*Nanofluid Flow in Porous Media*

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

**Figure 2.**

*Effect of magnetic field parameter on (a) the velocity (radial, axial and tangential) profile, (b) the temperature profile, (c) the concentration and (d) the pressure profile.*

**Figure 3.**

*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.*

*Ws*

**108**

*F|***(0)** **Frusteri**

**Osalusi et al.**

**Maleque et al.**

**Present**

**Frusteri**

**Osalusi et al.**

**Maleque et al.**

**Present**

**Frusteri**

**Osalusi et al.**

**Maleque**

**Present**

**study**

**[23]**

**et al. [24]**

**[23]**

**[24]**

**study**

**et al. [15]**

**[23]**

**[24]**

**study**

**et al. [15]**

**et al. [15]**

*γ* = 0, *ε* = 0

0.0

�2.0

�4.0

�5.0

�10.0

**Table 1.** *Comparison*

*S* ¼ 0*:*0, *α* ¼ 0*:*0, *N*

 *of the current and recent numerical values of the radial and tangential skin-friction*

¼ 0*:*0, *J* ¼ 0*:*0, *δ* ¼ 0*:*0, *Pr* ¼ 0*:*71, *β* ¼ 0*:*0, *Rd* ¼ 109, *Du* ¼ 0*:*0, *S*0 ¼ 0*:*0, *Sc* ¼ 1*:*0 and

 0.0499

 0.0499

 0.0506

 0.0502

 10.0003

 10.0003

 10.0017

 *coefficients*

 *and the rate of heat transfer coefficient for various of*

*M*

¼ 0*.*

 10.0019

 7.1002

 7.1002

 7.1020 *Ws with*

 7.1018

 0.0999

 0.0999

 0.0999

 0.0998

 5.0029

 5.0029

 5.0031

 5.0032

 3.5541

 3.5541

 3.5541

 3.5539

 0.1246

 0.1246

 0.1248

 0.1245

 4.0065

 4.0065

 4.0054

 4.0052

 2.8520

 2.8520

 2.8447

 2.8432

 0.2324

 0.2324

 0.2425

 0.2435

 2.0687

 2.0687

 2.0391

 2.0452

 1.5196

 1.5196

 1.4421

 1.4746

 0.4241

 0.4241

 0.5102

 0.4627

 0.6514

 0.6514

 0.6160

 0.6241

 0.5387

 0.5387

 0.3258

 0.3171

*Nanofluid Flow in Porous Media*

*–G|***(0)**

*–θ|***(0)**

**Figure 4.**

*Effect of temperature buoyancy parameter on (a) the velocity (radial, axial and tangential) profile, (b) the temperature profile, (c) the concentration and (d) the pressure profile.*

components of the velocity and pressure profiles increase with increase of temperature buoyancy parameter and increasing concentration buoyancy parameter, while the tangential component of the velocity, the temperature and the concentration profiles decrease with increasing temperature buoyancy parameter and concentra-

*Effect of Prandtl number on (a) the velocity (radial and axial) profile, (b) the temperature profile, (c) the*

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

The effects of *Pr* on the (radial and axial) components of the velocity, the temperature, concentration and pressure profiles are shown in **Figure 6a–d**, respectively. It is observed that both the (radial and axial) components of the velocity, the temperature and pressure profiles decrease with the increase of Prandtl number. While, the concentration profile increase. Physically, it means that thermal boundary layer thickness gets decreased. In fact, it is well known that the thermal boundary layer thickness is inversely proportional to the square root of Prandtl number. Hence, the decrease of temperature profile with increasing *Pr* is straight-

**Figures 7** and **10a–d** show the effects of radiation parameter and heat source parameter on the velocity (radial and axial), temperature, concentration and pressure profiles, also, we found that the (radial and axial) components of the velocity, temperature and pressure profiles increase with the increase of radiation parameter and increasing heat source parameter. While the concentration profile decrease. The effects of *Du* and *S*<sup>0</sup> on the velocity (radial and axial), temperature, concentration and pressure profiles, are shown in **Figure 8a–d**, respectively. It is observed that the (radial and axial) components of the velocity, temperature, concentration and pressure profiles increase with the decreasing the Dufour's number and

tion buoyancy parameter.

*concentration and (d) the pressure profile.*

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

increasing Soret number.

forward.

**111**

**Figure 6.**

#### **Figure 5.**

*Effect of concentration buoyancy parameter on (a) the velocity (radial, axial and tangential) profile, (b) the temperature profile, (c) the concentration and (d) the pressure profile.*

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

**Figure 6.** *Effect of Prandtl number on (a) the velocity (radial and axial) profile, (b) the temperature profile, (c) the concentration and (d) the pressure profile.*

components of the velocity and pressure profiles increase with increase of temperature buoyancy parameter and increasing concentration buoyancy parameter, while the tangential component of the velocity, the temperature and the concentration profiles decrease with increasing temperature buoyancy parameter and concentration buoyancy parameter.

The effects of *Pr* on the (radial and axial) components of the velocity, the temperature, concentration and pressure profiles are shown in **Figure 6a–d**, respectively. It is observed that both the (radial and axial) components of the velocity, the temperature and pressure profiles decrease with the increase of Prandtl number. While, the concentration profile increase. Physically, it means that thermal boundary layer thickness gets decreased. In fact, it is well known that the thermal boundary layer thickness is inversely proportional to the square root of Prandtl number. Hence, the decrease of temperature profile with increasing *Pr* is straightforward.

**Figures 7** and **10a–d** show the effects of radiation parameter and heat source parameter on the velocity (radial and axial), temperature, concentration and pressure profiles, also, we found that the (radial and axial) components of the velocity, temperature and pressure profiles increase with the increase of radiation parameter and increasing heat source parameter. While the concentration profile decrease. The effects of *Du* and *S*<sup>0</sup> on the velocity (radial and axial), temperature, concentration and pressure profiles, are shown in **Figure 8a–d**, respectively. It is observed that the (radial and axial) components of the velocity, temperature, concentration and pressure profiles increase with the decreasing the Dufour's number and increasing Soret number.

**Figure 4.**

*Nanofluid Flow in Porous Media*

**Figure 5.**

**110**

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

*Effect of concentration buoyancy 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.*

The effects of *J* on the velocity (radial and axial) and pressure profiles, are shown in **Figure 9a** and **b**, respectively. We found that the (radial and axial) components of the velocity and pressure profiles decrease with the increase Joule heating parameter. In **Figure 12a–c**, it is clear that the (radial and axial) components of the velocity, concentration and pressure profiles decrease with increase of chemical reaction parameter. **Figure 13a–d** displays the velocity (radial, axial and tangential), temperature, concentration and pressure profiles under the suction

*Effect of heat source parameter on (a) the velocity (radial and axial) profile, (b) the temperature profile, (c)*

*Effect of Joule heating parameter on (a) the velocity (radial and axial) profile, and (b) the pressure profile.*

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

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

**Figure 9.**

**Figure 10.**

**113**

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

#### **Figure 7.**

*Effect of radiation parameter on (a) the velocity (radial and axial) profile, (b) the temperature profile, (c) the concentration and (d) the pressure profile.*

#### **Figure 8.**

*Effect of Soret and Dufour's number on (a) the velocity (radial and axial) profile, (b) the temperature profile, (c) the concentration and (d) the pressure profile.*

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

**Figure 9.** *Effect of Joule heating parameter on (a) the velocity (radial and axial) profile, and (b) the pressure profile.*

The effects of *J* on the velocity (radial and axial) and pressure profiles, are shown in **Figure 9a** and **b**, respectively. We found that the (radial and axial) components of the velocity and pressure profiles decrease with the increase Joule heating parameter. In **Figure 12a–c**, it is clear that the (radial and axial) components of the velocity, concentration and pressure profiles decrease with increase of chemical reaction parameter. **Figure 13a–d** displays the velocity (radial, axial and tangential), temperature, concentration and pressure profiles under the suction

#### **Figure 10.**

*Effect of heat source parameter on (a) the velocity (radial and axial) profile, (b) the temperature profile, (c) the concentration and (d) the pressure profile.*

**Figure 7.**

**Figure 8.**

**112**

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

*concentration and (d) the pressure profile.*

*Nanofluid Flow in Porous Media*

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

*Effect of Soret and Dufour's number on (a) the velocity (radial and axial) profile, (b) the temperature profile,*

**Figure 11.**

*Effect of Schmidt number on (a) the velocity (radial, axial and tangential) profile, (b) the temperature profile, (c) the concentration and (d) the pressure profile.*

**Figure 13.**

**Figure 14.**

**115**

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

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

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

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

*Effect of slip parameter on (a) the velocity (radial, axial and tangential) profile, and (b) the pressure profile.*

**Figure 12.** *Effect of chemical reaction parameter on (a) the velocity (radial and axial) profile, (b) the concentration and (c) the pressure profile.*

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

**Figure 13.**

**Figure 11.**

**Figure 12.**

**114**

*(c) the pressure profile.*

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

*Nanofluid Flow in Porous Media*

*Effect of Schmidt number on (a) the velocity (radial, axial and tangential) profile, (b) the temperature profile, (c)*

*Effect of chemical reaction parameter on (a) the velocity (radial and axial) profile, (b) the concentration and*

*Effect of suction parameter on (a) the velocity (radial, axial and tangential) profile, (b) the temperature profile, (c) the concentration and (d) the pressure profile.*

**Figure 14.** *Effect of slip parameter on (a) the velocity (radial, axial and tangential) profile, and (b) the pressure profile.*


*M*

**117**

0.1

0.1

0.1

0.1

0.1

**Table 2.** *Numerical*

 *of the values skin-friction*

 *coefficient*

*Re*1*l*2*Cf* 1*;*

*Re*1*l*2*Cf* 2

 *, Nusselt number Nu and Sherwood number Sh at the surface with M, S, α, N, Pr, Rd, Du*&*S*0*, J, δ, Sc, β, Wsand γ.*

 2.0

 0.5

 0.5

 1.0

 5.0

 0.25 & 0.2

 0.5

 0.0

 0.64

 0.5

1.0

 0.2 0.5

0.8

 0.1757

 0.8264

 0.3162

 1.0613

*'s Effects on MHD Flow over*

*…*

 0.2403

 1.0968

 0.3156

 1.0573

 0.3857

 1.6257

 0.3144

 1.0486

 2.0

 0.5

 0.5

 1.0

 5.0

 0.25 & 0.2

 0.5

 0.0

 0.64

 0.5

1.0 1.5 2.0

0.3590

 1.9095

 0.3764

 1.5404

0.3737

 1.7684

 0.3445

 1.2854

0.3857

 1.6257

 0.3144

 1.0486

 2.0

 0.5

 0.5

 1.0

 5.0

 0.25 & 0.2

 0.5

 0.0

 0.64

 0.2 0.5 1.0

 2.0

 0.5

 0.5

 1.0

 5.0

 0.25 & 0.2

 0.5

 0.0

 0.22 0.64 0.78

 2.0

 0.5

 0.5

 1.0

 5.0

 0.25 & 0.2

 0.5

0.5

0.0 0.5

 *S*

*α*

*N*

 *Pr*

*Rd*

*Du*&*S***0**

*J* 0.5 1.5

*δ*

*Sc*

*β*

*Ws*

*γ*

*Re***1***l***2***Cf***1**

0.3857 0.3854 0.3795

0.3857

0.3931

0.6083

0.3857

0.3777 0.3927

0.3857

0.3770

 1.6244

 0.3075

 1.2368

 1.6257

 0.3144

 1.0486

 1.6268

 0.3193

 0.9130

1.6278

 0.3084

 1.2094

 1.6257

 0.3144

 1.0486

*Thermal Radiation and Thermal Diffusion for Soret and Dufour*

 1.4121

 0.3451

 0.5590

 1.6270

 0.2239

 1.0625

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

 1.6257

 0.3144

 1.0486

 1.6247

 0.3938

 1.0370

*Re***1***l***2***Cf***2**

*Nu* 0.3144

0.3111

 1.0485

 1.0486

*Sh*


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

> **Table**  *Numerical*

 *of the values skin-friction*

 *coefficient*

*Re*

*Cf* 1*;* 

*Re*

*Cf* 2

*Nu and Sherwood number Sh at the surface with*

*M, S, α, N, Pr, Rd, Du*&*S*0*, J, δ, Sc, β, Wsand γ.*

1*l*2

 *, Nusselt number*

1*l*2

*M*

**116**

0.1

2.0 3.0 0.1

 0.2 1.0 2.0

0.1

 2.0

 0.0 0.5 0.8

0.1

 2.0

 0.5

 0.0 0.5 0.9

0.1

 2.0

 0.5

 0.5

 0.71

1.0 2.3

0.1

0.1

0.1

 2.0

 0.5

 0.5

 1.0

 5.0

 0.25 & 0.2

 0.1

 2.0

 0.5

 0.5

 1.0

 5.0

 1.0 & 0.05

0.25 & 0.2 0.2 & 0.25

 2.0

 0.5

 0.5

 1.0

 1.0

5.0 10.0

 2.0

 0.5

 0.5

 1.0

 5.0

 0.25 & 0.2

 0.5

 0.0

 0.64

 0.5

1.0

 0.2

 0.3857 0.3323 0.3112 0.5534

0.4629

0.3857

0.1970

0.3857

0.4953

0.2510

0.3857

0.4894

0.3879

0.3857

0.3757

0.3683

0.3857

0.3892

0.3910

0.3857

0.3856

0.3858

 1.6256

 0.3195 0.3157

 1.0487

 1.0454

 1.6257

 0.3144

 1.0486

 1.6266

 0.2324

 1.0591

 1.6263

 0.2828

 1.0541

 1.6257

 0.3144

 1.0486

 1.6229

 0.5022

 1.0194

 1.6241

 0.4155

 1.0326

 1.6257

 0.3144

 1.0486

 1.6261

 0.2944

 1.0522

 1.6377

 0.3180

 1.0667

 1.6257

 0.3144

 1.0486

 1.6096

 0.3093

 1.0247

 1.6397

 0.3198

 1.0725

 1.6257

 0.3144

 1.0486

1.6005

 0.3039

 1.0056

 1.6257

 0.3144

 1.0486

 1.5071

 0.3185

 1.0663

 1.3978

 0.3220

 1.0870

1.7732

 0.3105

 1.0319

*Nanofluid Flow in Porous Media*

1.7275

 0.3116

 1.0367

 1.6257

 0.3144

 1.0486

 *S*

*α*

*N*

 *Pr*

*Rd*

*Du*&*S***0**

*J*

*δ*

*Sc*

*β*

*Ws*

*γ*

*Re*

**1***l***2***Cf***1**

*Re*

**1***l***2***Cf***2**

*Nu*

*Sh*

parameter. The (radial, axial and tangential) components of the velocity, temperature, concentration and pressure profiles decrease with increase of suction parameter. The effects of *γ* on the velocity (radial, axial and tangential) and pressure profiles are shown in **Figure 14a** and **b**, respectively. It is observed that the (radial and axial) components of the velocity, and pressure profiles increase with the increasing slip parameter. While the tangential component of the velocity profile decrease.

friction �*G<sup>=</sup>*

ð Þ 0

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

, heat transfer rate �*θ<sup>=</sup>*

the concentration buoyancy parameter *N*.

with increasing value of *δ*.

Sherwood number increases.

*B* external uniform magnetic field *B*<sup>0</sup> constant magnetic flux density

*cp* specific heat at constant pressure *cs* concentration susceptibility *Cf* <sup>1</sup> tangential skin-friction coefficient *Cf* <sup>2</sup> radial skin-friction coefficient *D* molecular diffusion coefficient

*<sup>g</sup>* gravitational acceleration, m s�<sup>2</sup> ½ �

<sup>1</sup> permeability of the porous medium *k*<sup>∗</sup> Rosseland mean absorption coefficient

sionless velocity *<sup>M</sup>* magnetic field parameter <sup>¼</sup> *<sup>σ</sup> <sup>B</sup>*<sup>2</sup>

*n* normal direction to the wall

*k* thermal conductivity *k*<sup>1</sup> rate of chemical reaction *KT* thermal-diffusion rate

*Nu* Nusselt number

*Du* Dufour number ¼ *D kT*ð Þ *Cw* � *C*<sup>∞</sup> *=ν cscp*ð Þ *Tw* � *T*<sup>∞</sup>

*F, G, H* radial ð Þ *F* , tangential ð Þ *G* and axial ð Þ *H* components of dimen-

*<sup>N</sup>* concentration buoyancy parameter <sup>¼</sup> *<sup>g</sup>βc*ð Þ *Cw* � *<sup>C</sup>*<sup>∞</sup> *<sup>=</sup>LR*Ω<sup>2</sup>

<sup>0</sup>*=ρ*<sup>Ω</sup>

*b* induced magnetic field *C* concentration distribution *Cw* uniform concentration *C*<sup>∞</sup> constant concentration

**Nomenclature**

*K*∗

**119**

�*φ=*ð Þ <sup>0</sup> , while an increase in the radial skin friction *<sup>F</sup>=*ð Þ <sup>0</sup> .

2. The (radial and axial) components of the velocity and pressure increase with increasing of temperature buoyancy parameter and concentration buoyancy parameter, while the tangential component of the velocity, the temperature and the concentration profiles decrease with increasing temperature buoyancy parameter and concentration buoyancy parameter. We found that the increase for all radial skin friction, tangential skin friction, heat transfer rate and mass transfer rate, with increasing of the temperature buoyancy parameter *α* and

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

3. The (radial and axial) components of the velocity, temperature and pressure profiles increase with the increase of heat source parameter. While the concentration profile decrease. And also, the radial skin friction, tangential skin friction and mass transfer rate increase while heat transfer rate decrease

4.The (radial and axial) components of the velocity, concentration and pressure profiles decrease with increase of chemical reaction parameter. And in **Table 2**, an increase the chemical reaction parameter, results in an decrease in the (radial and tangential) Skin-friction coefficient and Nusselt number, while

ð Þ <sup>0</sup> and mass transfer rate

The radial and tangential skin frictions and the heat and mass transfer coefficients are tabulated in **Table 2** for various values of *M*, *S*, *α*, *N*, *Pr*, *Rd*, *Du*&*S*0, *J*, *δ*, *Sc*, *β*, *Ws* and *γ*. We observed that increase for all magnetic field parameter *M* and porosity parameter *S* leads to an decrease in the all tangential skin friction �*G<sup>=</sup>* ð Þ 0 , heat transfer rate �*θ<sup>=</sup>* ð Þ <sup>0</sup> and mass transfer rate �*φ=*ð Þ <sup>0</sup> , while an increase in the radial skin friction *<sup>F</sup>=*ð Þ <sup>0</sup> , the increase for all radial skin friction, tangential skin friction, heat transfer rate and mass transfer rate, with increasing of the temperature buoyancy parameter *α* and the concentration buoyancy parameter *N*. We found that the radial skin friction, tangential skin friction, mass transfer rate decreases while heat transfer rate increase with increasing of Prandtl number, Dufour number decreases and Soret number increases. It can that be seen that the radial skin friction, tangential skin friction and mass transfer rate increase while heat transfer rate decrease with increasing values of *Rd* and *δ*. It is observed that an increase in the Joule heating parameter, results in a decrease in the tangential Skinfriction coefficient, Nusselt number and Sherwood number. The tangential skin friction and heat transfer rate decrease but the radial skin friction and mass transfer rate increase with increasing the Schmidt number. It also can be seen from this table that increasing the chemical reaction parameter to decrease in the radial skin friction, tangential skin friction and heat transfer rate while increase the mass transfer rate. We found also the tangential skin friction increase but the radial skin friction, Nusselt number and Sherwood number decrease with increasing the suction parameter. Finally, the radial skin friction and the tangential skin friction decrease but Nusselt number and Sherwood number increase with increasing slip parameter.
