Coupled Burgers equations: A model of polydispersive sedimentation

Esipov, Sergei E.

doi:10.1103/physreve.52.3711

Cited by 160 publications

(96 citation statements)

References 32 publications

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“…They have the same convective and diffusion form as the incompressible NavierStokes equations.VBE is a simple model for the understanding of physical flows and problems, such as hydrodynamic turbulence, shock wave theory, wave processes in thermo-elastic medium, vorticity transport, dispersion in porous media, see references. [1][2][3] Simulation of Burgers' equation is a natural and first step towards developing methods for the computation of the complex flows. In past a few decades, it has become customary to test new approaches in computational fluid dynamics by applying them to Burgers' equation, which has resulted in various finite-differences, finite volume, finite-element and boundary element methods.…”

Section: Introductionmentioning

confidence: 99%

Numerical solutions of coupled Burgers’ equations by an implicit finite-difference scheme

2013

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In this paper, an implicit exponential finite-difference scheme (Expo FDM) has been proposed for solving two dimensional nonlinear coupled viscous Burgers’ equations (VBEs) with appropriate initial and boundary conditions. The accuracy of the method has been illustrated by taking two numerical examples. Results are compared with exact solution and those already available in the literature by finding the L1, L2, L∞ and ER errors. Excellent numerical results indicate that the proposed scheme is efficient, reliable and robust technique for the numerical solutions of Burgers’ equation.

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Section: Introductionmentioning

confidence: 99%

Numerical solutions of coupled Burgers’ equations by an implicit finite-difference scheme

2013

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“…We mention that nonlinear diffusion in polydisperse suspensions has been considered by Esipov [43] and is postulated as part of a general "competition" mechanism for multispecies granular mixtures by Braun [13]. However, the terms considered in [43] account for hydrodynamic diffusion, and the consequences of the nonlinearity do not appear, since (apparently, for simplicity) these diffusivities are replaced by constants, and cross-diffusivities (e.g., the dependence of the flux of particle species 1 on the flux of species 2) are ignored, while in [13] the nonlinearity is retained, but cross-diffusivities are equally neglected, and no physical interpretation of the origin of nonlinear diffusion is given.…”

Section: Sediment Diffusivitymentioning

confidence: 99%

Strongly Degenerate Parabolic-Hyperbolic Systems Modeling Polydisperse Sedimentation with Compression

Tory¹,

Karlsen²,

Bürger³

et al. 2003

SIAM J. Appl. Math.

138

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Abstract. We show how existing models for the sedimentation of monodisperse flocculated suspensions and of polydisperse suspensions of rigid spheres differing in size can be combined to yield a new theory of the sedimentation processes of polydisperse suspensions forming compressible sediments ("sedimentation with compression" or "sedimentation-consolidation process"). For N solid particle species, this theory reduces in one space dimension to an N × N coupled system of quasilinear degenerate convection-diffusion equations. Analyses of the characteristic polynomials of the Jacobian of the convective flux vector and of the diffusion matrix show that this system is of strongly degenerate parabolic-hyperbolic type for arbitrary N and particle size distributions. Bounds for the eigenvalues of both matrices are derived. The mathematical model for N = 3 is illustrated by a numerical simulation obtained by the Kurganov-Tadmor central difference scheme for convectiondiffusion problems. The numerical scheme exploits the derived bounds on the eigenvalues to keep the numerical diffusion to a minimum.Key words. polydisperse suspensions, sedimentation, systems of conservation laws, strongly degenerate parabolic-hyperbolic systems, central difference approximation AMS subject classifications. 35K65, 35L40, 35L65, 65M06, 76T05 DOI. 10.1137/S00361399024081631. Introduction. Mathematical models for the (controlled) sedimentation of polydisperse suspensions of small particles, which belong to a finite number of species differing in size or density and are suspended in a viscous fluid, are important to many applications such as the chemical engineering, ceramic, pulp and paper, and food industries, mineral processing, wastewater treatment, and medicine [3,50,88,89,101,122]. The characteristic behavior of such mixtures is differential sedimentation, which leads to areas of different composition if an initially homogeneous suspension is allowed to settle. In this paper, we consider the additional property that the solid particles possibly form a compressible sediment layer. A mathematical model for polydisperse suspensions forming compressible sediments is developed, analyzed, and simulated, focusing on three different aspects.First, we show how two existing sedimentation models-one for monodisperse flocculated suspensions, which are described by scalar strongly degenerate parabolichyperbolic equations, and one for polydisperse suspensions of rigid spheres differing

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“…͑2͒ with the concentration dependent velocity from Eq. ͑8͒ is related to the Burgers equation, 17 which was originally proposed to understand turbulence and has applications in traffic flow, 18 surface growth dynamics, 19 sedimentation, 20 and other traveling shock phenomena. However, it is important to note that the shock behavior seen in SBA-LKMC arises from an artificial bias in the algorithm and does not correspond to a physically relevant phenomenon, because no hydrodynamic interactions between particles are considered in this work.…”

Section: ͑8͒mentioning

confidence: 99%

Lattice kinetic Monte Carlo simulations of convective-diffusive systems

Flamm

Diamond

Sinno

2009

The Journal of Chemical Physics

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Diverse phenomena in physical, chemical, and biological systems exhibit significant stochasticity and therefore require appropriate simulations that incorporate noise explicitly into the dynamics. We present a lattice kinetic Monte Carlo approach to simulate the trajectories of tracer particles within a system in which both diffusive and convective transports are operational. While diffusive transport is readily accounted for in a kinetic Monte Carlo simulation, we demonstrate that the inclusion of bulk convection by simply biasing the rate of diffusion with the rate of convection creates unphysical, shocklike behavior in concentrated systems due to particle pile up. We report that elimination of shocklike behavior requires the proper passing of blocked convective rates along nearest-neighbor chains to the first available particle in the direction of flow. The resulting algorithm was validated for the Taylor-Aris dispersion in parallel plate flow and multidimensional flows. This is the first generally applicable lattice kinetic Monte Carlo simulation for convection-diffusion and will allow simulations of field-driven phenomena in which drift is present in addition to diffusion.

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Coupled Burgers equations: A model of polydispersive sedimentation

Cited by 160 publications

References 32 publications

Numerical solutions of coupled Burgers’ equations by an implicit finite-difference scheme

Numerical solutions of coupled Burgers’ equations by an implicit finite-difference scheme

Strongly Degenerate Parabolic-Hyperbolic Systems Modeling Polydisperse Sedimentation with Compression

Lattice kinetic Monte Carlo simulations of convective-diffusive systems

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