Electro-deformation of a moving boundary: a drop interface and a lipid bilayer membrane

This dissertation focuses on the deformation of a viscous drop and a vesicle immersed in a (leaky) dielectric fluid under an electric field. A number of mathematical tools, both analytical and numerical, are developed for these investigations. The dissertation is divided into three parts. First, a large-deformation model is developed to capture the equilibrium deformation of a viscous spheroidal drop covered with non-diffusing insoluble surfactant under a uniform direct current (DC) electric field. The large- deformation model predicts the dependence of equilibrium spheroidal drop shape on the permittivity ratio, conductivity ratio, surfactant coverage, and the elasticity number. Results from the model are carefully compared against the small-deformation (quasispherical) analysis, experimental data and numerical simulation results in the literature. Moreover, surfactant effects, such as tip stretching and surface dilution effects, are greatly amplified at large surfactant coverage and high electric capillary number. These effects are well captured by the spheroidal model, but cannot be described in the second-order small-deformation theory. The large-deformation spheroidal model is then extended to study the equilibrium deformation of a giant unilamellar vesicle (GUV) under an alternating current (AC) electric field. The vesicle membrane is modeled as a thin capacitive spheroidal shell and the equilibrium vesicle shape is computed from balancing the mechanical forces between the fluid, the membrane and the imposed electric field. Detailed comparison against both experiments and small-deformation theory shows that the spheroidal model gives better agreement with experiments in terms of the dependence on fluid conductivity ratio, electric field strength and frequency, and vesicle size. Asymptotic analysis is conducted to compute the crossover frequency where a prolate vesicle crosses over to an oblate shape, and comparisons show the spheroidal model gives better agreement with experimental observations. Finally, a numerical scheme based on immersed interface method for two-phase fluids is developed to simulate the time-dependent dynamics of an axisymmetric drop in an electric field. The second-order immersed interface method is applied to solving both the fluid velocity field and the electric field. To date this has not been done before in the literature. Detailed numerical studies on this new numerical scheme shows numerical convergence and good agreement with the large-deformation model. Dynamics of an axisymmetric viscous drop under an electric field is being simulated using this novel numerical code

On the Numerical Solution of the Cylindrical Poisson Equation for Isolated Self -Gravitating Systems.

This dissertation addresses the need for an accurate and efficient technique which solves the Poisson equation for arbitrarily complex, isolated, self-gravitating fluid systems. Generally speaking, a potential solver is composed of two distinct pieces: a boundary solver and an interior solver. The boundary solver computes the potential, phi(xB) on a surface which bounds some finite volume of space, V, and contains an isolated mass-density distribution, rho(x). Given rho(x) and phi(xB), the interior solver computes the potential phi(x) everywhere within V. Herein, we describe the development of a numerical technique which efficiently solves Poisson\u27s equation in cylindrical coordinates on massively parallel computing architectures. First, we report the discovery of a compact cylindrical Green\u27s function (CCGF) expansion and show how the CCGF can be used to efficiently compute the exact numerical representation of phi(xB). As an analytical representation, the CCGF should prove to be extremely useful wherever one requires the isolated azimuthal modes of a self-gravitating system. We then discuss some mathematical consequences of the CCGF expansion, such as it\u27s applicability to all nine axisymmetric coordinate systems which are R -separable for Laplace\u27s equation. The CCGF expansion, as applied to the spherical coordinate system, leads to a second addition theorem for spherical harmonics. Finally, we present a massively parallel implementation of an interior solver which is based on a data-transpose technique applied to a Fourier-ADI (Alternating Direction Implicit) scheme. The data-transpose technique is a parallelization strategy in which all communication is restricted to global 3D data-transposition operations and all computations are subsequently performed with perfect load balance and zero communication. The potential solver, as implemented here in conjunction with the CCGF expansion, should prove to be an extremely useful tool in a wide variety of astrophysical studies, particularly those requiring an accurate determination of the gravitational field due to extremely flattened or highly elongated mass distributions

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

Fast, High-Order Accurate Integral Equation Methods and Application to PDE-Constrained Optimization

Over the last several decades, the development of fast, high-order accurate, and robust integral equation methods for computational physics has gained increasing attention. Using integral equation formulation as a global statement in contrast to a local partial differential equation (PDE) formulation offers several unique advantages. For homogeneous PDEs, the boundary integral equation (BIE) formulation allows accurate handling of complex and moving geometries, and it only requires a mesh on the boundary, which is much easier to generate as a result of the dimension reduction. With the acceleration of fast algorithms like the Fast Multipole Method (FMM), the computational complexity can be reduced to O(N), where N is the number of degrees of freedom on the boundary. Using standard potential theory decomposition, inhomogeneous PDEs can be solved by evaluating a volume potential over the inhomogeneous source domain, followed by a solution of the homogeneous part. Despite the advantages of BIE methods in easy meshing, near-optimal efficiency, and well conditioning, the accurate evaluation of nearly singular integrals is a classical problem that needs to be addressed to enable simulations for practical applications. In the first half of this thesis, we develop a series of product integration schemes to solve this close evaluation problem. The use of differential forms provides a dimensional-agnostic way of integrating the nearly singular kernels against polynomial basis functions analytically. So the problem of singular integration gets reduced to a matter of source function approximation. In 2D, this procedure has been traditionally portrayed by building a connection to complex Cauchy integral, then supplemented by a complex monomial approximation. In $3$D, the closed differential form requirement leads to the design of a new function approximation scheme based on harmonic polynomials and quaternion algebra. Under a similar framework, we develop a high-order accurate product integration scheme for evaluating singular and nearly singular volume integral equations (VIE) in complex domains using regular Cartesian grids discretization. A high-order accurate source term approximation scheme matching smooth volume integrals on irregular cut cells is developed, which requires no function extension. BIE methods have been widely used for studying Stokes flows, incompressible flows at low Reynolds' number, in both biological systems and microfluidics. In the second half of this thesis, we employ the BIE methods to simulate and optimize Stokes fluid-structure interactions. In 2D, a hybrid computational method is presented for simulating cilia-generated fluid mixing as well as the cilia-particle hydrodynamics. The method is based on a BIE formulation for confining geometries and rigid particles, and the method of regularized Stokeslets for the cilia. In 3D, we use the time-independent envelop model for arbitrary axisymmetric microswimmers to minimize the power loss while maintaining a target swimming speed. This is a quadratic optimization problem in terms of the slip velocity due to the linearity of Stokes flow. Under specified reduced volume constraint, we find prolate spheroids to be the most efficient micro-swimmer among various families of shapes we considered. We then derive an adjoint-based formulation for computing power loss sensitivities in terms of a time-dependent slip profile by introducing an auxiliary time-periodic function, and find that the optimal swimmer displays one or multiple traveling waves, reminiscent of the typical metachronal waves observed in ciliated microswimmers.PHDApplied and Interdisciplinary MathematicsUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/169695/1/hszhu_1.pd

Analytical and Numerical Studies of Several Fluid Mechanical Problems

In this thesis, three parts, each with several chapters, are respectively devoted to hydrostatic, viscous and inertial fluids theories and applications. In the hydrostatics part, the classical Maclaurin spheroids theory is generalized, for the first time, to a more realistic multi-layer model, which enables the studies of some gravity problems and direct numerical simulations of flows in fast rotating spheroidal cavities. As an application of the figure theory, the zonal flow in the deep atmosphere of Jupiter is investigated for a better understanding of the Jovian gravity field. High viscosity flows, for example Stokes flows, occur in a lot of processes involving low-speed motions in fluids. Microorganism swimming is such typical a case. A fully three dimensional analytic solution of incompressible Stokes equation is derived in the exterior domain of an arbitrarily translating and rotating prolate spheroid, which models a large family of microorganisms such as cocci bacteria. The solution is then applied to the magnetotactic bacteria swimming problem and good consistency has been found between theoretical predictions and laboratory observations of the moving patterns of such bacteria under magnetic fields. In the analysis of dynamics of planetary fluid systems, which are featured by fast rotation and very small viscosity effects, three dimensional fully nonlinear numerical simulations of Navier-Stokes equations play important roles. A precession driven flow in a rotating channel is studied by the combination of asymptotic analyses and fully numerical simulations. Various results of laminar and turbulent flows are thereby presented. Computational fluid dynamics requires massive computing capability. To make full use of the power of modern high performance computing facilities, a C++ finite-element analysis code is under development based on PETSc platform. The code and data structures will be elaborated, along with the presentations of some preliminary results

Finite difference method in prolate spheroidal coordinates for freely suspended spheroidal particles in linear flows of viscous and viscoelastic fluids

A finite difference scheme is used to develop a numerical method to solve the flow of an unbounded viscoelastic fluid with zero to moderate inertia around a prolate spheroidal particle. The equations are written in prolate spheroidal coordinates, and the shape of the particle is exactly resolved as one of the coordinate surfaces representing the inner boundary of the computational domain. As the prolate spheroidal grid is naturally clustered near the particle surface, good resolution is obtained in the regions where the gradients of relevant flow variables are most significant. This coordinate system also allows large domain sizes with a reasonable number of mesh points to simulate unbounded fluid around a particle. Changing the aspect ratio of the inner computational boundary enables simulations of different particle shapes ranging from a sphere to a slender fiber. Numerical studies of the latter particle shape allow testing of slender body theories. The mass and momentum equations are solved with a Schur complement approach allowing us to solve the zero inertia case necessary to isolate the viscoelastic effects. The singularities associated with the coordinate system are overcome using L'Hopital's rule. A straightforward imposition of conditions representing a time-varying combination of linear flows on the outer boundary allows us to study various flows with the same computational domain geometry. {For the special but important case of zero fluid and particle inertia we obtain a novel formulation that satisfies the force- and torque-free constraint in an iteration-free manner.} The numerical method is demonstrated for various flows of Newtonian and viscoelastic fluids around spheres and spheroids (including those with large aspect ratio). Good agreement is demonstrated with existing theoretical and numerical results.Comment: 32 pages, 12 figures. Accepted at Journal of Computational Physic

Methods in wave propagation and scattering

Supervised by Jin A. Kong.Also issued as Thesis (Ph.D.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 2001.Includes bibliographical references (p. 195-213).by Henning Braunisch

Methods in wave propagation and scattering

Thesis (Ph.D.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 2001.Vita.Includes bibliographical references (p. 195-213).This electronic version was submitted by the student author. The certified thesis is available in the Institute Archives and Special Collections.Aspects of wave propagation and scattering with an emphasis on specific applications in engineering and physics are examined. Frequency-domain methods prevail. Both forward and inverse problems are considered. Typical applications of the method of moments to rough surface three-dimensional (3-D) electromagnetic scattering require a truncation of the surface considered and call for a tapered incident wave. It is shown how such wave can be constructed as a superposition of plane waves, avoiding problems near both normal and grazing incidence and providing clean footprints and clear polarization at all angles of incidence. The proposed special choice of polarization vectors removes an irregularity at the origin of the wavenumber space and leads to a wave that is optimal in a least squared error sense. Issues in the application to 3-D scattering from an object over a rough surface are discussed. Approximate 3-D scalar and vector tapered waves are derived which can be evaluated without resorting to any numerical integrations. Important limitations to the accuracy and applicability of these approximations are pointed out. An analytical solution is presented for the electromagnetic induction problem of magnetic diffusion into and scattering from a permeable, highly but not perfectly conducting prolate spheroid under axial excitation, expressed in terms of an infinite matrix equation. The spheroid is assumed to be embedded in a homogeneous non-conducting medium as appropriate for low-frequency, high-contrast scattering governed by magnetoquasistatics. The solution is based on separation of variables and matching boundary conditions where the prolate spheroidal wavefunctions with complex wavenumber parameter are expanded in terms of spherical harmonics. For small skin depths, an approximate solution is constructed, which avoids any reference to the spheroidal wavefunctions. The problem of long spheroids and long circular cylinders is solved by using an infinite cylinder approximation. In some cases, our ability to evaluate the spheroidal wavefunctions breaks down at intermediate frequencies. To deal with this, a general broadband rational function approximation technique is developed and demonstrated. We treat special cases and provide numerical reference data for the induced magnetic dipole moment or, equivalently, the magnetic polarizability factor. The magnetoquasistatic response of a distribution of an arbitrary number of interacting small conducting and permeable objects is also investigated. Useful formulations are provided for expressing the magnetic dipole moment of conducting and permeable objects of general shape. An alternative to Tikhonov regularization for deblurring and inverse diffraction, based on a local extrapolation scheme, is described, analyzed, and illustrated numerically for the cases of continuation of fields obeying Laplace and Helmholtz equations. At the outset of the development, a special deconvolution problem, where a parameter describes the degree of additive blurring, is considered. No a priori knowledge on the unblurred data is assumed. A standard solution based on an output least squares formulation includes a regularization parameter into a linear, shift-invariant filter. The proposed alternative approach takes advantage of the analyticity of the smoothing process with respect to the blurring parameter. Here a simple local extrapolation scheme is employed. The problem is encountered in applications involving potential theories dealing with magnetostatics, electrostatics, and gravity data. As a generalization to the dynamic case, inverse diffraction of scalar waves is considered. Examples are presented and the two methods compared numerically. The problem of inferring unknown geometry and material parameters of a waveguide model from noisy samples of the associated modal dispersion curves is considered. In a significant reduction of the complexity of a common inversion methodology, the inner of two nested iterations is eliminated: The approach described does not employ explicit fitting of the data to computed dispersion curves. Instead, the unknown parameters are adjusted to minimize a cost function derived directly from the determinant of the boundary condition system matrix. This results in a very efficient inversion scheme that, in the case of noise-free data, yields exact results. Multi-mode data can be simultaneously processed without extra complications. Furthermore, the inversion scheme can accommodate an arbitrary number of unknown parameters, provided that the data have sufficient sensitivity to these parameters. As an important application, the sonic guidance condition for a fluid-filled borehole in an elastic, homogeneous, and isotropic rock formation is considered for numerical forward and inverse dispersion analysis. The parametric inversion with uncertain model parameters and the influence of bandwidth and noise are investigated numerically. The cases of multi-frequency and multi-mode data are examined. Finally, the borehole leaky-wave modes are classified according to the location of the roots of the characteristic equation on a multi-sheeted Riemann surface. A comprehensive set of dipole leaky-wave modal dispersions is computed. In an independent numerical experiment the excitation of some of these modes is demonstrated. The utilization of leaky-wave dispersion data for inversion is discussed.by Henning Braunisch.Ph.D

Glosarium Matematika

273 p.; 24 cm