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

Shear induced grain boundary motion for lamellar phases in the weakly nonlinear regime

We study the effect of an externally imposed oscillatory shear on the motion of a grain boundary that separates differently oriented domains of the lamellar phase of a diblock copolymer. A direct numerical solution of the Swift-Hohenberg equation in shear flow is used for the case of a transverse/parallel grain boundary in the limits of weak nonlinearity and low shear frequency. We focus on the region of parameters in which both transverse and parallel lamellae are linearly stable. Shearing leads to excess free energy in the transverse region relative to the parallel region, which is in turn dissipated by net motion of the boundary toward the transverse region. The observed boundary motion is a combination of rigid advection by the flow and order parameter diffusion. The latter includes break up and reconnection of lamellae, as well as a weak Eckhaus instability in the boundary region for sufficiently large strain amplitude that leads to slow wavenumber readjustment. The net average velocity is seen to increase with frequency and strain amplitude, and can be obtained by a multiple scale expansion of the governing equations

Lagrangian Based Methods for Coherent Structure Detection

There has been a proliferation in the development of Lagrangian analytical methods for detecting coherent structures in fluid flow transport, yielding a variety of qualitatively different approaches. We present a review of four approaches and demonstrate the utility of these methods via their application to the same sample analytic model, the canonical double-gyre flow, highlighting the pros and cons of each approach. Two of the methods, the geometric and probabilistic approaches, are well established and require velocity field data over the time interval of interest to identify particularly important material lines and surfaces, and influential regions, respectively. The other two approaches, implementing tools from cluster and braid theory, seek coherent structures based on limited trajectory data, attempting to partition the flow transport into distinct regions. All four of these approaches share the common trait that they are objective methods, meaning that their results do not depend on the frame of reference used. For each method, we also present a number of example applications ranging from blood flow and chemical reactions to ocean and atmospheric flows. (C) 2015 AIP Publishing LLC.ONR N000141210665Center for Nonlinear Dynamic

Asymmetric collapse by dissolution or melting in a uniform flow

An advection--diffusion-limited dissolution model of an object being eroded by a two-dimensional potential flow is presented. By taking advantage of the conformal invariance of the model, a numerical method is introduced that tracks the evolution of the object boundary in terms of a time-dependent Laurent series. Simulations of a variety of dissolving objects are shown, which shrink and then collapse to a single point in finite time. The simulations reveal a surprising exact relationship whereby the collapse point is the root of a non-analytic function given in terms of the flow velocity and the Laurent series coefficients describing the initial shape. This result is subsequently derived using residue calculus. The structure of the non-analytic function is examined for three different test cases, and a practical approach to determine the collapse point using a generalized Newton--Raphson root-finding algorithm is outlined. These examples also illustrate the possibility that the model breaks down in finite time prior to complete collapse, due to a topological singularity, as the dissolving boundary overlaps itself rather than breaking up into multiple domains (analogous to droplet pinch-off in fluid mechanics). In summary, the model raises fundamental mathematical questions about broken symmetries in finite-time singularities of both continuous and stochastic dynamical systems.Comment: 20 pages, 11 figure