Elasto-inertial rectification of oscillatory flow in an elastic tube

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 Mechanic

Quantifying microbubble streaming and its applications

The growing interest in microfluidics in the last two decades has resulted in new and exciting ways in which to drive microfluidic flows. A simple and powerful flow actuation method involves the use of acoustically excited microbubbles. For ease of manufacture and flow control, setups have largely focused on microbubbles of semi-cylindrical shape, attached to a wall of the microchannel. The application of an ultrasound field drives oscillations of the bubble interface, which then become rectified into strong secondary steady currents in the fluid, termed ``streaming''. While several researchers have used such setups in experiments, a theoretical quantification of the bubble streaming flows, crucial for the systematic design of practical microfluidics applications, has lagged behind. In the first part of the dissertation, we resolve both the primary oscillatory and secondary steady flow components. We begin by developing an asymptotic theory describing the oscillatory response of the bubble to the applied acoustic field. We show that the presence of viscous boundary layers and pinned contact lines at the walls (i) strongly couples volume oscillations of the bubble to shape oscillations of the interface, and (ii) results in much wider surface-mode frequency resonance peaks than is nominally predicted by potential flow theory. The oscillatory dynamics then feed into a calculation of the secondary flow, which rigorously accounts for boundary layers over the bubble and the wall. We show that the two-dimensional steady vortical streaming flows observed in experiment are governed at low frequencies by surface mode dynamics, but undergo a reversal of orientation at higher frequencies, where volume oscillations dominate. The theory therefore connects the oscillatory dynamics to the steady streaming, reproducing the entire spectrum of steady flow patterns observed in experiments, with no adjustable parameters. The 2D theory is then modified to include 3D flow effects, in the light of recent collaborative experimental measurements. We show that these flows arise due to the axial confinement of the bubble by no-slip walls, and can be modeled by a perturbation of the 2D streaming solutions by additional (axial) Stokes solutions. The 3D theory explains the experimentally observed flow kinematics over a wide range of time scales, showing that the 2D trajectories typically observed in experiments are in fact sections of a higher three-dimensional flow structure that becomes apparent only on much longer time scales. We then develop a Hamiltonian formalism that governs the long time 3D motion and is applicable to any perturbed 2D flow under confinement. Having now systematically developed a theoretical description of the flow field, the second part of the dissertation deals with its application to practically useful situations in microfluidics. We first analyze the micromixing between two fluid streams continuously transported through the channel by a Poiseuille flow, whose mixing properties are enhanced by an array of acoustically excited bubbles located at the channel walls. We argue that in order to achieve exponentially fast fluid mixing, it is necessary to introduce a temporal modulation in the flow field, achieved here through a duty cycling of the streaming flow (i.e., of the driving ultrasound). It is then shown using numerical simulations that the mixing is optimized at specific duty cycles that can be understood from global transport properties of the Poiseuille flow and the streaming vortices, thus forming the first protocol for open-flow mixing that is optimized from first principles. Finally, we analyze the motion of rigid spherical microparticles within streaming flows, with the intention of designing a size-sensitive sorting device. We show that assuming a short-range hard-core interaction to prevent penetration of particle and bubble surfaces is sufficient to explain a drift of particles across streamlines close to the bubble. This drift ultimately results in the size-dependent sorting behavior observed in experiments, provided that 3D flow effects are properly accounted for

Three-dimensional phenomena in microbubble acoustic streaming

Ultrasound-driven oscillating micro-bubbles have been used as active actuators in microfluidic devices to perform manifold tasks such as mixing, sorting and manipulation of microparticles. A common configuration consists on side-bubbles, created by trapping air pockets in blind channels perpendicular to the main channel direction. This configuration consists of acoustically excited bubbles with a semi-cylindrical shape that generate significant streaming flow. Due to the geometry of the channels, such flows have been generally considered as quasi two-dimensional. Similar assumptions are often made in many other microfluidic systems based on \emph{flat} micro-channels. However, in this paper we show that microparticle trajectories actually present a much richer behavior, with particularly strong out-of-plane dynamics in regions close to the microbubble interface. Using Astigmatism Particle Tracking Velocimetry, we reveal that the apparent planar streamlines are actually projections of a \emph{streamsurface} with a pseudo-toroidal shape. We therefore show that acoustic streaming cannot generally be assumed as a two-dimensional phenomenon in confined systems. The results have crucial consequences for most of the applications involving acoustic streaming as particle trapping, sorting and mixing.Comment: 5 pages, 4 high quality figures. Accepted for Publication in Phys. Rev. Applied, March 201

Rotation of an immersed cylinder sliding near a thin elastic coating

It is known that an object translating parallel to a soft wall in a viscous fluid produces hydro- dynamic stresses that deform the wall, which, in turn, results in a lift force on the object. Recent experiments with cylinders sliding under gravity near a soft incline, which confirmed theoretical arguments for the lift force, also reported an unexplained steady-state rotation of the cylinders [Saintyves et al. PNAS 113(21), 2016]. Motivated by these observations, we show, in the lubrication limit, that an infinite cylinder that translates in a viscous fluid parallel to a soft wall at constant speed and separation distance must also rotate in order to remain free of torque. Using the Lorentz reciprocal theorem, we show analytically that for small deformations of the elastic layer, the angular velocity of the cylinder scales with the cube of the sliding speed. These predictions are confirmed numerically. We then apply the theory to the gravity-driven motion of a cylinder near a soft incline and find qualitative agreement with the experimental observations, namely that a softer elastic layer results in a greater angular speed of the cylinder.Comment: 16 pages, 4 figure

Three-dimensional streaming flow patterns in confinement

Steady streaming flow exited by oscillating bubbles is an intriguing tool for transport, mixing, sorting, or force actuation applications in microfluidics. Often the geometry of the set-up is intended to encourage two-dimensional (2D) flows, keeping the flow pattern across the channel depth uniform. This condition cannot always be ideally fulfilled, and three-dimensional (3D) streaming effects may be greatly beneficial, e.g., in mixing applications. We demonstrate that a weak 3D streaming component can be combined with existing 2D streaming theory, resulting in a systematic description of 3D streaming flow patterns. We show that these patterns can indeed be observed in bubble microstreaming, using 3D trajectory tracking by astigmatic particle tracking velocimetry

Manipulation and size sorting of microparticles in streaming flow

When driven by an acoustic pressure field at ultrasound frequencies, microbubbles adsorbed at a solid boundary establish strong steady vortical flows. Microbubble steady streaming flows represent a unique type of actuating mechanism for microfluidics and have demonstrated great potential in handling micro-objects, controlling liquid transport as well as deforming biological objects (e.g., cells and vesicles). We demonstrate that the geometry of this type of flow can easily be changed interactively and noninvasively, shaping the local flow environment with micron accuracy. Microparticles thus experience tunable forces that alter their trajectories, e.g., depending on their size. Using a combination of bubble streaming and Poiseuille channel flows, we demonstrate devices for size dependent trapping, sorting, and focusing of microparticles. We further emphasize that these streaming flow fields can be described analytically by asymptotic methods, taking much of the guesswork out of the development of new devices

A reciprocal theorem for the prediction of the normal force induced on a particle translating parallel to an elastic membrane

When an elastic object is dragged through a viscous fluid tangent to a rigid boundary, it experiences a lift force perpendicular to its direction of motion. An analogous lift mechanism occurs when a rigid symmetric object translates parallel to an elastic interface or a soft substrate. The induced lift force is attributed to an elastohydrodynamic coupling that arises from the breaking of the flow reversal symmetry induced by the elastic deformation of the translating object or the interface. Here we derive explicit analytical expressions for the quasi-steady state lift force exerted on a rigid spherical particle translating parallel to a finite-sized membrane exhibiting a resistance toward both shear and bending. Our analytical approach proceeds through the application of the Lorentz reciprocal theorem so as to obtain the solution of the flow problem using a perturbation technique for small deformations of the membrane. We find that the shear-related contribution to the normal force leads to an attractive interaction between the particle and the membrane. This emerging attractive force decreases quadratically with the system size to eventually vanish in the limit of an infinitely-extended membrane. In contrast, membrane bending leads to a repulsive interaction whose effect becomes more pronounced upon increasing the system size, where the lift force is found to diverge logarithmically for an infinitely-large membrane. The unphysical divergence of the bending-induced lift force can be rendered finite by regularizing the solution with a cut-off length beyond which the bending forces become subdominant to an external body force.Comment: 15 pages, 4 figures, 80 references. Under revie

Size-dependent particle migration and trapping in 3D microbubble streaming flows

Acoustically actuated sessile bubbles can be used as a tool to manipulate microparticles, vesicles and cells. In this work, using acoustically actuated sessile semi-cylindrical microbubbles, we demonstrate experimentally that finite-sized microparticles undergo size-sensitive migration and trapping towards specific spatial positions in three dimensions with high reproducibility. The particle trajectories are successfully reproduced by passive advection of the particles in a steady three-dimensional streaming flow field augmented with volume exclusion from the confining boundaries. For different particle sizes, this volume exclusion mechanism leads to three regimes of qualitatively different migratory behavior, suggesting applications for separating, trapping, and sorting of particles in three dimensions.Comment: 12 pages, 7 figure