**3. Solution of the Problem**

The above obtained Equations (12)–(15) display the nonlinear ordinary differential equations in which ψ, θ, and ϕ are mutually dependent. Such types of problems cannot be handled by exact techniques. Therefore, we chose a more appropriate solution procedure, the homotopy perturbation method (HPM) [16,17] to solve the current highly complicated boundary value problems. The deformation equations for ψ, θ, and ϕ can be constructed as

$$\mathcal{L}\left(1-q'\right)\mathcal{L}\_1\Big(\hat{\psi}-\psi\_0\Big)+q'\Big[\frac{\partial^2}{\partial y^2}\Big(\frac{\partial^2\hat{\psi}}{\partial y^2}+\mathcal{M}\varepsilon\Big(\frac{\partial^2\hat{\psi}}{\partial y^2}\Big)^2-M^2\hat{\psi}\Big)+G\_r\frac{\partial\hat{\partial}}{\partial y}+G\_c\frac{\partial\phi}{\partial y}\Big]=0,\tag{18}$$

$$\mathcal{L}\left(1-q'\right)\mathcal{L}\_2\Big(\hat{\theta}-\theta\_0\Big)+q'\Big[\mathcal{N}\_b\frac{\partial\hat{\theta}}{\partial y}\frac{\partial\phi}{\partial y}+\mathcal{N}\_t\Big(\frac{\partial\hat{\theta}}{\partial y}\Big)^2+\frac{\partial^2\hat{\theta}}{\partial y^2}+Br\left(\frac{\partial^2\hat{\psi}}{\partial y^2}\right)^2+\mathcal{W}c\Big(\frac{\partial^2\hat{\psi}}{\partial y^2}\Big)^3\Big]=0,\tag{19}$$

$$(1 - q')\mathcal{L}z(\hat{\boldsymbol{\varphi}} - q\boldsymbol{\varphi}\_0) + q' \left[ \frac{\partial^2 \phi}{\partial \boldsymbol{y}^2} + \frac{N\_t}{N\_b} \frac{\partial^2 \varPhi}{\partial \boldsymbol{y}^2} \right] = \boldsymbol{0},\tag{20}$$

where £1 and £2 are linear operators which are picked as £

$$\mathfrak{E}\_1 = \frac{\partial^4}{\partial y^4} \text{ and } \mathfrak{E}\_2 = \frac{\partial^2}{\partial y^2} \tag{21}$$

and ψ0, <sup>θ</sup>0, and ϕ<sup>0</sup> are the initial approximations which must satisfy the boundary conditions as well as differential operator. The initial approximations for ψ, θ, and ϕ are elected as

$$\widehat{\psi\psi\_0} = \frac{\begin{pmatrix} \left(h\_{11} - h\_{12} - 2y\right) \ \left(-2\left(h\_{11} - h\_{12}\right) \ \left(h\_{11} - y\right) \ \left(h\_{12} - y\right)\right)}{2\left(h\_{11} - h\_{12}\right)^2} + F\left(\frac{h\_{11}^2 - 4h\_{11}h\_{12} + h\_{12}^2 + 2\left(h\_{11} + h\_{12}\right) \ y - 2y^2}{2\left(h\_{11} - h\_{12}\right)^2}\right) \tag{22}$$

$$\widehat{\theta\_0} = \frac{B\_i h\_{12} - B\_i y}{1 - B\_i h\_{11} + B\_i h\_{12}} \tag{23}$$

$$
\widehat{\varphi\_0} = \frac{-h\_{12} + y}{h\_{11} - h\_{12}} \tag{24}
$$

Applying perturbation on small embedding parameters *F* ∈ [0, 1], we suggest the following series solutions

$$
\dot{\psi} = \psi\_0 + q'\psi\_1 + q'^2\psi\_2 \dots \tag{25}
$$

$$
\widehat{\theta} = \theta \mathbf{o} + \boldsymbol{q}' \theta\_1 + \boldsymbol{q}'^2 \theta\_2 \dots \tag{26}
$$

$$
\widehat{\!\!\!\!\!\!\!\!\/} \widehat{\!\!\!\!\/} = \!\!\!\!\!\!\!\!\!/ \!\!\!\/+ \!\!\!\!\!\/) \hat{\!\!\!\!\/+} \hat{\!\!\!\!\/) \hat{\!\!\!\!\/} \hat{\!\!\!\/) \hat{\!\!\!\/} \hat{\!\!\!\/] \hat{\!\!\!\/} \hat{\!\!\!\/] \hat{\!\!\!\/} \hat{\!\!\!\/] \hat{\!\!\!\/} \hat{\!\!\!\/] \hat{\!\!\!\/} \hat{\!\!\!\/] \hat{\!\!\!\/} \hat{\!\!\!\/] \hat{\!\!\!\/} \hat{\!\!\!\/] \hat{\!\!\/} \hat{\!\!\!\/] \hat{\!\!\/} \hat{\!\!\!\/] \hat{\!\!\/} \hat{\!\!\/] \hat{\!\!\/} \hat{\!\!\/] \hat{\!\!\/} \hat{\!\!\/] \hat{\!\!\/}} \hat{\!\!\/] \hat{\!\!\/} \hat{\!\!\/] \hat{\!\!\/} \hat{\!\!\/] \hat{\!\!\/} \hat{\!\!\/] \hat{\!\!\/} \hat{\!\!\/] \hat{\!\!\/}} \hat{\!\!\/] \hat{\!\!\/} \hat{\!\!\/] \hat{\!\!\/} \hat{\!\!\/] \hat{\!\!\/} \hat{\!\!\/] \hat{\!\!\/} \hat{\!\!\/] \hat{\!\!\/} \hat{\!\!\/] \hat{\!\!\/} \hat{\!\!\/} \hat{\!\!\/] \hat{\!\!\/} \hat{\!\!\/} \hat{\!\!\/} \hat{\!\!\/$$

After substituting the above series solutions in Equations (18)–(20), we get the two systems for ψ, θ, and ϕ.

• Zeroth Order System

$$\begin{aligned} \mathcal{E}\_1[\psi\_0 - \widehat{\psi}\_0] &= 0, \\ \psi\_0 = \frac{F}{2}, \frac{\partial \psi\_0}{\partial y} = -1, \text{ at } y = h\_{1\prime} \text{ } \psi\_0 = -\frac{F}{2}, \frac{\partial \psi\_0}{\partial y} = -1, \text{ at } y = h\_{2\prime} \end{aligned} \tag{28}$$

$$\begin{aligned} \mathcal{E}\_2[\theta\_0 - \widehat{\theta\_0}] &= 0, \\ \theta\_0(h\_1) - B\_i \theta\_0(h\_1) &= -B\_i \text{ at } y = h\_i \text{ and } \theta\_0 = 0 \text{ at } y = h\_2 \end{aligned} \tag{29}$$

$$\begin{aligned} \mathcal{E}\_2[q\_0 - \widehat{q\_0}] &= 0, \\ q\_0 = 1, \text{ at } y = h\_1 \text{ and } q\_0 = 0 \text{ at } y = h\_{2, \epsilon} \end{aligned} \tag{30}$$

• First Order System

$$\begin{cases} \mathcal{L}\_1[\psi] + \frac{\partial^2}{\partial y^2} \left[ \frac{\partial^2 \psi\_0}{\partial y^2} + \mathcal{W} \mathcal{e} \left( \frac{\partial^2 \psi\_0}{\partial y^2} \right)^2 - \mathcal{M}^2 \psi\_0 \right] + G\_r \frac{\partial \mathcal{O}\_0}{\partial y} + G\_\mathbf{c} \frac{\partial \psi\_0}{\partial y} = 0, \\ \psi\_1 = 0, \ \frac{\partial \psi\_1}{\partial y} = 0, \ \text{at } y = h\_1 \text{ and } \psi\_1 = 0, \ \frac{\partial \psi\_1}{\partial y} = 0, \ \text{at } y = h\_2 \end{cases} \tag{31}$$

$$\begin{aligned} \left\{ \begin{aligned} \left[ \mathcal{E}\_2(\theta\_1) + N\_b \right] \frac{\partial \psi\_0}{\partial y} \cdot \frac{\partial \theta\_0}{\partial y} \right\} + N\_l \Big( \frac{\partial \theta\_0}{\partial y} \Big)^2 + \frac{\partial^2 \theta\_0}{\partial y^2} + Br \Big[ \left( \frac{\partial^2 \psi\_0}{\partial y^2} \right)^2 + \mathcal{W} \epsilon \Big( \frac{\partial^2 \psi\_0}{\partial y^2} \Big)^3 \right] = 0, \\ \theta\_1'(h\_1) - B\_l \theta\_1(h\_1) = 0 \text{ at } y = h\_1 \text{ and } \theta\_1 = 0 \text{ at } y = h\_2 \end{aligned} \tag{32}$$

$$\begin{aligned} \mathfrak{E}\_2(\wp\_1) + \frac{\partial^2 \wp\_0}{\partial y^2} + \frac{N\_t}{N\_b} \frac{\partial^2 \wp\_0}{\partial y^2} &= 0, \\ \wp\_1 = 0, \text{ at } y = h\_1, \text{ } \wp\_1 = 0 \text{ at } y = h\_2. \end{aligned} \tag{33}$$

• Zeroth Order Solutions

By solving zeroth order systems by built-in technique in mathematical software, we obtain

$$\begin{split} \psi\_{0} = \widehat{\psi}\_{0} &= \frac{\begin{pmatrix} h\_{11} + h\_{12} - 2y \ \left( \begin{array}{c} -2\left( h\_{11} - h\_{12} \right) \ (h\_{11} - y \end{array} \right) h\_{12} - y \end{pmatrix}}{2(h\_{11} - h\_{12})} + \\ & \frac{\begin{array}{c} F\left( h\_{11}^{2} - 4h\_{11}h\_{12} + h\_{12}^{2} + 2\left( h\_{11} + h\_{12} \right) \ y \right) - 2y \end{array}}{2\left(h\_{11} - h\_{12}\right)^{2}} \end{split} \tag{34}$$

$$\hat{\theta}\_0 = \hat{\theta}\_0 = \frac{B\_i h\_2 - B\_i y}{1 - B\_i h\_1 + B\_i h\_2} \tag{35}$$

$$
\varphi\_0 = \widehat{\varphi\_0} = \frac{-h\_{12} + y}{h\_{11} - h\_{12}} \tag{36}
$$

2 3

*Wey*<sup>5</sup> + *L*<sup>15</sup> + *yL*<sup>16</sup>

• First Order Solutions

*<sup>h</sup>*12*We*))*y*4)] <sup>−</sup> <sup>432</sup>

<sup>5</sup> *Br*-

The first order system has acquired the following general solutions

ψ<sup>1</sup> = <sup>−</sup><sup>1</sup> <sup>6</sup>(−1+*Bi*(*h*11−*h*12) 7 ) [1/4*Gc*(−1 + *Bi*(*h*<sup>11</sup> − *h*12) <sup>6</sup> <sup>−</sup> <sup>6</sup>(*<sup>F</sup>* <sup>+</sup> *<sup>h</sup>*<sup>11</sup> <sup>−</sup> *<sup>h</sup>*12)(−*h*11*M*<sup>2</sup> <sup>+</sup> <sup>2</sup>*h*<sup>3</sup> 11 *<sup>h</sup>*12*M*<sup>2</sup> <sup>−</sup> <sup>2</sup>*h*11*h*<sup>3</sup> <sup>12</sup>*M*<sup>2</sup> <sup>+</sup> *<sup>h</sup>*<sup>4</sup> <sup>12</sup>*M*<sup>2</sup> <sup>+</sup> <sup>48</sup>*We*(*<sup>F</sup>* <sup>+</sup> *<sup>h</sup>*<sup>11</sup> <sup>−</sup> *<sup>h</sup>*12) <sup>+</sup> *Bi*(*h*<sup>11</sup> <sup>−</sup> *<sup>h</sup>*12)(*Gr*(*h*<sup>11</sup> <sup>−</sup> *<sup>h</sup>*12) 5 + 6(*F* + *h*<sup>11</sup> − *h*12) - <sup>−</sup>*h*<sup>4</sup> <sup>11</sup>*M*<sup>2</sup> <sup>+</sup> <sup>2</sup>*h*<sup>3</sup> 11*h*12*M*<sup>2</sup> + +*h*<sup>4</sup> <sup>12</sup>*M*<sup>2</sup> <sup>+</sup> <sup>48</sup>*We*(*<sup>F</sup>* <sup>+</sup> *<sup>h</sup>*<sup>11</sup> <sup>−</sup> *<sup>h</sup>*12) ))*y*4+ 3 5 - −1 + *Bi*(*h*<sup>11</sup> − *h*12) 4 (*<sup>F</sup>* + *<sup>h</sup>*<sup>11</sup> <sup>−</sup> *<sup>h</sup>*12) *<sup>M</sup>*<sup>2</sup> *<sup>y</sup>*<sup>5</sup> ] + *L*<sup>11</sup> + *yL*<sup>12</sup> + *y*2*L*<sup>13</sup> + *y*3*L*<sup>14</sup> (37) θ<sup>1</sup> = <sup>1</sup> (−1+*Bi*(*h*11−*h*12))<sup>2</sup> (*h*11−*h*12) <sup>9</sup> [1/2*Bi*(*h*<sup>11</sup> − *h*12) 8 ((−1 + *Bi*(*h*<sup>11</sup> − *h*12)) *Nb*+ *Bi*(*h*<sup>11</sup> − *h*12) *Nt*)Pr + 36*Br*(−1 + *Bi*(*h*<sup>11</sup> − *h*12)) 2 (*F* + *h*<sup>11</sup> − *h*12) 2 (*h*<sup>11</sup> − *h*12) 2 (*h*<sup>3</sup> <sup>11</sup>− 3*h*<sup>2</sup> <sup>12</sup>(*h*<sup>12</sup> <sup>−</sup> <sup>2</sup>*We*) + <sup>3</sup>*h*11- *h*2 <sup>12</sup> <sup>+</sup> <sup>2</sup>*FWe* <sup>−</sup> *<sup>h</sup>*12- *h*2 <sup>12</sup> <sup>−</sup> <sup>6</sup>*We* <sup>+</sup> <sup>6</sup>*h*12*We* ))*y*<sup>2</sup> <sup>−</sup> <sup>24</sup>*Br* (−<sup>1</sup> <sup>+</sup> *Bi*(*h*<sup>11</sup> <sup>−</sup> *<sup>h</sup>*12))<sup>2</sup> (*F* + *h*<sup>11</sup> − *h*12) 2 (*h*<sup>11</sup> + *h*12) *h*<sup>3</sup> <sup>11</sup> <sup>−</sup> <sup>3</sup>*h*<sup>2</sup> <sup>11</sup>(*h*<sup>12</sup> <sup>−</sup> <sup>3</sup>*We*) <sup>+</sup> <sup>3</sup>*h*<sup>12</sup> - *h*2 <sup>12</sup> <sup>+</sup> <sup>3</sup>*FWe* (−*h*12(*h*<sup>2</sup> <sup>12</sup> <sup>−</sup> <sup>9</sup>*FWe* <sup>+</sup> <sup>9</sup>*h*12*We*))*y*<sup>3</sup> <sup>+</sup> <sup>12</sup>*Br*(−<sup>1</sup> <sup>+</sup> *Bi*(*h*<sup>11</sup> <sup>−</sup> *<sup>h</sup>*12))2(*F*<sup>+</sup> *<sup>h</sup>*<sup>11</sup> <sup>−</sup> *<sup>h</sup>*12)2(*h*<sup>3</sup> <sup>11</sup> <sup>−</sup> <sup>3</sup>*h*<sup>2</sup> <sup>11</sup>(*h*<sup>12</sup> <sup>−</sup> <sup>6</sup>*We*) <sup>+</sup> <sup>3</sup>*h*11- *h*2 <sup>12</sup> <sup>+</sup> <sup>6</sup>*FWe* <sup>−</sup> *<sup>h</sup>*12(*h*<sup>2</sup> <sup>12</sup> − 18*FWe* + 18 (38)

−1 + *Bi*(*h*<sup>11</sup> − *h*12) (*F* + *h*<sup>11</sup> − *h*12)

ϕ<sup>1</sup> = <sup>1</sup> (−1+*Bi*(*h*11−*h*12))<sup>2</sup> (*h*11−*h*12) 9 *Nb Nt*[1/3*Bi*(*h*<sup>11</sup> − *h*12) 8 ((−1 + *Bi*(*h*<sup>11</sup> − *h*12)) *Nb*+ *Bi*(*h*<sup>11</sup> <sup>−</sup> *<sup>h</sup>*12) *Nt*)Pr <sup>+</sup> <sup>36</sup>*Br*(−<sup>1</sup> <sup>+</sup> *Bi*(*h*<sup>11</sup> <sup>−</sup> *<sup>h</sup>*12))<sup>2</sup> (*F* + *h*<sup>11</sup> − *h*12) 2 (*h*<sup>11</sup> − *h*12) 2 (*h*<sup>3</sup> 11 <sup>−</sup>3*h*<sup>2</sup> 11- *h*<sup>12</sup> − 2*We* + 3*h*<sup>1</sup> - *h*2 <sup>12</sup> <sup>+</sup> <sup>2</sup>*FWe* <sup>−</sup> *<sup>h</sup>*12- *h*2 <sup>12</sup> <sup>−</sup> <sup>6</sup>*FWe* <sup>+</sup> <sup>6</sup>*h*12*We*)*Y*<sup>2</sup> <sup>−</sup> <sup>24</sup>*Br*(−<sup>1</sup> +*Bi*(*h*<sup>11</sup> <sup>−</sup> *<sup>h</sup>*12))2(*<sup>F</sup>* + *<sup>h</sup>*<sup>11</sup> <sup>−</sup> *<sup>h</sup>*12) 2 (*h*<sup>11</sup> + *h*12) *h*<sup>3</sup> <sup>11</sup> <sup>−</sup> <sup>3</sup>*h*<sup>2</sup> <sup>12</sup>(*h*<sup>12</sup> <sup>−</sup> <sup>3</sup>*We*) <sup>+</sup> *<sup>h</sup>*11- *h*2 <sup>12</sup> <sup>+</sup> <sup>3</sup>*FWe* - <sup>−</sup>*h*12- *h*2 <sup>12</sup> <sup>−</sup> <sup>9</sup>*FWe* <sup>+</sup> <sup>9</sup>*h*2*We <sup>Y</sup>*<sup>3</sup> + <sup>12</sup>*Gc*(−<sup>1</sup> + *Bi*(*h*<sup>11</sup> <sup>−</sup> *<sup>h</sup>*12) 2 (*F* + *h*<sup>11</sup> − *h*12) 2 - *h*3 <sup>11</sup> <sup>−</sup> <sup>3</sup>*h*<sup>2</sup> <sup>11</sup>(*h*<sup>12</sup> <sup>−</sup> <sup>6</sup>*We*) <sup>+</sup> <sup>3</sup>*h*<sup>2</sup> <sup>11</sup>(*h*<sup>12</sup> <sup>−</sup> <sup>6</sup>*We*) <sup>+</sup> <sup>3</sup>*h*11- *h*2 <sup>11</sup> <sup>+</sup> <sup>6</sup>*FWe* <sup>−</sup> *<sup>h</sup>*12- *h*2 <sup>12</sup> <sup>−</sup> <sup>18</sup>*FWe <sup>y</sup>*4] <sup>−</sup> - 432 <sup>5</sup> *Br*(−<sup>1</sup> <sup>+</sup> *Bi*(*h*<sup>11</sup> <sup>−</sup> *<sup>h</sup>*12))<sup>2</sup> (*F* + *h*<sup>11</sup> − *h*12) 2 3 *Wey*<sup>5</sup> + *L*<sup>17</sup> + *yL*<sup>18</sup> (39)

The final solutions according to the concept of HPM are given by using *q* → 1 in Equations (2)–(27).

$$
\psi = \psi\_0 + \psi\_1 + \dots \tag{40}
$$

$$
\theta = \theta\_0 + \theta\_1 + \dots \tag{41}
$$

$$
\varphi = \varphi\_0 + \varphi\_1 + \dots \tag{42}
$$

where constants *Lij*, *i* = 1, *j* = 1 − 8 can be found by routine calculation. The complete solutions of ψ, θ, and ϕ can be obtained by supposed solutions. The solution for pressure gradient *dp*/*dx* can be found by simply substituting the values in Equation (12). The mathematical formula for the pressure increase function Δ*p* can been visualized in next equation that has been solved numerically by built-in technique numerical integration on Mathematica.

$$
\Delta p = \int\_0^1 \left(\frac{dp}{d\mathbf{x}}\right) d\mathbf{x} \tag{43}
$$
