Steady advection–diffusion around finite absorbers in two-dimensional potential flows

Choi, Jaehyuk; Margetis, Dionisios; Squires, Todd M.; Bazant, Martin Z.

doi:10.1017/s0022112005005008

Cited by 32 publications

(59 citation statements)

References 46 publications

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“…We expect that an analogous equation to Eq. (14), relating G ∞ (w) and |G ′ ∞ (e iθ )| −2 , will hold in a channel geometry, and Saffman-Taylor fingers should be exact solutions to the mean-field approximation of that equation.…”

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confidence: 99%

Average Shape of Transport-Limited Aggregates

2005

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We study the relation between stochastic and continuous transport-limited growth models, which generalize conformal-mapping formulations of diffusion-limited aggregation (DLA) and viscous fingering, respectively. We derive a nonlinear integro-differential equation for the asymptotic shape (average conformal map) of stochastic aggregates, whose mean-field approximation is the corresponding continuous equation, where the interface moves at its local expected velocity. Our equation accurately describes advection-diffusion-limited aggregation (ADLA), and, due to nonlinear averaging over fluctuations, the average ADLA cluster is similar, but not identical, to an exact solution of the mean-field dynamics. Similar results should apply to all models in our class, thus explaining the known discrepancies between average DLA clusters and viscous fingers in a channel geometry. Straightforward derivation of such approximate theories typically involves uncontrolled assumptions required to "close" an infinite hierarchy of equations for moments of the probability distribution of the stochastic field. An alternative approach consists of deriving asymptotic solutions to a deterministic version of the original stochastic dynamics, assuming that such solutions capture the behavior of ensemble average of the original stochastic field [2]. Such approach, however, might turn out to be unreliable as well, since it underestimates the possible effects of noise on the asymptotic evolution of a stochastic field.A nontrivial example in which such approach has been advanced over the last two decades is the fractal morphology of patterns observed in computer simulations of the celebrated diffusion limited aggregation (DLA) model [3]. Since the relation between the mathematical formulations of the stochastic DLA process and the continuous viscous fingering dynamics was established [4], the striking similarity between patterns observed in both processes has triggered various attempts to interpret viscous fingering dynamics as a mean field for DLA [5][6][7][8].In this Letter, we study the connection between a broad class of stochastic transport-limited aggregation processes and their continuous counterparts [9]. In our models, growth is fuelled by nonlinear, non-Laplacian transport processes, such as advection-diffusion and electrochemical conduction, which satisfy conformally invariant equations [10]. Stochastic and continuous dynamics are defined by generalizing conformal-mapping formulations of DLA [11] and viscous fingering [12,13], respectively. We show that the continuous dynamics is a selfconsistent mean-field approximation of the stochastic dynamics, which, nevertheless, does not accurately predict the average shape of a random ensemble of aggregates.We consider a set of two-dimensional scalar fields, ϕ = {ϕ 1 , ϕ 2 , . . . , ϕ M }, whose gradients produce quasi-static, conserved "flux densities",in Ω z (t), the exterior of a singly connected domain that represents a growing aggregate at time t. (The coefficients, {C ij } may be nonlinea...

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confidence: 99%

Average Shape of Transport-Limited Aggregates

2005

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“…The flux profile σ(θ, Pe) on the absorber has been studied extensively, and very accurate asymptotic approximations are available [27]. (A numerical code in matlab is also at http://advection-diffusion.net.)…”

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“…Mathematically, this is a general consequence of the conformal invariance of the transport process [7], which causes the total flux (Nusselt number) to depend only on the conformal radius and not the asymmetric shape of the particle [27]. Physically, the enhancement of dissolution on the upstream side of the solid is cancelled by the reduction in dissolution on the downstream side.…”

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Interfacial dynamics in transport-limited dissolution

Bazant

2006

Phys. Rev. E

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Various model problems of "transport-limited dissolution" in two dimensions are analyzed using time-dependent conformal maps. For diffusion-limited dissolution (reverse Laplacian growth), several exact solutions are discussed for the smoothing of corrugated surfaces, including the continuous analogs of "internal diffusion-limited aggregation" and "diffusion-limited erosion". A class of non-Laplacian, transport-limited dissolution processes are also considered, which raise the general question of when and where a finite solid will disappear. In a case of dissolution by advectiondiffusion, a tilted ellipse maintains its shape during collapse, as its center of mass drifts obliquely away from the background fluid flow, but other initial shapes have more complicated dynamics.The analysis of interfacial dynamics using conformal maps ("Loewner chains") is a classical subject, which is finding unexpected applications in physics [1,2,3]. For dynamics controlled by Laplacian fields, there is a vast literature on continuous models of viscous fingering [4,5], and stochastic models of diffusion-limited aggregation (DLA) [3,6] and fractal curves in critical phenomena [1,2]. Conformal-map dynamics has also been formulated for a class of non-Laplacian growth phenomena of both types [7], driven by conformally invariant transport processes [8]. For growth limited by advection-diffusion in a potential flow, the connection between continuous and stochastic growth patterns has been elucidated [9], and the continuous dynamics has also been studied in cases of freezing in flowing liquids [10,11,12,13].In all of these examples, the moving interface separates a "solid" region, where singularities in the map reside, a "fluid" region, where the driving transport processes occur and the map is univalent. (In viscous fingering, these are the inviscid and viscous fluid regions, respectively.) Most attention has been paid to problems of "transportlimited growth", where the solid region grows into the fluid region, since the dynamics is unstable and typically leads to cusp singularities in finite time [14,15,16] (without surface tension [4]). Here, we consider various time-reversed problems of "transport-limited dissolution" (TLD), where the solid recedes from the fluid region, e.g. driven by advection-diffusion in a potential flow. These are stable processes, so we focus on continuous dynamics, without surface tension.Stochastic diffusion-limited dissolution (DLD), sometimes called "diffusion-limited erosion" (DLE) or "anti-DLA", has been simulated by allowing random walkers in the fluid to annihilate particles of the solid upon contact [17,19,20,21], rather than aggregating as in DLA [18]. Outward radial DLE on a lattice, or "internal DLA" (IDLA), where the random walkers start at the origin and cause a fluid cavity to grow in an infinite solid, has also been studied by mathematicians, who proved that the asymptotic shape is a sphere in any dimension [22].We begin our analysis by summarizing some exact solutions for continuous DLD, which cou...

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“…20 If the scalar field has a time-independent solution then the net mass transport, associated with the boundary value problem, is proportional 19 to the diffusion coefficient D, or more precisely to the inverse of the Péclet number ͑1/Pe͒. 23 This was exploited by Witting 24 and Szeri 25 to consider the effect of capillary waves on the enhancement of transport across a wavy interface. An example satisfying the above conditions is mass transport across the interface of a spherical bubble undergoing volume oscillations in an incompressible liquid.…”

Section: B Lagrangian Coordinatesmentioning

confidence: 99%

Diffusion and Brownian motion in Lagrangian coordinates

Fyrillas

Nomura

2007

The Journal of Chemical Physics

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In this paper we consider the convection-diffusion problem of a passive scalar in Lagrangian coordinates, i.e., in a coordinate system fixed on fluid particles. Both the convection-diffusion partial differential equation and the Langevin equation are expressed in Lagrangian coordinates and are shown to be equivalent for uniform, isotropic diffusion. The Lagrangian diffusivity is proportional to the square of the relative change of surface area and is related to the Eulerian diffusivity through the deformation gradient tensor. Associated with the initial value problem, we relate the Eulerian to the Lagrangian effective diffusivities (net spreading), validate the relation for the case of linear flow fields, and infer a relation for general flow fields. Associated with the boundary value problem, if the scalar transport problem possesses a time-independent solution in Lagrangian coordinates and the boundary conditions are prescribed on a material surface/interface, then the net mass transport is proportional to the diffusion coefficient. This can be also shown to be true for large Peclet number and time-periodic flow fields, i.e., closed pathlines. This agrees with results for heat transfer at high Peclet numbers across closed streamlines.

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Steady advection–diffusion around finite absorbers in two-dimensional potential flows

Cited by 32 publications

References 46 publications

Average Shape of Transport-Limited Aggregates

Average Shape of Transport-Limited Aggregates

Interfacial dynamics in transport-limited dissolution

Diffusion and Brownian motion in Lagrangian coordinates

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