**3. Exact Solution of Problem**

Exact solution of Equation (19) satisfying boundary conditions (22) can be deduced as:

$$\begin{array}{l} \mathbf{V} = \frac{1}{\Gamma\_{m}} \Big( 2 \cosh((h\_{1} - h\_{2}) \, m) \Big( (-h\_{1} - h\_{2} + 2y) \Big( F\eta\_{1} m^{2} + 1 \Big) - \eta\_{2} m^{2} (F + h\_{1} - h\_{2}) \Big) \\ + 2 (\eta\_{2} m^{2} (F - h\_{2}) (\cosh(m(y - h\_{1})) + \cosh(m(y - h\_{2})) - 1) - F\eta\_{1} m (\sinh(m(y - h\_{1})) \\ + \sinh(m(y - h\_{2}))) + F\cosh(m(y - h\_{1})) - F\cosh(m(y - h\_{2})) + h\_{1} (\eta\_{2} m^{2} (\cosh(m(y - h\_{1})) \\ + \cosh(m(y - h\_{2})) - 1) - \eta\_{1} m (\sinh(m(y - h\_{1})) + \sinh(m(y - h\_{2}))) + \cosh(m(y - h\_{1})) \\ - \cosh(m(y - h\_{2})) + 1) + h\_{2} (\eta\_{1} m (\sinh(m(y - h\_{1})) + \sinh(m(y - h\_{2}))) - \cosh(m(y - h\_{1})) \\ + \cosh(m(y - h\_{2})) + 1) - 2y \Big) + m(h\_{1} + h\_{2} - 2y) (F\eta\_{2}^{2} m^{4} - \eta\_{1} \Big( F\eta\_{1} m^{2} + 2 \Big) \\ - F\rfloor \sinh((h\_{1} - h\_{2}) \, m \Big), \end{array} \tag{25}$$

where *<sup>m</sup>* <sup>=</sup> *<sup>M</sup>* cos <sup>Θ</sup>, <sup>η</sup><sup>1</sup> <sup>=</sup> <sup>η</sup><sup>∗</sup> 1 (1+λ<sup>1</sup> ),η<sup>2</sup> <sup>=</sup> <sup>η</sup><sup>∗</sup> 1 (1+λ<sup>1</sup> ) and *<sup>L</sup>*<sup>∞</sup> is a function of *<sup>x</sup>* defined in the Appendix A. Now making use of Equation (25) in Equation (20), the exact solution of Equation (20) is derived as:

$$\begin{array}{l} \theta = & \frac{m^2 \text{Fe} \{\text{F} + \text{h}\_1 - \text{h}\_2\}}{l\_0} \Big( 2 \Big( 2m^2 y^2 (\cosh((\text{h}\_1 - \text{h}\_2) \, m) - 1) + \cosh(m(-\text{h}\_1 - \text{h}\_2 + 2y)) \Big) \\ & + m(\eta\_1 (\eta\_1 \, m(4m^2 y^2 (\cosh((\text{h}\_1 - \text{h}\_2) \, m) + 1) - \cosh(2m(y - \text{h}\_1)) - \cosh(2m(y - \text{h}\_2))) \\ & - 2 \cosh(m(-\text{h}\_1 - \text{h}\_2 + 2y))) + 2 (4m^2 y^2 \sinh((\text{h}\_1 - \text{h}\_2) \, m) + \sinh(2m(y - \text{h}\_1))) \\ & - \sinh(2m(y - \text{h}\_2)))) + \eta\_2^2 m^2 (-(2(2m^2 y^2 (\cosh((\text{h}\_1 - \text{h}\_2) \, m) + 1) \\ & + \cosh(m(-\text{h}\_1 - \text{h}\_2 + 2y)))) + \cosh(2m(y - \text{h}\_1)) + \cosh(2m(y - \text{h}\_2))) \\ & + 2 \eta\_2 m (\eta\_1 \, m(\sinh(2m(y - \text{h}\_1)) + \sinh(2m(y - \text{h}\_2)) + 2 \sinh(m(-\text{h}\_1 - \text{h}\_2 + 2y))) \\ & - \cosh(2m(y - \text{h}\_1)) + \cosh(2m(y - \text{h}\_2))) - \cosh(2m(y - \text{h}\_1)) \\ & - \cosh(2m(y - \text{h}\_2))) + A\_1 y + A\_0. \end{array} \tag{26}$$

where *A*<sup>0</sup> ,*A*<sup>1</sup> are functions of x and their values are computed by means of Equation (23) as:

$$\begin{array}{lcl} A\_0 &=& \frac{1}{8\left(2\beta + h\_1 - h\_2\right)\left(\lambda\_1 + 1\right)} \left(8\left(\beta + h\_1\right)\left(\lambda\_1 + 1\right) + \\ & \frac{1}{\lambda\_3} \left(m^2 \text{EcPr}\left(F + h\_1 - h\_2\right)^2 L\_9 L\_6^{-2} - 2m^2 \text{EcPr}L\_{11}\left(\beta - h\_2\right) \left(F + h\_1 - h\_2\right)^2 L\_6 L\_4 \\ & + m^2 \text{EcPr}\left(F + h\_1 - h\_2\right)^2 L\_{10} L\_4^{-2}\right)), \\\\ A\_1 &=& \frac{4m^4 \text{EcPr}\left(F + h\_1 - h\_2\right)^2}{8\left(\lambda\_1 + 1\right)\left(2\beta + h\_1 - h\_2\right)^2} \left(\eta\_2^2 \left(h\_1 + h\_2\right) m^4 \left(2\beta + h\_1 - h\_2\right) \left(\cosh\left(\left(h\_1 - h\_2\right)m\right) + 1\right)\right) \end{array} \tag{27}$$

$$\begin{array}{ll} A\_{1} = & \frac{\sinh\left(\frac{1}{2} + \frac{h\_{1} - h\_{2}}{h\_{2}}\right)}{8\left(\lambda\_{1} + 1\right)\left(2\beta + h\_{1} - h\_{2}\right)} \frac{1}{L\delta} \left(\eta\_{2}^{2} \left(h\_{1} + h\_{2}\right) m^{4} \left(2\beta + h\_{1} - h\_{2}\right) \left(\cosh\left(\left(h\_{1} - h\_{2}\right)m\right) + 1\right) \right.\\ & \left. - 2\left(h\_{1} + h\_{2}\right) \left(2\beta + h\_{1} - h\_{2}\right) \left(\eta\_{1}m \cosh\left(\frac{1}{2}\left(h\_{1} - h\_{2}\right)m\right) + \sinh\left(\frac{1}{2}\left(h\_{1} - h\_{2}\right)m\right)\right)^{2} \\ & - 4\eta\_{2} \cosh\left(\frac{1}{2}\left(h\_{1} - h\_{2}\right)m\right) \left(2\beta m \cosh\left(\left(h\_{1} - h\_{2}\right)m\right) + \sinh\left(\left(h\_{1} - h\_{2}\right)m\right)\right) \\ & \left. \left(\eta\_{1}m \cosh\left(\left(\left(h\_{1} - h\_{2}\right)m\right)\right) + \sinh\left(\frac{1}{2}\left(h\_{1} - h\_{2}\right)m\right)\right) \right) - \frac{8\left(\lambda\_{1} + 1\right)}{8\left(\lambda\_{1} + 1\right)\left(2\beta + h\_{1} - h\_{2}\right)}. \end{array} \tag{28}$$

With the help of Equation (26), exact solution of the concentration profile in Equation (21) is concluded as:

$$\begin{array}{l} \Phi = & \frac{n^2 \text{Pr} \text{S} \text{sinc} \text{E} \left( \frac{\text{s} + \text{h}\_1 - \text{h}\_2}{\text{L}} \right)^2}{L\_0} \Big( -2 (2m^2 y^2 (\cosh((\text{h}\_1 - \text{h}\_2) \, m) - 1) + \cosh(m(-\text{h}\_1 - \text{h}\_2 + 2y))) \Big) \\ & + m(\eta\_1 (\eta\_1 m (2(\cosh(m(\text{h}\_1 - \text{h}\_2 + 2y)) - 2m^2 y^2 (\cosh((\text{h}\_1 - \text{h}\_2) \, m) + 1)) + \cosh(2m(y - \text{h}\_1))) \\ & + \cosh(2m(y - \text{h}\_2))) - 2 (4m^2 y^2 \sinh((\text{h}\_1 - \text{h}\_2) \, m) + \sinh(2m(y - \text{h}\_1)) - \sinh(2m(y - \text{h}\_2)))) \\ & + \eta\_2^2 m^3 (2(2m^2 y^2 (\cosh((\text{h}\_1 - \text{h}\_2) \, m) + 1) + \cosh(m(-\text{h}\_1 - \text{h}\_2 + 2y)))) \\ & + \cosh(2m(y - \text{h}\_2))) - 2 \eta\_2 m (\eta\_1 m (\sinh(2m(y - \text{h}\_1)) + \sinh(2m(y - \text{h}\_2))) \\ & + 2 \sinh(m(-\text{h}\_1 - \text{h}\_2 + 2y))) - \cosh(2m(y - \text{h}\_1)) + \cosh(2m(y - \text{h}\_2)))) \\ & + \cosh(2m(y - \text{h}\_1)) + \cosh(2m(y - \text{h}\_2)))) \end{array} \tag{29}$$

where *A*<sup>2</sup> ,*A*<sup>3</sup> are functions of x and their values are computed by means of Equation (24) as:

$$\begin{array}{llll} A\_{2} = & \frac{1}{8\left(2\gamma + h\_{1} - h\_{2}\right)\left(\lambda\_{1} + 1\right)} \left(8\left(\gamma + h\_{1}\right)\left(\lambda\_{1} + 1\right) + \frac{1}{L\_{3}}\right) \left(m^{2} \text{PrScSrErc}\left(\mathcal{F} + h\_{1} - h\_{2}\right)^{2}L\_{2}L\_{4}\right) \\ & + m^{2} \text{PrScSrErc}\left(\mathcal{F} + h\_{1} - h\_{2}\right)^{2}L\_{1}L\_{6}^{2} - 2m^{2} \text{PrScSrErc}L\_{7}\left(\mathcal{F} + h\_{1} - h\_{2}\right)^{2}L\_{3}L\_{4}\right) \end{array} \tag{30}$$

$$\begin{array}{ll} A\_{3} = & \frac{4m^{4} \text{PScSrEr}\left(\frac{\mathbf{r} + \mathbf{h}\_{1} - \mathbf{h}\_{2}}{2}\right)^{2}}{8\left(\lambda\_{1} + 1\right)\left(2\gamma + h\_{1} - h\_{2}\right)L\_{8}} \left(\eta\_{2}^{2}\left(h\_{1} + h\_{2}\right)m^{4}\left(2\gamma + h\_{1} - h\_{2}\right)\left(-\left(\cosh\left(\left(h\_{1} - h\_{2}\right)m\right) + 1\right)\right)}\\ & + 2\left(h\_{1} + h\_{2}\right)\left(2\gamma + h\_{1} - h\_{2}\right)\left(\eta\_{1}m \cosh\left(\frac{1}{2}\left(h\_{1} - h\_{2}\right)m\right) + \sinh\left(\frac{1}{2}\left(h\_{1} - h\_{2}\right)m\right)\right)^{2}}{4\eta\_{2}\cosh\left(\frac{1}{2}\left(h\_{1} - h\_{2}\right)m\right)\left(2\gamma m \cosh\left(\left(h\_{1} - h\_{2}\right)m\right) + \sinh\left(\left(h\_{1} - h\_{2}\right)m\right)\right)}\\ & + 4\eta\_{2}\cosh\left(\frac{1}{2}\left(h\_{1} - h\_{2}\right)m\right)\left(2\gamma m \cosh\left(\left(h\_{1} - h\_{2}\right)m\right) + \sinh\left(\left(h\_{1} - h\_{2}\right)m\right)\right) & -\frac{8\left(\lambda\_{1} + 1\right)}{8\left(\lambda\_{1} + 1\right)\left(2\gamma + h\_{1} - h\_{2}\right)}.\end{array} \tag{31}$$

It should be noted that *L*<sup>0</sup> − *L*<sup>11</sup> appeared in Equation (26) through (31) and are functions of *x* defined in the Appendix A.
