Unsteady suspended sediment distribution in an ice-covered channel through fractional advection–diffusion equation

Abstract

Despite several applications of the fractional advection–diffusion equation (fADE) in studying sediment transport in an open channel flow, its application is limited to apprehending the non-local movement of sediment particles in an ice-covered channel with a steady, uniform flow field. An unsteady fADE is considered where the space term is non-local with a non-integer order and the mathematical model with Caputo fractional derivative is able to estimate the variation of sediment concentration along a vertical as well as with time in the ice-covered channel. An eddy viscosity expression is used, which includes the variation in roughness between the channel bed and ice cover surface. The Chebyshev collocation method and the Euler backward method are used to solve the fADE with the initial and boundary conditions and the convergence of the methods is established. The temporal variation of concentration shows that for a zero initial condition, the concentration profile first increases and then becomes stable after a certain time; for a non-zero initial concentration, the profile decreases with an increase in time and eventually a steady state is achieved. The effect of the order of the fractional derivative on the vertical variation of concentration at different times for zero and non-zero initial concentrations is studied and it is found that the order of the fractional derivative has a greater impact at smaller times. The impact of several parameters on concentration profiles is studied at different times and the validation of the model is done by comparing it with experimental studies under restricted conditions.