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 the contact between the wall and particle. In the numerical simulations we consider both rigid body and linear elasticity models for the falling particles. In the first case, we compare results obtained with the well established Arbitrary Lagrangian Eulerian (ALE) approach and a moving domain CutFEM method together with a simple and common approach for contact avoidance. For the full fluid-structure interaction (FSI) problem with contact, we use a fully Eulerian approach in combination with a unified FSI-contact treatment using Nitsche's method. For higher computational efficiency we use the geometrical symmetry of the experimental set up to reformulate the FSI system into two spatial dimensions. Finally, we show full three dimensional ALE computations to study the effects of small perturbations in the initial state of the particle to investigate deviations from a perfectly vertical fall observed in the experiment. The methods are implemented in open-source finite element libraries and the results are made freely available to aide reproducibility.
A stationary viscous jet falling from an oriented nozzle onto a moving surface is studied, both theoretically and experimentally. We distinguish three flow regimes and classify them by the convexity of the jet shape (concave, vertical and convex). The fluid is modeled as a Newtonian fluid, and the model for the flow includes viscous effects, inertia and gravity. By studying the characteristics of the conservation of momentum for a dynamic jet, the boundary conditions for each flow regime are derived, and the flow regimes are characterized in terms of the process and material parameters. The mo…
This fluid dynamics video presents experiments and simulations of gravity-driven particulate jets in viscous fluids at low Reynolds number. An initially straight jet is shown to develop varicose modulations of its diameter as it sediments under the action of gravity. While this instability is qualitatively reminiscent of the classical Rayleigh-Plateau instability for immiscible fluids, its mechanism has yet to be understood as neither inertia nor surface tension play a role in the case of a dilute suspension at Re=0.
Dynamics of regular clusters of many non-touching particles falling under gravity in a viscous fluid at low Reynolds number are analysed within the point-particle model. Evolution of two families of particle configurations is determined: 2 or 4 regular horizontal polygons (called `rings') centred above or below each other. Two rings fall together and periodically oscillate. Four rings usually separate from each other with chaotic scattering. For hundreds of thousands of initial configurations, a map of the cluster lifetime is evaluated, where the long-lasting clusters are centred around period…
A new method of accurate calculation of the coefficient of viscosity of a test liquid from experimentally measured terminal velocity of a ball falling in the test liquid contained in a narrow tube is described. The calculation requires the value of a multiplicative correction factor to the apparent coefficient of viscosity calculated by substitution of terminal velocity of the falling ball in Stokes formula. This correction factor, the so-called viscosity ratio, a measure of deviation from Stokes limit, arises from non-vanishing values of the Reynolds number and the ball/tube radius ratio. The…
We analyze the stationary flow of a jet of Newtonian fluid that is drawn by gravity onto a moving surface. The situation is modeled by a third-order ODE on a domain of unknown length and with an additional integral condition; by solving part of the equation explicitly we can reformulate the problem as a first-order ODE, again with an integral constraint. We show that there are two flow regimes, and characterize the associated regions in the three-dimensional parameter space in terms of an easily calculable quantity. In a qualitative sense the results from the model are found to correspond with…
We examine the interaction of two-dimensional solitary pulses on falling liquid films. We make use of the second-order model derived by Ruyer-Quil and Manneville [Eur. Phys. J. B 6, 277 (1998); Eur. Phys. J. B 15, 357 (2000); Phys. Fluids 14, 170 (2002)] by combining the long-wave approximation with a weighted residuals technique. The model includes (second-order) viscous dispersion effects which originate from the streamwise momentum equation and tangential stress balance. These effects play a dispersive role that primarily influences the shape of the capillary ripples in front of the solitar…
This paper describes a simple and inexpensive method of measuring viscosity of a Newtonian fluid using the ball drop technique and an inexpensive point and shoot ~1000 frame per second camera. We successfully measured the viscosity of glycerol and glycerol-water mixture with high precision. We used three different size copper balls of diameters 0.8 mm, 1.59 mm, and 2.38 mm to check the accuracy of the measured viscosity in different concentrations of glycerol-water mixer solutions ranging from 50% to 100% (pure glycerol). Our measurements are in excellent agreement with the measurements conduc…
We investigate nonlinear periodic and solitary two-dimensional rolling waves in a falling two-layer liquid film in the regime of non-zero Reynolds numbers. At any flow rate, a falling two-layer liquid film is known to be linearly unstable with respect to long-wave deformations of the liquid-air surface and liquid-liquid interface. Two different types of zero-amplitude neutrally stable waves propagate downstream without growing or shrinking: a zig-zag surface mode and a thinning varicose interface mode. Using a boundary-layer reduction of the Navier-Stokes equation, we investigate the onset, po…
Thin viscous fluid threads falling onto a moving belt behave in a way reminiscent of a sewing machine, generating a rich variety of periodic stitch-like patterns including meanders, W-patterns, alternating loops, and translated coiling. These patterns form to accommodate the difference between the belt speed and the terminal velocity at which the falling thread strikes the belt. Using direct numerical simulations, we show that inertia is not required to produce the aforementioned patterns. We introduce a quasi-static geometrical model which captures the patterns, consisting of three coupled OD…
From an obliquely falling rod in a viscous fluid to the motion of suspended magnetic bead chains that are driven by a gradient magnetic field and that make an arbitrary angle with the magnetic force vector: A Stokes flow study
We study experimentally the dynamics of one and two ball-chains settling under gravity in a very viscous fluid at a Reynolds number much smaller than unity. We demonstrate that single ball-chains in most cases do not tend to be planar and often rotate, not keeping the ends at the same horizontal level. Shorter ball-chains usually form shapes resembling distorted U, and longer ones in the early stage of the evolution form a shape resembling distorted W, and later deform non-symmetrically and significantly out of plane. This behaviour is reproduced in our numerical simulations for a single very …
On the impact of clusters of rigid balls on the motion of a viscous fluid
Institute of Mathematics of the Academy of Sciences of the Czech Republic（捷克科学院数学研究所）；Nečas Center for Mathematical Modelling（内恰斯数学建模中心）；BCAM（巴斯克计算数学中心）
A thread of viscous fluid falling onto a moving surface exhibits a spectacular variety of types of motion as the surface speed and nozzle height are varied. For modest nozzle heights, four clear regimes are observed. For large surface speed, the thread is dragged into a stretched centenary configuration which is confined to a plane. As the surface speed is lowered, this exhibits a supercritical bifurcation to a meandering state. At very low surface speeds, the state resembles the usual coiling motion of a viscous thread falling on a stationary surface. In between the meandering and coiling reg…
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…
When a viscous fluid, like oil or syrup, streams from a small orifice and falls freely under gravity, it forms a long slender thread, which can be maintained in a stable, stationary state with lengths up to several meters. We shall discuss the shape of such liquid threads and their surprising stability. It turns out that the strong advection of the falling fluid can almost outrun the Rayleigh-Plateau instability. Even for a very viscous fluid like sirup or silicone oil, the asymptotic shape and stability is independent of viscosity and small perturbations grow with time as $\exp({{\rm C} t^{1/…
The case of a rotating object traveling through viscous fluid appears in many phenomena like the banana ball and missile movement. In this work, we build a model to predict the trajectory of such rotating objects with near-cylinder geometry. The analytical expression of Magnus force is given and a wind tunnel experiment is carried out, which shows the Magnus force is well proportional to the product of angular velocity and centroid velocity. The trajectory prediction is consistent with the trajectory record experiment of Magnus glider, which implies the validity and robustness of this model.
We present a control-volume approach for deriving a simplified model for the gravity-driven flow of an axisymmetric liquid film along a vertical fiber. The model accounts for gravitational, viscous, inertial and surface tension effects and results in a pair of coupled one-dimensional nonlinear partial differential equations for the film profile and average downward velocity as functions of time and axial distance along the fiber. Two versions of the model are obtained, one assuming a plug-flow velocity profile and a constant thin boundary layer thickness to model the drag force on the fluid, t…
The range and speed of direct ink writing, the workhorse of 3d and 4d printing, is limited by the practice of liquid extrusion from a nozzle just above the surface to prevent instabilities to cause deviations from the required print path. But what if could harness the ``fluid rope trick", whence a thin stream of viscous fluid falling from a height spontaneously folds or coils, to write specified patterns on a substrate? Using Deep Reinforcement Learning we control the motion of the extruding nozzle and thence the fluid patterns that are deposited on the surface. The learner (nozzle) repeatedly…
A thin thread of viscous fluid falling onto a moving belt generates a surprising variety of patterns depending on the belt speed, fall height, flow rate, and fluid properties. Here we simulate this experiment numerically using the Discrete Viscous Threads method that can predict the non-steady dynamics of thin viscous filaments, capturing the combined effects of inertia, stretching, bending and twisting. Our simulations successfully reproduce nine out of ten different patterns previously seen in the laboratory, and agree closely with the experimental phase diagram of Morris et al.\ (2008). We …
The free motion of balls is investigated experimentally in continously stratified fluid in a finite container. The oscillation frequency is found to be very close to the local Brunt-Vaisala frequency. The effect of added mass proves to be practically negligible. The evolution of rear jets is demonstrated, and a kind of long term levitation is found. We show that the classical viscous drag would lead to a much stronger damping than observed in the experiment. This is interpreted as a consequence of the feedback from the previously excited internal waves following their reflection from the bound…
The gravity-driven motion of rigid particles in a viscous fluid is relevant in many natural and industrial processes, yet this has mainly been investigated for spherical particles. We therefore consider the sedimentation of non-spherical (spheroidal) isolated and particle pairs in a viscous fluid via numerical simulations using the Immersed Boundary Method. The simulations performed here show that the critical Galileo number for the onset of secondary motions decreases as the spheroid aspect ratio departs from 1. Above this critical threshold, oblate particles perform a zigzagging motion where…
When submillimetric particles are confined in a fluid such that a compact cluster of particles lie above the clear fluid, particles will detach from the lower boundary of the cluster and form an unstable separation front giving rise to growing fingers of falling particles. We study this problem using both experiments and hybrid granular/fluid mechanics models. In the case of particles from 50 to 500 microns in diameter falling in air, we study the horizontal density fluctuations at early times: the amplitude of the density difference between two points at a certain horizontal distance grows as…
