Coating with Newtonian fluid by a two-parameter blade family leading to exact dynamic and thermal solutions

Abstract

This study utilizes mathematical analysis to explore the classic phenomenon of coating by a thin fluid layer between a blade and a moving substrate, with the aim of uncovering new blade functions. Through fluid flow analysis, we derive a two-parameter family of blade geometries encompassing the widely studied cases of simple planar and exponential blades. By solving the associated momentum, thermal, and pressure fields in closed-form, we obtain explicit expressions for key engineering quantities of interest, including pressure gradient, load, film thickness, drag, and Nusselt numbers on both the substrate and the blade. These analytical formulas provide powerful tools for designing blades that effectively influence the physical and mechanical processes of coating, ultimately determining the desired film thickness. For instance, our analysis reveals that convex blade geometries generate lower load and drag, while concave geometries offer smoother fluid contact, leading to thinner film thicknesses and enhanced heat transfer.