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Research Article Open access CC BY 4.0

A Rigorous Homogenization for a Two-Scale Convergence Approach to Piping Flow Erosion with Deposition in a Spatially Heterogeneous Soil

Adu Sakyi, Peter Amoako-Yirenkyi, Isaac Kwame Dontwi

Journal of Advances in Mathematics and Computer Science · pp. 9–25 · Published 27 May 2020

10.9734/jamcs/2020/v35i330256

Abstract

We present a rigorous homogenization approach to modelling piping flow erosion in a spatially heterogeneous soil. The aim is to provide a justication to a formal homogenization approach to piping flow erosion with deposition in a spatially heterogeneous soil. Under the assumption that the soil domain is perforated periodically with cylindrical repeating microstructure, we begin by proving that a solution to the proposed set of microscopic equations exist. Two-scale convergence is then used to study the asymptotic behaviour of solutions to the microscopic problem as the microscopic length scale approaches zero(0). We thus derive rigorously a homogenized model or macro problem as well as explicit formula for the eective coecients. A strong observation from the numerical simulation was that, soil particle concentration in the water/soil mixture decreases but at a decreasing rate whereas soil particle deposition increases at regions with increasing amount of particle concentration in the flow causing a reduction in bare pore spaces across the soil domain.

Homogenization piping flow erosion deposition entrainment microscopic scale macro- scopic scale.

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