Date of Award
2025
Document Type
Dissertation
Degree Name
Doctor of Philosophy (PhD)
Department
Mathematical Sciences
Committee Chair
S. S. Ravindran
Committee Member
Ian W. Knowles
Committee Member
Shan Zhao
Committee Member
Timothy S. Newman
Committee Member
Summer Atkins
Research Advisor
S. S. Ravindran
Subject(s)
Electro-osmosis--Mathematical models, Electrokinetics--Mathematical models
Abstract
Electroosmotic (EO) flow has attracted much attention in recent years due to applications in microfluidics, such as in lab-on-a-chip type systems. Because of the inherent difficulties in measuring and monitoring microfluidic channels, analysis of EO flow is often performed by numerical simulation. The Navier-Stokes-Debye-Hückel model — a simplification of the Nernst-Planck model which couples conservation of the ionic concentrations to the Navier-Stokes equations — is the preferred model for studying such flows when the zeta potential is small. In this work, we study the well-posedness of the unsteady Debye-Huckël model, and propose decoupled time-stepping schemes for numerical simulation. We demon- strate the well-posedness of the evolutionary Debye-Hückel model by Galerkin approximation, a priori estimates, and the Aubin-Simon compactness theorem for Bochner spaces. In particular, we prove the existence of weak solutions in two and three spatial dimensions, as well as uniqueness of the solution in two dimensions. We also propose and study two second-order accurate decoupled time stepping schemes based on mixed finite element spatial discretization. The time stepping schemes employ an extrapolation in time of the nonlinear terms such that their skew-symmetry properties are preserved. We prove that the both of these decoupled time stepping schemes are unconditionally stable. We also establish optimal error bounds for the velocity, potentials, and pressure under a mild restriction on the time-step size for the first scheme, and without any restriction for the second. In addition we study the long-time stability of the discrete scheme and show that in two spatial dimensions it is stable for all time in both 𝐿2 and 𝐻1 norms. We conclude with numerical results that support our theoretical claims and demonstrate the practical value of our proposed schemes.
Recommended Citation
Bosma, Craig S., "Analysis of second-order decoupled time-stepping schemes for the Navier-Stokes-Debye-Hückel model of electroosmotic flow" (2025). Dissertations. 458.
https://louis.uah.edu/uah-dissertations/458
Comments
Submitted ... in the Joint Applied Mathematics Program.