A Fourier–Chebyshev collocation method for the mass transport in a layer of power-law fluid mud

Abstract

A Fourier–Chebyshev collocation spectral method is employed in this work to compute the Lagrangian drift or mass transport due to periodic surface pressure loading in a thin layer of non-Newtonian fluid mud, which is modeled as a power-law fluid. Because of the non-Newtonian rheology, these problems are nonlinear and must be solved numerically. On assuming that the solutions are of the same permanent waveform as the pressure loading, the governing equations are made time-independent by referring to a horizontal axis that moves at the same speed as the wave. The solutions are periodic in the horizontal direction, but are non-periodic in the vertical direction, and the computational domain is therefore discretized according to the Fourier–Chebyshev spectral collocation scheme. In this study, the spatial derivatives are computed with a differentiation matrix. In order to incorporate the boundary conditions, the matrix diagonalization technique is used to solve the matrix equation, and all the definite integrals in the vertical direction based on the collocation points are performed by the modified Clenshaw–Curtis quadrature rule. The developed method is applied to compute the first- and second-order motion of the mud. The comparison between the numerical results and the analytical solution in the Newtonian limit shows the good accuracy of the spectral method.