Mathematical Modeling of Lift and Drainage Problems for Thin Film Flow of a Third-Grade Fluid

Abstract

Thin film flow is a significant problem in fluid dynamics because of its wide applications in coating processes, lubrication systems, wire coating, polymer processing, food processing, chemical engineering, and related industrial operations. In many practical situations, the working fluids exhibit non-Newtonian behavior and cannot be accurately described by classical Newtonian models. Third-grade fluids are especially important because they can represent nonlinear stress–strain behavior, viscoelastic effects, and shear-dependent characteristics of complex fluids. This study develops mathematical models for lift and drainage problems involving the thin film flow of an incompressible third-grade fluid. In the lift configuration, a vertical moving belt carries a thin layer of fluid upward against gravity, whereas in the drainage configuration, the fluid flows down an inclined plane under the action of gravity. The governing equations are derived from the continuity, momentum, and energy equations under steady, laminar, and thin film assumptions. The effects of dynamic viscosity (\mu), density (\rho), film thickness (h), belt velocity (U), gravitational acceleration (g), inclination angle (\alpha), thermal conductivity (k), and third-grade material parameters (\beta_{2}) and (\beta_{3}) are considered. The main dependent quantities include velocity profile, temperature distribution, flow rate, and average velocity. The expected physical trends indicate that viscosity, density, film thickness, and inclination angle significantly influence fluid transport, while thermal conductivity controls heat distribution in the film. The proposed models provide useful insight for coating, lubrication, wire coating, polymer processing, food processing, and chemical engineering applications.