Effects of surface tension on a floating body in two dimensions

Abstract

A model for calculating the force profile and the moment profile of a floating body in two dimensions with an arbitrary cross-section is proposed. Three types of cross-sections with different contact angles and densities are calculated by using the model to determine the vertical and rotational equilibria and their stabilities. Results show that the model can be applied to convex floating bodies with finitely many sharp edges. The study is then extended to investigate the surface tension effects on the vertical and rotational stabilities by varying the following parameters: the radii of curvature of the solid surface at the contact lines and the size of floating body. In general, the smaller the radii of curvature the better the vertical and rotational stabilities. However, for the contact angle $\unicode[STIX]{x1D703}=0$ (or $\unicode[STIX]{x1D703}=\unicode[STIX]{x03C0}$) the radii of curvature have no effect on the vertical stability of the floating body. By varying the size of the floating body, it is found that the vertical and rotational stabilities of mesoscale floating bodies vary continuously between the stabilities of the macroscale and microscale floating bodies with other parameters remaining unchanged.

Surface tension force on a partially submerged horizontal concave cylinder

“…Zhang et al. (2018) has shown that, when the cylinder has a convex shape, the meniscus can be determined uniquely by its height . This conclusion can be easily explained by (2.5).…”

“…its minor axis is parallel to the water line) with the same shape. However, the rotational equilibrium of the standing cylinder is unstable, while the lying cylinder is rotationally stable (Zhang, Zhou & Zhu 2018). This implies that the floating elliptical cylinder will automatically adjust its orientation due to the total moment, and therefore its load-bearing capacity is evaluated based on the configuration of the lying cylinder.…”

“…Zhang et al. (2018) also studied cylinders with more irregular shapes by developing a mathematical model for calculating the force profile and the moment profile of a cylinder with an arbitrary cross-section. In this model, it is assumed that the cross-section of the cylinder is non-concave to fluids so that the meniscus (if it exists) can be determined uniquely by the cylinder's position.…”

Capillary surfaces in and around exotic cylinders with application to stability analysis

“…It is known that the meridional curvature of the exotic cylinder with is given by (3.20). This expression can also be deduced from the relation between the variations and (see appendix B in Zhang, Zhou & Zhu (2018)): where all the quantities are dimensionless, denotes the curvature of solid at the contact point, denotes the inclination angle of the meniscus at the contact point and and denote the variations of the contact point height and the vertical displacement of floating body, respectively. This equation provides quantitative relations between the infinitesimal vertical displacement of a two-dimensional floating body and the change of the contact point height.…”

“…It is known that the meridional curvature of the exotic cylinder with K in = 0 is given by (3.20). This expression can also be deduced from the relation between the variations δH and δU (see appendix B in Zhang, Zhou & Zhu (2018)…”