Stationary flow driven by non-sinusoidal time-periodic pressure gradients in wavy-walled channels

• Published in: Applied mathematical modelling 2023-10, Vol.122, p.693-705
• Main Authors: Alaminos-Quesada, J., Gutiérrez-Montes, C., Coenen, W., Sánchez, A.L.
• Format: Article
• Language: English
• Subjects: Non-sinusoidal pressure; Oscillatory flow; Steady streaming
• ISSN: 0307-904X
• DOI: 10.1016/j.apm.2023.06.013

•Most oscillatory physiological flows display a periodic time dependence characterized by a complex waveform.•Steady streaming driven by non-sinusoidal pressure gradients in wall-bounded flows is investigated by asymptotic and numerical methods.•The asymptotic predictions, derived for small stroke lengths, are in excellent agreement with the numerical results.•The steady streaming induced by pressure gradients derived from in-vivo MRI measurements in the spinal canal is computed.•Inter-frequency interactions are found to be more pronounced in cardiac-driven flow than in respiratory-driven flow. [Display omitted] The classical problem of secondary flow driven by a sinusoidally varying pressure gradient is extended here to address periodic pressure gradients of complex waveform, which are present in many oscillatory physiological flows. A slender two-dimensional wavy-walled channel is selected as a canonical model problem. Following standard steady-streaming analyses, valid for small values of the ratio ε of the stroke length of the pulsatile motion to the channel wavelength, the spatially periodic flow is described in terms of power-law expansions of ε, with the Womersley number assumed to be of order unity. The solution found at leading order involves a time-periodic velocity with a zero time-averaged value at any given point. As in the case of a sinusoidal pressure gradient, effects of inertia enter at the following order to induce a steady flow in the form of recirculating vortices with zero net flow rate. An improved two-term asymptotic description of this secondary flow is sought by carrying the analysis to the following order. It is found that, when the pressure gradient has a waveform with multiple harmonics, the resulting velocity corrections display a nonzero flow rate, not present in the single-frequency case, which enables stationary convective transport along the channel. Direct numerical simulations for values of ε of order unity are used to investigate effects of inertia and delineate the range of validity of the asymptotic limit ε≪1. The comparisons of the time-averaged velocity obtained numerically with the two-term asymptotic description reveals that the latter remains remarkably accurate for values of ε exceeding 0.5. As an illustrative example, the results of the model problem are used to investigate the cerebrospinal-fluid flow driven along the spinal canal by the cardiac and respiratory cycles, characterized by markedly non-sinusoidal w