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2012 Homogenization of the PoissonNernstPlanck Equations for Ion Transport
 in Charged Porous Media. arXiv:1202.1916
"... Abstract. Effective PoissonNernstPlanck (PNP) equations are derived for macroscopic ion transport in charged porous media. Homogenization analysis is performed for a twocomponent periodic composite consisting of a dilute electrolyte continuum (described by standard PNP equations) and a continuou ..."
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Abstract. Effective PoissonNernstPlanck (PNP) equations are derived for macroscopic ion transport in charged porous media. Homogenization analysis is performed for a twocomponent periodic composite consisting of a dilute electrolyte continuum (described by standard PNP equations) and a continuous dielectric matrix, which is impermeable to the ions and carries a given surface charge. Three new features arise in the upscaled equations: (i) the effective ionic diffusivities and mobilities become tensors, related to the microstructure; (ii) the effective permittivity is also a tensor, depending on the electrolyte/matrix permittivity ratio and the ratio of the Debye screening length to mean pore size; and (iii) the surface charge per volume appears as a continuous “background charge density”. The coefficient tensors in the macroscopic PNP equations can be calculated from periodic reference cell problem, and several examples are considered. For an insulating solid matrix, all gradients are corrected by a single tortuosity tensor, and the Einstein relation holds at the macroscopic scale, which is not generally the case for a polarizable matrix. In the limit of thin double layers, Poisson’s equation is replaced by macroscopic electroneutrality (balancing ionic and surface charges). The general form of the macroscopic PNP equations may also hold for concentrated solution theories, based on the localdensity and meanfield approximations. These results have broad applicability to ion transport in porous electrodes, separators, membranes, ionexchange resins, soils, porous rocks, and biological tissues.
A continuum theory with longrange forces for solids
, 2005
"... A nonlocal integraltype continuum theory, introduced by Silling [61] as peridynamic theory, is analyzed. The theory differs from classical (local) continuum theories in that it involves longrange forces that act between continuum particles. In contrast to existing nonlocal theories, surface trac ..."
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A nonlocal integraltype continuum theory, introduced by Silling [61] as peridynamic theory, is analyzed. The theory differs from classical (local) continuum theories in that it involves longrange forces that act between continuum particles. In contrast to existing nonlocal theories, surface tractions are excluded. As consequence, the theory is applicable off and on defects and interfaces: cracks and phase boundaries are a natural part of the body, and do not require special treatment. Furthermore, as the model loses "local stiffness", the resistance to the formation of discontinuities is reduced. The theory can be viewed in two ways: It is a multiscale model in that it can be applied simultaneously on the atomistic level and on the macroscopic level with a smooth interface between the two regimes. Alternatively, it can be used as a purely phenomenological macroscopic model that still works off and on defects. Forces of microscopic range may represent the interaction of an underlying microstructure such as discrete particles. Forces of macroscopic range are not necessarily physical and are
Generalized Iteration Method for FirstKind Integral Equations
"... An iteration method is described to solve onedimensional, firstkind integral equations with finite integration limits and difference kernel, K(x − x ′), that decays exponentially. The method relies on deriving via the Wiener–Hopf factorization and solving by suitable iterations in the Fourier comp ..."
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An iteration method is described to solve onedimensional, firstkind integral equations with finite integration limits and difference kernel, K(x − x ′), that decays exponentially. The method relies on deriving via the Wiener–Hopf factorization and solving by suitable iterations in the Fourier complex plane a pair of integral relations, where each iteration accounts for all end point singularities in x of the exact solution. For even and odd kernels, this pair reduces to decoupled, 2ndkind Fredholm equations, and the iteration yields Neumann series subject to known convergence criteria. This formulation is applied to a classic problem of steady advectiondiffusion in the twodimensional (2D) potential flow of concentrated fluid. The remarkable overlap of recently derived asymptotic expansions for the flux in this case is shown to be intimately related to the analyticity of the kernel Fourier transform. 1.
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"... Abstract. We compute the mean first passage time (MFPT) for a Brownian particle inside a twodimensional disk with reflective boundaries and a small interior trap that is rotating at a constant angular velocity. The inherent symmetry of the problem allows for a detailed analytic study of the situati ..."
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Abstract. We compute the mean first passage time (MFPT) for a Brownian particle inside a twodimensional disk with reflective boundaries and a small interior trap that is rotating at a constant angular velocity. The inherent symmetry of the problem allows for a detailed analytic study of the situation. For a given angular velocity, we determine the optimal radius of rotation that minimizes the average MFPT over the disk. Several distinct regimes are observed, depending on the ratio between the angular velocity ω and the trap size ε, and several intricate transitions are analyzed using the tools of asymptotic analysis and Fourier series. For ω ∼ O(1), we compute a critical value ωc> 0 such that the optimal trap location is at the origin whenever ω < ωc and is off the origin for ω> ωc. In the regime 1 ω O(ε−1) the optimal trap path approaches the boundary of the disk. However, as ω is further increased to O(ε−1), the optimal trap path “jumps ” closer to the origin. Finally, for ω O(ε−1) the optimal trap path subdivides the disk into two regions of equal area. This simple geometry provides a good test case for future studies of MFPT with more complex trap motion.
STEADILY TRANSLATING PARABOLIC DISSOLUTION FINGERS∗
"... Abstract. Dissolution fingers (or wormholes) are formed during the dissolution of a porous rock as a result of nonlinear feedback between the flow, transport, and chemical reactions at pore surfaces. We analyze the shapes and growth velocities of such fingers within the thinfront approximation, in ..."
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Abstract. Dissolution fingers (or wormholes) are formed during the dissolution of a porous rock as a result of nonlinear feedback between the flow, transport, and chemical reactions at pore surfaces. We analyze the shapes and growth velocities of such fingers within the thinfront approximation, in which the reaction is assumed to take place instantaneously with reactants fully consumed at the dissolution front. We concentrate on the case when the main flow is driven by a constant pressure gradient far from the finger, and the permeability contrast between the inside and the outside of the finger is finite. Using Ivantsov ansatz and conformal transformations we find the family of steadily translating fingers characterized by a parabolic shape. We derive the reactant concentration field and the pressure field inside and outside of the fingers and show that the flow within them is uniform. The advancement velocity of the finger is shown to be inversely proportional to its radius of curvature in the small Péclet number limit and independent of the radius of curvature for large Péclet numbers.
Biomechanical Information Transfer: Maximum Caliber, λ
, 2010
"... genome ejection dynamics, and the formation of otoliths in zebrafish ..."
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LIBRARIES Membraneless Hydrogen Bromine Laminar Flow Battery for LargeScale Energy Storage
, 2013
"... Electrochemical energy storage systems have been considered for a range of potential largescale energy storage applications. These applications vary widely, both in the order of magnitude of energy storage that is required and the rate at which energy must be charged and discharged. One such applic ..."
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Electrochemical energy storage systems have been considered for a range of potential largescale energy storage applications. These applications vary widely, both in the order of magnitude of energy storage that is required and the rate at which energy must be charged and discharged. One such application aids the integration of renewable energy technologies onto the electrical grid by shifting the output from renewable energy resources to periods of high demand, relaxing transmission and distribution requirements and reducing the need for fossil fuel burning plants. Although the market need for such solutions is well known, existing technologies are still too expensive to compete with conventional combustionbased solutions. In this thesis, the hydrogen bromine laminar flow battery (HBFLB) is proposed and examined for its potential to provide low cost energy storage using the rapid reaction kinetics of hydrogenbromine reaction pairs and a membraneless laminar flow battery architecture. In this architecture, fluid reactants and electrolyte flow through a small channel at sufficiently low Reynolds number that laminar flow is