**4. Graphical Results and Discussion**

The thin film motion of a micropolar fluid through porous media with the impact of energy radiation and thermophoresis through a stretching plate is investigated. The non-linear coupled differential Equations (13)–(16) with physical conditions (17) and (18) were determined through HAM. The effects of all the embedded constants on the dimensionless velocity field, dimensionless microrotation, dimensionless temperature field, and concentration fields—*f*(η), *g*(η), θ(η), and *φ*(η), respectively—are observed. The physical geometry of the modeled problem is demonstrated by Figure 1. Liao [33–35] presented *h* curves to measure the convergence of the series solution for accurate

results of the system, so suitable *h*-curves are drawn for the velocity profile *f*(η), microrotation profile *g*(η), temperature profile θ(η), and concentration profile *φ*(η) in range of −2.0 ≤ *hf* ≤ 0.1, −2 ≤ *hg* ≤ 0, −2.1 ≤ *h*<sup>θ</sup> ≤ 0.1, and −2 ≤ *h<sup>φ</sup>* ≤ 0, respectively, in Figures 2–5. The influence of permeability parameter *Mr* on the velocity field is described in Figure 6. The permeability parameter should be increased at a very small level because of the small thickness of the liquid film because higher values of *Mr*, that is , *Mr* → ∞ correspond to the case in which there is no porous medium. The increasing values of *Mr* respond to the large opening of the porous space, which reduces retardation of the flow; so for increasing values of *Mr*, the velocity increases in this region. The larger values of the inertia coefficient parameter *Nr* increase the velocity of fluid as a result of its direct relation with fluid motion, deliberated in Figure 7. The influence of Δ versus motion of liquid film is represented in Figure 8. As Δ has an inverse relation with viscosity, the viscosity falls for larger values of Δ, while the velocity of the liquid film is raised. Figures 9 and 10 indicate the relationship between β with the fluid velocity profile *f*(η) and microrotation profile *g*(η). The fluid motion reduces with the increase in the liquid film thickness. The reason is clear, because larger values of β dominate the viscous forces and, as a result, the fluid velocity decreases. In other words, the thickness of the liquid film shows resistance to liquid flow, and fluid velocity causes retardation towards the free surface—this effect is very clear in the rotation velocity field *g*(η). The microrotation profile *g*(η) of the liquid film rises with the increasing microrotation *Gr*, as displayed in Figure 11, because the microrotation parameter has an inverse relation with the viscosity parameter. As a result, the viscosity reduces with the rising values of *Gr*; therefore, larger values of *Gr* offer low resistance to the flow and the velocity of fluid increases. Figure 12 demonstrates the variation of the inertia parameter *Nr* on the non-dimensional microrotation profile *g*(η). It is observed that the rise in the inertia parameter *Nr* material parameter reduces the microrotation profile. The inclusion of thermal radiation in the equation of energy is always used as a special case and, in most of the problems in the existing literature, the energy equation is used without radiation. If the thermal radiation parameter *R* becomes zero, the temperature field θ(η) in Abo-Eldahab and Ghonaim [17], Rashidi et al. [18,19], and Heydari et al. [20] becomes meaningless, so it is not clear when the thermal radiation parameter *R* becomes zero in these papers. Therefore, our case of thermal radiation is reciprocal to the above published work, and is the same as Khan [27], Qasim et al. [29], and Mahmood and Khan [32]. Therefore, the temperature rises with the larger values of thermal radiation parameter, as shown in Figure 13, because the thickness of the boundary layer (thin film) is directly related to thermal radiation. Physically, the rate of energy transport increases and, as a result, the temperature of the fluid rises. The dimensionless fluid thickness β has a vital role in temperature distribution. θ(η) decreases with increasing values of β, which is obvious from Figure 14. The size of thin film absorbing heat, and thus the temperature of the fluid, decreases and, as a result, a cooling effect is produced. In other words, the thickness of the fluid decreases with the increasing temperature. Figure 15 represents the comparison of temperature and Prandtl number *Pr*. The temperature falls with growing values of *Pr*. In fact, the larger values of *Pr* enhance the viscous diffusion more than the thermal diffusion and, as a result, the temperature profile declines. Schmidt number verses concentration is deliberated in Figure 16. The rising values of Schmidt number *Sc* decrease the concentration field, because molecular diffusivity is inversely related to *Sc*. The contribution of the Soret number *Sr* is represented in Figure 17, showing that *φ*(η) rises when the Soret number *Sr* increases. In fact, the larger Soret number increases the viscosity and, therefore, *φ*(η) accelerates. Figure 18 shows the relationship between thermophoretic parameter τ and *φ*(η). They are inversely related to each other. Rising values of τ reduce the size of the boundary layer. The concentration field rises as thickness β increases, as shown in Figure 19, because of cohesive forces between molecules dominated by the increasing value of the parameter β, which result a rise in friction force and cause the fluid flow.

**Figure 2.** *hf* curves for the velocity field.

**Figure 3.** *hg* curves for the velocity field in rotation.

**Figure 4.** *h*θ curves for the temperature field.

**Figure 5.** *hφ* curves for the concentration field.

**Figure 6.** Effect of permeability parameter *Mr* on the velocity.

**Figure 7.** The comparison of dimensionless velocity with inertia coefficient parameter *Nr*.

**Figure 8.** Velocity verses vortex–viscosity parameter Δ.

**Figure 9.** Variation of dimensionless velocity with dimensionless fluid thickness β.

**Figure 10.** Variation of dimensionless microrotation profile with fluid thickness β.

**Figure 11.** Microrotation profile under the effect of microrotation parameter *Gr*.

**Figure 12.** Variation of dimensionless microrotation profile with inertial parameter *Nr*.

**Figure 13.** Temperature verses radiation parameter *R*.

**Figure 14.** Temperature verses film thickness parameter β.

**Figure 15.** Temperature versus Prandtl number *Pr*.

**Figure 16.** Variation of dimensionless concentration with Schmidt number *Sc*.

**Figure 17.** Variation of dimensionless concentration with Soret number *Sr*.

**Figure 18.** Concentration versus thermophoretic parameter τ.

**Figure 19.** Variation of dimensionless concentration with dimensionless fluid thickness β.
