*2.2. Boundary Conditions*

The set of boundary conditions that describe the interaction of lubricated walls with the bi-phase flow are:

• Boundary conditions at lower wall

$$u\_f(y) = \beta \left\{ \frac{\partial u\_f}{\partial y} - \frac{\eta\_1}{\mu\_s} \left( \frac{\partial^3 u\_f}{\partial y^3} \right) \right\}\_{\prime} \text{ when } y = -h \tag{13}$$

$$
u''\_{\,f}(y) = 0, \text{ when } y = -h \tag{14}$$

$$\Theta(y) = \Theta\_{0\prime} \text{ when } y = -h \tag{15}$$

• Boundary conditions at upper wall

$$u\_f(y) = -\beta \left\{ \frac{\partial u\_f}{\partial y} - \frac{\eta\_1}{\mu\_s} \left( \frac{\partial^3 u\_f}{\partial y^3} \right) \right\}, \text{ when } y = h \tag{16}$$

$$
\mu''\_{\,\,f}(y) = 0, \text{ when } y = h \tag{17}
$$

$$\Theta(y) = \Theta\_{l\prime} \text{ when } y = h \tag{18}$$

By using the dimensionless quantities:

$$\begin{aligned} \mu\_f &= \text{l\u1}\_f; \ \mu\_p = \text{l\u1}\_p; \ \ y = hy; \ \mathbf{x} = h\mathbf{x}; \ p = \frac{\mu\_\ast l\mathbf{I}}{h} p; \ B\_r = \frac{l\mathbf{1}^2 \mu\_\ast}{k(\Theta\_l - \Theta\_0)};\\ \gamma^2 &= \frac{h^2 \mu\_\ast}{\eta\_1}; \ M^2 = \frac{a\Theta\_0^2 h^2}{\mu\_\ast}; \ \ \beta\_1 = \frac{\beta}{h}; \ m = \frac{\mu\_\ast}{h^2 \mathcal{S}}; \ \ \Theta(\Theta\_l - \Theta\_0) = \Theta - \Theta\_0 \end{aligned} \tag{19}$$

Equations (10)–(12), after dropping the bars, can be obtained as:

$$\frac{1}{\gamma^2} \frac{\partial^4 u\_f}{\partial y^4} - \frac{\partial^2 u\_f}{\partial y^2} - \frac{\mathbb{C}}{m(1-\mathbb{C})} \left(u\_p - u\_f\right) + \frac{M^2}{(1-\mathbb{C})} u\_f + \frac{\partial p}{\partial \mathbf{x}} = 0 \tag{20}$$

$$\frac{\partial^2 \Theta}{\partial y^2} = \frac{B\_r}{\chi^2} \left(\frac{\partial u\_f}{\partial y}\right) \left(\frac{\partial^3 u\_f}{\partial y^3}\right) - B\_r \left(\frac{\partial u\_f}{\partial y}\right)^2 \tag{21}$$

where

$$
\mu\_p = \mu\_f - m \left(\frac{\partial p}{\partial \mathbf{x}}\right) \tag{22}
$$

As the original source of magnetized and heated bi-phase is on slippery walls and constant pressure gradient. Therefore, by taking of d*p*/d*x* = *P*, Equations (20) and (21) can be obtained as:

$$\frac{d^4 u\_f}{dy^4} - \chi^2 \frac{d^2 u\_f}{dy^2} + \frac{\chi^2 M^2}{(1 - \mathcal{C})} u\_f + \frac{\chi^2}{(1 - \mathcal{C})} P = 0 \tag{23}$$

$$\frac{d^2\Theta}{dy^2} + B\_r \left(\frac{du\_f}{dy}\right)^2 - \frac{B\_r}{\chi^2} \left(\frac{du\_f}{dy}\right) \left(\frac{d^3u\_f}{dy^3}\right) = 0\tag{24}$$

Similarly, in view of Equation (19), the corresponding boundary conditions given in Equations (13)–(18) in the dimensionless form are:

$$u\_f(y) = \beta\_1 \left\{ \frac{du\_f}{dy} - \frac{1}{\gamma^2} \left( \frac{d^3 u\_f}{dy^3} \right) \right\}, \text{ when } y = -1 \tag{25}$$

$$
u''\_{\,f}(y) = 0, \text{ when } y = -1\tag{26}$$

$$
\Theta(y) = 0, \text{ when } y = -1 \tag{27}
$$

$$u\_f(y) = -\beta\_1 \left\{ \frac{du\_f}{dy} - \frac{1}{\gamma^2} \left( \frac{d^3 u\_f}{dy^3} \right) \right\}, \text{ when } y = 1 \tag{28}$$

$$
u''\_{\,f}(y) = 0, \text{ when } y = 1\tag{29}$$

$$\Theta(y) = 1,\ when\ y = 1\tag{30}$$
