**3. Solution by HAM**

The approximate solution of the Equations (23)–(26) corresponding to the Equations (27) and (28) are treated with Homotopy Analysis Method (HAM). The auxiliary parameters encircled the solution which normalize and switch to conjunction of the solutions. Let us take the initial guesses:

$$
\hat{f}\mathfrak{o}(\eta) = \eta,\quad \hat{\mathfrak{o}}\mathfrak{o}(\eta) = 1,\quad \hat{\mathfrak{o}}\mathfrak{o}(\eta) = 1\tag{32}
$$

Let us denote the linear operators by *Lf* , *Lθ*, and *L<sup>φ</sup>* defined as:

$$L\_f(f) = f'' \,, L\_\emptyset(\theta) = \theta'' \,, \ L\_\phi(\phi) = \phi'' \, \tag{33}$$

with the property

$$L\_f(\mathcal{\alpha}\_1 + \mathcal{\alpha}\_2 \eta + \mathcal{\alpha}\_3 \eta^2) = 0,\\ L\_\theta(\mathcal{\alpha}\_4 + \mathcal{\alpha}\_5 \eta) = 0,\\ L\_\theta(\mathcal{\alpha}\_6 + \mathcal{\alpha}\_7 \eta) = 0 \tag{34}$$

where *<sup>j</sup>* for *j* ∈ {1, 2, ..., 7} are the general solution coefficients. The fundamental procedure of the solution by using HAM is explained in [72,73,76].

#### **4. Results and Discussion**

The current analysis is carried out on the thin film flow of Reiner-Philippoff fluid of boundary layer type over a time-dependent stretching plate. The aim of this subsection is to study the velocity distribution, temperature distribution, and concentration profile with physical effects of different embedding parameters, which are discussed in Figures 2–16.

Figure 2 demonstrates the thin film thickness *β* during the fluid motion. In the performance of coating, thin film thickness play a key role. Physically, the thickness of the films is directly related to the velocity. Figure 2b reflects a more rapid variations in *β* under smaller values of and *S* as compared to the results observed in part (a). The velocity of the fluid decreases with an increase in the thickness parameter *β*. It happens because for larger values *β* of the fluid viscosity increases, and as a result a gradual fall can be observe in the gradient of the velocity profile.

**Figure 2.** (**a**) Impact of *β* on *f* (*η*), when = 0.4, *S* = 0.2 and (**b**) Impact of *β* on *f* (*η*), when = 0.01, *S* = 0.05.

Figure 3 illustrates the effect of the stretching parameter . The velocity profile shows an increase with the increasing values of , because the lower plate always behaves directly to the flow fluid velocity. Physically, for > 0, the surface accelerating rises, < 0 decelerating the surface, while = 0 depict the random motion of the surface. Figure 3b reflects the sensitivity of , under smaller values of *β* and *S*.

**Figure 3.** (**a**) Impact of on *f* (*η*), when *β* = 0.1, *S* = 0.2 and (**b**) Impact of *β* on *f* (*η*), when *β* = 0.05, *S* = 0.005.

Figure 4 reveals the behavior of unsteady constraint *S* over *f* (*η*) for dissimilar values of the embedded parameters. It is observed that the velocity profile *f* (*η*) directly varies with unsteadiness parameter *S*. The velocity profile climes up with the increasing behavior of *S*. Furthermore, it is observed that the solution is possible only in the closed interval [0, 1] for *S*. Also, the increasing values of *S* increases the motion of the nanofluid.

**Figure 4.** (**a**) Impact of *S* on *f* (*η*), when *β* = 0.1, = 0.2 and (**b**) Impact of *β* on *f* (*η*), when *β* = 0.01, = 0.01.

The impact of *Pr* on *θ*(*η*) is shown in Figure 5. An inverse relation has been observed between the temperature and the Prandlt number. Physically, for small values of *Pr* these fluids have larger thermal conductivity and vice versa. As a result, for larger values of *Pr* the thermal boundary layer declines.

**Figure 5.** Impact of *Pr* on *θ*(*η*), when *S* = 0.7, *β* = 1.0, = 0.4, *Nb* = 0.5, *Nt* = 0.3, *Sc* = 0.7.

The effect of the thin film thickness *β* on temperature profile for different values of the embedded parameters is shown in Figure 6. A similar effect in velocity profile is observed for *β*. The larger the thickness of the liquid film, the lesser the heat transfer. In other words, the flow of heat in the larger thickness film faces more more difficulty, as compared to a lesser thickness film.

Figure 7 illustrates the temperature distribution under Brownian motion parameter *Nb*. In general, due to the irregular motion of the particles, this causes a collision between these particles. An increase in heat of the fluid can be seen with the ascending order of the Brownian motion parameter *Nb*, consequently, free surface nanoparticle volume friction decreases.

The impact of the unsteadiness parameter *S* on the heat profile *θ*(*η*) is presented in Figure 8. It is observed that *θ*(*η*) varies directly with *S*. An increase in *S* increases the temperature of the fluid, which further increases the kinetic energy of the fluid, and results in increment of the liquid film motion.

**Figure 6.** Impact of *β* on *θ*(*η*), when *S* = 0.6, = 0.5, *Nb* = 0.5, *Nt* = 0.3, *Sc* = 0.1, *Pr* = 0.7.

**Figure 7.** Impact of *Nb* on *θ*(*η*), when *S* = 0.6, = 0.5, *β* = 0.4, *Nt* = 0.3, *Sc* = 0.1, *Pr* = 0.7.

**Figure 8.** Impact of *S* on *θ*(*η*), when *Nb* = 0.5, = 0.5, *β* = 1.0, *Nt* = 0.3, *Sc* = 0.9, *Pr* = 0.6.

Figure 9 illustrates the effect of thermophoresis parameter *Nt* on temperature profile. The limitations thermophoresis helps in the increase of a surface temperature. The irregularity in motion (Brownian motion), causes a temperature increase due to the kinetic energy produced by nano suspended particles, which results in thermophoretic force generation. The intensity produced by this force compels the fluid to move away from the stretching sheet. As a result, larger values of *Nt* cause an increase in temperature, due to which the surface temperature also increases.

Figure 10 describes the Schmidt number effect over temperature profile. Schmidt number physically relates the boundary-layer of mass transfer to the hydrodynamics layer. Increasing rate of the viscous diffusion keeping the mass flux constant increases the Schmidt number, which as a result decreases the heat profile, as shown in the figure.

**Figure 9.** Impact of *Nt* on *θ*(*η*), when *Nb* = 0.5, = 0.4, *β* = 1.0, *S* = 0.3, *Sc* = 0.6, *Pr* = 0.5.

**Figure 10.** Impact of *Sc* on *θ*(*η*), when *Nb* = 0.5, = 0.1, *β* = 0.1, *S* = 0.3, *Nt* = 0.3, *Pr* = 0.6.

The effect of Brownian motion parameter *Nb* on *φ*(*η*) is shown in Figure 11. Brownian motion is the irregular motion fluid particles. At molecular level Brownian motion of micropoler nanofluid leading the thermal conductivity of nanofluids. The figure describes an inverse relation between the concentration profile and *Nb*. The boundary-layer thicknesses diminishes due to an increase in *Nb*, which results in reducing the concentration.

**Figure 11.** Impact of *Nb* on *φ*(*η*), when *Sc* = 0.7, = 0.1, *β* = 0.9, *S* = 0.7, *Nt* = 0.3, *Pr* = 0.6.

Figure 12 describes the effect of thermophoresis parameter *Nt* on concentration field. It is clear from the figure that an increase in *Nt* increases the concentration field. This is because higher values of *Nt* increase the nanofluid molecules kinetic energy, and as a result the concentration increases.

**Figure 12.** Impact of *Nt* on *φ*(*η*), when *Sc* = 0.7, = 0.3, *β* = 0.9, *S* = 0.5, *Nb* = 0.5, *Pr* = 0.6.

The impact of *Pr* on *φ* is shown in Figure 13. Larger values of the Prandtl number *Pr* cause the concentration to falls down. The information from the figure reveals that large values of *Pr* cause the concentration profile to fall down. Physically, the thermal boundary-layer vanishes with greater values of *Pr* and as a result the concentration profile falls. The same phenomenon is observed for the heat profile.

**Figure 13.** Impact of *Pr* on *φ*(*η*), when *Sc* = 0.6, = 0.2, *β* = 0.6, *S* = 0.5, *Nb* = 0.6, *Nt* = 0.6.

Figure 14 shows the concentration profile *φ*(*η*) behavior, under the effect of the unsteadiness parameter *S*. A direct relation has been observed between the unsteadiness parameter *S* and *φ*(*η*) the concentration profile . Increasing the unsteadiness parameter *S*, causes an increase in the temperature to be observed, that blows the kinetic energy off the fluid, which leads to an increase in the concentration of the liquid film.

Figure 15 reveals the opposite information as discussed in the temperature distribution under different parameters. The above diagram shows that the concentration profile decreases due to an increase in Schmidt number *Sc*, which as a result reduces the boundary-layer thickness. This is because of the physical significance of the Schmidt number, which relates both the mass and hydrodynamic layer.

Figure 16 illustrates the effect of thin film thickness *β* on *φ*(*η*) for the different values of the embedded parameters. It is clear that the concentration profile falls with higher values of *β*. The same effect has been observed for *β* in the velocity distribution as well as in temperature distribution.

**Figure 14.** Impact of *S* on *φ*(*η*), when *Sc* = 0.7, = 0.8, *β* = 0.9, *Pr* = 0.5, *Nb* = 0.8, *Nt* = 0.4.

**Figure 15.** Impact of *Sc* on *φ*(*η*), when *S* = 0.7, = 0.8, *β* = 0.9, *Pr* = 0.6, *Nb* = 0.8, *Nt* = 0.4.

**Figure 16.** Impact of *β* on *φ*(*η*), when *S* = 0.7, = 0.8, *Sc* = 0.9, *Pr* = 0.6, *Nb* = 0.8, *Nt* = 0.4.

#### **5. Tables Discussion**

Table 1 depicts the influence of Nusselt number. The effects of *Nt*, *S*, *β*, and *Pr* on Θ (0) are shown. It is clear that larger values of *Nt* and *S* decrease Θ (0), while the unsteadiness parameter *S* and thickness parameter *β* increase Θ (0). The influence of stretching parameters and *λ* of Reiner-Philippoff fluid and unsteadiness parameter *S* on skin friction *Cf* is presented in Table 2. It is observed that the increasing values of stretching parameters of Reiner-Philippoff fluid and *λ* decrease *Cf* , while unsteadiness parameter *S* increases *Cf* . The effects of *Nb*, *Nt*, *Pr*, *Sc*, and *S* on the Sherwood number Φ (0) are demonstrated in Table 3. It is observed that local Sherwood number values increases due to an increase in thermoporetic parameter *Nt*. Increasing values of Schmidt

number *Sc* decreases the Sherwood number, while increasing unsteady parameter and Prandtl number decreases the Sherwood number.


**Table 1.** Variation in Nusselt number with different values of the parameters *Nt*, *Pr*, *β*, and *S*.

**Table 2.** Variation in skin friction with different values of the parameters , *λ*, and *S*.


**Table 3.** Variation in Sherwood number with different values of the parameters *Nb*, *Nt*, *Sc*, *S* and *Pr*.

