Derivation of the nonlinear equations for surface of fluid adhering to a moving plate withdrawn from a liquid pool

TL;DR

This paper derives nonlinear equations governing the surface profile of a viscous liquid film adhering to a slowly withdrawn plate, aiding in understanding and simulating industrial coating processes.

Contribution

It develops a mathematical model using Navier-Stokes equations and polynomial approximation to analyze nonlinear film flow on a moving plate.

Findings

Derived equations describe the nonlinear behavior of the film surface.

The model enables simulation of film adherence quality.

Applicable to processes like metal coating and wire adhering.

Abstract

Many technological processes include preparing some special materials adhering to a product surface. For example, this problem is important for the magnetic tape producing, wire adhering, etc. For a surface withdrawn from the molten metal or the other liquid material there is a problem to determine a profile of a film surface. It is subject of this paper. We developed the mathematical model for the simulation of the adhering process of viscous liquid film to a slowly moving plate, which is vertically withdrawn from the molten metal or the other fluid capacity. The Navier-Stokes equations for a film flow on a surface of the withdrawn plate are considered with the corresponding boundary conditions, and the polynomial approximation is used for the film flow profile. The equations after integration across the layer of a film flow result in the system of partial differential equations for the wavy surface of a film flow, of flow rate and of flow energy.The derived equations are used for analysis of the nonlinear film flow that determines the quality of a fluid adhering on a surface of the withdrawn plate.