Document Type

Dissertation

Date of Award

8-12-2020

Degree Name

Doctor of Philosophy in Mathematical Sciences - (Ph.D.)

Department

Mathematical Sciences

First Advisor

Michael R. Booty

Second Advisor

Michael Siegel

Third Advisor

Shahriar Afkhami

Fourth Advisor

Linda Jane Cummings

Fifth Advisor

Pushpendra Singh

Abstract

Numerical methods are developed for accurate solution of two-phase flow in the zero Reynolds number limit of Stokes flow, when surfactant is present on a drop interface and in its bulk phase interior. The methods are designed to achieve high accuracy when the bulk Péclet number is large, or equivalently when the bulk phase surfactant has small diffusivity

In the limit of infinite bulk Péclet number the advection-diffusion equation that governs evolution of surfactant concentration in the bulk is singularly perturbed, indicating a separation of spatial scales. A hybrid numerical method based on a leading order asymptotic reduction in this limit, that scales out the Péclet number dependence, is adapted to resolve the drop interior flow, the bulk surfactant evolution, and the transfer of surfactant between the bulk and surface phases.

A more traditional numerical method that solves the full governing equations without the asymptotic reduction is also developed. This is designed to achieve high accuracy at large Péclet number by use of complex variable techniques that map the evolving drop shape and flow velocity onto the fixed domain of the unit disk, where a Chebyshev-Fourier spectral method is developed to resolve the bulk phase surfactant evolution.

Results of the two methods are compared for 2D simulations of drop dynamics, when the drop is stretched or deformed in either a strain flow or in a shear flow. Recirculation of the interior flow and surfactant exchange on the interior of the drop induce more intricate dynamics than when bulk surfactant is present in the exterior phase.

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