Impact on liquids : void collapse and jet formation

A spectacular example of free surface flow is the impact of a solid object on a liquid: At\ud impact a “crown” splash is created and a surface cavity (void) emerges which\ud immediately starts to collapse due to the hydrostatic pressure of the surrounding liquid.\ud Eventually the cavity closes in a single point about halfway down its length and shoots\ud out a fast and extremely slender water jet. Here we impact thin circular discs a few\ud centimeters in radius with velocities of a few meters per second. Combining high-speed\ud imaging with sophisticated boundary-integral simulations we elucidate various aspects of\ud this fascinating process.\ud First we show that the mechanism behind the formation of the fast, almost needle-like\ud liquid jet is reminiscent of the violent jets of fluidized metal created during the explosion\ud “of lined cavities” in military and mining operations. We obtain quantitative agreement\ud between our simulations, experiments, and analytical model.\ud Next we use visualization experiments to measure the air flow as it is squeezed out of\ud the shrinking impact cavity. Together with numerical simulations we show that even in\ud our simple system of a 2 cm disc impacting at merely 1 m/s the air flow easily attains\ud supersonic velocities.\ud A long-standing controversy in the fluid dynamics community has been until recently the\ud pinch-off behavior of a bubble inside a liquid. Our observation of different time scales for\ud the onset of the predicted final regime reconciles the different views expressed in recent literature about bubble pinch-off.\ud Next we replace the impacting disc by a long, smooth cylinder and find that the closure\ud position of the cavity displays distinct regimes separated by discrete jumps which are\ud consistently observed in experiment and numerical simulations.\ud Finally, we simulate the collapse of nanobubbles nucleating from small (50 nm) pits\ud drilled into a silicon wafer. We find that just prior to final collapse a jet very similar in\ud appearance to those after solid object impact forms and penetrates deep into the hole

Hydrodynamic interaction between particles near elastic interfaces

We present an analytical calculation of the hydrodynamic interaction between two spherical particles near an elastic interface such as a cell membrane. The theory predicts the frequency dependent self- and pair-mobilities accounting for the finite particle size up to the 5th order in the ratio between particle diameter and wall distance as well as between diameter and interparticle distance. We find that particle motion towards a membrane with pure bending resistance always leads to mutual repulsion similar as in the well-known case of a hard-wall. In the vicinity of a membrane with shearing resistance, however, we observe an attractive interaction in a certain parameter range which is in contrast to the behavior near a hard wall. This attraction might facilitate surface chemical reactions. Furthermore, we show that there exists a frequency range in which the pair-mobility for perpendicular motion exceeds its bulk value, leading to short-lived superdiffusive behavior. Using the analytical particle mobilities we compute collective and relative diffusion coefficients. The appropriateness of the approximations in our analytical results is demonstrated by corresponding boundary integral simulations which are in excellent agreement with the theoretical predictions.Comment: 16 pages, 7 figures and 109 references. Manuscript accepted for publication in J. Chem. Phy

Brownian motion near an elastic cell membrane: A theoretical study

Elastic confinements are an important component of many biological systems and dictate the transport properties of suspended particles under flow. In this chapter, we review the Brownian motion of a particle moving in the vicinity of a living cell whose membrane is endowed with a resistance towards shear and bending. The analytical calculations proceed through the computation of the frequency-dependent mobility functions and the application of the fluctuation-dissipation theorem. Elastic interfaces endow the system with memory effects that lead to a long-lived anomalous subdiffusive regime of nearby particles. In the steady limit, the diffusional behavior approaches that near a no-slip hard wall. The analytical predictions are validated and supplemented with boundary-integral simulations.Comment: 16 pages, 7 figures and 161 references. Contributed chapter to the flowing matter boo

Slow rotation of a spherical particle inside an elastic tube

In this paper, we present an analytical calculation of the rotational mobility functions of a particle rotating on the centerline of an elastic cylindrical tube whose membrane exhibits resistance towards shearing and bending. We find that the correction to the particle rotational mobility about the cylinder axis depends solely on membrane shearing properties while both shearing and bending manifest themselves for the rotational mobility about an axis perpendicular to the cylinder axis. In the quasi-steady limit of vanishing frequency, the particle rotational mobility nearby a no-slip rigid cylinder is recovered only if the membrane possesses a non-vanishing resistance towards shearing. We further show that for the asymmetric rotation along the cylinder radial axis, a coupling between shearing and bending exists. Our analytical predictions are compared and validated with corresponding boundary integral simulations where a very good agreement is obtained.Comment: 23 pages, 7 figures and 107 references. Revised manuscript resubmitted to Acta Mec

Long-lived anomalous thermal diffusion induced by elastic cell membranes on nearby particles

The physical approach of a small particle (virus, medical drug) to the cell membrane represents the crucial first step before active internalization and is governed by thermal diffusion. Using a fully analytical theory we show that the stretching and bending of the elastic membrane by the approaching particle induces a memory in the system which leads to anomalous diffusion, even though the particle is immersed in a purely Newtonian liquid. For typical cell membranes the transient subdiffusive regime extends beyond 10 ms and can enhance residence times and possibly binding rates up to 50\%. Our analytical predictions are validated by numerical simulations.Comment: 13 pages and 5 figures. The Supporting Information is included. Manuscript accepted for publication in Phys. Rev.

Creeping motion of a solid particle inside a spherical elastic cavity

On the basis of the linear hydrodynamic equations, we present an analytical theory for the low-Reynolds-number motion of a solid particle moving inside a larger spherical elastic cavity which can be seen as a model system for a fluid vesicle. In the particular situation where the particle is concentric with the cavity, we use the stream function technique to find exact analytical solutions of the fluid motion equations on both sides of the elastic cavity. In this particular situation, we find that the solution of the hydrodynamic equations is solely determined by membrane shear properties and that bending does not play a role. For an arbitrary position of the solid particle within the spherical cavity, we employ the image solution technique to compute the axisymmetric flow field induced by a point force (Stokeslet). We then obtain analytical expressions of the leading order mobility function describing the fluid-mediated hydrodynamic interactions between the particle and confining elastic cavity. In the quasi-steady limit of vanishing frequency, we find that the particle self-mobility function is higher than that predicted inside a rigid no-slip cavity. Considering the cavity motion, we find that the pair-mobility function is determined only by membrane shear properties. Our analytical predictions are supplemented and validated by fully-resolved boundary integral simulations where a very good agreement is obtained over the whole range of applied forcing frequencies.Comment: 15 pages, 5 figures, 90 references. To appear in Eur. Phys. J.

Particle mobility between two planar elastic membranes: Brownian motion and membrane deformation

We study the motion of a solid particle immersed in a Newtonian fluid and confined between two parallel elastic membranes possessing shear and bending rigidity. The hydrodynamic mobility depends on the frequency of the particle motion due to the elastic energy stored in the membrane. Unlike the single-membrane case, a coupling between shearing and bending exists. The commonly used approximation of superposing two single-membrane contributions is found to give reasonable results only for motions in the parallel, but not in the perpendicular direction. We also compute analytically the membrane deformation resulting from the motion of the particle, showing that the presence of the second membrane reduces deformation. Using the fluctuation-dissipation theorem we compute the Brownian motion of the particle, finding a long-lasting subdiffusive regime at intermediate time scales. We finally assess the accuracy of the employed point-particle approximation via boundary-integral simulations for a truly extended particle. They are found to be in excellent agreement with the analytical predictions.Comment: 14 pages, 8 figures and 96 references. Revised version resubmitted to Phys. Fluid

Hydrodynamic mobility of a solid particle nearby a spherical elastic membrane. II. Asymmetric motion

In this paper, we derive analytical expressions for the leading-order hydrodynamic mobility of a small solid particle undergoing motion tangential to a nearby large spherical capsule whose membrane possesses resistance towards shearing and bending. Together with the results obtained in the first part (Daddi-Moussa-Ider and Gekle, Phys. Rev. E {\bfseries 95}, 013108 (2017)) where the axisymmetric motion perpendicular to the capsule membrane is considered, the solution of the general mobility problem is thus determined. We find that shearing resistance induces a low-frequency peak in the particle self-mobility, resulting from the membrane normal displacement in the same way, although less pronounced, to what has been observed for the axisymmetric motion. In the zero frequency limit, the self-mobility correction near a hard sphere is recovered only if the membrane has a non-vanishing resistance towards shearing. We further compute the particle in-plane mean-square displacement of a nearby diffusing particle, finding that the membrane induces a long-lasting subdiffusive regime. Considering capsule motion, we find that the correction to the pair-mobility function is solely determined by membrane shearing properties. Our analytical calculations are compared and validated with fully resolved boundary integral simulations where a very good agreement is obtained.Comment: 17 pages, 9 figures and 64 references. Manuscript accepted for publication in Phys. Rev.

Generation and Breakup of Worthington Jets After Cavity Collapse

Helped by the careful analysis of their experimental data, Worthington (1897) described roughly the mechanism underlying the formation of high-speed jets ejected after the impact of an axisymmetric solid on a liquid-air interface. In this work we combine detailed boundary-integral simulations with analytical modeling to describe the formation and break-up of such Worthington jets in two common physical systems: the impact of a circular disc on a liquid surface and the release of air bubbles from an underwater nozzle. We first show that the jet base dynamics can be predicted for both systems using our earlier model in Gekle, Gordillo, van der Meer and Lohse. Phys. Rev. Lett. 102 (2009). Nevertheless, our main point here is to present a model which allows us to accurately predict the shape of the entire jet. Good agreement with numerics and some experimental data is found. Moreover, we find that, contrarily to the capillary breakup of liquid cylinders in vacuum studied by Rayleigh, the breakup of stretched liquid jets at high values of both Weber and Reynolds numbers is not triggered by the growth of perturbations coming from an external source of noise. Instead, the jet breaks up due to the capillary deceleration of the liquid at the tip which produces a corrugation to the jet shape. This perturbation, which is self-induced by the flow, will grow in time promoted by a capillary mechanism. We are able to predict the exact shape evolution of Worthington jets ejected after the impact of a solid object - including the size of small droplets ejected from the tip due to a surface-tension driven instability - using as the single input parameters the minimum radius of the cavity and the flow field before the jet emerges