical results and discussion**

The system of non-linear ordinary differential Eqs. (20)–(25) together with the boundary conditions (26) are locally similar and solved numerically by using the Keller-box method. In order to gain physical insight, the velocity (radial, axial and tangential), temperature, concentration and pressure profiles have been discussed by assigning numerical values to the parameter, encountered in the problem which the numerical results are tabulated and displayed with the graphical illustrations.

In order to verify the accuracy of our present method, we have compared our results with those of Frusteri and Osalusi [15], Osalusi and Sibanda [23] and Maleque and Sattar [24]. **Table 1** shows the values of *F<sup>=</sup>* ð Þ <sup>0</sup> *,* � *<sup>G</sup><sup>=</sup>* ð Þ <sup>0</sup> and � *<sup>θ</sup><sup>=</sup>* ð Þ 0 for various values of *Ws*. The comparisons in all above cases are found to be excellent and agreed, so it is good.

**Figures 2**, **3** and **11a–d** display the velocity (radial, axial and tangential), temperature, concentration and pressure profiles under the magnetic field parameter, porosity parameter and Schmidt number. The (radial, axial and tangential) components of the velocity and pressure profile decrease with increase of magnetic field due to the inhibiting influence of the Lorentz force and increasing of all porosity parameter and Schmidt number, while the temperature and the concentration profiles increase with increasing of all magnetic field parameter, porosity parameter and Schmidt number. In **Figures 4** and **5a–d**, it is clear that the (radial and axial)


**Table 1.** *Comparison of the current and recent numerical values of the radial and tangential skin-friction coefficients and the rate of heat transfer coefficient for various of Ws withS*¼0*:*0,*α*¼0*:*0,*N*¼0*:*0,*J*¼0*:*0,*δ*¼0*:*0,*Pr*¼0*:*71,*β*¼0*:*0,*Rd*¼109,*Du*¼0*:*0,*S*0¼0*:*0,*Sc*¼1*:*0and*M*¼0*.*

**Figure 2.**

**Figure 3.**

**109**

*Effect of magnetic field parameter on (a) the velocity (radial, axial and tangential) profile, (b) the*

*Thermal Radiation and Thermal Diffusion for Soret and Dufour's Effects on MHD Flow over…*

*DOI: http://dx.doi.org/10.5772/intechopen.82186*

*Effect of porosity parameter on (a) the velocity (radial, axial and tangential) profile, (b) the temperature*

*profile, (c) the concentration and (d) the pressure profile.*

*temperature profile, (c) the concentration and (d) the pressure profile.*
