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We provide a numerical algorithm for numerically approximating a centrally located floating ball. We give examples of equilibria, and we present non-unique cases for the same physical parameters when the density of the ball is either greater than the supporting liquid (heavy) or lighter than the density of the vapor above (light). We classify the non-uniqueness by analyzing a function related to the force balance. We derive the potential energy of these states, and make comparisons of the non-unique cases. In the cases of both the light and heavy floating balls, the evidence presented supports the conjecture that when there are two equilibria, the one with lower energy corresponds to the location of triple junction (between the ball, the vapor and the liquid) that is closer to the equator of the ball.

Consider a ball of density

The energies considered in this model are due to the surface tension and gravity. Surface tension energy is taken, as usual, to be proportional to the area of the free surface with proportionality constant

with wetting coefficients

with “capillary constant”

The natural physical setting for these problems is in a bounded container in

The goal of this note is to provide numerically computed approximations to some sample configurations when the ball is centrally located in the container and the fluid is assumed to be symmetric about the central axis. We will focus on the cases of the density

With this setting established, we are able to proceed to the configurations of interest. Assume that the ball floats centrally, and so that the center of the ball is at a height d. Thus the boundary of the ball can be described by its azimuthal angle

where

where

when the surface is a graph over some base domain. However, we will not restrict ourselves to these limited configurations. We have boundary conditions

where

In [

energy due to gravity is measured as

Under the assumption of symmetry about the vertical axis, the PDE with boundary conditions can be reduced to

This representation of the equation allows for the parameterized curve to pass through both inflection points and vertical points. Various authors have studied this system with differing boundary conditions and also as a family of solution curves. Solutions are sometimes known as Euler elastica. See Aspley, He, and McCuan [

Then (2) becomes

and we define

We seek solutions to (3) that satisfy (4) and

If we replace the bounded container with an infinite sea of liquid, then we follow Bhatnagar and Finn [

Before moving on to these details we would like to mention some other authors and their works on floating objects: Bemelmans, Galdi, Kyed [

We will consider first the bounded container in

Next we illustrate how the new free boundary condition can be used to determine when there is non uni- queness of the equilibria. We give numerical examples. Finally, in the case of the bounded container, we calculate the potential energy of each configuration and compare the energy of the two configurations with the same physical properties.

We first need to measure the volume of fluid held in the container. One may compute

Our formula does not require that the height u is a function over a base domain.

In what follows we will assign

The numerical procedure is a shooting method, where we integrate the system of ODEs

with initial conditions

to some ending arc-length

which say that the configuration satisfies the volume constraint and the force balance condition as well as the need for the fluid interface to extend to the wall of the container with the prescribed contact angle.

The parameters

This leads to a solution of the initial value problem where we can evaluate the conditions (8)-(11). We use Matlab’s fsolve to then vary

We next proceed with a few examples. In

As an approach to explain the behavior in the examples in the second pair, we turn to a study of the function

and we use Matlab’s fsolve to then vary

We do not attempt to evaluate

We show the plot of F for

With these solutions to the partial floating ball in hand, we are able to numerically calculate the energy of the configuration. It is simpler to use the scaled mathematical energy

energies are also trivial to compute:

The gravitational potential energy of the liquid is more convenient to compute in sections. First we will treat the portion directly below the interface, but due to the lack of a closed form expression and also due to the variable step size of the data we do this numerically with the trapezoid rule:

where

where h is the maximum step size taken in the irregularly spaced data, and

The case where

which matches (15) upon subtracting

We next consider the unbounded container problem in

that satisfies

satisfying

See [

We proceed to the full problem of computing the floating ball configuration in the unbounded

with inclination angle

with

Our results can be compared to Bhatnagar and Finn [

Our examples use

We begin here with the floating ball in a bounded container, and we ask the following question: what values of

It should be stated that while this current work focuses more directly on the influence of the contact angles on the nonuniqueness criterion, the influence of R and a on the nonuniqueness criterion is just as important. To see this impact, compare

With the energy computed as in Section 2.1, we first use it in conjunction with the graph of

plotted with that of

Then the energy is computed with this value of

the smaller value of

We then included the above analysis into the program that generated

Next, we trace the energy values from a sampling of balls with given, fixed densities as the semi-equilibrium are computed through the range

a | |||
---|---|---|---|

0.25 | verified | verified | |

0.5 | verified | verified | |

0.75 | verified | verified | |

1.0 | verified | verified | |

1.25 | verified | verified | |

1.5 | verified | verified | |

1.75 | verified | ||

2 | verified |

examined. Here we fix a few densities and in contrast, in the previous discussions the densities would vary with the value of

be a discontinuous change in d. Then we move through the parameter

Finally, we turn to a study of

We compute F over

rigorous stability analysis. The results from this section then can be seen as purely an analysis of the non- uniqueness criteria.

Conjecture 1. The centrally located floating ball has at most two equilibria in a bounded container in 2D, and the centrally located floating ball has at most two equilibria in the unbounded configuration in 2D. In both cases, this will occur at some value or values of the azimuthal angle

We have developed a robust numerical solver for finding the equilibria of a centrally located floating ball in both bounded and unbounded problems in

Ray Treinen, (2016) Examples of Non-Uniqueness of the Equilibrium States for a Floating Ball. Advances in Materials Physics and Chemistry,06,177-194. doi: 10.4236/ampc.2016.67019