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Time multiscale of sorption kinetics

Nel documento PROCEEDINGS OF SIMAI 2020+21 (pagine 195-199)

Clarissa Astuto

Università degli Studi di Catania Viale Andrea Doria, 6, 95125 - Catania Italy

clarissa.astuto@unict.it Giovanni Russo

Università degli Studi di Catania Viale Andrea Doria, 6, 95125 - Catania Italy

russo@dmi.unict.it Mohammed Lemou Universitè de Rennes

Rennes

mohammed.lemou@univ-rennes1.fr Armando Coco

Oxford Brookes University Oxford

acoco@brookes.ac.uk Antonio Raudino

Università degli Studi di Catania Viale Andrea Doria, 6, 95125 - Catania Italy

araudino@unict.it

We consider numerical methods for solving a 2D transport-diffusion equation in the highly oscillatory regime. This work is part of a long project for the multiscale simulation of surfactant diffusion in presence of a moving trap. The domain of interest is typically a region inside a cylinder exterior to a bubble, which is an oscillatory spheroid. Bubble oscillations of the order of a few nanometers are selectively excited, and surfactant transport is accurately measured [4]. The oscillation frequency is of the order of hundreths Hz while the diffusing time is of the order of hours, thus the different scales in time introduce a multiscale challenge.

We start by the assumption that the motion of the bubble and the fluid are periodic in time, with a period proportional to a small parameter ε (so that it is much shorter than typical diffusion time). An approximate model is then derived, based on the asymptotic expansion of the solution in the small parameter ε.

The computational domain is defined with a 3D axisymmetric geometry, so it can be discretized on a 2D Cartesian mesh in cylindrical coordinates [3]. A detailed solution of the Stokes (or Navier-Stokes) governing the fluid motion is numerically pre-computed over a period, by a suitably developed second order scheme based on a ghost-point level-set method on a regular Cartesian grid [2]. The drift-diffusion equation on the moving fluid are then solved. The effect of the attractive-repulsive potential of the bubble is treated by a suitable boundary condition derived in [3].

Parma, 30 Aug – 3 Sep 2021 160 Back to Table of Contents

Minisymposium MS-15

We obtain an efficient and robust integrator for the detailed computation of the solution of the surfactant diffusion in presence of an oscillating trap, which is able to solve the problem without resolving the small oscillation time scale.

References

[1] P. Chartier, M. Lemou, F. Mèhats, G. Vilmart, A New Class of Uniformly Accurate Numerical Schemes for Highly Oscillatory Evolution Equations, Foundations of Computational Mathematics, 2020.

[2] A. Coco, A multigrid ghost-point level-set method for incompressible Navier-Stokes equations on moving domains with curved boundaries. Journal of Computational Physics, 2020.

[3] A. Raudino, A. Grassi, G. Lombardo, G. Russo, C. Astuto, M. Corti, Anomalous sorption kinetics of self-interacting particles by a spherical trap. Communication in Computational Physics, submitted.

[4] A. Raudino, D. Raciti, A. Grassi, M. Pannuzzo, M. Corti Oscillations of Bubble Shape Cause Anomalous Surfactant Diffusion: Experiments, Theory, and Simulations.

Langmuir, 2016.

Minisymposium MS-15

Modeling and numerics of sorption kinetics

Giovanni Russo

Università degli Studi di Catania Viale Andrea Doria, 6, 95125 - Catania Italy

russo@dmi.unict.it Clarissa Astuto

Università degli Studi di Catania Viale Andrea Doria, 6, 95125 - Catania Italy

clarissa.astuto@unict.it Armando Coco Oxford Brookes University

Oxford

acoco@brookes.ac.uk Antonio Raudino

Università degli Studi di Catania Viale Andrea Doria, 6, 95125 - Catania Italy

araudino@unict.it

The treatment of diffusion of surfactants (anions and cations) in presence of an oscillating bubble is an interesting interdisciplinary problem.

The system is described by drift-diffusion equations, in which the effect of the bubble is modelled through an attractive-repulsive potential acting on the anions [1, 2]. In spite of the apparent simplicity of the model, from the computational point of view the problem is a challenging one, for various reasons.

First, the problem is intrinsically multiscale in space, because the range of interaction between the bubble and the anions (nanometers) is orders of magnitude smaller than the size of the domain of interest (millimeters). Second, if the bubble oscillates then the diffusion of the ions is coupled with the motion of the fluid which has to be computed in a moving domain. Third, in typical experiments with an oscillating bubble, the period of oscillations (few milliseconds) is orders of magnitude smaller than the diffusion time (hours).

The aim of the talk is to describe a model that solves the challenge of the multiple scales in space. A multiscale single carrier model has been derived, which describes the interaction of the bubble on the anions by a suitable boundary condition of the diffusion equation for the ions, derived by mass conservation and asymptotic analysis in the region near the trap. The interaction potential is assumed to be of a small thickness δ, still with a non-negligible effect on the diffusant. The solution is formally expanded in terms of the small parameter δ. It is shown that to lowest order in δ the concentration is in local equilibrium, with a shape given by a Boltzmann distribution. From this one can derive an effective boundary condition relating the time dependence of the boundary concentration to the flux (and therefore to the normal gradient of the concentration).

Parma, 30 Aug – 3 Sep 2021 162 Back to Table of Contents

Minisymposium MS-15

The multiscale model is extended to take into account the effect of saturation (i.e.

non negligible concentration), which may be relevant near the trap.

Numerical solutions are obtained by discretizing the equations on a regular Cartesian grid by a ghost-point level-set method. The resulting large sparse linear system is efficiently solved by a multigrid technique [3].

Both models are validated by comparison with accurate solutions of the drift-diffusion equation in one and two space dimensions. It is observed that as δ decreases, the discrep-ancy between the fully resolved model and the reduced multiscale one is approximately proportional to δ.

References

[1] A. Raudino, D. Raciti, A. Grassi, M. Pannuzzo, M. Corti Oscillations of Bubble Shape Cause Anomalous Surfactant Diffusion: Experiments, Theory, and Simulations.

Langmuir, 2016.

[2] A. Raudino, A. Grassi, G. Lombardo, G. Russo, C. Astuto, M. Corti, Anomalous sorption kinetics of self-interacting particles by a spherical trap. Communication in Computational Physics, submitted.

[3] A. Coco, A multigrid ghost-point level-set method for incompressible Navier-Stokes equations on moving domains with curved boundaries. Journal of Computational Physics, 2020.

Minisymposium MS-15

An improved Rational EXponential Integrator for

Nel documento PROCEEDINGS OF SIMAI 2020+21 (pagine 195-199)

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