# A Mathematical Model of Fluid Flow in Evaporating Droplets Using Wedge Geometry

## Abstract

As a droplet of liquid evaporates, particles within the droplet are often pulled to the edge and deposited in a ring-shaped pattern. This is known as the coffee-ring effect. The coffee-ring effect is largely due to evaporation taking place at a pinned contact line. Since this formation is an adverse outcome in many practical applications, methods to counteract the coffee-ring effect have become of interest, including the application of an electric field. In this research project, we discuss the first stage in constructing a coupled mathematical model of colloidal transport within an evaporating droplet under the influence of an electric field. The first stage is to simply capture the fluid flow within a pinned evaporating droplet without any consideration of particles or an electric field. We consider a thin axisymmetric droplet of a Newtonian solvent in contact with ambient air that is undergoing diffusion-limited evaporation. Away from the contact line, we model the droplet dynamics by applying the lubrication approximation to simplify the Navier-Stokes equations. To characterize the flow near the contact line, we assume the droplet shape is a wedge and derive analytical solutions of evaporative-driven Stokes flow near a pinned contact line. We connect the lubrication model and the wedge model by specifying height and flux conditions at the boundary between the two regions. We solve for the position of the droplet interface by implementing a method of lines approach in MATLAB. We find that for our specified conditions, the two regions evaporate on different time scales. However, if either the evaporation rate or the rate of contact angle decrease is appropriately adjusted, the two regions evaporate simultaneously.

## Library of Congress Subject Headings

Evaporation--Mathematical models; Fluid dynamics--Mathematical models; Microfluidics

8-15-2018

Thesis

## Degree Name

Applied and Computational Mathematics (MS)

## Department, Program, or Center

School of Mathematical Sciences (COS)

Kara L. Maki

Michael Schertzer

Steven Weinstein

## Campus

RIT – Main Campus

ACMTH-MS

COinS