Determination of the Thickness of a Power-Law Fluid Driven by the Penetration of a Long Gas Bubble in a Rectangular Channel Using a Singular Perturbation Method

C Escobar,E Luna,F Méndez

doi:10.2174/1874123100802010089

Abstract

In the present work an analytical and numerical study is presented in order to determine the residual fluid film thickness of a power-law fluid on the walls of a rectangular horizontal channel when it is displaced by another immiscible fluid of negligible viscosity. The mathematical model describes the motion of the displaced fluid and the interface between both fluids. In order to obtain the residual film thickness, m , we used a singular perturbation technique: the matching asymptotic method; in the limit of small capillary number, Ca . The main results indicated that the residual film thickness of the non-Newtonian fluid decreases for decreasing values of the power-law index, which is in qualitative agreement with experimental results.

Highlights

The steady displacement of Newtonian and nonNewtonian fluids by long bubbles, confined in vertical and horizontal cylindrical ducts or parallel plates has received considerable attention during the past decades, due to the fundamental and practical importance of this process in many industrial applications
Typical examples appear in film coating, bubble columns, gas-assisted injection molding, lubrication theory, oil recovery in naturally fractured reservoirs, etc
The theory predicted that the residual fraction increases with increasing values of the power-law index which is in qualitative agreement with the experimental observations of previous investigators

Summary

Introduction

The steady displacement of Newtonian and nonNewtonian fluids by long bubbles, confined in vertical and horizontal cylindrical ducts or parallel plates has received considerable attention during the past decades, due to the fundamental and practical importance of this process in many industrial applications. For Newtonian fluids this problem has been widely studied for different cases and parting from the pioneer analytical works of Bretherton [1] and Cox [2] and the experimental work of Taylor [3], the basic mechanics of deposition films are well understood In this direction, we can emphasize the following works: Wilson [4] studied analytically the film coating in a plate when it is drawn steadily out of a bath of the liquid: the drag-out problem. Using the method of matched asymptotic expansions to predict the thickness of the film for small values of the capillary number Ca , this author showed that the analysis of Landau [5] for this same problem, represents the leading

Methods

Results

Conclusion