Spatio-temporal frequency response analysis of forced slip velocity effect on solute concentration oscillations in a reverse osmosis membrane channel

•A spatio-temporal frequency response analysis is developed.•A multi-dimensional pulse input is designed to contain multiple wavenumbers and frequencies.•It is used to study effects of forced slip velocity on solute concentration oscillations.•It helps the design of control input profiles that decrease fouling in membrane systems. A spatio-temporal frequency response analysis is developed based on the multi-dimensional Fourier transform, which decomposes spatio-temporal input and output signals into travelling waves of different spatial wavenumbers and temporal frequencies. A spatio-temporal pulse test is also developed to allow for simultaneous input of multiple wavenumber–frequency combinations. This analysis is applied to a membrane channel, where the input is an effective streamwise fluid velocity at the wall that varies along the membrane length and in time. The combined effect of the temporal frequencies and spatial wavenumbers of the slip velocity on the resulting solute concentration oscillations, which have the potential for reducing the susceptibility of the membrane to fouling, is analysed at Reynolds numbers of 280 and 560, and Schmidt numbers of 600 and 1200. Frequencies up to 800Hz and wavenumbers up to 500m−1 are studied. It is found that for each wavenumber there is a corresponding temporal frequency that results in a maximum amplitude ratio. Developing a spatio-temporal frequency response profile for such a system helps the design of control input profiles that decrease fouling in membrane systems.