2,094 research outputs found
Conformal mapping methods for interfacial dynamics
The article provides a pedagogical review aimed at graduate students in
materials science, physics, and applied mathematics, focusing on recent
developments in the subject. Following a brief summary of concepts from complex
analysis, the article begins with an overview of continuous conformal-map
dynamics. This includes problems of interfacial motion driven by harmonic
fields (such as viscous fingering and void electromigration), bi-harmonic
fields (such as viscous sintering and elastic pore evolution), and
non-harmonic, conformally invariant fields (such as growth by
advection-diffusion and electro-deposition). The second part of the article is
devoted to iterated conformal maps for analogous problems in stochastic
interfacial dynamics (such as diffusion-limited aggregation, dielectric
breakdown, brittle fracture, and advection-diffusion-limited aggregation). The
third part notes that all of these models can be extended to curved surfaces by
an auxilliary conformal mapping from the complex plane, such as stereographic
projection to a sphere. The article concludes with an outlook for further
research.Comment: 37 pages, 12 (mostly color) figure
Simulation of cell movement through evolving environment: a fictitious domain approach
A numerical method for simulating the movement of unicellular organisms which respond to chemical signals is presented. Cells are modelled as objects of finite size while the extracellular space is described by reaction-diffusion partial differential equations. This modular simulation allows the implementation of different models at the different scales encountered in cell biology and couples them in one single framework. The global computational cost is contained thanks to the use of the fictitious domain method for finite elements, allowing the efficient solve of partial differential equations in moving domains. Finally, a mixed formulation is adopted in order to better monitor the flux of chemicals, specifically at the interface between the cells and the extracellular domain
Advection-diffusion equations with random coefficients on evolving hypersurfaces
We present the analysis of advection-diffusion equations with random coefficients on moving hypersurfaces. We define weak and strong material derivative, that take into account also the spacial movement. Then we define the solution space for these kind of equations, which is the Bochner-type space of random functions defined on moving domain. Under suitable regularity assumptions we prove the existence and uniqueness of solutions of the concerned equation, and also we give some regularity results about the solution
Chaotic Mixing in Three Dimensional Porous Media
Under steady flow conditions, the topological complexity inherent to all
random 3D porous media imparts complicated flow and transport dynamics. It has
been established that this complexity generates persistent chaotic advection
via a three-dimensional (3D) fluid mechanical analogue of the baker's map which
rapidly accelerates scalar mixing in the presence of molecular diffusion. Hence
pore-scale fluid mixing is governed by the interplay between chaotic advection,
molecular diffusion and the broad (power-law) distribution of fluid particle
travel times which arise from the non-slip condition at pore walls. To
understand and quantify mixing in 3D porous media, we consider these processes
in a model 3D open porous network and develop a novel stretching continuous
time random walk (CTRW) which provides analytic estimates of pore-scale mixing
which compare well with direct numerical simulations. We find that chaotic
advection inherent to 3D porous media imparts scalar mixing which scales
exponentially with longitudinal advection, whereas the topological constraints
associated with 2D porous media limits mixing to scale algebraically. These
results decipher the role of wide transit time distributions and complex
topologies on porous media mixing dynamics, and provide the building blocks for
macroscopic models of dilution and mixing which resolve these mechanisms.Comment: 36 page
A least-squares implicit RBF-FD closest point method and applications to PDEs on moving surfaces
The closest point method (Ruuth and Merriman, J. Comput. Phys. 227(3):1943-1961, [2008]) is an embedding method developed to solve a variety of partial differential equations (PDEs) on smooth surfaces, using a closest point representation of the surface and standard Cartesian grid methods in the embedding space. Recently, a closest point method with explicit time-stepping was proposed that uses finite differences derived from radial basis functions (RBF-FD). Here, we propose a least-squares implicit formulation of the closest point method to impose the constant-along-normal extension of the solution on the surface into the embedding space. Our proposed method is particularly flexible with respect to the choice of the computational grid in the embedding space. In particular, we may compute over a computational tube that contains problematic nodes. This fact enables us to combine the proposed method with the grid based particle method (Leung and Zhao, J. Comput. Phys. 228(8):2993-3024, [2009]) to obtain a numerical method for approximating PDEs on moving surfaces. We present a number of examples to illustrate the numerical convergence properties of our proposed method. Experiments for advection-diffusion equations and Cahn-Hilliard equations that are strongly coupled to the velocity of the surface are also presented.NSERC Canada Grant (RGPIN 2016-04361),
Hong Kong Research Grant Council GRF Grant,
Hong Kong Baptist University FRG Gran
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