This study presents the theoretical investigation of a roll-over thin layer formation under the lubrication approximation theory. The set of differential equations derived by lubrication approximation is solved by the optimal homotopy asymptotic method (OHAM) to obtain precise expressions for pressure and velocity gradients. Critical quantities such as velocity, pressure gradient, and coating layer depth are numerically estimated. The impact of parameters affecting the coating and layer formation is revealed in detail. Results indicate that the transport properties of the higher-grade fluid play an essential role in regulating velocity, pressure, and the final coated region. Moreover, couple stress effects on the properties of fluid particles to be coated on roller-surface have also been studied.

In this article, Optimal Homotopy Asymptotic Method (OHAM) is used to approximate results of time-fractional order Fokker-Planck equations. In this work, 3rd order results obtained through OHAM are compared with the exact solutions. It was observed that results from OHAM have better convergence rate for time-fractional order Fokker-Planck equations. The solutions are plotted and the relative errors are tabulated.

In present learning, surface protection layer progression of a third-grade fluid (TGF) is examined. Fluid transport within the micro passage made by the firm bladehas beenpresented. Main system of equations of fluidity have been narrated and streamlined by means of lubrication approximation theory (LAT). Here, approximate solutions of velocity, pressure gradient, and coating depth have been presented. Results of coating and layer forming have been tabulated and discussed as well. It is observed that the transport properties of third-order fluid delivers an instrument to regulate flow velocity, pressure, and affect the final coated region.