A model for the electric field-driven deformation of a drop or vesicle in strong electrolyte solutions

A model is constructed to describe the arbitrary deformation of a drop or vesicle that contains and is embedded in an electrolyte solution, where the deformation is caused by an applied electric field. The applied field produces an electrokinetic flow or induced charge electro-osmosis. The model is based on the coupled Poisson-Nernst-Planck and Stokes equations. These are reduced or simplified by forming the limit of strong electrolytes, for which ion densities are relatively large, together with the limit of thin Debye layers. Debye layers of opposite polarity form on either side of the drop interface or vesicle membrane, together forming an electrical double layer. Two formulations of the model are given. One utilizes an integral equation for the velocity field on the interface or membrane surface together with a pair of integral equations for the electrostatic potential on the outer faces of the double layer. The other utilizes a form of the stress-balance boundary condition that incorporates the double layer structure into relations between the dependent variables on the layer's outer faces. This constitutes an interfacial boundary condition that drives an otherwise unforced Stokes flow outside the double layer. For both formulations relations derived from the transport of ions in each Debye layer give additional boundary conditions for the potential and ion concentrations outside the double layer.Comment: 35 pages including 2 figures and reference

Level Set Jet Schemes for Stiff Advection Equations: The SemiJet Method

Many interfacial phenomena in physical and biological systems are dominated by high order geometric quantities such as curvature. Here a semi-implicit method is combined with a level set jet scheme to handle stiff nonlinear advection problems. The new method offers an improvement over the semi-implicit gradient augmented level set method previously introduced by requiring only one smoothing step when updating the level set jet function while still preserving the underlying methods higher accuracy. Sample results demonstrate that accuracy is not sacrificed while strict time step restrictions can be avoided

Electrohydrodynamic Simulations of Capsule Deformation Using a Dual Time-Stepping Lattice Boltzmann Scheme

Capsules are fluid-filled, elastic membranes that serve as a useful model for synthetic and biological membranes. One prominent application of capsules is their use in modeling the response of red blood cells to external forces. These models can be used to study the cell’s material properties and can also assist in the development of diagnostic equipment. In this work we develop a three dimensional model for numerical simulations of red blood cells under the combined influence of hydrodynamic and electrical forces. The red blood cell is modeled as a biconcave-shaped capsule suspended in an ambient fluid domain. Cell deformation occurs due to fluid motion and electrical forces that arise due to differences in the electrical properties between the internal fluid, external fluid, and cell membrane. The electrostatic equations are solved using the immersed interface method. A finite element method is used to compute the membrane’s elastic forces and the membrane’s bending resistance is described by the Helfrich bending energy functional. The membrane forces are coupled to the fluid equations through the immersed boundary method, where the elastic, bending, and electric forces appear as force densities in the Navier-Stokes equations. The fluid equations are solved using a novel dual time-stepping (DTS) lattice Boltzmann method (LBM), which decouples the fluid and capsule discretizations. The computational efficiency of the DTS scheme is studied for capsules in shear flow where it is found that the newly proposed scheme decreases computational time by a factor of 10 when compared to the standard LBM capsule model. The method is then used to study the dynamics of spherical and biconcave capsules in a combined shear flow and DC electric field. For spherical capsules the effect of field strength, shear rate, membrane capacitance, and membrane conductance are studied. For biconcave capsules the effect of the electric field on the tumbling and tank-treading modes of biconcave capsules is discussed

Electrohydrodynamic Simulations of the Deformation of Liquid-Filled Capsules

A comprehensive two- and three-dimensional framework for the electrohydrodynamic simulation of deformable capsules is provided. The role of a direct current (DC) electric field on the deformation and orientation of a liquid-filled capsule is thoroughly considered numerically. This framework is based on lattice Boltzmann method for the fluid, finite element method for the membrane structure of the capsule, fast immersed interface method for the electric field and immersed boundary method being used to consider the fluid-structure-electric interaction. Under the effect of electric field, two different types of equilibrium states, prolate or oblate are obtained. The numerical algorithm is also applied to study the interfacial tension droplet and red blood cell under shear flow. The capsules are more deformed and arrive at equilibrium status more quickly under stronger electric field. Bending stiffness will suppress the deformation and cause transition from tank-treading to tumbling for the red blood cell. However, the applied electric field will slow down the transition from tank-treading to the tumbling motion or even stay in the tank-treading motion with stronger electric field

A General, Mass-Preserving Navier-Stokes Projection Method

The conservation of mass is common issue with multiphase fluid simulations. In this work a novel projection method is presented which conserves mass both locally and globally. The fluid pressure is augmented with a time-varying component which accounts for any global mass change. The resulting system of equations is solved using an efficient Schur-complement method. Using the proposed method four numerical examples are performed: the evolution of a static bubble, the rise of a bubble, the breakup of a thin fluid thread, and the extension of a droplet in shear flow. The method is capable of conserving the mass even in situations with morphological changes such as droplet breakup.Comment: Submitted to Computer Physics Communication

Spectrally-Accurate Close Evaluation Schemes for Stokes Boundary Integral Operators

Dense particulate flow simulations using integral equation methods demand accurate evaluation of Stokes layer potentials on arbitrarily close interfaces. In this thesis, two spectrally-accurate integration schemes for close evaluation of 2D Stokes layer potentials are developed -- a global quadrature for the moving particles (e.g., blood cells, vesicles) represented as smooth closed curves, and an adaptive panel quadrature for the stationary boundaries (e.g., vessel walls, microfluidic channels) which are more complex curves that can be non-smooth. Both schemes rely on expressing Stokes layer potentials in terms of Laplace potentials and related complex contour integrals, which are then evaluated accurately either through a singularity cancellation technique or using analytic expressions. Numerical examples are presented to demonstrate the robustness and super-algebraic convergence of both schemes. Finally, as an application of the integration schemes, we investigate the electrohydrodynamic interactions between (possibly deflated) vesicles, where interesting behaviors unique to vesicles, such as circulatory and oscillatory motions, are observed and analyzed.PHDApplied and Interdisciplinary MathematicsUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttps://deepblue.lib.umich.edu/bitstream/2027.42/153404/1/boweiwu_1.pd