*Similarity Transformations*

The dimensionless variable *f* and similarity variable η for transformation as

$$\begin{aligned} \psi\_r f(\eta) &= \psi(x, y, t) \Big(\frac{\text{tr}}{1 - \text{ht}}\Big)^{-\frac{1}{2}}, \eta = \sqrt{\frac{c}{v(1 - \text{ht})}} y, h(t) = \sqrt{\frac{v(1 - \text{ht})}{c}};\\ \theta(\eta) &= T\_0 - T(x, y, t) \Big(\frac{\text{tr}^2}{2v(1 - \text{nt})^{-\frac{3}{2}}} \Big(T\_{ref}\Big)\Big)^{-1} \end{aligned} \tag{25}$$

Here ψ(*<sup>x</sup>*..., *y*...,*t*...) indicate the stream function which identically satisfying Equation (16), *<sup>h</sup>*(*t*). identifies the thin film thickness. The velocity components in term of stream function are obtained as

$$
\mu = \psi\_y = \left(\frac{c\mathbf{x}}{1 - bt}\right) f'(\eta), \\
\upsilon = -\psi\_\mathbf{x} = -\left(\frac{\upsilon c}{1 - bt}\right)^{\frac{1}{2}} f(\eta), \tag{26}
$$

Inserting the similarity transformation Equation (25) into Equations (16)–(18) and Equation (23) fulfills the continuity Equation (16)

$$\left(\varepsilon f^{\prime\prime\prime} + f f^{\prime\prime} + n \xi (-f^{\prime\prime})^{n-1} f^{\prime\prime\prime} - (f^{\prime})^2 - St \big(\frac{1}{2} \eta f^{\prime\prime} + f^{\prime}\big) - M f^{\prime} = 0\_{\ldots\prime} \tag{27}$$

$$(1+Rd)\theta'' - \Pr\left(\frac{S}{2}(3\theta+\eta\theta') + 2f'\theta - \theta'f\right) = \,\_0\tag{28}$$

The boundary constrains of the problem are:

$$\begin{array}{lclcl}f'(0)&=1, \ f(0)&=0, \theta(0)&=1,\\ f''(\beta)&=0, \theta'(\beta)&=0,\\ f(\beta)&=\frac{\text{S}\theta}{\text{T}}\end{array} \tag{29}$$

The dimensionless film thickness β = *b* <sup>υ</sup>(<sup>1</sup>−α*<sup>t</sup>*) *h*(*t*) which gives

$$h\_t = -\frac{\alpha\beta}{2} \sqrt{\frac{\nu}{b(1-\alpha t)}}\tag{30}$$

The physical constraints after generalization are obtained as, St = β*c* is the non-dimensional measure of unsteadiness, ε = *a*ρν is a Sisko fluid parameter, ξ = *b*ρν ⎛⎜⎜⎜⎜⎝- *cx* (<sup>1</sup>−*bt*) √ν 32 ⎞⎟⎟⎟⎟⎠*n*−1 is stretching parameter, and *M* = <sup>σ</sup>*f B*20 *<sup>b</sup>*ρ*f* represents the magnetic, Pr = ρυ*cp k* = μ*cp k* is the Prandtl number and Rd =4σ*T*3*<sup>s</sup> kk*∗ represent the radiation parameter.

The Skin friction is defined as

$$\mathcal{L}\_f = \frac{\left(\mathcal{S}\_{xy}\right)\_{y=0}}{\rho \mathcal{U}\_w^2},$$

where

$$S\_{xy} = \mu\_0 (a + b \left| u\_y \right|^n u\_y) y = 0 \tag{32}$$

$$
\mathbb{C}\_f \sqrt{\text{Re}\_x} = \left[ \varepsilon f^{\prime \prime}(0) - (-f^{\prime \prime}(0))^m \right] \tag{33}
$$

where *Rex* is known as the local Reynolds number defined as *Rex* = *Uwx* ν . The Nusselt number is defined as *u* = δ*Qw* ˆ *<sup>k</sup>*(*<sup>T</sup>*−*T*δ), in, which *Qw* is the heat flux, where *Qw* = − ˆ *k*( ∂*T*∂*y* )η = 0. After the dimensionalization the *u* is gotten the below as

$$
\mu = -\left(1 + \frac{4}{3}Rd\right)\Theta'(0),
\tag{34}
$$

#### **4. Application of Homotopy Analysis Method**

In this section HAM is applied to Equations (27)–(29) to ge<sup>t</sup> an approximate analytical solution of MHD Sisko fluid flow over unsteady sheet in a following way:

$$f\_0(\eta) = \eta\_\prime \theta\_0(\eta) \,=\, 1,\tag{35}$$

*Processes* **2019**, *7*, 369

> The linear operators are

$$\mathcal{L}\_f(f) = \frac{d^3 f}{d\eta^{3'}} , \mathcal{L}\_\theta(\theta) \, \, = \, \frac{d^2 \theta}{d\eta^2} \tag{36}$$

The above differential operators' contents are shown below as

$$\begin{aligned} L\_f(\psi\_1 + \psi\_2 \eta + \psi\_3 \eta^2) &= 0, \\ L\_0(\psi\_4 + \psi \5 \eta) &= 0 \end{aligned} \tag{37}$$

where 5*i* = 1 ψ*<sup>i</sup>*, *i* = 1, 2, 3 ... are considered as arbitrary constant.

#### *4.1. Zeroth rder Deformation Problem*

Expressing ∈ [01]. as an embedding parameter with associate parameters *f* , and θ where - - 0. Then in case of zero order distribution the problem will be in the following form:

$$(1 -)L\_f\{f(\eta\_\prime) - f\_0(\eta)\} = \, \_fN\_f\{f(\eta\_\prime)\}.\tag{38}$$

$$(1 -)L\_{\theta} (\Theta(\eta\_{\prime}) - \Theta\_{0}(\eta)) = \hbar\_{\theta} \mathcal{N}\_{\theta} (\hat{f}(\eta\_{\prime}), \hat{\theta}(\eta\_{\prime}))\_{\prime}$$

The subjected boundary conditions are obtained as

$$\begin{array}{rclclcl}f(\eta;P)\Big|\_{\eta=0} &=& 0, & \frac{\partial f(\eta;P)}{\partial \eta}\Big|\_{\eta=0\_{\text{un}}} &=& 1, & \frac{\partial^2 f(\eta;P)}{\partial \eta^2}\Big|\_{\eta=\beta} &=& 0, \\ & & \theta(\eta;P)\Big|\_{\eta=0} &=& 1, & \frac{\partial \theta(\eta;P)}{\partial \eta}\Big|\_{\eta=\beta} &=& 0,\end{array} \tag{39}$$

The resultant nonlinear operators have been mentioned as:

$$\begin{array}{rcl} \mathcal{N}\_f\Big(f(\eta;\mathsf{y})\Big) &=& \varepsilon f\_{\eta\eta\eta} + n\xi \Big(-f\_{\eta\eta}\Big)^{n-1} f\_{\eta\eta\eta} + f\_{\eta\eta}f \\ & - \Big(f\_{\eta}\Big)^2 - St \Big(f\_{\eta} + \frac{\eta}{2}f\_{\eta\eta}\Big) - \mathcal{M}f\_{\eta\prime} \end{array} \tag{40}$$

$$\left[N\_{\theta\dots}[f(\eta\_{\uparrow}), \theta(\eta\_{\uparrow})]\right] = (1 + Rd)\theta\_{\eta\eta} - \Pr\left(\frac{S}{2}(3\theta + \eta\theta\_{\eta}) + 2f\_{\eta}\theta - \theta\_{\eta}f\right) \tag{41}$$

Expanding ˆ <sup>f</sup>(η;), <sup>θ</sup><sup>ˆ</sup>(η;) in term of with use of Taylor's series expansion we get:

$$\begin{array}{rcl}f\_{\cdots}(\eta;P)&=&f\_{0}(\eta)+\sum\_{i=1}^{\infty}f\_{i}(\eta)P^{i}.\\\theta(\eta;P)&=&\theta\_{0}(\eta)+\sum\_{i=1}^{\infty}\theta\_{i}(\eta)P^{i}.\end{array} \tag{42}$$

where

$$\left. f\_i(\eta) \right|\_{\mathbb{P}} = \left. \frac{1}{i!} \frac{\partial f(\eta; P)}{\partial \eta} \right|\_{P=0}, \left. \theta\_i(\eta) \right|\_{P=0} = \left. \frac{1}{i!} \frac{\partial \theta(\eta; P)}{\partial \eta} \right|\_{P=0} \tag{43}$$

Here *f* , θ are selected in a way that the Series (43) converges at *P* = 1, switching *P* = 1 in (43), we obtain:

$$\begin{aligned} f(\eta\_\prime) &= f\_0(\eta) + \sum\_{i=1}^\infty f\_i(\eta), \\ \theta(\eta\_\prime) &= \theta\_0(\eta) + \sum\_{i=1}^\infty \theta\_i(\eta), \end{aligned} \tag{44}$$

#### *4.2. ith-Order Deformation Problem*

Differentiating zeroth order equations *ith* time we obtained the *ith* order deformation equations with respect to . Dividing by *i*! and then inserting = 0, so *ith* order deformation equations

$$\begin{array}{ll}L\_f(f\_i(\eta) - \underline{\xi}\_i f\_{i-1}(\eta)) &= h\_f \mathfrak{R}\_i^f(\eta), \\ L\_\partial(\theta\_i(\eta) - \underline{\xi}\_i \theta\_{i-1}(\eta)) &= h\_\partial \mathfrak{R}\_i^\partial(\eta). \end{array} \tag{45}$$

The resultant boundary conditions are:

$$\begin{array}{rcl} f\_i(0) &= f\_i'(0) = f\_i''(\beta) = 0, \\ \theta\_i(0) &= \theta\_i(\beta) = 0. \end{array} \tag{46}$$

$$\begin{aligned} \Re\_i^f(\eta) &= \varepsilon f\_{i-1}^{\prime\prime} + n\xi \sum\_{j=0}^{i-1} \left( (-f^{\prime\prime})^{n-1}{}\_{i-1-j} f\_j^{\prime\prime} \right) - \sum\_{j=0}^{i-1} f\_{i-1-j} f^{\prime\prime} \\ &\sum\_{j=0}^{i-1} f\_{i-1-j}^{\prime} f\_j^{\prime} - \text{St} \left( f\_{i-1}^{\prime} + \frac{\eta}{2} f^{\prime i}{}\_{i-1} \right) \end{aligned} \tag{47}$$

$$R\_i^0(\eta) = (1+Rd)\theta\_{i-1}^{\prime\prime} - Pr\left[\left(\frac{S}{2}(3\theta\_{i-1}+\eta\theta\_{i-1}^{\prime})\right) - \sum\_{j=-0}^{i-1} f\_{i-1-j}\theta\_j^{\prime} + 2\sum\_{j=-0}^{i-1} f\_{i-1-j}^{\prime}\theta\_j\right].\tag{48}$$

where

$$\xi\_i = \begin{cases} 1, \text{ if P} > 1 \\ 0, \text{ if P} \le 1 \end{cases} \tag{49}$$

#### *4.3. Convergence of Solution*

After using the HAM method to calculate these solutions of the modelled function as velocity and temperature, these parameters *hf* , *h*θ are seen. The responsibility of the computed parameters is to regulate of convergence of the series results. In the conceivable region of *h*, *h* -curves of *f* (0) and θ(0) for 12*th* order approximation are plotted in Figure 2 for different values of numbers. The *h*-curves consecutively display the valid area. Table 1 values shows the numerical results of HAM solutions at dissimilar approximation. Its shows that the HAM method is fast convergent.

**Table 1.** The Homotopy Analysis Method (HAM) convergence table up to 25th order approximations when ε = 0.5, β = 0.5, ξ = *St* = *M* = 0.1.


**Figure 2.** The *h*-curve graphs for velocity profile for *n* = 0, 1, 2, 3 where β = 0.5, ξ = *St* = *M* = 0.1, ε = 1.

#### **5. Results and Discussion**

The current work investigates the MHD and radiative flow of Sisko thin film flow having unsteady stretching sheet. The purpose of the subsection is to inspect the physical outcomes of dissimilar implanting on the velocity distributions *f*(η) and temperature distribution <sup>Θ</sup>(η) which are described in Figures 3–12. Figure 3a–d shows the influence of unsteady constraint *St* on the velocity profile for dissimilar values of power index *n* = 0, 1, 2 and 3. Increasing *St* increase the velocity field *f*(η). It is clear that varying power indices x having similar response to the time dependent parameter, that is the increase value of the power index rise the velocity distribution. The impact of the unsteadiness parameter *St* on the heat profile <sup>θ</sup>(η) is shown in Figure 4. It is observed that <sup>θ</sup>(η) directly proportional to *St*. augmented *St* rises the temperature, which in turn rises the kinetic energy of the fluid, so the fluid motion increased. It is perceived that the effect of *St* for the different value of power index *n* = 0, 1, 2 and 3 having a similar effect on heat profile <sup>θ</sup>(η). Figure 5a–d demonstrate the effect of the film thickness β for dissimilar values of power index *n* = 0, 1, 2 and 3. It is perceived that the velocity profile falling down with higher values. Figure 6a–d describe the characteristics for *M* for changed values of power index *n* = 0, 1, 2 and 3. When *M* increase on the surface of the sheet during the flow, the flow rate falls, which in results decrease the velocity profiles. This significance of *M* on velocity field is due to the rise in the *M* progresses the friction force of the movement, which is called the Lorentz force. It is the fact that fluid velocity reduce in the boundary layer sheet. Figure 7 demonstrates the effect of film thickness β on temperature profile. It is observed that the large values of film thickness β rise the temperature, and actually the higher value of β speed up the molecular motion of the liquids which in turn increases the internal energy and the temperature increase. The effect of stretching parameter ξ for each changed values of power index *n* = 0, 1, 2 and 3 on velocity profile is shown in Figure 8a–c. In case of *n* = 1, it is clear from Figure 8a that velocity profile increase for large value of stretching parameter ξ. When values of power index are varied and the effect of stretching parameter ξ become changed and for *n* = 3 this effect is totally opposite that is the velocity field *f*(η) reduces. For *n* = 0 the stretching parameter ξ becomes zero. The effect of Sisko fluid parameter ε for each changed values of power index *n* = 0, 1, 2 and 3 on velocity profile is shown in Figure 9a–d. The large values of Sisko fluid parameter ε rise the fluid motion, but when the power index goes toward increase then this effect is seen changed and in case of *n* = 3 the velocity field reduces (Figure 9d). The impact of *Pr* on <sup>Θ</sup>(η) is shown in Figure 10. Both temperature and concentration distributions vary in reverse form with *Pr*. When the *Pr* . number increasing it decrease the temperature distribution, and when the *Pr* number decreases it increases the temperature distribution. Same effect of *Pr* on concentration distribution is shown in (32). For increasing values of *Pr* power index *n* = 0, 1, 2 and 3 thermal radiation increases rapidly. The effect of *Rd* (thermal radiation parameter) on <sup>θ</sup>(η) is presented in Figure 11. When the heat transmission coefficient is minor, then the thermal radiation plays an important role in the heat transfer of comprehensive surface. By increasing thermal radiation parameter *Rd*, its show that it augments the temperature in the fluid film. Due to this rising the rate of cooling is going down. For increasing values of power index *n* = 0, 1, 2 and 3 thermal radiation increases rapidly. The HAM and numerical comparison has been displayed in in Figure 12a–d. Excellent agreemen<sup>t</sup> is found between HAM and numerical method.

**Figure 4.** Effect of *St* on <sup>θ</sup>(η).

**Figure 6.** Effect of *M* on *f*(η) for *n* = 0, 1, 2, 3.

**Figure 7.** Effect of *M* on <sup>θ</sup>(η).

**Figure 8.** Effect of ξ on *f*(η) for *n* = 1, 2, 3.

**Figure 9.** Effect of ε on *f*(η) for *n* = 0, 1, 2, 3.

**Figure 10.** Effect of *Pr* on <sup>θ</sup>(η).

**Figure 11.** Effect of *Rd* on <sup>θ</sup>(η).

**Figure 12.** Comparison graphs between Homotopy Analysis Method (HAM) & numerical solution for velocity profiles *f*(η) when β = ε = 1, *h* = −0.6, *St* = *M* = 0.1, ξ = 0.8.
