**3. Analytical Results**

Analytical solutions of nonlinear and coupled Equations (9)–(13) subject to (14)–(16) are obtained as:

$$F(z,t) = f\_1(t)z^7 + f\_2(t)z^2 + f\_3(t)z^3 + f\_4(t)z^4 + f\_5(t)z^5 + f\_6(t)z^6 + f\_7(t)z^7 + f\_8(t)z^8$$

$$\begin{array}{c c c c} + f \bullet (t)z^9 + f\_{10}(t)z^{10} + f\_{11}(t)z^{11} + f\_{12}(t)z^{12} + f\_{13}(t)z^{13} + f\_{14}(t)z^{14} \\ + f\_{15}(t)z^{15} \end{array} \tag{53}$$

$$\begin{aligned} G(z,t) &= 1 + \mathcal{g}\_1(t)z^2 + \mathcal{g}\_2(t)z^3 + \mathcal{g}\_3(t)z^4 + \mathcal{g}\_5(t)z^5 + \mathcal{g}\_6(t)z^6 + \mathcal{g}\_7(t)z^7 \\ &+ \mathcal{g}\_8(t)z^8 + \mathcal{g}\_9(t)z^9 \end{aligned} \tag{54}$$

$$\mathcal{W}(z,t) = w\_1(t)z^2 + w\_2(t)z^3 + w\_3(t)z^4 + w\_4(t)z^5 + w\_5(t)z^6 + w\_6(t)z^7 + w\_7(t)z^8$$

$$\begin{array}{c} + w\_9(t)z^9 + w\_{10}(t)z^{10} + w\_{11}(t)z^{11} + w\_{12}(t)z^{12} \\ + w\_{13}(t)z^{13} + w\_{14}(t)z^{14} + w\_{15}(t)z^{15} + w\_{16}(t)z^{16} \end{array} \tag{55}$$

$$\begin{aligned} \Gamma(z,t) &= 1 + m\_1(t)z^2 + m\_2(t)z^2 + m\_3(t)z^3 + m\_4(t)z^4 + m\_5(t)z^5 + m\_6(t)z^6 + m\gamma(t)z^7 \\ &\quad + m\_8(t)z^8 \end{aligned} \tag{56}$$

$$\begin{aligned} \tau(z,t) &= n\_1(t)z \quad + n\_2(t)z^2 + n\_3(t)z^3 + n\_4(t)z^4 + n\_5(t)z^5 + n\_6(t)z^6 + n\_7(t)z^7 \\ &\quad + n\_8(t)z^8 + n\_9(t)z^9 + n\_{10}(t)z^{10} \end{aligned} \tag{57}$$

where the expressions *f*1, *f*<sup>2</sup> ... *f*15, *g*1, *g*<sup>2</sup> ... *g*9, *m*1, *m*<sup>2</sup> ... *m*8, *n*1, *n*<sup>2</sup> ... *n*<sup>10</sup> and *w*1, *w*<sup>2</sup> ... *w*<sup>16</sup> are given in the Appendix A.

#### **4. Discussion**

The process of coating heavily depends upon the time taken by any fluid to settle down on the surface of the material; a fluid can only be considered more suitable for the coating if it takes less time to leave its effects on the surface. Moreover, the engaged nanoparticles are of very small size and of a concentration of at most 2%. The effects on viscosity, thermal conductivity, density and heat capacity are evaluated experimentally in many communications. It is now a well-established fact that in the presence of such a small quantity of nanosized particles, the nature of fluid does not change but changes in physical properties are evident. For that, many correlations are presented for different situations and particles. To serve the purpose of this study, four different kinds of Newtonian fluids having diverse physical and chemical properties are considered instead of non-Newtonian fluids because coatings with such types of fluids would have a tremendous impact on the cost, volume, weight, and mechanical properties of electronic, optoelectronic, and photovoltaic devices; thus, this portion is dedicated to the parametric study of the proposed model in which four kinds of Newtonian fluids, such as water, ethanol, methanol and ethylene-glycol are opted for as the base fluids. The gold and silver nanoparticles are used to furnish the thin metallic and shiny coating on the surface of the

rotating disk. The main reason to carry out this graphical work is to confirm whether or not the obtained mathematical results are in complete coherence with the physical expectation of the spin coatings. Moreover, the graphic illustrations will help to make a sound judgement about the role and contribution of field variables. Major parameters which have been comprehensively focused on are the concentration of the metallic particles and the thermocapillary parameter. Furthermore, the presented parametric study unlike the customary results and discussion have been delicately divided into three following sub sections to make this comparative analysis more clear and fathomable.
