**3. Results and discussion**

Due to limitation of convergence of the classical DTM which is only valid near *η* ¼ 0, the multi-step transformation is used. Carrying out multi-step differential

> *η Hi* � �*<sup>i</sup> F <sup>i</sup>*ð Þ*<sup>k</sup>*

*η Hi* � �*<sup>i</sup>*

*η Hi* � �*<sup>i</sup> C*

*<sup>θ</sup>i*ð Þ*<sup>k</sup>*

*<sup>i</sup>*ð Þ*<sup>k</sup>* (25)

*Z*3ð Þ*i Z*5ð Þþ *k* � *i N:Z*7ð Þ*k cosα*

9

>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>=

>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>;

(26)

*th* sub-domain; *k* = 0, 1, 2, … m represents

*Fi*ð Þ¼ *<sup>η</sup>* <sup>X</sup> *k*

*<sup>θ</sup>i*ð Þ¼ *<sup>η</sup>* <sup>X</sup> *k*

*Ci*ð Þ¼ *<sup>η</sup>* <sup>X</sup>

*<sup>i</sup>*ð Þ*<sup>k</sup>* are the transformed functions, respectively. The transformation of the associated boundary conditions follows as:

Where *i* = 0, 1, 2, 3 … . n indicates the *i*

equations, and the results obtained are shown in **Table 1**.

ð Þ *Z*3ð Þ*i :Z*2ð Þ *k* � *i*

2 X *k*

*i*¼0

½ � *Z*3ð Þ*i Z*7ð Þ *k* � *i*

<sup>4</sup> *<sup>θ</sup>*<sup>0</sup> ð Þ*<sup>η</sup>*¼<sup>0</sup>

*<sup>y</sup>*<sup>2</sup> *Ra*<sup>∗</sup> ð Þ*<sup>y</sup>* <sup>3</sup>*=*<sup>4</sup>

<sup>4</sup> *<sup>C</sup>*<sup>0</sup> ð Þ*<sup>η</sup>*¼<sup>0</sup>

*<sup>u</sup>*<sup>2</sup> *<sup>y</sup>*<sup>2</sup> 2 *f* <sup>00</sup> � � *η*¼0

*Pr*<sup>1</sup>*=*<sup>4</sup> *f* <sup>00</sup> � � *η*¼0

*<sup>Z</sup>*1ð Þ*<sup>k</sup> cos<sup>α</sup>* � <sup>1</sup>

X *k*

*i*¼0

*i*¼0

*i*¼0

*k*

*i*¼0

the number of terms of the power series. *Hi*represents the sub-domain interval, and

*Z*1ð Þ¼ 0 1, *Z*2ð Þ¼ 0 *a*, *Z*3ð Þ¼ 0 0, *Z*4ð Þ¼ 0 0, *Z*5ð Þ¼ 0 *b*, *Z*6ð Þ¼ 0 1, *Z*7ð Þ¼ 0 *c* where *a*, *b*, and *c* are obtained by solving the system of algebraic simultaneous

*<sup>Z</sup>*4ð Þ*<sup>i</sup> <sup>Z</sup>*4ð Þþ *<sup>k</sup>* � *<sup>i</sup>* <sup>3</sup>

4 X *k*

" #

*i*¼0

transformations, we have:

*Computational Fluid Dynamics Simulations*

*F*

*<sup>i</sup>*ð Þ*<sup>k</sup>* , *<sup>θ</sup>i*ð Þ*<sup>k</sup>* ,*<sup>C</sup>*

*<sup>Z</sup>*1ð Þ¼ *<sup>k</sup>* <sup>þ</sup> <sup>1</sup> *<sup>Z</sup>*<sup>2</sup> ð Þ*<sup>k</sup>*

*<sup>Z</sup>*3ð Þ¼ *<sup>k</sup>* <sup>þ</sup> <sup>1</sup> *<sup>Z</sup>*<sup>4</sup> ð Þ*<sup>k</sup>*

*<sup>Z</sup>*4ð Þ¼ *<sup>k</sup>* <sup>þ</sup> <sup>1</sup> *<sup>Z</sup>*<sup>5</sup> ð Þ*<sup>k</sup>*

*<sup>Z</sup>*6ð Þ¼ *<sup>k</sup>* <sup>þ</sup> <sup>1</sup> *<sup>Z</sup>*<sup>7</sup> ð Þ*<sup>k</sup>*

Parameters Domain:

*Z*2ð Þ¼ *k* þ 1

*Z*5ð Þ¼ *k* þ 1

*Z*7ð Þ¼ *k* þ 1

**Table 1.**

**30**

*k* þ 1

*k* þ 1

*k* þ 1

1 *k* þ 1

*k* þ 1

*Pr* < 1, *Le* ≫ 1, *Le* ≪ 1, and *N* ≥ 0

The local Nusselt number: *Nu* � ½ � *Gr* <sup>∗</sup> ð Þ*<sup>y</sup>* <sup>1</sup>

The shearing stress on the plate: *<sup>τ</sup><sup>w</sup>* <sup>¼</sup> <sup>∝</sup>

*Important dimensionless parameters of interest.*

The Local Sherwood number: *Sh* ¼ �½ � *Gr* <sup>∗</sup> ð Þ*<sup>y</sup>* <sup>1</sup>

The coefficient of skin friction: *Cf* <sup>¼</sup> <sup>∝</sup>*ϑPr*1*=*4*Ra*<sup>∗</sup> ð Þ*<sup>y</sup>* <sup>3</sup>*=*<sup>4</sup>

� 3 <sup>4</sup> *:*Pr*:Le k* þ 1

X *k*

*i*¼0

� 3 4 Pr *k* þ 1

**Table 1** shows the expressions for the dimensionless parameters that are of interest in this work. The solutions to Eqs. (13)–(15) subjected to Eq. (16) solved using the multi-step DTM method are presented graphically in the figures below. The results shown are for Prandtl numbers of 0.01, 0.1, 0.5, and 0.72, respectively.

**Figure 2** shows the profiles of local skin friction against Prandtl number for various angles of inclination at a constant Lewis number. It could be observed from the plots that the shearing stress decreases as the Prandtl number increases for all the plate angles considered. More importantly, it is illustrated by the graphs that the local skin friction also decreases with increase in the buoyancy ratio and also with respect to the angle of inclination of the wall.

**Figure 3** shows the plots of local Nusselt number against Prandtl number for various angles of inclination at a constant Lewis number. It is clearly seen from the results that the rate of heat transfer (Nusselt number) increases as the Pr increases for all the wall inclination angles. Also to note is the fact that Nusselt number increases as the buoyancy ratio is increased.

**Figure 4** shows the results of how the local Sherwood number changes as the Lewis number is increased for various angles of inclination at a constant Prandtl number. It is noted from the graphs that the local Sherwood number increases as the Lewis number increases for all angles of inclination. Also, it is observed from the figures that the rate of increase of Sherwood number is dependent on N as well as the angle of inclination of the wall.

In **Figure 5**, the similarity profiles of the effect of the buoyancy ratio and the angles of inclination on the dimensionless velocities at fixed Prandtl and Lewis numbers is presented. It could be interpreted from the results for N = 0 that a maximum velocity is obtained for a vertical wall while at an angle of 60°, the minimum velocity. Also, it is clearly seen from the plots that aside from the vertical wall possessing the maximum velocity, its vertical velocity value for N = 1 is higher when compared to the case when N = 0. When the buoyancy ratio N is further increased to 1.5, the velocity for the vertical wall also increases. The trend in the figure also shows that increasing the inclination angle increases the velocity boundary layer thickness for all buoyancy ratio.

In **Figure 6**, the similarity profiles of dimensionless temperature for the wall angles of inclinations considered in this study at constant Lewis number are presented. The temperature profiles show that despite the varying buoyancy ratio, the thermal boundary layer thickness increases as the inclination angle of the wall increases.

**Figure 7** presents the similarity profiles of dimensionless species concentration distributions for the angles of inclination of the wall at a constant Lewis number. It is clearly seen that as the buoyancy ratio is increased, the concentration boundary layer thickness has a decreasing trend under this same flow configuration but as the angle of inclination increases, *δ<sup>c</sup>* increases.

**Figure 8** shows the plots of the coupled similarity profiles of dimensionless temperature, species concentration, and velocity for wall inclination angle of 60° under the constant Lewis number of 10. It can be seen from the results that increasing N has negligible effect on the trio of velocity, concentration and temperature boundary layer thicknesses for a fixed wall angle. However, the effect of N is very noticeable in the vertical velocity values which is clearly higher when N = 1.5 compared to when N = 0.

**Figure 2.**

*Similarity profiles of effects of Prandtl number on skin friction for (a) N = 0; (b) N = 1; (c) N = 1.5 at constant Lewis number of 10.*

**Figure 3.**

**33**

*constant Lewis number of 10.*

*Similarity profiles of effects of Prandtl number on Nusselt number for (a) N = 0; (b) N = 1; (c) N = 1.5 at*

*Scaling Investigation of Low Prandtl Number Flow and Double Diffusive Heat and Mass…*

*DOI: http://dx.doi.org/10.5772/intechopen.90896*

*Scaling Investigation of Low Prandtl Number Flow and Double Diffusive Heat and Mass… DOI: http://dx.doi.org/10.5772/intechopen.90896*

**Figure 3.** *Similarity profiles of effects of Prandtl number on Nusselt number for (a) N = 0; (b) N = 1; (c) N = 1.5 at constant Lewis number of 10.*

**Figure 2.**

**32**

*Lewis number of 10.*

*Computational Fluid Dynamics Simulations*

*Similarity profiles of effects of Prandtl number on skin friction for (a) N = 0; (b) N = 1; (c) N = 1.5 at constant*

**Figure 5.**

**35**

*Similarity profiles of dimensionless velocity for (a) N = 0, (b) N = 1, (c) N = 1.5 at Pr = 0.1 and Le = 10.*

*Scaling Investigation of Low Prandtl Number Flow and Double Diffusive Heat and Mass…*

*DOI: http://dx.doi.org/10.5772/intechopen.90896*

**Figure 4.** *Similarity profiles of effects of Lewis number on Sherwood number for (a) N = 0; (b) N = 1; (c) N = 1.5 at constant Pr of 0.1.*

*Scaling Investigation of Low Prandtl Number Flow and Double Diffusive Heat and Mass… DOI: http://dx.doi.org/10.5772/intechopen.90896*

**Figure 5.** *Similarity profiles of dimensionless velocity for (a) N = 0, (b) N = 1, (c) N = 1.5 at Pr = 0.1 and Le = 10.*

**Figure 4.**

**34**

*constant Pr of 0.1.*

*Computational Fluid Dynamics Simulations*

*Similarity profiles of effects of Lewis number on Sherwood number for (a) N = 0; (b) N = 1; (c) N = 1.5 at*

i. The velocity boundary layer thickness is maximum when the plate is in the vertical position, while it is minimum when the plate is at the inclined angle of 60° to the vertical for any value of Lewis number but within a certain

*Scaling Investigation of Low Prandtl Number Flow and Double Diffusive Heat and Mass…*

ii. The thermal boundary thickness is maximum when the wall is inclined at 60° to the vertical and minimum when in a vertical position while keeping

iii. Thermal boundary thickness decreases with increase in Prandtl number for all angles of inclination while keeping both the Lewis number and

*Similarity profiles of dimensionless concentration for (a) N = 0, (b) N = 1.5 at Pr = 0.1 and Le = 10.*

buoyancy ratio.

*DOI: http://dx.doi.org/10.5772/intechopen.90896*

N constant.

**Figure 7.**

**37**

buoyancy ratio constant.

**Figure 6.** *Similarity profiles of dimensionless temperature for (a) N = 0, (b) N = 1.5 at Pr = 0.1 and Le = 10.*
