Fractal derivative model for the transport of the suspended sediment in unsteady flows

Abstract

This paper makes an attempt to develop a Hausdorff fractal derivative model for describing the vertical distribution of suspended sediment in unsteady flow. The index of Hausdorff fractal derivative depends on the spatial location and the temporal moment in sediment transport. We also derive the approximate solution of the Hausdorff fractal derivative advection-dispersion equation model for the suspended sediment concentration distribution, to simulate the dynamics procedure of suspended concentration. Numerical simulation results verify that the Hausdorff fractal derivative model provides a good agreement with the experimental data, which implies that the Hausdorff fractal derivative model can serve as a candidate to describe the vertical distribution of suspended sediment concentration in unsteady flow.

A review of applications of fractional advection–dispersion equations for anomalous solute transport in surface and subsurface water

“…It should also be noticed that the applications of FADEs in hydrology are mainly focused on the simulation of tracer experiments. Previous studies have shown that FADEs can also be used to simulate the change of subsurface water tables (Tu, Ercan, & Levent, 2018) and the transport of suspended sediment in an unsteady flow (Nie et al, 2018), to describe the vertical distribution of suspended sediment in a steady flow (Nie et al, 2017), and to forecast pollutant drift on coastal water surfaces (Qin et al, 2017). Yu et al (2018) also proved that the fractional derivate model could be used to simulate colloid transport in the dense vegetation and soil systems.…”

“…Simulation of contaminant transport processes faces additional challenges due, in part, to the high cost associated with tracer studies, especially in a large river. New techniques based on satellite observations (Neal et al, 2009;L. C. Smith & Pavelsky, 2008) as well as advances in GPS (Global Position System) and Doppler Current Profiling technology (Piotukh, Baranov, Kuklev, & Podymov, 2016;Shen et al, 2010;Wang & Gao, 2013) have the potential to improve the future application of FADEs.…”

“…In recent years, substantial effort has been devoted to the development of FADE theories for describing anomalous solute transport, such as those dealing with nonlocal boundary conditions (BCs; Baeumer, Kovács, Meerschaert, & Sankaranarayanan, 2018;, equations in the improved forms (Jannelli, Ruggieri, & Speciale, 2019;Nie, Sun, Liu, Ze, & Xie, 2018), as well as analytical and numerical solving schemes (Angstmann,Henry,Jacobs,& 10 2 10 3 10 4 10 5 10 6…”

“…In recent years, substantial effort has been devoted to the development of FADE theories for describing anomalous solute transport, such as those dealing with nonlocal boundary conditions (BCs; Baeumer, Kovács, Meerschaert, & Sankaranarayanan, 2018; Y. Zhang et al, 2019), equations in the improved forms (Jannelli, Ruggieri, & Speciale, 2019; Nie, Sun, Liu, Ze, & Xie, 2018), as well as analytical and numerical solving schemes (Angstmann, Henry, Jacobs, & McGann, 2019; Bohaienko, 2019; Cartalade, Younsi, & Neel & M. C., 2019; Cheng, Duan, & Li, 2019; Hussain & Haq, 2019; Jannelli et al, 2019; Singh, Chatterjee, & Singh, 2017; Singh & Chatterjee, 2017; Yu, Zhang, Sun, & Zheng, 2018). However, there is still a certain gap for FADEs to be an effective prediction tool.…”

Entropy and fractal nature

“…Direct analogy to Cantor set is the dust that floats in the air. Brian Kaye devoted fractality of different dusts: mine dust, fumed silica, radioactive dust, and powders for ceramics industry [4][5][6].…”