**3. Governing equations**

The governing equations which are derived in Chapter-II under the above assumptions yields.

Region-1:

$$\frac{\text{dU}\_1}{\text{dY}} = 0 \text{ [Law of conservation of mass]} \tag{1}$$

$$\rho\_1 = \rho\_0[1 - \beta\_{1T}(T\_1 - T\_0) - \beta\_{1C}(C\_1 - C\_0)] \text{ [Physicalstate]} \tag{2}$$

$$\frac{\rho\_1 + K\,d^2 U\_1}{\rho\_1} + \frac{K}{\rho\_1} \frac{\text{dn}}{\text{dY}} + \text{g}\beta\_{1T}(T\_1 - T\_0) + \text{g}\beta\_{1C}(\text{C}\_1 - \text{C}\_0) - \frac{\sigma B\_0}{\rho\_1} \text{l} = \text{0 [Momentum]} \tag{3}$$

$$\gamma \frac{d^2 n}{d\mathbf{Y}^2} - K \left[ 2n + \frac{\mathbf{d} \mathbf{U}\_1}{\mathbf{d} \mathbf{Y}} \right] = \mathbf{0} \text{ [Conversion of Angular Momentum]} \tag{4}$$

where *<sup>γ</sup>* <sup>¼</sup> *<sup>μ</sup>*<sup>1</sup> <sup>þ</sup> *<sup>K</sup>* 2 � �*j*

$$\frac{k\_1}{\rho\_1 \mathbf{C}\_p} \frac{d^2 T\_1}{d \mathbf{Y}^2} + \frac{\mathbf{1}}{\rho\_1 \mathbf{C}\_p} \left[ \mu\_1 \left( \frac{\mathbf{d} \mathbf{U}\_1}{d \mathbf{Y}} \right)^2 + \frac{\rho\_1 D\_1 K\_{T1}}{\mathbf{C}\_{S1}} \frac{d^2 \mathbf{C}\_1}{d \mathbf{Y}^2} \right] = \mathbf{0} \tag{5}$$

$$D\_1 \frac{d^2 \mathbf{C}\_1}{d \mathbf{Y}^2} + \frac{D\_1 K\_{T1}}{T\_M} \frac{d^2 T\_1}{d \mathbf{Y}^2} = \mathbf{0} \text{ [Diffusion]} \tag{6}$$

Region-2:

$$\frac{\text{dU}\_2}{\text{dY}} = \mathbf{0} \left[ \text{Continuuity} \right] \tag{7}$$

$$
\rho\_2 = \rho\_0 \left[ 1 - \beta\_{2T} (T\_2 - T\_0) - \beta\_{2C} (C\_2 - C\_0) \right] \text{ [State]} \tag{8}
$$

$$\frac{\mu\_2 \, d^2 U\_2}{\rho\_2 \, \text{d} \mathbf{Y}^2} + \text{g} \boldsymbol{\beta}\_{2T} (T\_2 - T\_0) + \text{g} \boldsymbol{\beta}\_{2C} (\text{C}\_2 - \text{C}\_0) - \frac{\sigma B\_0 \,^2 U\_2}{\rho\_2} = \mathbf{0} \text{ [Momentum]} \tag{9}$$

$$\frac{k\_2}{\rho\_2 \mathbf{C}\_p} \frac{d^2 T\_2}{d \mathbf{Y}^2} + \frac{\mathbf{1}}{\rho\_2 \mathbf{C}\_p} \left[ \mu\_2 \left( \frac{\mathbf{d} \mathbf{U}\_2}{d \mathbf{Y}} \right)^2 + \frac{\rho\_2 D\_2 K\_{T2}}{\mathbf{C}\_{S2}} \frac{d^2 \mathbf{C}\_2}{d \mathbf{Y}^2} \right] = \mathbf{0} \tag{10}$$

$$D\_2 \frac{d^2 \mathbf{C}\_2}{d\mathbf{Y}^2} + \frac{D\_2 K\_{T2}}{T\_M} \frac{d^2 T\_2}{d\mathbf{Y}^2} = \mathbf{0} \text{ [Diffusion]} \tag{11}$$

*Convective Heat and Mass Transfer of Two Fluids in a Vertical Channel DOI: http://dx.doi.org/10.5772/intechopen.94529*

The above equation models (1) to (11) are solved by the following boundary and interface parameters.

$$\begin{array}{l} U\_1 = 0 \text{ at } Y = -h\_1 \text{, } U\_2 = 0 \quad \text{at } Y = h\_2, U\_1(0) = U\_2(0),\\ T = T\_1 \text{ at } Y = -h\_1, T = T\_2 \text{ at } Y = h\_2, T\_1(0) = T\_2(0),\\ C = C\_1 \text{ at } Y = -h\_1, C = C\_2 \text{ at } Y = h\_2, C\_1(0) = C\_2(0),\\ n = 0 \text{ at } Y = -h\_1, (\mu\_1 + K) \frac{\text{dU}\_1}{\text{dY}} + \text{Kn} = \mu\_2 \frac{\text{dU}\_2}{\text{dY}} \text{ at } Y = 0,\\ \frac{\text{dn}}{\text{dY}} = \mathbf{0} \text{ at } Y = \mathbf{0}, \ k\_1 \frac{\text{dT}\_1}{\text{dY}} = k\_2 \frac{\text{dT}\_1}{\text{dY}} \text{ at } Y = \mathbf{0}, D\_1 \frac{\text{dC}\_1}{\text{dY}} = D\_2 \frac{\text{dC}\_2}{\text{dY}} \text{ at } Y = \mathbf{0}.\\ \mathbf{T1} \text{ at } Y = 1 \qquad \mathbf{v} \quad \text{at } \mathbf{1} \qquad \mathbf{v} \quad \text{and} \qquad \mathbf{1} \qquad \mathbf{v} \quad \text{and} \qquad \mathbf{1} \qquad \mathbf{v} \quad \text{and} \qquad \mathbf{1} \end{array}$$

The following non dimensional variables form the equation systems (1) to (11) in to dimensionless form:

 $y = \frac{V}{h\_1}$ (region-1),  $y = \frac{V}{h\_2}$ (region-2),  $u\_1 = \frac{U\_1}{U\_0}$ ,  $u\_2 = \frac{U\_1}{U\_0}$ ,  $\theta\_1 = \frac{T\_1 - T\_0}{\Delta T}$ ,  $\theta\_2 = \frac{T\_2 - T\_0}{\Delta T}$ ,  $N = \frac{h\_1}{U\_0}$ ,  $j = h^2$  (Characteristic length),  $K' = \frac{K}{\mu\_1}$ ,  $c\_1 = \frac{C\_1 - C\_0}{\Delta C}$ ,  $c\_2 = \frac{C\_2 - C\_0}{\Delta T}$ ,  $\textbf{Gr} = \frac{g \beta\_1 \Delta T h\_1^3}{\nu\_1^2}$ ,  $\textbf{Gr} = \frac{g \beta\_0 \Delta C h\_1^3}{\nu\_1^2}$ ,  $R = \frac{U\_0 h\_1}{\nu\_1}$ ,  $\textbf{Sr} = \frac{D\_1 K\_{T1} \Delta T}{T\_{M4} C\_{U} h\_1}$ ,  $\textbf{Sc} = \frac{D\_1}{D\_1}$ ,  $\textbf{D} = \frac{D\_1}{C\_{D}} \frac{\Delta C}{\Delta T}$ ,  $\textbf{Pr} = \frac{\mu\_1 \text{Cp}}{k\_1}$ ,  $\textbf{Pr} = \frac{U\_0^2}{k\_1}$ ,  $\textbf{Pr} = \frac{U\_0^2}{k\_1}$ ,  $\textbf{Pr} = \frac{U\_0}{k\_1}$ ,  $\textbf{Pr} = \frac{U\_0}{C\_P \Delta T}$ , 
$$C\_{D\_1} = \frac{C\_{D\_1}}{\nu\_1}$$
,  $K\_{T1} = D\_1 \cdot 1 \quad k\_1 = \frac{\mu\_1}{\nu\_1}$ ,  $\rho\_1 = 1 \quad \rho\_{T1} = \frac{\rho\_{T1}}{\nu\_1}$ 

*CS* <sup>¼</sup> *CS*<sup>1</sup> *CS*<sup>2</sup> , *KT* <sup>¼</sup> *KT*<sup>1</sup> *KT*<sup>2</sup> , *<sup>D</sup>* <sup>¼</sup> *<sup>D</sup>*<sup>1</sup> *D*<sup>2</sup> , *<sup>h</sup>* <sup>¼</sup> *<sup>h</sup>*<sup>1</sup> *h*2 , *<sup>m</sup>* <sup>¼</sup> *<sup>μ</sup>*<sup>1</sup> *μ*2 , *<sup>α</sup>* <sup>¼</sup> *<sup>k</sup>*<sup>1</sup> *k*2 , *<sup>ρ</sup>* <sup>¼</sup> *<sup>ρ</sup>*<sup>1</sup> *ρ*2 , *<sup>b</sup>*<sup>1</sup> <sup>¼</sup> *<sup>β</sup>*1*<sup>T</sup> β*2*<sup>T</sup>* , *<sup>b</sup>*<sup>2</sup> <sup>¼</sup> *<sup>β</sup>*1*<sup>C</sup> β*2*<sup>C</sup>* ,*<sup>ν</sup>* <sup>¼</sup> *<sup>ν</sup>*<sup>1</sup> *ν*2 . The dimensionless forms of governing equations thus obtained are:

Region-1:

$$\frac{d^2N}{d\mathbf{y}^2} - \frac{2K'}{2+K'} \left(2N + \frac{d\mathbf{u}\_1}{d\mathbf{y}}\right) = \mathbf{0} \tag{12}$$

$$(\mathbf{1} + K')\frac{d^2 u\_1}{d\mathbf{y}^2} + K'\frac{d\mathbf{N}}{d\mathbf{y}} + \frac{\mathbf{G}\mathbf{r}}{R}\theta\_1 + \frac{\mathbf{G}\mathbf{c}}{R}\boldsymbol{\varepsilon}\_1 - \mathbf{M}\mathbf{u}\_1 = \mathbf{0} \tag{13}$$

$$\frac{1}{\text{Pr}R}\frac{d^2\theta\_1}{\text{dy}^2} + \frac{\text{Ec}}{R}\left(\frac{\text{du}\_1}{\text{dy}}\right)^2 + \frac{\text{Du}}{R}\frac{d^2c\_1}{\text{dy}^2} = \mathbf{0} \tag{14}$$

$$\frac{1}{\text{Sc}R} \frac{d^2 c\_1}{\text{dy}^2} + \text{Sr} \frac{d^2 \theta\_1}{\text{dy}^2} = 0 \tag{15}$$

Region �2

$$\frac{d^2 u\_2}{d\mathbf{y}^2} + \frac{m}{b\_1 \rho h^2} \frac{\mathbf{Gr}}{R} \theta\_2 + \frac{m}{b\_2 \rho h^2} \frac{\mathbf{Gr}}{R} c\_2 - \frac{\mathbf{mM}}{h^2} u\_2 = 0 \tag{16}$$

$$\frac{\rho h}{a} \frac{\mathbf{1}}{\text{Pr}R} \frac{d^2 \theta\_2}{\text{d}y^2} + \frac{\rho h}{m} \frac{\text{Ec}}{R} \left(\frac{\text{du}\_2}{\text{dy}}\right)^2 + \frac{c\_r h}{\text{DK}\_T} \frac{D\_\mu}{R} \frac{d^2 c\_2}{\text{dy}^2} = \mathbf{0} \tag{17}$$

$$\frac{h}{D} \left( \frac{1}{\text{Sc}R} \right) \frac{d^2 c\_2}{\text{d} \mathbf{y}^2} + \frac{h}{K\_T D} \text{Sr} \frac{d^2 \theta\_2}{\text{d} \mathbf{y}^2} = \mathbf{0} \tag{18}$$

The dimensionless boundary and interface conditions thus formed are:

$$\begin{aligned} u\_1 &= 0 \text{ at } y = -1, u\_2 = 0 \text{ at } y = 1, u\_1(0) = u\_2(0), \\ \theta\_1 &= 1 \text{ at } y = -1, \theta\_2 = 0 \text{ at } y = 1, \theta\_1(0) = \theta\_2(0), \\ c\_1 &= 1 \text{ at } y = -1, c\_2 = 0 \text{ at } y = 1, c\_1(0) = c\_2(0), \\ N &= 0 \text{ at } y = -1, \frac{\text{du}\_1}{\text{dy}} + \frac{K'}{1 + K'} N = \frac{1}{\text{mh}(1 + K')} \frac{\text{du}\_2}{\text{dy}} \text{ at } y = 0, \\ \frac{\text{dN}}{\text{dy}} &= 0 \text{ at} y = 0, \frac{d\theta\_1}{\text{dy}} = \frac{1}{\text{ha}} \frac{d\theta\_2}{\text{dy}} \text{ at } y = 0, \frac{\text{dc}\_1}{\text{dy}} = \frac{1}{\text{hD}} \frac{\text{dc}\_2}{\text{dy}} \text{ at } y = 0. \end{aligned} \tag{19}$$

### **4. Solution of the problem**

The finite element method as described in Chapter-II is applied in solving the dimensionless coupled differential equations generated by the fluid flows. For the problem discussed here, it is considered that each region is classified into 100 linear elements and each element is 3 nodded.

The element equations associated with Eqs. (12) to (18) is as follows:

$$\int\_{y\_i}^{y\_{i+1}} \left( \frac{d^2N}{\mathrm{d}\mathbf{y}^2} \cdot \frac{2K'}{2+K'} \left[2N + \frac{\mathrm{d}\mathbf{u}\_1}{\mathrm{d}\mathbf{y}} \right] \right) \eta\_k \mathrm{d}\mathbf{y} = \mathbf{0} \tag{20}$$

$$\int\_{y\_i}^{y\_{i+1}} \left( (\mathbf{1} + K') \frac{d^2 u\_1}{d\mathbf{y}^2} + K' \frac{d\mathbf{N}}{d\mathbf{y}} + \frac{\mathbf{Gr}}{R} \theta\_1 + \frac{\mathbf{Gc}}{R} c\_1 - \mathbf{M} \mathbf{u}\_1 \right) \eta\_k d\mathbf{y} = \mathbf{0} \tag{21}$$

$$\int\_{y\_i}^{y\_{i+1}} \left( \frac{\mathbf{1}}{\text{Pr}R} \frac{d^2 \theta\_1}{\text{d}y^2} + \frac{\text{Ec}}{R} \left( \frac{\text{du}\_1}{\text{d}y} \right)^2 + \frac{\text{Du}}{R} \frac{d^2 c\_1}{\text{d}y^2} \right) \eta\_k \text{d}y = \mathbf{0} \tag{22}$$

$$\int\_{y\_i}^{y\_{i+1}} \left( \frac{1}{\text{Sc}R} \frac{d^2 c\_1}{\text{d}y^2} + \text{Sr} \frac{d^2 \theta\_1}{\text{d}y^2} \right) \eta\_k \text{dy} = 0 \tag{23}$$

$$\int\_{y\_i}^{y\_i} \left( \frac{d^2 u\_2}{d\mathbf{y}^2} + \frac{m}{b\_1 \rho h^2} \frac{\mathbf{G}\mathbf{r}}{R} \theta\_2 + \frac{m}{b\_2 \rho h^2} \frac{\mathbf{G}\mathbf{c}}{R} c\_2 - \frac{\mathbf{mM}}{h^2} u\_2 \right) \chi\_k \mathbf{d} \mathbf{y} = \mathbf{0} \tag{24}$$

$$\int\_{\mathcal{Y}\_i}^{\eta\_{i+1}} \left( \frac{\rho h}{a} \frac{\mathbf{1}}{\text{Pr}R} \frac{d^2 \theta\_2}{\text{d}\mathbf{y}^2} + \frac{\rho h}{m} \frac{\text{Ec}}{R} \left( \frac{\text{du}\_2}{\text{d}\mathbf{y}} \right)^2 + \frac{c\_s h}{\text{DK}\_T} \frac{D\_\mathbf{u}}{R} \frac{d^2 c\_2}{\text{d}\mathbf{y}^2} \right) \chi\_k \text{d}\mathbf{y} = \mathbf{0} \tag{25}$$

$$\int\_{y\_i}^{y\_{i+1}} \left( \frac{h}{D} \left( \frac{1}{\text{Sc} \, R} \right) \frac{d^2 c\_2}{\text{d} \mathbf{y}^2} + \frac{h}{K\_T D} \text{Sr} \frac{d^2 \theta\_2}{\text{d} \mathbf{y}^2} \right) \chi\_k \text{dy} = \mathbf{0} \tag{26}$$

Where *η<sup>k</sup>* and *χ<sup>k</sup>* denotes the shape functions of a typical element *yi* , *yi*þ<sup>1</sup> � � in the region 1 and 2 correspondingly.

On integrating the above equations and by replacing the finite element Galerkin calculations,

$$\begin{aligned} u^i\_1 &= \sum\_{j=1}^3 u^i\_j \eta^i\_j, \ c^i\_1 = \sum\_{j=1}^3 c^i\_j \eta^i\_j, \ N^i = \sum\_{j=1}^3 N^i\_j \eta^i\_j, \ \theta^i\_1 = \sum\_{j=1}^3 \theta^i\_j \eta^i\_j, \\ u^i\_2 &= \sum\_{j=1}^3 u^i\_j \chi^i\_j, \ c^i\_2 = \sum\_{j=1}^3 c^i\_j \chi^i\_j, \ \theta^i\_2 = \sum\_{j=1}^3 \theta^i\_j \chi^i\_j. \\ \text{From Eq. (20) we get} \end{aligned}$$

$$\int\_{\mathcal{Y}\_i}^{\mathcal{Y}\_{i+1}} \frac{\mathrm{dN}^i}{\mathrm{dy}} \frac{d\eta\_k}{\mathrm{dy}} \mathrm{d}\mathbf{y} + \frac{2K'}{2+K'} \int\_{\mathcal{Y}\_i}^{\mathcal{Y}\_{i+1}} \left[2\mathrm{N}^i \eta\_k \mathrm{d}\mathbf{y} \cdot \frac{d\eta\_k}{\mathrm{dy}} \boldsymbol{u}\_1^i\right] \mathrm{d}\mathbf{y} = \left[\eta\_k \frac{\mathrm{dN}^i}{\mathrm{dy}} + \eta\_k \boldsymbol{u}\_1^i\right]\_{\mathcal{Y}\_i}^{\mathcal{Y}\_{i+1}}$$

The stiffness matrix equation corresponding to the above is

*Convective Heat and Mass Transfer of Two Fluids in a Vertical Channel DOI: http://dx.doi.org/10.5772/intechopen.94529*

$$
\begin{bmatrix} a\_{\mathbf{k}}^i \end{bmatrix} \begin{bmatrix} N\_{\mathbf{k}}^i \end{bmatrix} + \begin{bmatrix} b\_{\mathbf{k}j}^i \end{bmatrix} \begin{bmatrix} u\_h^i \end{bmatrix} = \begin{bmatrix} Q\_{1j}^i \end{bmatrix} \tag{27}
$$

where  $a\_{\mathbf{k}j}^i = \int\_{\mathcal{Y}\_j}^{\mathcal{Y}\_{i+1}} \frac{dp\_h}{d\mathbf{y}} \frac{dv\_j^i}{d\mathbf{y}} \,\mathrm{d}\mathbf{y} + \frac{2K'}{2+K'} \int\_{\mathcal{Y}\_i}^{\mathcal{Y}\_{i+1}} \left[2\eta\_k \eta\_j^i \right] \mathrm{d}\mathbf{y}$ 

$$b\_{\mathbf{k}j}^i = -\frac{2K'}{2+K'} \int\_{\mathcal{Y}\_j}^{\mathcal{Y}\_{i+1}} \left[\frac{d\eta\_j^i}{d\mathbf{y}} \eta\_k \right] \mathrm{d}\mathbf{y}$$

$$Q\_{1j}^i = \left[\eta\_k \frac{d\mathbf{N}^j}{d\mathbf{y}} + \eta\_k u\_1^i \right]\_{\mathcal{Y}\_i}^{\mathcal{Y}\_{i+1}}$$

From Eq. (21) we get

$$\begin{split} \int\_{\mathcal{Y}\_{i}}^{\mathcal{Y}\_{i+1}} (\mathbf{1} + K') \frac{d\eta\_{k}}{\mathbf{d}\mathbf{y}} \frac{\mathbf{d}\mathbf{u}^{i}\_{1}}{\mathbf{d}\mathbf{y}} \, \mathrm{d}\mathbf{y} + \int\_{\mathcal{Y}\_{i}}^{\mathcal{Y}\_{i+1}} K' \frac{d\eta\_{k}}{\mathbf{d}\mathbf{y}} N^{i} \mathrm{d}\mathbf{y} \cdot \frac{\mathrm{Gr}}{R} \int\_{\mathcal{Y}\_{i}}^{\mathcal{Y}\_{i+1}} \eta\_{k} \theta\_{1}^{i} \mathrm{d}\mathbf{y} \\ \frac{\mathrm{Gr}}{R} \int\_{\mathcal{Y}\_{i}}^{\mathcal{Y}\_{i+1}} \eta\_{k} c\_{1}^{i} \, \mathrm{d}\mathbf{y} + M \int\_{\mathcal{Y}\_{i}}^{\mathcal{Y}\_{i+1}} \eta\_{k} u\_{1}^{i} \, \mathrm{d}\mathbf{y} = \left[ \cdot (\mathbf{1} + K') \eta\_{k} \, \frac{\mathrm{d}\mathbf{u}\_{1}^{i}}{\mathrm{d}\mathbf{y}} - K' \eta\_{k} N^{i} \right]\_{\mathcal{Y}\_{i}}^{\mathcal{Y}\_{i+1}} \end{split}$$

The stiffness matrix equation corresponding to the above is

*c i* kj h i *ui k* � � <sup>þ</sup> *di* kj h i *<sup>N</sup><sup>i</sup> k* � � <sup>þ</sup> *<sup>ℯ</sup><sup>i</sup>* kj h i *<sup>θ</sup><sup>i</sup> k* � � <sup>þ</sup> *<sup>f</sup> i* kj h i *<sup>c</sup> i k* � � <sup>¼</sup> *<sup>Q</sup>*<sup>2</sup> *i j* h i (28) where *c<sup>i</sup>* kj <sup>¼</sup> <sup>Ð</sup> *yi*þ<sup>1</sup> *yi* <sup>1</sup> <sup>þ</sup> *<sup>K</sup>*<sup>0</sup> ð Þ *<sup>d</sup>η<sup>i</sup> j* dy *dη<sup>k</sup>* dy dy <sup>þ</sup> *<sup>M</sup>* <sup>Ð</sup> *yi*þ<sup>1</sup> *yi ηi j ηk*dy, *di* kj <sup>¼</sup> *<sup>K</sup>*<sup>0</sup> <sup>Ð</sup> *yi*þ<sup>1</sup> *yi dη<sup>i</sup> j* dy *η<sup>k</sup>* � �dy *ℯi* kj ¼ � Gr *R* ð *yi*þ<sup>1</sup> *yi ηi j ηk* h idy, *<sup>f</sup> i* kj ¼ � Gc *R* ð *yi*þ<sup>1</sup> *yi ηi j ηk* h idy *Q*2 *i <sup>j</sup>* <sup>¼</sup> ‐ <sup>1</sup> <sup>þ</sup> *<sup>K</sup>*<sup>0</sup> ð Þ*η<sup>k</sup>* du*<sup>i</sup>* 1 dy � *<sup>K</sup>*<sup>0</sup> *ηkN<sup>i</sup>* " #*yi*þ<sup>1</sup> *yi*

From Eq. (22) we get

$$\begin{split} & \frac{1}{\Pr R} \int\_{\mathcal{Y}\_{i}}^{\mathcal{Y}\_{i+1}} \int\_{\mathcal{Y}} \frac{d\eta\_{k}}{\text{dy}} \frac{d\theta^{i}\_{1}}{\text{dy}} \text{dy} - \frac{\text{Ec}}{R} \int\_{\mathcal{Y}\_{i}}^{\mathcal{Y}\_{i+1}} \eta\_{k} \left( \frac{\text{du}^{i}\_{1}}{\text{dy}} \right)^{2} \text{dy} \text{-} \frac{\text{Du}}{R} \int\_{\mathcal{Y}\_{i}}^{\mathcal{Y}\_{i+1}} \frac{d\eta\_{k}}{\text{dy}} \frac{\text{dc}^{i}\_{1}}{\text{dy}} \text{dy} \\ &= \left[ \frac{1}{\Pr R} \eta\_{k} \frac{d\theta^{i}\_{1}}{\text{dy}} - \frac{\text{Du}}{R} \eta\_{k} \frac{\text{dc}^{i}\_{1}}{\text{dy}} \right]\_{\mathcal{Y}\_{i}}^{\mathcal{Y}\_{i+1}} \end{split}$$

The stiffness matrix equation corresponding to the above is

$$
\begin{bmatrix} \mathbf{g}\_{\mathbf{k}\mathbf{j}}^{i} \end{bmatrix} \begin{bmatrix} \theta\_{\mathbf{k}}^{i} \end{bmatrix} + \begin{bmatrix} \boldsymbol{u}\_{\mathbf{k}}^{i} \end{bmatrix}^{T} \begin{bmatrix} \boldsymbol{h}\_{\mathbf{k}\mathbf{j}}^{i} \end{bmatrix} \begin{bmatrix} \boldsymbol{u}\_{\mathbf{k}}^{i} \end{bmatrix} + \begin{bmatrix} \boldsymbol{m}\_{\mathbf{k}\mathbf{j}}^{i} \end{bmatrix} \begin{bmatrix} \mathbf{c}\_{\mathbf{k}}^{i} \end{bmatrix} = \begin{bmatrix} \mathbf{Q}\_{\mathbf{3}\mathbf{j}}^{i} \end{bmatrix} \tag{29}
$$

where

*gi* kj <sup>¼</sup> <sup>1</sup> Pr*R* ð *yi*þ<sup>1</sup> *yi dη<sup>k</sup>* dy *dη<sup>i</sup> j* dy dy, *<sup>h</sup><sup>i</sup>* kj <sup>¼</sup> �Ec *R* ð *yi*þ<sup>1</sup> *yi dη<sup>i</sup>* 1 dy � �<sup>2</sup> *<sup>d</sup>η<sup>i</sup>* 1 dy � � *<sup>d</sup>η<sup>i</sup>* 2 dy � � *<sup>d</sup>η<sup>i</sup>* 1 dy � � *<sup>d</sup>η<sup>i</sup>* 3 dy � � *dη<sup>i</sup>* 2 dy � � *<sup>d</sup>η<sup>i</sup>* 1 dy � � *<sup>d</sup>η<sup>i</sup>* 2 dy � �<sup>2</sup> *<sup>d</sup>η<sup>i</sup>* 2 dy � � *<sup>d</sup>η<sup>i</sup>* 3 dy � � *dη<sup>i</sup>* 3 dy � � *<sup>d</sup>η<sup>i</sup>* 1 dy � � *<sup>d</sup>η<sup>i</sup>* 3 dy � � *<sup>d</sup>η<sup>i</sup>* 2 dy � � *<sup>d</sup>η<sup>i</sup>* 3 dy � �<sup>2</sup> 2 6 6 6 6 6 6 6 6 6 4 3 7 7 7 7 7 7 7 7 7 5 *η<sup>k</sup>* ½ �dy*: mi* kj <sup>¼</sup> �Du *R* ð *yi*þ<sup>1</sup> *yi dη<sup>k</sup>* dy *dη<sup>i</sup> j* dy dy, *<sup>Q</sup>*<sup>3</sup> *i <sup>j</sup>* <sup>¼</sup> <sup>1</sup> Pr*<sup>R</sup> <sup>η</sup><sup>k</sup> dθ<sup>i</sup>* 1 dy � Du *<sup>R</sup> <sup>η</sup><sup>k</sup>* dc*i* 1 dy " #*yi*þ<sup>1</sup> *yi :*

From Eq. (23) we get

$$\frac{1}{\mathbf{Sc}R} \int\_{\mathcal{Y}\_i}^{\mathcal{Y}\_{i+1}} \frac{d\eta\_k}{\mathbf{d}\mathbf{y}} \frac{\mathbf{d}\mathbf{c}\_1^i}{\mathbf{d}\mathbf{y}} \mathbf{d}\mathbf{y} + \text{Sr} \int\_{\mathcal{Y}\_i}^{\mathcal{Y}\_{i+1}} \frac{d\eta\_k}{\mathbf{d}\mathbf{y}} \frac{d\theta\_1^i}{\mathbf{d}\mathbf{y}} \mathbf{d}\mathbf{y} = \left[ \frac{\mathbf{1}}{\mathbf{Sc}R} \eta\_k \frac{\mathbf{d}\mathbf{c}\_1^i}{\mathbf{d}\mathbf{y}} + \text{Sr} \eta\_k \frac{d\theta\_1^i}{\mathbf{d}\mathbf{y}} \right]\_{\mathcal{Y}\_i}^{\mathcal{Y}\_{i+1}}$$

The stiffness matrix equation corresponding to the above is

$$
\begin{bmatrix} n\_{\mathbf{k}\mathbf{j}}^i \end{bmatrix} \left[ c\_{\mathbf{k}}^i \right] + \left[ p\_{\mathbf{k}\mathbf{j}}^i \right] \left[ \theta\_k^i \right] = \left[ Q\_{\mathbf{v}}^i \right] \tag{30}
$$

$$
\text{where } n\_{\mathbf{k}\mathbf{j}}^i = \frac{1}{\mathbf{S} \mathbf{c} \cdot \mathbf{R}} \int\_{\mathbf{y}\_i}^{\mathbf{y}\_{i+1}} \frac{d\eta\_i}{d\mathbf{y}} \frac{d\eta\_i^i}{d\mathbf{y}} \,\mathrm{d\mathbf{y}}, \, p\_{\mathbf{k}\mathbf{j}}^i = \mathrm{S\mathbf{r}} \int\_{\mathbf{y}\_i}^{\mathbf{y}\_{i+1}} \frac{d\eta\_i}{d\mathbf{y}} \frac{d\eta\_i^i}{d\mathbf{y}} \,\mathrm{d\mathbf{y}}
$$

$$
Q\_{\mathbf{v}}{}\_j^i = \left[ \frac{\mathbf{1}}{\mathbf{S} \mathbf{c} \, R} \eta\_k \frac{\mathbf{d}\mathbf{c}\_1^i}{\mathbf{d}\mathbf{y}} + \mathrm{S\mathbf{r}} \,\eta\_k \frac{d\mathbf{o}\_1^i}{d\mathbf{y}} \right]\_{\mathbf{y}\_i}^{\mathbf{y}\_{i+1}}
$$

From Eq. (24) we get

$$\begin{aligned} \int\_{\mathcal{Y}\_i}^{\mathcal{Y}\_{i+1}} \frac{d\chi\_k}{\text{dy}} \frac{\text{du}\_2^i}{\text{dy}} \text{dy} \cdot \frac{m}{b\_1 \rho h^2} \frac{\text{Gr}}{R} \int\_{\mathcal{Y}\_i}^{\mathcal{Y}\_{i+1}} \chi\_k \theta\_2^i \text{dy} \cdot \frac{m}{b\_2 \rho h^2} \frac{\text{Gr}}{R} \int\_{\mathcal{Y}\_i}^{\mathcal{Y}\_{i+1}} \chi\_k c\_2^i \text{ dy} \\ + \frac{\text{mM}}{h^2} \int\_{\mathcal{Y}\_i}^{\mathcal{Y}\_{i+1}} \chi\_k u\_2^i \text{ dy} = \left[ \begin{aligned} \chi\_k \frac{\text{du}\_2^i}{\text{dy}} \end{aligned} \right]\_{\mathcal{Y}\_i}^{\mathcal{Y}\_{i+1}} \end{aligned}$$

The stiffness matrix equation corresponding to the above is

$$
\mathbb{E}\left[\mathbf{C}\_{\mathbf{k}\mathbf{j}}^{i}\right]\left[\boldsymbol{u}\_{\mathbf{k}}^{i}\right] + \left[\boldsymbol{D}\_{\mathbf{k}\mathbf{j}}^{i}\right]\left[\boldsymbol{\theta}\_{\mathbf{k}}^{i}\right] + \left[\boldsymbol{E}\_{\mathbf{k}\mathbf{j}}^{i}\right]\left[\boldsymbol{c}\_{\mathbf{k}}^{i}\right] = \left[\boldsymbol{Q}\_{\mathbf{S}}\boldsymbol{i}\right] \tag{31}
$$

*Convective Heat and Mass Transfer of Two Fluids in a Vertical Channel DOI: http://dx.doi.org/10.5772/intechopen.94529*

$$\begin{split} \text{where } & \mathbf{C}^{i}\_{\mathbf{k}j} = \int\_{\mathbf{y}\_{i}}^{\mathbf{y}\_{i+1}} \frac{d\mathbf{y}^{i}\_{j}}{d\mathbf{y}} \frac{d\mathbf{y}\_{i}}{d\mathbf{y}} \, \mathrm{d}\mathbf{y} + \frac{\mathrm{m}\mathbf{M}}{\boldsymbol{\hbar}^{2}} \int\_{\mathbf{y}\_{i}}^{\mathbf{y}\_{i+1}} \boldsymbol{\chi}^{i}\_{j} \boldsymbol{\chi}\_{k} \, \mathrm{d}\mathbf{y}, \, D^{i}\_{\mathbf{k}j} = -\underset{\mathbf{b}\_{1} \boldsymbol{\mu}^{i}}{\operatorname{d}\mathbf{y}^{i}} \frac{\mathrm{Gr}}{\boldsymbol{\hbar}} \int\_{\mathbf{y}\_{i}}^{\mathbf{y}\_{i+1}} \Big[\boldsymbol{\chi}^{i}\_{j} \boldsymbol{\chi}\_{k}\Big] \, \mathrm{d}\mathbf{y}, \\ \boldsymbol{E}^{i}\_{\mathbf{k}j} = -\underset{\mathbf{b}\_{2} \boldsymbol{\mu}^{i}}{\operatorname{d}\mathbf{y}^{i}} \frac{\mathrm{Gr}}{\boldsymbol{\hbar}} \int\_{\mathbf{y}\_{i}}^{\mathbf{y}\_{i+1}} \Big[\boldsymbol{\chi}^{i}\_{j} \boldsymbol{\chi}\_{k}\Big] \, \mathrm{d}\mathbf{y}, \, Q\_{\mathbf{s}j} = \left[ \boldsymbol{\chi}\_{k} \frac{\mathrm{d}\mathbf{u}^{i}\_{j}}{\mathrm{d}\mathbf{y}} \right]\_{\mathbf{y}\_{i}}^{\mathbf{y}\_{i+1}}. \end{split}$$
  $\text{From Eq. (25) we get}$ 

From Eq. (25) we get

$$\frac{\rho h}{a} \frac{1}{\text{Pr}R} \int\_{\mathcal{Y}\_i}^{\mathcal{Y}\_{i+1}} \frac{d\chi\_k}{\text{dy}} \frac{d\theta\_2^i}{\text{dy}} \text{d\mathbf{y}} - \frac{\rho h}{m} \frac{\text{Ec}}{R} \int\_{\mathcal{Y}\_i}^{\mathcal{Y}\_{i+1}} \chi\_k \left(\frac{\text{du}\_2^i}{\text{dy}}\right)^2 \text{dy} \cdot \frac{c\_i h}{\text{DK}\_T} \frac{\text{Du}}{R} \int\_{\mathcal{Y}\_i}^{\mathcal{Y}\_{i+1}} \frac{d\chi\_k}{\text{dy}} \frac{\text{dc}\_2^i}{\text{dy}} \text{dy} = 0$$
 
$$\left[\frac{\rho h}{\alpha} \frac{\text{1}}{\text{Pr}R} \chi\_k \frac{d\theta\_2^i}{\text{dy}} - \frac{c\_i h}{\text{DK}\_T} \frac{\text{Du}}{R} \chi\_k \frac{\text{dc}\_2^i}{\text{dy}}\right]\_{\mathcal{Y}\_i}^{\mathcal{Y}\_{i+1}}$$

The stiffness matrix equation corresponding to the above is

$$
\begin{bmatrix} F\_{\mathbf{k}\mathbf{j}}^{i} \end{bmatrix} \begin{bmatrix} \theta\_{\mathbf{k}}^{i} \end{bmatrix} + \begin{bmatrix} u\_{\mathbf{k}}^{i} \end{bmatrix}^{T} \begin{bmatrix} G\_{\mathbf{k}\mathbf{j}}^{i} \end{bmatrix} \begin{bmatrix} u\_{\mathbf{k}}^{i} \end{bmatrix} + \begin{bmatrix} H\_{\mathbf{k}\mathbf{j}}^{i} \end{bmatrix} \begin{bmatrix} \mathbf{c}\_{\mathbf{k}}^{i} \end{bmatrix} = \begin{bmatrix} Q\_{\mathbf{G}\mathbf{j}}^{i} \end{bmatrix} \tag{32}
$$

where *F<sup>i</sup>* kj <sup>¼</sup> *<sup>ρ</sup><sup>h</sup> α* 1 Pr *R* Ð *yi*þ<sup>1</sup> *yi dχ<sup>k</sup>* dy *dχ<sup>i</sup> j* dy dy, *Gi* kj ¼ � *<sup>ρ</sup><sup>h</sup> m* Ec *R* Ð *yi*þ<sup>1</sup> *yi dχ<sup>i</sup>* 1 dy � �<sup>2</sup> *<sup>d</sup>χ<sup>i</sup>* 1 dy � � *<sup>d</sup>χ<sup>i</sup>* 2 dy � � *<sup>d</sup>χ<sup>i</sup>* 1 dy � � *<sup>d</sup>χ<sup>i</sup>* 3 dy � � *dχ<sup>i</sup>* 2 dy � � *<sup>d</sup>χ<sup>i</sup>* 1 dy � � *<sup>d</sup>χ<sup>i</sup>* 2 dy � �<sup>2</sup> *<sup>d</sup>χ<sup>i</sup>* 2 dy � � *<sup>d</sup>χ<sup>i</sup>* 3 dy � � *dχ<sup>i</sup>* 3 dy � � *<sup>d</sup>χ<sup>i</sup>* 1 dy � � *<sup>d</sup>χ<sup>i</sup>* 3 dy � � *<sup>d</sup>χ<sup>i</sup>* 2 dy � � *<sup>d</sup>χ<sup>i</sup>* 3 dy � �<sup>2</sup> 2 6 6 6 6 6 6 6 6 6 4 3 7 7 7 7 7 7 7 7 7 5 *χ<sup>k</sup>* ½ �dy *Hi* kj <sup>¼</sup> ‐ *csh* DK*<sup>T</sup>* Du *R* Ð *yi*þ<sup>1</sup> *yi dχ<sup>k</sup>* dy *dχ<sup>i</sup> j* dy dy, *Q*<sup>6</sup> *i <sup>j</sup>* <sup>¼</sup> *<sup>ρ</sup><sup>h</sup> α* 1 Pr *<sup>R</sup> χ<sup>k</sup> dθ<sup>i</sup>* 2 dy � *csh* DK*<sup>T</sup>* Du *<sup>R</sup> χ<sup>k</sup>* dc*i* 2 dy h i*yi*þ<sup>1</sup> *yi* .

From Eq. (26) we get

$$\frac{h}{D} \frac{1}{\operatorname{Sc} R} \int\_{\gamma\_i}^{\gamma\_{i+1}} \frac{d\chi\_k}{\operatorname{dy}} \frac{\operatorname{dc}\_2^i}{\operatorname{dy}} \operatorname{dy} + \frac{h}{K\_T D} \operatorname{Sr} \int\_{\gamma\_i}^{\gamma\_{i+1}} \frac{d\chi\_k}{\operatorname{dy}} \frac{d\theta\_2^i}{\operatorname{dy}} \operatorname{dy} = \left[ \frac{h}{D} \frac{1}{\operatorname{Sc} R} \chi\_k \frac{\operatorname{dc}\_2^i}{\operatorname{dy}} + \frac{h}{K\_T D} \operatorname{Sr} \chi\_k \frac{d\theta\_2^i}{\operatorname{dy}} \right]\_{\gamma\_i}^{\gamma\_{i+1}}$$

The stiffness matrix equation corresponding to the above is

$$
\begin{bmatrix} \left[ L\_{\mathbf{k}\mathbf{j}}^{i} \right] \left[ \mathcal{E}\_{\mathbf{k}}^{i} \right] + \left[ \mathbf{M}\_{\mathbf{k}\mathbf{j}}^{i} \right] \left[ \theta\_{\mathbf{k}}^{i} \right] = \left[ Q\_{\mathcal{T}j}^{\,i} \right] \tag{33}
$$

where \$L\_{\mathbf{kj}}^{i} = \frac{h}{D} \frac{1}{\mathrm{Sc} \mathcal{R}} \int\_{\mathcal{Y}\_{i}}^{\mathcal{Y}\_{i+1}} \frac{d\chi\_{k}}{\mathrm{d}\mathbf{y}} \frac{d\chi\_{j}^{i}}{\mathrm{d}\mathbf{y}} \,\mathrm{d}\mathbf{y}, \,\mathcal{M}\_{\mathrm{kj}}^{i} = \frac{h}{K\_{TD}} \mathrm{Sr} \int\_{\mathcal{Y}\_{i}}^{\mathcal{Y}\_{i+1}} \frac{d\chi\_{k}}{\mathrm{d}\mathbf{y}} \frac{d\chi\_{j}^{i}}{\mathrm{d}\mathbf{y}} \,\mathrm{d}\mathbf{y} 
$$\mathcal{Q}\_{\mathcal{T}\_{j}}^{i} = \left[ \frac{h}{D} \frac{1}{\mathrm{Sc} \mathcal{R}} \chi\_{k} \frac{\mathrm{d}\mathbf{c}\_{2}^{i}}{\mathrm{d}\mathbf{y}} + \frac{h}{K\_{T}D} \mathrm{Sr} \chi\_{k} \frac{d\theta\_{2}^{i}}{\mathrm{d}\mathbf{y}} \right]\_{\mathcal{Y}\_{i}}^{\mathcal{Y}\_{i+1}}$$

The Langrange's interpolation polynomials are used as the shape functions at each of the nodes are considered as follows:

$$\begin{split} \eta\_{i}^{1} &= \frac{\left(y - \left(\frac{2i - 101}{100}\right)\right)\left(y - \left(\frac{2i - 100}{100}\right)\right)}{\left(\left(\frac{2i - 102}{100}\right) - \left(\frac{2i - 101}{100}\right)\right)\left(\left(\frac{2i - 102}{100}\right) - \left(\frac{2i - 100}{100}\right)\right)}, \\ \eta\_{i}^{2} &= \frac{\left(y - \left(\frac{2i - 102}{100}\right)\right)\left(y - \left(\frac{2i - 100}{100}\right)\right)}{\left(\left(\frac{2i - 101}{100}\right) - \left(\frac{2i - 102}{100}\right)\right)\left(\left(\frac{2i - 101}{100}\right) - \left(\frac{2i - 100}{100}\right)\right)}, \\ \eta\_{i}^{3} &= \frac{\left(y - \left(\frac{2i - 101}{100}\right)\right)\left(y - \left(\frac{2i - 102}{100}\right)\right)}{\left(\left(\frac{2i - 100}{100}\right) - \left(\frac{2i - 102}{100}\right)\right)\left(\left(\frac{2i - 100}{100}\right) - \left(\frac{2i - 101}{100}\right)\right)}. \end{split}$$

and similarly for *χ*<sup>1</sup> *<sup>i</sup>* , *χ*<sup>2</sup> *<sup>i</sup>* , *χ*<sup>3</sup> *i* .

The shear stress values, heat (Nusselt number) and mass transfer rate (Sherwood number) are calculated at both walls as per the following relations:

$$\begin{split} \mathbf{St}\_{1} &= \left[\frac{\partial \mathbf{u}\_{1}}{\partial \boldsymbol{\mathcal{V}}}\right]\_{\mathbf{y}=-1}, \mathbf{St}\_{2} = \left[\frac{\partial \mathbf{u}\_{2}}{\partial \boldsymbol{\mathcal{V}}}\right]\_{\mathbf{y}=1}, \mathbf{Nu}\_{1} = \left[\frac{\partial \boldsymbol{\theta}\_{1}}{\partial \boldsymbol{\mathcal{V}}}\right]\_{\mathbf{y}=-1}, \mathbf{Nu}\_{2} = \left[\frac{\partial \boldsymbol{\theta}\_{2}}{\partial \boldsymbol{\mathcal{V}}}\right]\_{\mathbf{y}=1}, \mathbf{Sh}\_{1} \\ &= \left[\frac{\partial \mathbf{c}\_{1}}{\partial \boldsymbol{\mathcal{V}}}\right]\_{\mathbf{y}=-1}, \mathbf{Sh}\_{2} = \left[\frac{\partial \mathbf{c}\_{2}}{\partial \boldsymbol{\mathcal{V}}}\right]\_{\mathbf{y}=1}. \end{split}$$

#### **5. Results and discussion**

The numerical solution of the system of equations is analyzed for several values of the governing factors and its corresponding graphical representations are resulted. Thermal Grashof (Gr), Molecular Grashof (Gc) and Reynolds numbers (R), Magnetic field (M) and Material parameters (K<sup>0</sup> ) and Dufour (Du), Schmidt (Sc), Soret (Sr) and Eckert numbers (Ec) are fixed as Gr = 5, Gc = 5, R = 3, M = 3,

#### **Figure 2.**

*(a) Represents behavior of u. (b) Represents behavior of N for Gr. (c) Represents behavior of θ. (d) Represents behavior of c for Gr.*

*Convective Heat and Mass Transfer of Two Fluids in a Vertical Channel DOI: http://dx.doi.org/10.5772/intechopen.94529*

K0 = 0.1, Du = 0.08, Sr. = 0.1, Sc = 0.66, Sr. = 0.001 for all the profiles excepting the varying parameter.

The profiles of all the governing parameters are depicted from **Figures 2**–**10**. The flow in micropolar region is found to be more than the flow in viscous region. The variations of linear momentum and angular momentum are clear for each and every governing parameter. The variation of temperature and diffusion are very narrow except for the parameters R, Du, Sr., and Sc. The temperature and diffusion are uniform across the channel and are found to be significant at the mid region of

**Figure 3.**

*(a) Represents behavior of u. (b) Represents behavior of N for Gc. (c) Represents behavior of θ. (d) Represents behavior of c for Gc.*

#### **Figure 4.**

*(a) Represents behavior of u. (b) Represents behavior of N for R. (c) Represents behavior of θ. (d) Represents behavior of c for R.*

**Figure 5.** *(a) Represents behavior of u. (b) Represents behavior of N for M. (c) Represents behavior of θ. (d) Represents behavior of c for M.*

**Figure 6.**

*(a) Represents behavior of u. (b) Represents behavior of N for K*<sup>0</sup> *. (c) Represents behavior of θ.*

*.*

*(d) Concentration profiles for K*<sup>0</sup>

the channel. The diffusion is slightly effected at the interface due to two fluids. Hence the two fluid flow model has much importance in the real time systems. All our results are compared with earlier studies and they are validated.

**Figure 2(a)**–**(d)** illustrate the effect of Grashof numbers on velocity, angular velocity, temperature and diffusion. As Gr increases the velocity and angular velocity increases substantially. The buoyancy enhances the flow in both regions i.e. thermal buoyancy force dominates the viscous force in both regions of the channel and it is found to be more in micropolar region. The lowest velocity corresponds to

#### *Convective Heat and Mass Transfer of Two Fluids in a Vertical Channel DOI: http://dx.doi.org/10.5772/intechopen.94529*

**Figure 7.**

*(a) Represents behavior of u. (b) Represents behavior of N for Du. (c) Represents behavior of θ. (d) Represents behavior of c for Du.*

**Figure 8.**

*(a) Represents behavior of u. (b) Represents behavior of N for Sr. (c) Represents behavior of θ. (d) Represents behavior of c for Sr.*

Gr = 2. Higher Gr values boost up the flow in both regions. As Gr increases the minute enhancement of temperature and diffusion are observed. Similar observations are noticed with all the variations of Gc which are plotted in **Figure 3(a)**–**(d)**.

**Figure 4(a)**–**(d)** describe the Reynolds number (R) impact on velocity, angular velocity, temperature and diffusion. The reduction of velocity is found with increase of Reynolds number due to domination of inertial force on viscous force in both regions of the channel and found more drastic in viscous region. Also reduces the micro rotation with increase of Reynolds number. The effect of inertial forces enhances the temperature and reduction of the diffusion is shown with increase of R.

**Figure 9.** *(a) Represents behavior of u. (b) Represents behavior of N for Sc. (c) Represents behavior of θ. (d) Represents behavior of c for Sc.*

**Figure 10.** *(a) Represents behavior of u. (b) Represents behavior of N for Ec. (c) Represents behavior of θ. (d) Represents behavior of c for Ec.*

**Figure 5(a)**–**(d)** describe the magnetic field (M) effect on velocity, angular velocity, temperature and diffusion. They portray that there could be seen reduction in velocity and angular velocity as M increases. It shows that magnetic field has a tendency to retard fluid velocity and angular velocity due to the formation of resistive Lorentz force, where when magnetic effect is applied to the fluid, it tends to retard the fluid motion. The magnetic field parametric impact on temperature and diffusion is minute.

**Figure 6(a)**–**(d)** explain the effect of material parameter (K<sup>0</sup> ) on velocity, angular velocity, temperature and diffusion. The effect of this parameter is very significant in both velocity and angular velocity, as K<sup>0</sup> increases the velocity decreases significantly and this is reversed with respect to angular velocity. Minute effect is observed for both temperature and diffusion.

**Figure 7(a)**–**(d)** explain the effect of Dufour number (Du) on velocity, angular velocity, temperature and diffusion. **Figure 7(a)** depict the fact that as Du increases, i.e. when molecular diffusivity increases, the velocity is reduced and leads to reduction of micro rotation also from **Figure 7(b)** it clearly indicates the influence of the concentration gradients to the thermal energy flux in the flow. **Figure 7(c)** specifies that the temperature reduces with increase of molecular diffusivity over the thermal diffusivity. It is clear that the diffusion profiles increase with increase of dufour number as observed in **Figure 7(d).**

**Figure 8(a)**–**(d)** relates the Soret number (Sr) impact on velocity, angular velocity, temperature and diffusion. **Figure 8(a)** shows that as Sr. increases i.e. thermal diffusivity increases the decrease in velocity is found and micro rotation also decreases with increase of Sr. from **Figure 8(b)**. Soret number states the impact of temperature gradients stimulating considerable mass diffusion effects. Here, as Soret number increases it leads to rise in temperature and shows the decay in the fluid concentration from **Figure 8(c)** and **(d)**.

Figure (a)–(d) specify the Schmidt number (Sc) effect on velocity, angular velocity, temperature and diffusion. **Figure 9(a)** and **(b)** show that as Sc increases, velocity and angular velocity decreases significantly. From **Figure 9(c)** and **(d)** it is found that the temperature increases with increase of Sc and fluid concentration reduces with increase in Schmidt number.

**Figure 10(a)**–**(d)** define the effect of Eckert number (Ec) on velocity, angular velocity, temperature and diffusion. From all the Figures it is concluded that the enthalpy is not having much influence over the flow for small variation of enthalpy. **Figure 10(a)** and **(b)** show that the velocity and angular velocity increase when Eckert number increases. So it is observed that momentum and angular momentum are inversely proportional to enthalpy. From **Figure 10(c)** the temperature increases with increase of kinetic energy. The kinetic energy reduces the concentration of the fluid as shown from **Figure 10(d)**.

**Table 1** shows the Shear stress and Nusselt and Sherwood numbers values with all the effects of all governing functions. From this table, it is observed that the absolute Shear stress enhances with increase in Gr on both the boundaries *y* ¼ �1 and *y* ¼ 1 because of buoyancy forces and similar nature is observed for Gc also. For increase of Reynolds number, magnetic field and Material parameter and Dufour, Soret and Schmidt numbers, the stress reduces on both the boundaries. This case is reversed for dissipation effect. The Nusselt number i.e. rate of heat transfer decreases on the boundary at *y* ¼ �1 and increases on the other boundary *y* ¼ 1 for the parameters Gr, Gc, R, Sr., Sc and Ec. The rise in convection is leading to reduction of heat transfer rate on the plate bounding the region �1, the reverse effect is observed for boundary of the region-2. Drastic heat transfer rate is observed for the variations of the Reynolds Number. The increase in the Reynolds number decreases the heat transfer rate on the left plate and enhances on the right plate. For the other parameters M, K<sup>0</sup> , Du the effect is reversal. The Sherwood number i.e. rate of mass transfer increase on the boundary at *y* ¼ �1 and decrease at the boundary *y* ¼ 1 for the parameters Gr, Gc, R, Sr., Sc, Ec. This is because the rise in convection and inertial forces leading to enhance the concentration. For the other parameters M, K<sup>0</sup> , Du the effect is reversal i.e. mass transfer increases at the left boundary and decreases at the right.



*Convective Heat and Mass Transfer of Two Fluids in a Vertical Channel DOI: http://dx.doi.org/10.5772/intechopen.94529*

**Table 1.**

*Shear stress, Nusselt number, Sherwood numbers.*
