Document Type
Dissertation
Date of Award
12-31-2017
Degree Name
Doctor of Philosophy in Mathematical Sciences - (Ph.D.)
Department
Mathematical Sciences
First Advisor
Michael R. Booty
Second Advisor
Michael Siegel
Third Advisor
Shidong Jiang
Fourth Advisor
Yuan N. Young
Fifth Advisor
Petia Mladenova Vlahovska
Abstract
Electrokinetic flow typically occurs when an electric field is applied to a fluid electrolyte in the presence of a boundary or interface. The electric field induces a force that causes oppositely charged ions to migrate in opposite directions while simultaneously diffusing due to Brownian motion. In an unbounded medium this process, which is called electrodiffusion, does not separate bulk electric charge into separate regions and causes no bulk motion of the host fluid solvent. However, in the presence of a boundary or interface that either impedes or is impervious to the movement of ions in the normal direction, the normal component of the electric field causes ions to accumulate there in an ion cloud, and a local separation of charge occurs. The electric field then exerts a coherent force on the ion cloud along the local field lines, which induces a bulk flow of the host fluid. This type of electrokinetic phenomenon is sometimes described as induced charge electro-osmosis.
In this study, the motion of ions is impeded by a fluid-fluid interface between two immiscible electrolyte fluids. Electric charge separates into ion clouds that occur in spatially narrow layers, which are called Debye layers, that are of opposite sign on either side of the interface. Of the net force that the electric field exerts on the Debye layers, the normal component can cause the interface to deform and the tangential component can induce a tangential fluid velocity at the fluid-fluid interface and a shear in the neighboring Debye layers.
The main theme of this thesis is to formulate and implement an accurate numerical method to solve the full system of governing equations for electrokinetic flow. These are the Poisson-Nernst-Planck equations for ion conservation and the electrostatic field, with the Stokes equation for incompressible flow subject to an electrostatic force, together with suitable interfacial and initial conditions. The numerical method is based on a Legendre polynomial pseudo-spectral technique in the case where the drop shape is spherical or, equivalently, interfacial forces are dominated by surface tension. A semi-analytical technique is developed to: (i) resolve the radial component of the electrostatic and flow fields that is designed to be highly accurate when the ratio of the Debye layer thickness to the drop radius is small, i.e., in the thin Debye layer limit, and (ii) to implement the far-field boundary conditions accurately. The thin Debye layer regime is typical of applications, but is also where most numerical methods such as traditional finite difference schemes achieve poor accuracy due to separation of spatial scales.
The numerical method is validated against asymptotic results in the thin Debye layer limit. These are of two types. The first is a model for steady-state electrophoresis of a spherical bubble due to Schnitzer et al. (J. Fluid Mech. 753, 49-79, 2014). The second is a model for the time-dependent evolution of a two-electrolyte viscous spherical drop that is derived in this thesis. The results of numerical simulations for the two-electrolyte drop are presented, and comparison is made between the numerical and asymptotic results.
Recommended Citation
Cao, Rui, "Numerical methods and simulation for time dependent electrokinetic flow" (2017). Dissertations. 1793.
https://digitalcommons.njit.edu/dissertations/1793
