Capillary Floating and the Billiard Ball Problem

In a study of capillary floating, Finn (J Math Fluid Mech 11:443?C458, 2009) described a procedure for determining cross-sections of non-circular, infinite convex cylinders that float horizontally on a liquid surface in every orientation with contact angle ??/2. Finn??s procedure yielded incomplete results for other contact angles; he raised the question as to whether an analogous construction would be feasible in that case. In the note, Finn (J Math Fluid Mech 11:464?C465, 2009) pointed out a connection with an independent problem on billiard caustics citing the unpublished work (Gutkin in Proceedings of the Workshop on Dynamics and Related Questions, PennState University, 1993) of the present author. Here we present a solution of the billiard problem in full detail, thus settling Finn??s question in a surprising way. In particular, we show that such floating cylinders exist if and only if the contact angle lies in a certain, explicitly described countably dense set. Moreover, for each element ?? in this set we exhibit a family of convex, non-circular cylinders that float in every orientation with contact angle ??. Our discussion contains other material of independent interest for the billiard ball problem.     

The Floating Ball Paradox

In capillary theory there are two kinds of surface tension. There is the surface tension at the interface between two immiscible fluids. Thomas Young [9] also allowed for there to be a surface tension associated with a liquid-solid interface. He proceeded to use a balance of forces argument to derive the well-known contact angle condition along a liquid-liquid-solid intersection. The validity of this argument has recently been called into question by R. Finn [6]. A floating ball experiment discussed in that paper leads to an apparent paradox. We address this issue.      

Remarks on “Floating Bodies Subject to Capillary Attractions”

Admissible shapes of bodies are characterized, according to the range of configurations in which the body can be made to float in a bath with horizontal surface, in the presence of capillary attractions.     

A Revised Analytic Solution to the Linear Displacement Problem Including Capillary Pressure Effects

Fokas and Yortsos (SIAM J. Appl. Math. 42(2), 318–332, 1982) and Yortsos and Fokas (SPEJ 23(1), 115–124, 1983) presented the only published exact solution of the linear waterflood problem that includes capillary effects. Despite the importance of this breakthrough, their approach has largely been disregarded due to the perceived limitations that it presented in modeling real physical situations. In this article, we show that by appropriately normalizing relevant parameters of the governing equation involved, a substantial level of the limitations is taken care of. The resultant governing equation obtained is one in terms of a parameter N _M related to the mobility ratio and another parameter N _V, representing a ratio of the viscous to capillary forces. The results of the explicit solutions obtained indicate that these two parameters are indeed the controlling parameters of the flow, and that the capillary effects are practically non-existent even when N _V = 100. These analytical results serve a very useful utility in validating numerical simulators.     

Capillary fingering: Percolation and fractal dimension

We present experimental and theoretical results concerning immiscible displacements (drainage) in 2-dimensional permeable media. When capillary forces are predominant, the injected fluid presents very thin fingers and the Representative Elementary Volume concept cannot be used for describing the partial saturations. The purpose of this paper is to show how this classical concept can be replaced by a statistical approach based on fractal geometry.Communication presented at the International Symposium on the Stochastic Approach to Subsurface Flow, Montvillargenne 4–7 June 1985.     

Fluid Flows Around Floating Bodies, I: The Hydrostatic Case

We consider the hydrostatic configuration of a body floating freely on a liquid. Under the influence of gravitational and capillary forces there exists an equilibrium solution with contact angle π/2. This solution is the minimizer of a variational problem with an obstacle condition; the corresponding free boundary consists of the curve where the capillary surface meets the floating body.     

Capillary absorption in porous sheets and surfaces subject to evaporation

Capillary absorption by an initially dry porous sheet or surface is examined. The sheet absorbs moisture from one end while evaporation takes place across its surface. We are interested in the effect of evaporation on the progress and distribution of moisture in the sheet, including the equilibrium moisture profile. The process is modelled using a nonlinear diffusion equation with a linear sink. An algebraic analytical approximation is obtained relating the various physical parameters as well as an exact steady state solution for arbitrary properties. The complete problem is solved numerically. Good agreement is obtained between the analytical and numerical solutions.     

The Viscous Surface-Internal Wave Problem: Global Well-Posedness and Decay

We consider the free boundary problem for two layers of immiscible, viscous, incompressible fluid in a uniform gravitational field, lying above a general rigid bottom in a three-dimensional horizontally periodic setting. We establish the global well-posedness of the problem both with and without surface tension. We prove that without surface tension the solution decays to the equilibrium state at an almost exponential rate; with surface tension, we show that the solution decays at an exponential rate. Our results include the case in which a heavier fluid lies above a lighter one, provided that the surface tension at the free internal interface is above a critical value, which we identify. This means that sufficiently large surface tension stabilizes the Rayleigh–Taylor instability in the nonlinear setting. As a part of our analysis, we establish elliptic estimates for the two-phase stationary Stokes problem.     

Dynamic Contact Behavior of a Golf Ball during Oblique Impact: Effect of Friction between the Ball and Target

The oblique impact between a golf ball and a polymethyl methacrylate (PMMA) target with smooth transparent surfaces was studied using a high-speed video camera. Video images of the impact were recorded from the backside of the target and were used to determine the contact time, contact area, and the displacement and rotation of the ball along the target. The average tangential and angular velocities were determined as functions of the inbound ball velocity. As the inbound ball velocity increased, the contact area and average tangential and angular velocities also increased while the contact time decreased. An oiled PMMA target was used to study the effect of reduced friction between the ball and target. The results were compared with earlier data for a steel target with relatively rough surfaces. The contact area and time were unaffected by friction, but the average tangential velocity increased while the average angular velocity decreased as the friction decreased.     

Collagen Fiber Network Infiltration: Permeability and Capillary Infiltration

In dentin restoration, collagen fiber network infiltration is an issue. Using data from the literature, we have constructed a relevant numerical geometrical model of the network. The specificity of our model is that the fibers are taken into account implicitly using a regularized Heaviside function. This function is either used to set the viscosity or to localize the contact line where capillary forces are applied. A level set technique with respect to fluid infiltration front tracking in five fiber networks using the level set method and Navier–Stokes equations with capillary terms is used to point out efficient critical infiltration parameters. A variational formulation which can be implemented in a finite elements model is proposed both for the infiltration front and the contact line. Because of lack of knowledge on fiber orientation, different configurations were tested through permeability assessment of the whole network. Fiber orientation, interfibrillar space and contact angle influence were investigated.     

The Jacobi-Rosochatius Problem on an Ellipsoid: the Lax Representations and Billiards

In this article, we construct the Lax representations of the geodesic flow, the Jacobi-Rosochatius problem and its perturbations by means of separable polynomial potentials on an ellipsoid. We prove complete integrability in the case of a generic symmetric ellipsoid and describe analogous systems on complex projective spaces. Also, we consider billiards within an ellipsoid under the influence of the Hook and Rosochatius potentials between the impacts. A geometric interpretation of the integrability analogous to the classical Chasles and Poncelet theorems is given.     

Entire Minimal Parabolic Trajectories: The Planar Anisotropic Kepler Problem

We continue the variational approach to parabolic trajectories introduced in our previous paper (Barutello et al., Entire parabolic trajectories as minimal phase transitions. arXiv:1105.3358v1, 2011), which sees parabolic orbits as minimal phase transitions. We deepen and complete the analysis in the planar case for homogeneous singular potentials. We characterize all parabolic orbits connecting two minimal central configurations as free-time Morse minimizers (in a given homotopy class of paths). These may occur for at most one value of the homogeneity exponent. In addition, we link this threshold of existence of parabolic trajectories with the absence of collisions for all the minimizers of fixed-end problems, and also with the existence of action minimizing periodic trajectories with nontrivial homotopy type.