A Rigorous Homogenization for a Two-Scale Convergence Approach to Piping Flow Erosion with Deposition in a Spatially Heterogeneous Soil
Adu Sakyi *
Department of Mathematics, Kwame Nkrumah University of Science and Technology, Center for Scientific and Technical Computing, National Institute for Mathematical Sciences, Kumasi, Ghana.
Peter Amoako-Yirenkyi
Department of Mathematics, Kwame Nkrumah University of Science and Technology, Center for Scientific and Technical Computing, National Institute for Mathematical Sciences, Kumasi, Ghana.
Isaac Kwame Dontwi
Department of Mathematics, Kwame Nkrumah University of Science and Technology, Center for Scientific and Technical Computing, National Institute for Mathematical Sciences, Kumasi, Ghana.
*Author to whom correspondence should be addressed.
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.
Keywords: Homogenization, piping flow erosion, deposition, entrainment, microscopic scale, macro- scopic scale.