**5. Results and discussion**

<sup>1</sup> ( ) ( ) 1 11 ( ) ( ) ( )

 x

<sup>0</sup> ( ) 10 10 10 ( ) ( ) ( )

 qx

*kr r r*

**F F F**

{ } { }

= - -D ¢

**H H**

A diag diag ˆ ˆ ,

*r r*

{ } ( )

*D Ec D*

<sup>1</sup> A diag{ } 2 diag ˆ ˆ , 2 Pr

<sup>1</sup> A diag 2 diag ˆ ˆ , <sup>2</sup>

*A о B* denotes the Hadarmard product (element-wise multiplication) of matrices *A* and *B* of the same order, and *I* and *O* are the identity and zero matrices, respectively. Boundary conditions (33) and (34) of linear system (32) are in the same manner as done in [2]. This is followed by solution of linear system (32) to get approximations *f <sup>r</sup>*(*ξc*), *hr*(*ξc*), *θr*(*ξc*), *ϕr*(*ξc*) for each *r* =1, 2, ... and *c* =0, 1, 2, ..., *N* . However, this last step requires suitable initial approx‐

*D*

ì ü =- - + + - + í ý

**F F**

*r r r*

*r x rr*

ì ü =- + - + + - + í ý

**F F**

æ öì ü =- - + D + - + = - ¢ ¢ ¢ ç ÷ í ý è øî þ

12 13 1 14 2

{ } { } ( ) { }

**Θ Θ F**

*r r r*

= +D +¢¢ ¢ ¢ o o

{ } { }

**Φ Φ**

{ } ( )

*r r*

,

**R FF F** Q - Q,

*r r rr*

A diag diag , ˆ

22 3

*<sup>A</sup> <sup>A</sup> A BI D D*

A diag diag 2 1 diag , ˆ ˆ

= - + +D ¢ ¢¢

(1 )

*Ec*

*r rr*

**RF F** o o F F

=- + ¢ ¢

= - ¢

*A DA IA I*

l

 x

*r w Nk r N N r N r N*

+ + ++

ˆ , 1, , 1, 1,

0, 0, 0, 0,

{ } { } ( ) 2 3

( )

*D BD*

*r r r r rr r*

( ) <sup>2</sup>

*<sup>A</sup> K AI D D*

*<sup>A</sup> AQI D D*

ˆ,,, ,

<sup>3</sup> ˆ ˆ diag 2 diag , , 2 2

h

**F F R FH F H**

2

2

î þ

h

h

î þ

 l

*r rr*

**R FF F F**

o o

2 2

o o

2

*Sc*

1

<sup>1</sup> ˆ ˆˆ diag 2diag diag 1 , <sup>2</sup> *<sup>r</sup> <sup>r</sup> r r p <sup>A</sup> <sup>A</sup> A M ID D D K*

æ ö æ ö ì ü = + - - + + + - + +D ¢¢ ç ÷ ¢ ç ÷ í ý î þ è ø è ø

> l¢¢ ¢ =D = = = - ¢

+++

 x

> fx

å == = = (34)

= == = = å (33)

 f x

h

 xq

0

*Df h* x

=

21

32 (3)

41

42 (4)

imation for which we choose

x

*k f f Df h*

0

=

*k*

where

11

31

*N*

x *N*

298 Numerical Simulation - From Brain Imaging to Turbulent Flows

The non-linear differential Eqs. (11)–(14) with boundary conditions (15)–(16) depend on several parameters, such as micropolar Δ, unsteadiness *A*, thermal buoyancy *λ*1, solutal buoyancy *λ*2, non-dimensional material *λ*3, magnetic field M, local porous*K*p, non-dimensional parameter *B*, Eckert number *Ec*, heat generation and/or absorption and chemical reaction. All the SQLM results presented in this work were obtained using *N* = 50 collocation points, and we are glad to highlight that convergence was achieved in just about five iterations. We take the infinity value *η∞* to be 40. Unless otherwise stated, the default values for the parameters are taken as:

Pr = 0.71, *B* = 0.1, M = 1, *Ec* = 0.1, *Sc* = 0.22, *Δ* Δ = 0.1, λ1 = λ2 = 0.5, *Kp* = 1, *K* = 0.5, f\_w = 0.5, *λ*3. In order to validate our numerical method, it was compared to MATLAB routine bvp4c which is an adaptive Lobatto quadrature iterative scheme. This is depicted in **Table 1**. In **Table 1**, we observe that the current results completely agree with the results generated by bvp4c. It is worth noting that convergence of SQLM occurs as early as at the sixth iteration and the method is extremely faster, saving *cpu* time. This gives confidence to our proposed method. We also observe in **Table 1** that the rates of transfers are greatly affected by the micropolar parameter Δ.


**Table 1.** Comparison of the SQLM results of *f* −*f*″ (0), −*θ*′(0), *ϕ*′ (0) with those obtained by bvp4c for different values of the micropolar parameter *Δ*.

We observe in **Table 2** that the wall stresses, the Nusselt and Sherwood numbers are signifi‐ cantly affected by the changing values of the unsteadiness parameter. The skin-friction coefficient as expected increases with increasing values of the stretching parameter.


**Table 2.** The effects of the unsteadiness parameter on − *f* ″(0), *h* ′(0), −*θ*′(0), −*ϕ*′(0).

**Table 3** depicts the influence ofthe thermal buoyancy parameter on the skin friction coefficient, the local couple wall stress, the local Nusselt and Sherwood numbers. Both local wall stresses are reduced as the values of buoyancy parameters are increased but the Nusselt and Sherwood numbers increase with increasing values of the buoyancy parameters.


**Table 3.** The influence of the thermal buoyancy parameter on the skin friction coefficient, couple stress and rate of heat and mass transfer coefficient.


**Table 4.** The effects of the local porous parameter on the skin friction coefficient, couple stress, rate of heat and mass transfer coefficient.


**Table 5.** The influence of the magnetic field parameter on the skin friction coefficient, couple stress, rate of heat and mass transfer coefficient.

The effect of medium porosity on the wall stresses and the Nusselt and Sherwood numbers is depicted in **Table 4**. Porosity significantly affects the transferrates. The effect of magnetic field parameteris depicted in**Table 5**. As expected, the presence ofthe magnetic field has prominent effects on the skin-friction coefficient as well on the heat and mass transferrates. The drag force that is generated by the presence of magnetic field causes significant resistance to the velocity of the fluid thus increases the wall stresses but reduces the rates of heat and mass transfer.

**Table 6** shows the effects of the micropolar parameter and the non-dimensional material parameter on the wall stress. The micropolar parameterincreases the values of the wall couple stress, but the non-dimensional material parameter reduces the values of the wall stress.


**Table 6.** The effects of the micropolar parameter and the non-dimensional material parameter on the couple stress.

The influence of the micropolar parameter *Δ* on the axial velocity is depicted in **Figure 1**. It can be observed in **Figure 1** that axial velocity is an increasing function of the micropolar parameter. Physically, micropolar fluids show reduced drag compared to viscous fluids.

**Figure 1.** Velocity profile for various values of *Δ*.

*A* **−** *f* **″(0)** *h* **′(0) −***θ***′(0) −***θ***′(0)** 1.18860026 0.03277007 0.22581676 1.34959740 1.51530691 0.02767645 1.21557294 1.75188835 1.86832868 0.02329578 2.06905446 2.36549832

**Table 3** depicts the influence ofthe thermal buoyancy parameter on the skin friction coefficient, the local couple wall stress, the local Nusselt and Sherwood numbers. Both local wall stresses are reduced as the values of buoyancy parameters are increased but the Nusselt and Sherwood

**Table 3.** The influence of the thermal buoyancy parameter on the skin friction coefficient, couple stress and rate of heat

**Table 4.** The effects of the local porous parameter on the skin friction coefficient, couple stress, rate of heat and mass

**Table 5.** The influence of the magnetic field parameter on the skin friction coefficient, couple stress, rate of heat and

The effect of medium porosity on the wall stresses and the Nusselt and Sherwood numbers is depicted in **Table 4**. Porosity significantly affects the transferrates. The effect of magnetic field parameteris depicted in**Table 5**. As expected, the presence ofthe magnetic field has prominent

**Table 2.** The effects of the unsteadiness parameter on − *f* ″(0), *h* ′(0), −*θ*′(0), −*ϕ*′(0).

300 Numerical Simulation - From Brain Imaging to Turbulent Flows

numbers increase with increasing values of the buoyancy parameters.

and mass transfer coefficient.

transfer coefficient.

mass transfer coefficient.

*λ***<sup>1</sup> −** *f* **″(0)** *h* **′(0) −***θ***′(0) −** *ϕ***′(0)** −0.5 1.77407269 0.03036702 1.17144784 1.73796915 0 1.64314463 0.02897505 1.19450881 1.74511388 0.5 0.46281664 0.01845219 1.34838298 1.80239822

*Kp* **<sup>−</sup>** *f* **″(0)** *h* **′(0) <sup>−</sup>***θ′***(0) <sup>−</sup>***ϕ***′(0)** 0.1 −0.51158820 0.01764248 1.36287821 1.81467597 5 0.33757366 0.01499546 1.39049894 1.82865098 10 0.63059835 0.01202912 1.41759309 1.84519070

*M* **−** *f* **″(0)** *h* **′(0) −***θ***′(0) −** *ϕ***′(0)** 1.51550691 0.02767645 1.21557294 1.75188835 1.91685662 0.03972746 1.16026663 1.73369138 2.24935200 0.03274343 1.11815954 1.72039545

> In **Figure 2** we display the effect of unsteadiness parameter on the axial velocity *f* ' (*η*) Increasing the values of the unsteadiness parameter (*A*) causes the velocity boundary layer thickness to decrease, thereby reducing the velocity profiles. This is due to increased drag force on the surface. Surface stretching can therefore be used as a stabilizing mechanism in an effort to delay the transition from laminar flow to turbulent fluid flow.

**Figure 2.** Velocity profile with *A*.

The effect of the permeability of the porous medium parameter (*Kp*) on the translational velocity distribution profiles is depicted in **Figure 3**. The translational velocity increases with increasing values of the porosity parameter. Physically, increasing the porosity of the medium implies that the holes of the medium become larger, thereby reducing the resistivity of the medium.

**Figure 4** displays the effect of the thermal buoyancy parameter on the translational velocity distribution. The velocity profiles are reduced for the opposing flows (*λ*<sup>1</sup> <0). However *λ*<sup>1</sup> becomes more positive and favourable pressure gradients are enhanced, thereby accelerat‐ ing the fluid flow as can be clearly observed in **Figure 4**. It is interesting to note that for large values of the thermal buoyancy parameter, the translational velocity over-shoots nearthe wall over the moving speed of the sheet. This substantiates the notion that buoyancy accelerates

**Figure 3.** Variation of the porosity parameter on the axial velocity.

transition from laminar flow to turbulent flow; therefore, this must always be properly regulated in systems where turbulence is destructive. We also remark that solutal buoyancy as expected has the same effect as thermal buoyancy.

**Figure 4.** The influence of the thermal buoyancy on the axial velocity.

**Figure 2.** Velocity profile with *A*.

302 Numerical Simulation - From Brain Imaging to Turbulent Flows

**Figure 3.** Variation of the porosity parameter on the axial velocity.

medium.

The effect of the permeability of the porous medium parameter (*Kp*) on the translational velocity distribution profiles is depicted in **Figure 3**. The translational velocity increases with increasing values of the porosity parameter. Physically, increasing the porosity of the medium implies that the holes of the medium become larger, thereby reducing the resistivity of the

**Figure 4** displays the effect of the thermal buoyancy parameter on the translational velocity distribution. The velocity profiles are reduced for the opposing flows (*λ*<sup>1</sup> <0). However *λ*<sup>1</sup> becomes more positive and favourable pressure gradients are enhanced, thereby accelerat‐ ing the fluid flow as can be clearly observed in **Figure 4**. It is interesting to note that for large values of the thermal buoyancy parameter, the translational velocity over-shoots nearthe wall over the moving speed of the sheet. This substantiates the notion that buoyancy accelerates

**Figure 5.** Variation of angular velocity with *A*.

The influence of the unsteadiness parameter on the angular velocity *h* (*η*) is displayed in **Figure 5**. The unsteadiness parameter has pronounced influence on the angular velocity with values of *h* (*η*) picking up at *η* =1, as can be clearly seen in **Figure 5**. However, the angular velocity approaches zero as *η* increases infinitely and the unsteadiness parameter (*A*) increases. **Figure 6** shows the effect of the microrotation parameter (*B*) on the angular velocity. We observe that the microrotation effect is more pronounced as expected near the surface. Increasing values of *B* results in much increasing values of the angular velocity profiles.

**Figure 6.** Influence of *B* on the velocity.

We observe in **Figure 7** that the angular velocity is significantly affected by the micropolar parameter (*Δ*) The angular velocity is greatly induced due to the vortex viscosity effect as *Δ* increases.

**Figure 7.** Variation of the angular velocity with *Δ*.

The effect of thermal buoyancy parameter (*λ*1) is displayed in **Figure 8**. We observe that the angular velocity *h* (*η*) increases with increasing values of the thermal buoyancy parameter *λ*1. On a New Numerical Approach on Micropolar Fluid, Heat and Mass Transfer Over an Unsteady Stretching Sheet Through Porous Media in the Presence of a Heat Source/Sink and Chemical Reaction http://dx.doi.org/10.5772/63800 305

**Figure 8.** Angular velocity profiles for various values of thermal buoyancy.

**Figure 6** shows the effect of the microrotation parameter (*B*) on the angular velocity. We observe that the microrotation effect is more pronounced as expected near the surface. Increasing values of *B* results in much increasing values of the angular velocity profiles.

We observe in **Figure 7** that the angular velocity is significantly affected by the micropolar parameter (*Δ*) The angular velocity is greatly induced due to the vortex viscosity effect as *Δ*

The effect of thermal buoyancy parameter (*λ*1) is displayed in **Figure 8**. We observe that the angular velocity *h* (*η*) increases with increasing values of the thermal buoyancy parameter *λ*1.

**Figure 6.** Influence of *B* on the velocity.

304 Numerical Simulation - From Brain Imaging to Turbulent Flows

**Figure 7.** Variation of the angular velocity with *Δ*.

increases.

In **Figure 9**, we display the effect of the metrical parameter on the angular velocity. The angular velocity is greatly reduced by increasing value of *λ*3. This means that either the spin gradient coefficient increases or the microinertia density is reduced.

**Figure 9.** Variation of angular velocity with material parameter.

The effect of the unsteadiness parameteris displayed in **Figure 10**. The thermal boundary layer thickness is greatly reduced by increasing values of the unsteadiness parameter thus reduc‐ ing the fluid temperature distribution

**Figure 11** displays the effect of viscous dissipation on the temperature distribution resulting in increased Eckert number that causes heat energy to be stored in the region as a result of dissipation. This dissipation is caused by viscosity and elastic deformation, thus generating heat due to the frictional heating.

The effect ofthermal buoyancy parameteris depleted in **Figure 12**. The thermal boundary layer thickness is reduced when the value of the thermal buoyancy is increased. The fluid temper‐ ature is reduced at every point, except at the wall with increasing values of the thermal buoyancy parameter.

**Figure 10.** Temperature profile for various values of *A*.

**Figure 11.** Variation of temperature with the Eckert number.

On a New Numerical Approach on Micropolar Fluid, Heat and Mass Transfer Over an Unsteady Stretching Sheet Through Porous Media in the Presence of a Heat Source/Sink and Chemical Reaction http://dx.doi.org/10.5772/63800 307

**Figure 12.** Temperature profile for various values of thermal buoyancy.

dissipation. This dissipation is caused by viscosity and elastic deformation, thus generating

The effect ofthermal buoyancy parameteris depleted in **Figure 12**. The thermal boundary layer thickness is reduced when the value of the thermal buoyancy is increased. The fluid temper‐ ature is reduced at every point, except at the wall with increasing values of the thermal

heat due to the frictional heating.

306 Numerical Simulation - From Brain Imaging to Turbulent Flows

**Figure 10.** Temperature profile for various values of *A*.

**Figure 11.** Variation of temperature with the Eckert number.

buoyancy parameter.

**Figure 13** displays the variation of temperature distribution within the fluid flow for various values of the heat source/sink parameter. As expected, the fluid temperature increases with increasing values of heat at the source but decreases with increasing values of heat at a sink.

**Figure 13.** Temperature profile for various values of heat source/sink.

**Figure 14** shows the variation of the unsteadiness parameter on the concentration profiles. It is clearly observed that increasing values of *A* reduces both the solutal boundary layer thickness thus reduces the concentration distributions

**Figure 14.** Concentration profile for various values *A*

Lastly, the influence of a chemical reaction parameter on the concentration profiles is depict‐ ed in **Figure 15**. Physically, the concentration profiles decreases as the chemical reaction parameter increases.

**Figure 15.** Variation.

### **6. Conclusions**

The problem of MHD micropolar fluid, heat and mass transfer over unsteadiness stretching sheet through porous medium in the presence of a heat source/sink and chemical reaction is studied in this chapter. By applying suitable similarity transformations, we transformed the governing partial differential equations into a system of ordinary differential equations. We then applied the recently developed numerical technique known as the SQLM to solve the resultant set of non-linear ordinary differential equations. The accuracy of the SQLM was validated againstthe bvp4c routine method. We observed thatthe SQLMperforms much better than the bvp4c in terms of rate of convergence as well cpu time.

Based on the present study, the following conclusions are made:

