Alexander Panchenko | Washington State University (original) (raw)

Papers by Alexander Panchenko

arXiv (Cornell University), Jan 16, 2013

Molecules of a nematic liquid crystal respond to an applied magnetic field by reorienting themsel... more Molecules of a nematic liquid crystal respond to an applied magnetic field by reorienting themselves in the direction of the field. Since the dielectric anisotropy of a nematic is small, it takes relatively large fields to elicit a significant liquid crystal response. The interaction may be enhanced in colloidal suspensions of ferromagnetic particles in a liquid crystalline matrixferronematics-as proposed by Brochard and de Gennes in 1970. The ability of these particles to align with the field and, simultaneously, cause reorientation of the nematic molecules, greatly increases the magnetic response of the mixture. Essentially the particles provide an easy axis of magnetization that interacts with the liquid crystal via surface anchoring. We derive an expression for the effective energy of ferronematic in the dilute limit, that is, when the number of particles tends to infinity while

Revista Matemática Iberoamericana, 2003

We prove the existence of complex geometrical optics solutions for Lipschitz conductivities. More... more We prove the existence of complex geometrical optics solutions for Lipschitz conductivities. Moreover we show that, in dimensions n ≥ 3 that one can uniquely recover a W 3/2,∞ conductivity from its associated Dirichlet-to-Neumann map or voltage to current map.

Journal of Computational Physics, 2016

We present a novel formulation of the Pairwise Force Smoothed Particle Hydrodynamics Model (PF-SP... more We present a novel formulation of the Pairwise Force Smoothed Particle Hydrodynamics Model (PF-SPH) and use it to simulate two-and threephase flows in bounded domains. In the PF-SPH model, the Navier-Stokes equations are discretized with the Smoothed Particle Hydrodynamics (SPH) method and the Young-Laplace boundary condition at the fluid-fluid interface and the Young boundary condition at the fluid-fluid-solid interface are replaced with pairwise forces added into the Navier-Stokes equations. We derive a relationship between the parameters in the pairwise forces and the surface tension and static contact angle. Next, we demonstrate the accuracy of the model under static and dynamic conditions. Finally, to demonstrate the capabilities and robustness of the model we use it to simulate flow of three fluids in a porous medium.

Mathematical Methods in the Applied Sciences, 2005

The paper is devoted to study of acoustic wave propagation in a partially consolidated composite ... more The paper is devoted to study of acoustic wave propagation in a partially consolidated composite material containing loose particles. Friction of particles against the consolidated part of the material causes mechanical energy dissipation. This situation is modelled by assuming that the medium has a periodic microstructure changing rapidly on the small scale. Each of the periodic microscopic cells is composed of a viscoelastic matrix containing a rigid particle in frictional contact with the matrix. We use the methods of two-scale convergence to obtain effective acoustic equations for the homogenized material. The effective equations are history-dependent and contain the body force term, reminiscent of the well known Stokes drag force.

Applicable Analysis, 2013

, Abu Dhabi. Acoustic Propagation in a Random Saturated Medium: The Biphasic Case. Osteoporosis i... more , Abu Dhabi. Acoustic Propagation in a Random Saturated Medium: The Biphasic Case. Osteoporosis is characterized by a decrease in strength of the bone matrix. Since the loss of bone density and the destruction of the bone microstructure is most evident in osteoporosis cancellous bone, it is natural to consider developing accurate ultrasound models for the isonification of cancellous bone. We develop an effective model of acoustic wave propagation in a two-phase, non-periodic medium modeling a fine mixture of linear elastic solid and a viscous Newtonian fluid. Bone tissue is an important example of a composite material that can be modeled in this fashion. We extend known homogenization results for periodic geometries to the case a stationary random, scale-separated microstructure. The ratio ε of the macroscopic length scale and a typical size of the microstructural inhomogeneity is a small parameter of the problem. We employ stochastic two-scale convergence in the mean to pass to the limit ε → 0 in the governing equations. The effective model is a biphasic phase viscoelastic material with long time history dependence.

Soft matter, Jan 7, 2014

Many biological systems consist of self-motile and passive agents both of which contribute to ove... more Many biological systems consist of self-motile and passive agents both of which contribute to overall functionality. However, little is known about the properties of such mixtures. Here we formulate a model for mixtures of self-motile and passive agents and show that the model gives rise to three different dynamical phases: a disordered mesoturbulent phase, a polar flocking phase, and a vortical phase characterized by large-scale counter rotating vortices. We use numerical simulations to construct a phase diagram and compare the statistical properties of the different phases with observed features of self-motile bacterial suspensions. Our findings afford specific insights regarding the interaction of microorganisms and passive particles and provide novel strategic guidance for efficient technological realizations of artificial active matter.

Abstract. We study the closure problem for continuum balance equations that model mesoscale dynam... more Abstract. We study the closure problem for continuum balance equations that model mesoscale dynamics of large ODE systems. The underlying microscale model consists of classical Newton equations of particle dynamics. As a mesoscale model we use the balance equations for spatial averages obtained earlier by a number of authors: Murdoch and Bedeaux, Hardy, Noll and others. The momentum balance equation con-tains a flux (stress), which is given by an exact function of particle positions and velocities. We propose a method for approximating this function by a sequence of operators applied to average density and momen-tum. The resulting approximate mesoscopic models are systems in closed form. The closed from property allows one to work directly with the mesoscale equaitons without the need to calculate underlying particle trajectories, which is useful for modeling and simulation of large particle systems. The proposed closure method utilizes the theory of ill-posed problems, in particula...

SIAM Journal on Mathematical Analysis, 2005

We study suspensions of rigid particles (inclusions) in a viscous incompressible fluid. The parti... more We study suspensions of rigid particles (inclusions) in a viscous incompressible fluid. The particles are close to touching one another, so that the suspension is near the packing limit, and the flow at small Reynolds number is modeled by the Stokes equations. The objective is to determine the dependence of the effective viscosity μ on the geometric properties of the particle array. We study spatially irregular arrays, for which the volume fraction alone is not sufficient to estimate the effective viscosity. We use the notion of the interparticle distance parameter δ, based on the Voronoi tessellation, and we obtain a discrete network approximation of μ , as δ → 0. The asymptotic formulas for μ , derived in dimensions two and three, take into account translational and rotational motions of the particles. The leading term in the asymptotics is rigorously justified in two dimensions by constructing matching upper and lower variational bounds on μ. While the upper bound is obtained by patching together local approximate solutions, the construction of the lower bound cannot be obtained by a similar local analysis because the boundary conditions at fluid-solid interfaces must be resolved for all particles simultaneously. We observe that satisfying these boundary conditions, as well as the incompressibility condition, amounts to solving a certain algebraic system. The matrix of this system is determined by the total number of particles and their coordination numbers (number of neighbors of each particle). We show that the solvability of this system is determined by the properties of the network graph (which is uniquely defined by the array of particles) as well as by the conditions imposed at the external boundary.

Continuum Mechanics and Thermodynamics, 2010

A two-dimensional bistable lattice is a periodic triangular network of non-linear bistable rods. ... more A two-dimensional bistable lattice is a periodic triangular network of non-linear bistable rods. The energy of each rod is piecewise quadratic and has two minima. Consequently, a rod undergoes a reversible phase transition when its elongation reaches a critical value. We study an energy minimization problem for such lattices. The objective is to characterize the effective energy of the system when the number of nodes in the network approaches infinity. The most important feature of the effective energy is its "flat bottom". This means that the effective energy density is zero for all strains inside a certain three-dimensional set in the strain space. The flat bottom occurs because the microscopic discrete model has a large number of deformed states that carry no forces. We call such deformations still states. In the paper, we present a complete characterization of the "flat bottom" set in terms of the parameters of the network. This is done by constructing a family of still states whose average strains densely fill the set in question. The two-dimensional case is more difficult than the previously studied case of one-dimensional chains, because the elongations in two-dimensional networks must satisfy certain compatibility conditions that do not arise in the one-dimensional case. We derive these conditions for small and arbitrary deformations. For small deformations a complete analysis is provided.

arXiv: Fluid Dynamics, 2004

We study effective shear viscosity � ⋆ and effective extensional viscosity λ ⋆ of concentrated no... more We study effective shear viscosity � ⋆ and effective extensional viscosity λ ⋆ of concentrated non-colloidal suspensions of rigid spherical particles. The focus is on the spatially disordered arrays, but periodic arrays are considered as well. We use recently developed discrete network approximation techniques to obtain asymptotic formulas for � ⋆ and λ ⋆ as the typical interparticle distance δ tends to zero, assuming that the fluid flow is governed by Stokes equations. For disordered arrays, the volume fraction alone does not determine the effective viscosity. Use of the network approximation allows us to study the dependence of � ⋆ and λ ⋆ on variable distances between neighboring particles in such arrays. Our analysis, carried out for a two-dimensional model, can be characterized as global because it goes beyond the local analysis of flow between two particles and takes into account hydrodynamical interactions in the entire particle array. Previously, asymptotic formulas for � ⋆ ...

We obtain a kinetic description of spatially averaged dynamics of particle systems. Spatial avera... more We obtain a kinetic description of spatially averaged dynamics of particle systems. Spatial averaging is one of the three types of averaging relevant within the Irwing-Kirkwood procedure (IKP), a general method for deriving macroscopic equations from molecular models. The other two types, ensemble averaging and time averaging, have been extensively studied, while spatial averaging is relatively less understood. We show that the average density, linear momentum, and kinetic energy used in IKP can be obtained from a single average quantity, called the generating function. A kinetic equation for the generating function is obtained and tested numerically on Lennard-Jones oscillator chains.

We study quasi-static deformation of dense granular packings. The packing is deformed by imposing... more We study quasi-static deformation of dense granular packings. The packing is deformed by imposing external boundary conditions, which model engineering experiments such as shear and compression. We propose a two-dimensional network model of such deformations. The model takes into account elastic interparticle interactions and incorporates geometric impenetrability constraints. The effects of friction are neglected. In our model, a granular packing is represented by a spring-lattice network, whereby the particle centers correspond to vertices of the network, and interparticle contacts correspond to the edges. When the network deforms, each spring either preserves its length (this corresponds to a solid-like contact), or expands (this represents a broken contact). We prove that under certain geometric conditions on the network, in the energy-minimizing configuration there are at least two non-stretched springs attached to each node, which means that every particle has at least two sol...

A simple model for simulating flows of active suspensions is investigated. The approach is based ... more A simple model for simulating flows of active suspensions is investigated. The approach is based on dissipative particle dynamics. While the model is potentially applicable to a wide range of self-propelled particle systems, the specific class of self-motile bacterial suspensions is considered as a modeling scenario. To mimic the rod-like geometry of a bacterium, two dissipative particle dynamics particles are connected by a stiff harmonic spring to form an aggregate dissipative particle dynamics molecule. Bacterial motility is modeled through a constant self-propulsion force applied along the axis of each such aggregate molecule. The model accounts for hydrodynamic interactions between self-propelled agents through the pairwise dissipative interactions conventional to dissipative particle dynamics. Numerical simulations are performed using a customized version of the open-source LAMMPS (Large-scale Atomic/Molecular Massively Parallel Simulator) software package. Detailed studies of...

The paper introduces a general framework for derivation of continuum equations governing meso-sca... more The paper introduces a general framework for derivation of continuum equations governing meso-scale dynamics of large particle systems. The balance equations for spatial averages such as density, linear momentum, and energy were previously derived by a number of authors. These equations are not in closed form because the stress and the heat flux cannot be evaluated without the knowledge of particle positions and velocities. We propose a closure method for approximating fluxes in terms of other meso-scale averages. The main idea is to rewrite the non-linear averages as linear convolutions that relate micro- and meso-scale dynamical functions. The convolutions can be approximately inverted using regularization methods developed for solving ill-posed problems. This yields closed form constitutive equations that can be evaluated without solving the underlying ODEs. We test the method numerically on Fermi-Pasta-Ulam chains with two different potentials: the classical Lennard-Jones, and t...

Abstract. The paper introduces a general framework for derivation of continuum equations governin... more Abstract. The paper introduces a general framework for derivation of continuum equations governing meso-scale dynamics of large particle systems. The balance equations for spatial averages such as density, linear momentum, and energy were previously derived by a number of authors. These equations are not in closed form because the stress and the heat flux cannot be evaluated without the knowledge of particle positions and velocities. We propose a closure method for approximating fluxes in terms of other meso-scale averages. The main idea is to rewrite the non-linear averages as linear convolutions that relate micro- and meso-scale dynamical functions. The convolutions can be approximately inverted using regularization methods developed for solving ill-posed problems. This yields closed form constitutive equations that can be evaluated without solving the underlying ODEs. We test the method numerically on Fermi-Pasta-Ulam chains with two different potentials: the classical Lennard-Jo...

ArXiv, 2019

Gromov--Hausdorff distances measure shape difference between the objects representable as compact... more Gromov--Hausdorff distances measure shape difference between the objects representable as compact metric spaces, e.g. point clouds, manifolds, or graphs. Computing any Gromov--Hausdorff distance is equivalent to solving an NP-Hard optimization problem, deeming the notion impractical for applications. In this paper we propose polynomial algorithm for estimating the so-called modified Gromov--Hausdorff (mGH) distance, whose topological equivalence with the standard Gromov--Hausdorff (GH) distance was established in \cite{memoli12} (Memoli, F, \textit{Discrete \& Computational Geometry, 48}(2) 416-440, 2012). We implement the algorithm for the case of compact metric spaces induced by unweighted graphs as part of Python library \verb|scikit-tda|, and demonstrate its performance on real-world and synthetic networks. The algorithm finds the mGH distances exactly on most graphs with the scale-free property. We use the computed mGH distances to successfully detect outliers in real-world soc...

We study the effects of geometric impenetrability constraints in statics of frictionless granular... more We study the effects of geometric impenetrability constraints in statics of frictionless granular packings. The packing is deformed by imposing boundary conditions, modeling those in shear and compression experiments. In our two-dimensional model, the packing is represented by a spring-lattice network, whereby the particle centers correspond to vertices of the network, and interparticle contacts correspond to the edges. For the springs, we use a quadratic elastic interaction potential. Combined with the linearized impenetrability constraints, it provides a regularization of the hard-sphere potential for small displacements. When the network deforms, each spring either preserves its length (this corresponds to a solid-like contact), or expands (this represents a broken contact). Solid-like contacts are either sheared or stuck. A contact between two particles is sheared when the local motion of one particle relative to another is an infinitesimal shear. If a pair of particles in conta...

The paper is devoted to homogenization of two-phase incompressible viscoelastic flows with disord... more The paper is devoted to homogenization of two-phase incompressible viscoelastic flows with disordered microstructure. We study two cases. In the first case, both phases are modeled as Kelvin-Voight viscoelastic materials. In the second case, one phase is a Kelvin-Voight material, and the other is a viscous Newtonian fluid. The microscale system contains the conservation of mass and balance of momentum equations. The inertial terms in the momentum equation incorporate the actual interface advected by the flow. In the constitutive equations, a frozen interface is employed. The interface geometry is arbitrary: we do not assume periodicity, statistical homogeneity or scale separation. The problem is homogenized using G-convergence and oscillating test functions. Since the microscale system is not parabolic, previously known constructions of the test functions do not work here. The test functions developed in the paper are non-local in time and satisfy divergence-free constraint exactly....

Physics of Fluids

Starting from a fine-scale dissipative particle dynamics (DPD) model of self-motile point particl... more Starting from a fine-scale dissipative particle dynamics (DPD) model of self-motile point particles, we derive meso-scale continuum equations by applying a spatial averaging version of the Irving-Kirkwood-Noll procedure. Since the method does not rely on kinetic theory, the derivation is valid for highly concentrated particle systems. Spatial averaging yields a stochastic continuum equations similar to those of Toner and Tu. However, our theory also involves a constitutive equation for the average fluctuation force. According to this equation, both the strength and the probability distribution vary with time and position through the effective mass density. The statistics of the fluctuation force also depend on the fine scale dissipative force equation, the physical temperature, and two additional parameters which characterize fluctuation strengths. Although the self-propulsion force entering our DPD model contains no explicit mechanism for aligning the velocities of neighboring particles, our averaged coarse-scale equations include the commonly encountered cubically nonlinear (internal) body force density.

Eprint Arxiv Physics 0409050, Sep 1, 2004

We study effective shear viscosity µ ⋆ and effective extensional viscosity λ ⋆ of concentrated no... more We study effective shear viscosity µ ⋆ and effective extensional viscosity λ ⋆ of concentrated non-colloidal suspensions of rigid spherical particles. The focus is on the spatially disordered arrays, but periodic arrays are considered as well. We use recently developed discrete network approximation techniques to obtain asymptotic formulas for µ ⋆ and λ ⋆ as the typical interparticle distance δ tends to zero, assuming that the fluid flow is governed by Stokes equations. For disordered arrays, the volume fraction alone does not determine the effective viscosity. Use of the network approximation allows us to study the dependence of µ ⋆ and λ ⋆ on variable distances between neighboring particles in such arrays. Our analysis, carried out for a two-dimensional model, can be characterized as global because it goes beyond the local analysis of flow between two particles and takes into account hydrodynamical interactions in the entire particle array. Previously, asymptotic formulas for µ ⋆ and λ ⋆ were obtained via asymptotic analysis of lubrication effects in a single thin gap between two closely spaced particles. The principal conclusion in the paper is that, in general, asymptotic formulas for µ ⋆ and λ ⋆ obtained by global analysis are different from the formulas obtained from local analysis. In particular, we show that the leading term in the asymptotics of µ ⋆ is of lower order than suggested by the local analysis (weak blow up), while the order of the leading term in the asymptotics of λ ⋆ depends on the geometry of the particle array (either weak or strong blow up). We obtain geometric conditions on a random particle array under which the asymptotic order of λ ⋆ coincides with the order of the local dissipation in a gap between two neighboring particles, and show that these conditions are generic. We also provide an example of a uniformly closely packed particle array for which the leading term in the asymptotics of λ ⋆ degenerates (weak blow up).