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The interaction between deformable surfaces and oscillatory driving is known to yield complex secondary time-averaged flows due to inertial and elastic nonlinearities. Here, we revisit the problem of oscillatory flow in a cylindrical tube with a deformable wall, and analyze it under a long-wave }theory for small deformations, but for arbitrary Womersley numbers. We find that the oscillatory pressure does not vary linearly along the length of a deformable channel, but instead decays exponentially with spatial oscillations. We show that this decay occurs over an elasto-visco-inertial length scale that depends on the material properties of the fluid and the elastic walls, the geometry of the system, and the frequency of the oscillatory flow, but is independent of the amplitude of deformation. Inertial and geometric nonlinearities associated with the elastic deformation of the channel drive a time-averaged secondary flow. We quantify this flow using numerical solutions of our perturbation theory, and gain insight into these solutions with analytic approximations. The theory identifies a complex non-monotonic dependence of the time-averaged flux on the elastic compliance and inertia, including a reversal of the flow. Finally, we show that our analytic theory is in excellent quantitative agreement with the three-dimensional direct numerical simulations of \citet{pande2023oscillatory}., Comment: 17 pages, 6 figures. Submitted to Journal of Fluid Mechanics

The application of oscillatory flow around an obstacle drives a steady ``streaming'' due to inertial rectification, which has been used in a host of microfluidic applications. While theory has focused largely on two-dimensional (2D) flows, streaming in many practical microfluidic devices is three-dimensional (3D) due to confinement. We develop a three-dimensional streaming theory around an obstacle in a microchannel with a Hele-Shaw like geometry, where one dimension (depth) is much shorter than the other two dimensions. Utilizing inertial lubrication theory, we demonstrate that the time-averaged streaming flow has a three-dimensional structure. Notably, the flow changes direction across the depth of the channel, which is a feature not observed in less confined streaming setups. This feature is confirmed by our experiments of streaming around a cylinder sandwiched in a microchannel. Our theory also predicts that the streaming velocity decays as the inverse cube of the distance from the cylinder, faster than that expected from previous two-dimensional approaches. We verify this decay rate quantitatively using particle tracking measurements from experiments of streaming around cylinders with different aspect ratios at different driving frequencies.

Non-invasive detection of microvascular alterations in deep tissues in vivo provides critical information for clinical diagnosis and evaluation of a broad-spectrum of pathologies. Recently, the emergence of super-resolution ultrasound localization microscopy (ULM) offers new possibilities for clinical imaging of microvasculature at capillary level. Currently, the clinical utility of ULM on clinical ultrasound scanners is hindered by the technical limitations, such as long data acquisition time, and compromised tracking performance associated with low imaging frame-rate. Here we present an in-human ULM on a high frame-rate (HFR) clinical ultrasound scanner to achieve super-resolution microvessel imaging using a short acquisition time (<10s). Ultrasound MB data were acquired from different human tissues, (liver, kidney, pancreatic, and breast tumor) using an HFR clinical scanner. By leveraging the HFR and advanced processing techniques including sub-pixel motion registration, MB signal separation, and Kalman filter-based tracking, MBs can be robustly localized and tracked for successful ULM under the circumstances of relatively high MB concentration and limited data acquisition time in humans. Subtle morphological and hemodynamic information were demonstrated on data acquired with single breath-hold and free-hand scanning. Compared with contrast-enhanced power Doppler generated based on the same MB dataset, ULM showed a 5.7-fold resolution improvement in a vessel, and provided a wide-range flow speed measurement that is Doppler angle-independent. This study demonstrated the feasibility of ultrafast in-human ULM in various human tissues based on a clinical scanner that supports HFR imaging, and showed a great potential for the implementation of super-resolution ultrasound microvessel imaging in a myriad of clinical applications involving microvascular abnormalities and pathologies., Comment: 41 pages, 5 figures, 4 supplemental figures