The purpose of the present study is to extend the simple concept of apparent coefficient of restitution, widely approached in the literature for the case of a single-contact-point between a sphere and a wall, to the case of bouncing whose complexity is increased due to the shape of the contacting object.
Experiments are carried out with a finite-length cylinder, freely falling in a liquid at rest. Complex 3D motions of the cylinder with respect to the wall occur during bouncing, including multi-contact-points between the cylinder and the bottom as well as cavitation.
We investigate numerical modelling of an idealized situation (2D infinite cylinder falling parallel to the wall) with 2D simulations where the fluid equations of motion were coupled to the particle equation of motion through an Immersed Boundary Method. The particle equation of motion is coupled to an elastic force to model bouncing, requiring a parameterization of the cut-off length (interpreted as a roughness) and the contact time (associated with the contact elasticity) used here to capture the experimental observations. The simulations confirmed that i) the departure of the coefficient of restitution from 0 is strictly dependent on the apparent roughness and ii) the coefficient of restitution depends on the contact time.
Finally, we model the coefficient of restitution as the product of two contributions to the mechanical loss of energy: the collision-to-terminal velocity ratio ($V_c/V_t$) of the approach-phase and the rebound-to-collision velocity ratio ($-V_r/V_c$) of the contact-phase. This leads to a reasonably good prediction of the coefficient of restitution in the intermediate regime in $St$. This suggests the relevance of lumping the complex details of physical phenomena involved during contact into a simple concept based on the contact apparent roughness and elasticity.
In this technical note, we demonstrate the robustness of our numerical scheme of vorticity iteration in dealing with the dipole-wall interaction at small viscosity, with emphasis on mesh convergence, boundary vorticity as well as wall viscous dissipation. In particular, it is found that, among the four different dipole configurations, the processes of vortex-wall collision at no-slip surfaces are exceedingly complex and are functions of the initial conditions. The critical issue direct numerical simulations is to establish mesh convergence which appears to be case-dependent. Roughly speaking, …
Motivated by the dynamics of microbubbles near catalytic surfaces in bubble-powered microrockets, we consider theoretically the growth of a free spherical bubble near a flat no-slip surface in a Stokes flow. The flow at the bubble surface is characterised by a constant slip length allowing to tune the hydrodynamic mobility of its surface and tackle in one formulation both clean and contaminated bubbles as well as rigid shells. Starting with a bubble of infinitesimal size, the fluid flow and hydrodynamic forces on the growing bubble are obtained analytically. We demonstrate that, depending on t…
We study experimentally the collision between a sphere falling through a viscous fluid, and a solid plate below. It is known that there is a well-defined threshold Stokes number above which the sphere rebounds from such a collision. Our experiment tests for direct contact between the colliding bodies, and contrary to prior theoretical predictions, shows that solid-on-solid contact occurs even for Stokes numbers just above the threshold for rebounding. The dissipation is fluid-dominated, though details of the contact mechanics depend on the surface and bulk properties of the solids. Our experim…
The slow viscous flow through a doubly-periodic array of cylinders does not have an analytical solution. However, as a reduced model for the flow within fibrous porous media, this solution is important for many real-world systems. We asymptotically determine the flow around a doubly-periodic array of infinite slender cylinders, by placing doubly-periodic two-dimensional singularity solutions within the cylinder and expanding the no-slip condition on the cylinder's surface in powers of the cylinder radius. The asymptotic solution provides a closed-form estimate for the flow and forces as a func…
Thin, viscous liquid films subjected to impact events can deform. Here we investigate free surface oil film deformations that arise due to the air pressure buildup under the impacting and rebouncing water drops. Using Digital Holographic Microscopy, we measure the 3D surface topography of the deformed film immediately after the drop rebound, with a resolution down to 20 nm. We first discuss how the film is initially deformed during impact, as a function of film thickness, film viscosity, and drop impact speed. Subsequently, we describe the slow relaxation process of the deformed film after the…
We study the steady streaming between two infinitely long circular cylinders produced by small amplitude transverse vibrations of the inner cylinder about the axis of the outer cylinder. The Vishik-Lyusternik method is employed to construct an asymptotic expansion of the solution of the Navier-Stokes equations in the limit of high-frequency vibrations for Reynolds numbers of order of unity. The effect of the Stokes drift of fluid particles is also studied. It is shown that it is nonzero not only within the boundary layers but also in higher order terms of the expansion of the averaged outer fl…
Numerical simulations using the Lattice Boltzmann Method are presented of the following two-dimensional incompressible flow problem. Starting from configurations corresponding to translating inviscid equilibria, namely, the translating Föppl equilibria (counter-rotating point vortex pair, either fore or aft of a translating circular cylinder) and the translating Hill equilibria (counter-rotating point vortex pair, either fore or aft of an elliptic cylinder), viscosity is turned on for $t >0$ and the subsequent viscous interaction is simulated. The interaction is in a dynamically coupled settin…
Non-wetting substrates allow impacting liquid drops to spread, recoil, and takeoff, provided they are not too heavy (Biance et al. 2006) or too viscous (Jha et al. 2020). In this article, using direct numerical simulations with the volume of fluid method, we investigate how viscous stresses and gravity conspire against capillarity to inhibit drop rebound. Close to the bouncing to non-bouncing transition, we evidence that the initial spreading stage can be decoupled from the later retraction and takeoff, allowing to understand the rebound as a process converting the surface energy of the spread…
We study the hydrodynamic interaction between two closely-spaced waving elastic cylinders immersed within a viscous liquid, at the creeping flow regime. The cylinders are actuated by a forced oscillation of the slope at their clamped end and are free at the opposite end. We obtain an expression for the interaction force and apply an asymptotic expansion based on a small parameter representing the ratio between the elastic deflections and the distance between the cylinders. The leading-order solution is an asymmetric oscillation pattern at the two frequencies ($ω_1,ω_2$) in which the cylinders …
The stability of a two-dimensional viscous flow between two rotating porous cylinders is studied. The basic steady flow is the most general rotationally-invariant solution of the Navier-Stokes equations in which the velocity has both radial and azimuthal components, and the azimuthal velocity profile depends on the Reynolds number. It is shown that for a wide range of the parameters of the problem, the basic flow is unstable to small two-dimensional perturbations. Neutral curves in the space of parameters of the problem are computed. Calculations show that the stability properties of this flow…
This paper deals with the small oscillations of two circular cylinders immersed in a viscous stagnant fluid. A new theoretical approach based on an Helmholtz expansion and a bipolar coordinate system is presented to estimate the fluid forces acting on the two bodies. We show that these forces are linear combinations of the {\textcolor{black}{cylinder accelerations}} and velocities, through viscous fluid added coefficients. {\textcolor{black}{To assess the validity of this theory, we consider the case of two equal size cylinders, one of them being stationary while the other one is forced sinuso…
Institute for Theoretical Physics, University of Amsterdam（阿姆斯特丹大学理论物理研究所）；Dutch Institute for Emergent Phenomena (DIEP), University of Amsterdam（阿姆斯特丹大学荷兰涌现现象研究所）
The different types of instabilities of free cylinders (diameter $D$, length $L$) have been studied in a viscous flow (velocity $U$) between parallel vertical walls of horizontal width $W$ at a distance $H$: the influence of the confinement parameters $D/H$ and $L/W$ has been investigated. As $D/H$ increases, there is a transition from stable flow to oscillations transverse to the walls and then to a fluttering motion with oscillations of the angle of the axis with respect to the horizontal. The two types of oscillations may be superimposed in the transition domain. The frequency $f$ of the tr…
In 1982, in his classical work, L. Rayleigh considered the instability of a cylinder of viscous liquid under capillary force, the so-called Plateau-Rayleigh instability. In this work, in linear approximation, he obtained a dispersion equation describing the increment of this instability as a function of wavelength, the radius of cylinder, the mass density, surface tension, and viscosity of the liquid. Hundreds of authors referred to this work, but none of them used his dispersion equation in its complete form; they used only the asymptotic solutions of his equation for zero and infinitely larg…
Direct numerical simulation is performed to study compressible, viscous flow around a circular cylinder. The present study considers two-dimensional, shock-free continuum flow by varying the Reynolds number between 20 and 100 and the freestream Mach number between 0 and 0.5. The results indicate that compressibility effects elongate the near wake for cases above and below the critical Reynolds number for two-dimensional flow under shedding. The wake elongation becomes more pronounced as the Reynolds number approaches this critical value. Moreover, we determine the growth rate and frequency of …
In a wide range of applications, microfluidic channels are implemented in soft substrates. In such configurations, where fluidic inertia and compressibility are negligible, the propagation of fluids in channels is governed by a balance between fluid viscosity and elasticity of the surrounding solid. The viscous-elastic interactions between elastic substrates and non-Newtonian fluids are particularly of interest due to the dependence of viscosity on the state of the system. In this work, we study the fluid-structure interaction dynamics between an incompressible non-Newtonian fluid and a slende…
A novel mechanism for the transport of microscale particles in viscous fluids is demonstrated. The mechanism exploits the trapping of such particles by rotational streaming cells established in the vicinity of an oscillating cylinder, recently analyzed in previous work. The present work explores a strategy of transporting particles between the trapping points established by multiple cylinders undergoing oscillations in sequential intervals. It is demonstrated that, by controlling the sequence of oscillation intervals, an inertial particle is effectively and predictably transported between the …
The rebound of droplets impacting a deep fluid bath is studied both experimentally and theoretically. Millimetric drops are generated using a piezoelectric droplet-on-demand generator and normally impact a bath of the same fluid. Measurements of the droplet trajectory and other rebound metrics are compared directly to the predictions of a linear quasi-potential model, as well as fully resolved direct numerical simulations (DNS) of the unsteady Navier-Stokes equations. Both models resolve the time-dependent bath and droplet shapes in addition to the droplet trajectory. In the quasi-potential mo…
In this paper we have investigated a circular band formation of fluid-rigid particle mixtures in a fully filled cylinder horizontally rotating about its cylinder axis by direct numerical simulation. These phenomena are modeled by the Navier-Stokes equations coupled to the Euler-Newton equations describing the rigid solid motion of the non-neutrally particles. The formation of circular bands studied in this paper is not resulted by mutual interaction between the particles and the periodic inertial waves in the cylinder axis direction (as suggested in Phys. Rev. E, 72, 021407 (2005)), but due to…
When a liquid drop falls on a solid substrate, the air layer in between them delays the occurrence of liquid--solid contact. For impacts on smooth substrates, the air film can even prevent wetting, allowing the drop to bounce off with dynamics identical to that observed for impacts on superamphiphobic materials. In this article, we investigate similar bouncing phenomena, occurring on viscous liquid films, that mimic atomically smooth substrates, with the goal to probe their effective repellency. We elucidate the mechanisms associated to the bouncing to non-bouncing (floating) transition using …
Many microorganisms swim in a highly heterogeneous environment with obstacles such as fibers or polymers. To better understand how this environment affects microorganism swimming, we study propulsion of a cylinder or filament in a fluid with a sparse, stationary network of obstructions modeled by the Brinkman equation. The mathematical analysis of swimming speeds is investigated by studying an infinite-length cylinder propagating lateral or spiral displacement waves. For fixed bending kinematics, we find that swimming speeds are enhanced due to the added resistance from the fibers. In addition…
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 s…
In this paper I consider the self-excited rotation of an elliptical cylinder towed in a viscous fluid as a canonical model of nonlinear fluid structure interactions with possible applications in the design of sensors and energy extraction devices. First, the self-excited ellipse system is shown to be analogous to the forced bistable oscillators studied in classic chaos theory. A single variable, the distance between the pivot and the centroid, governs the system bifurcation into bi-stability. Next, fully coupled computational fluid dynamics simulations of the motion of the cylinder demonstrate…
The Navier-Stokes-Fourier model for a 3D thermoconducting viscous fluid, where the evolution equation for the temperature T contains a term proportional to the rate of energy dissipation, is investigated analitically at the light of the rotational invariance property. Two cases are considered: the Couette flow and a flow with a radial velocity between two rotating impermeable and porous coaxial cylinders, respectively. In both cases, we show the existence of a maximum value of T, T_max, when the difference of temperature Delta T=T_2-T_1 on the surfaces of the cylinders is assigned. The role of…
Viscous streaming flows generated by objects of constant curvature (circular cylinders, infinite plates) have been well understood. Yet, characterization and understanding of such flows when multiple body length-scales are involved has not been looked into, in rigorous detail. We propose a simplified setting to understand and explore the effect of multiple body curvatures on streaming flows, analyzing the system through the lens of bifurcation theory. Our setup consists of periodic, regular lattices of cylinders characterized by two distinct radii, so as to inject discrete curvatures into the …
We present an asymptotic theory for analytical characterization of the high-Reynolds-number incompressible flow of a Newtonian fluid past a shear-free circular cylinder. The viscosity-induced modifications to this flow are localized and except in the neighborhood of the rear stagnation point, behave like a linear perturbation of the inviscid flow. Our theory gives a highly accurate description of these modifications by including the contribution from the most significant viscous term in a correctional perturbation expansion about an inviscid base state. We derive the boundary layer equation fo…
We investigate the linear properties of the steady and axisymmetric stress-driven spin-down flow of a viscous fluid inside a spherical shell, both within the incompressible and anelastic approximations, and in the asymptotic limit of small viscosities. From boundary layer analysis, we derive an analytical geostrophic solution for the 3D incompressible steady flow, inside and outside the cylinder $\mathcal{C}$ that is tangent to the inner shell. The Stewartson layer that lies on $\mathcal{C}$ is composed of two nested shear layers of thickness $O(E^{2/7})$ and $O(E^{1/3})$. We derive the lowest…
In lab-scale Faraday experiments, meniscus waves respond harmonically to small-amplitude forcing without threshold, hence potentially cloaking the instability onset of parametric waves. Their suppression can be achieved by resorting to a contact line pinned at the container brim with static contact angle $θ_s=90^{\circ}$ (brimful condition). However, tunable meniscus waves are desired in some applications as those of liquid-based biosensors, where they can be controlled adjusting the shape of the static meniscus by slightly under/over-filling the vessel ($θ_s\ne90^{\circ}$) while keeping the c…
The article proposes an algorithm to model the collision between arbitrary ellipsoids in viscous fluid. It is composed of several steps, each improving upon the standard procedure employed in the current literature. First, an efficient contact detection algorithm is presented. Then, the semi-implicit Immersed Boundary Method of Tschisgale et al. (2018) is enhanced by the collision forces using the hard-sphere approach, so that the resulting model accounts for fluid forces throughout the entire collision process. Additionally, a new lubrication model is proposed that applies a constant lubricat…
Viscous streaming is an efficient mechanism to exploit inertia at the microscale for flow control. While streaming from rigid features has been thoroughly investigated, when body compliance is involved, as in biological settings, little is known. Here, we investigate body elasticity effects on streaming in the minimal case of an immersed soft cylinder. Our study reveals an additional streaming process, available even in Stokes flows. Paving the way for advanced forms of flow manipulation, we illustrate how gained insights may translate to complex geometries beyond circular cylinders.
A numerical benchmark for fluid--structure--contact interaction
Mathematical Institute, Faculty of Mathematics and Physics, Charles University（查理大学数学物理学院数学研究所）；University of Konstanz, Department of Mathematics & Statistics（康斯坦茨大学数学与统计系）；Leibniz University Hannover, Institute of Applied Mathematics（汉诺威莱布尼茨大学应用数学研究所）；Department of Mathematical Analysis, Faculty of Mathematics and Physics, Charles University（查理大学数学物理学院数学分析系）；Department of Mathematics, Uppsala University（乌普萨拉大学数学系）
We evaluate a number of different finite element approaches for fluid-structure (contact) interaction problems against data from physical experiments. For this we take the data from experiments by Hagemeier [Mendeley Data, doi: 10.17632/mf27c92nc3.1]. This consists of trajectories of single particles falling through a highly viscous fluid and rebounding off the bottom fluid tank wall. The resulting flow is in the transitional regime between creeping and turbulent flows. This type of configuration is particularly challenging for numerical methods due to the large change of the fluid domain and …
We study the decay of high-Reynolds number Taylor-Couette turbulence, i.e. the turbulent flow between two coaxial rotating cylinders. To do so, the rotation of the inner cylinder (Re$_i=2 \times 10^6$, the outer cylinder is at rest) is stopped within 12 s, thus fully removing the energy input to the system. Using a combination of laser Doppler anemometry and particle image velocimetry measurements, six decay decades of the kinetic energy could be captured. First, in the absence of cylinder rotation, the flow-velocity during the decay does not develop any height dependence in contrast to the we…
A comprehensive study of the classical linear spin-down of a constant density viscous fluid (kinematic viscosity ν) rotating rapidly (angular velocity Ω) inside an axisymmetric cylindrical container (radius L, height H) with rigid boundaries, that follows the instantaneous small change in the boundary angular velocity at small Ekman number $E=ν/H^2Ω\ll 1$, was provided by Greenspan & Howard (1963). $E^{1/2}$-Ekman layers form quickly triggering inertial waves together with the dominant spin-down of the quasi-geostrophic (QG) interior flow on the $O(E^{-1/2}Ω^{-1})$ time-scale. On the longer la…
A framework for diagnosing inertial lift generation in wall-bounded flows: application to eccentric rotating cylinders in Newtonian and shear-thinning fluids
Graduate School of Engineering Science, The University of Osaka（大阪大学工学研究科）；RIKEN（理化学研究所）
We analyze the stability of a cylindrical Couette flow under the imposition of a weak axial flow in case of a very short cylinder with a narrow annulus gap. We consider an incompressible viscous fluid which is contained in the narrow gap between two concentric short cylinders, where the inner cylinder rotates with constant angular velocity. The caps of the cylinders have narrow tubes conically tapering to super narrow slits which allow for an axial flow along the surface of the inner cylinder. The approximated solution for the Couette flow for short cylinders was found and used for the stabili…
The decay of Taylor-Couette turbulence, i.e~the flow between two coaxial and independently rotating cylinders, is numerically studied by instantaneously stopping the forcing from an initially statistically stationary flow field at a Reynolds number of $Re=3.5\times 10^4$. The effect of wall-friction is analysed by comparing three separate cases, in which the cylinders are either suddenly made no-slip or stress-free. Different life stages are observed during the decay. In the first stage, the decay is dominated by large-scale rolls. Counterintuitively, when these rolls fade away, if the flow in…
