Modelling of Transportation Process in Plane Flows with Stagnation Points

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Modelling of Transportation Process in Plane Flows with Stagnation Points Boris S. Maryshev1,2   · Michael A. Zaks3  Received: 30 April 2020 / Accepted: 12 August 2020 © Springer Nature B.V. 2020

Abstract We study transport of passive tracers by plane laminar flows with stagnation points. This setup serves as a simple model of transport in porous media. Close to stagnation points, the flow velocity is much lower than on the average. Many experimental data disclose that transport through porous media is slower than predictions of the standard advection-diffusion model. Commonly, in descriptions of flows through porous media the heterogeneity of flow is disregarded, and some average velocity is adopted. To model regular arrays of flow patterns with stagnation points, we employ the construction of special flow: a combination of mapping and a sort of ceiling function with singularity. We show that, depending on the type of stagnation points, many experimentally established effects can be reproduced. This includes the linear sorption, usually described by the standard MIM model, as well as subdiffusion, described by the fractional MIM model. We also show that slow molecular diffusion does not eliminate the effects of the transport slowdown. Keywords  Transport through porous media · Special flow · Subdiffusion

1 Introduction. From Fluid Mechanics to Special Flows Fluid motions in porous media commonly arise due to the pressure difference between the inlet and outlet of a porous massif, and often feature complex flow patterns. Presence of many obstacles on the way of the fluid particles results, on the microscopic level, in curvy streamlines and existence of stagnation points where fluid velocity turns into zero. Structure of the velocity field near a stagnation point is prescribed by the nature of this point (shape of the obstacle or presence of adjacent vortices). Complexity of the flow pattern

* Boris S. Maryshev [email protected] Michael A. Zaks [email protected]‑berlin.de 1

Institute of Continuous Media Mechanics UB RAS, Perm, Russia 614013

2

Department of Theoretical Physics, Perm State University, Perm, Russia 614990

3

Institute of Physics, Humboldt University, 12489 Berlin, Germany



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B. S. Maryshev, M. A. Zaks

is, in its turn, reflected in the unusual properties of transport processes: for example, flows through porous media are often characterized by subdiffusion. In the present work we consider passive tracers, transported across the domain filled with porous medium, and focus on the time intervals that the individual tracers need for the passage through the domain. The distribution of such passage times for the representative ensemble of tracers is an essential characteristics of the transport. Below, we model complex flows by arrays of simple elementary patterns: flows past single obstacles or systems of several vortices. To start with, we restrict ourselves to timeindependent two-dimensional flows of incompressible viscous fluid, generated in domains of simple geometry by s