Steady Streaming and Pumping Driven by Two Frequency Oscillations

Past experiments showed net transport of an object sliding on a surface undergoing two mode vibrations for certain frequency pairs. Specifically, when the vibrations satisfied a non-antiperiodic waveform, net motion was seen. Inspired by these experiments, we look for a fluid analogue and revisit the classical problem of steady streaming in fluids (i.e., nonzero mean flow produced by a periodic forcing) driven by multifrequency oscillations. Using numerical simulations, we examine the flow generated by two-frequency oscillations of a rigid object immersed in fluid. Like the experiments, our results show that net pumping occurs when these two-frequency pairs produce a non-antiperiodic driving force. Furthermore, we use small amplitude analysis and extend the past results of steady streaming to the two-frequency case. While steady streaming occurs at second order in amplitude, pumping is a higher order effect which is analytically challenging to compute. Because pumping is a higher order effect, we use a hybrid numerical and analytical method to explain the mechanism behind pumping of two-frequency oscillations. For example, when the frequency ratio is two, pumping is a third order effect in amplitude, and we further generalize our analysis to different frequency ratios.