Extended Annular Capillary Surfaces

Abstract

The height of the surface of a fluid in an annular tube is explored using a shooting method to solve a boundary value problem where the radii and the contact angles are given. The contact angles on the inner and outer tube surface need not be the same. These surfaces are then extended so that they are no longer graphs. The extended surfaces are shown to solve a boundary value problem over an annular base domain where given inclination angles are achieved at given radii.

Capillarity and Archimedes’ principle

“…We are able to give a partial answer under the assumption of rotational symmetry of the object and the interface. This symmetry appears to hold in the physical system of Figure 1, and similar symmetric interfaces have been shown to exist mathematically in [Treinen 2012] and [Elcrat et al 2004b]. For further discussion of this point, see Section 6.…”

“…The barrier to getting a more definitive result lies in the complicated nature of the system of ordinary differential equations determining the rotationally symmetric interface. For a survey of recent progress in understanding the family of solutions to these equations, see [Finn 1986;Vogel 1982;Siegel 2006;Siegel 1980;Nickolov 2002;Elcrat et al 2004a;Turkington 1980;Johnson and Perko 1968;Treinen 2012].…”

On the Symmetry of Solutions to Some Floating Drop Problems

“…Thus, either w ⊂ Γτ for some τ = τ 0 , or w does not extend to infinity without self-intersections. For a discussion of some of the other possible surfaces see [19].…”

“…The curve of horizontal points of the Γτ 0 's was studied in [22], in the related context of liquid bridges. Again, we will parametrize these surfaces with inclination angle ψ ∈ [−π, 0), and (13), (14), as in [19], which is little more than a notational change. The results in [22] may be interpreted so that the curve of horizontal points (r(−π, τ ), u(−π, τ )), which is parametrized by τ , is the graph of a monotonically increasing function of r. Consider the surface T obtained by revolving the curve of horizontal points (r(−π, τ ), u(−π, τ )) around the axis of symmetry.…”

A general existence theorem for symmetric floating drops

“…Defineṙ ≡ ∂r/∂h. The following two results are from [14]: Lemma 3.9. On −π ≤ ψ < 0,ṙ > 0, and on 0 < ψ ≤ π,ṙ < 0.…”