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

General exotic capillary tubes

“…The ECT has the boundary parameter χ 1 = χ * 1 corresponding to each meniscus because of its 'exotic' property, as discussed in § 1. We have verified that χ 1 = χ 1 * for exotic cylinders with ε = +1 by numerically solving (2.10) in a previous study (Zhang & Zhou 2020). The Jacobi equation (2.10) for ε = +1 is numerically solved using the spectral parameter power series method (Kravchenko & Porter 2010), which expresses the general solution of the Sturm-Liouville equation as a spectral parameter power series.…”

“…The solution curves for ε = +1 can also be determined by integrating a different parametric form of the Young-Laplace equation (see e.g. Zhang & Zhou 2020), where the parameter is not the arc length s but the inclination angle ψ. However, this parametric form is not suitable for the case of ε = −1, because of the singularity of (d/dψ) (i.e.…”

“…the instability points. This can be explained in that the envelopes can be regarded as ECTs with zero contact angle, and the menisci in the ECTs correspond to the instability points to pressure disturbances (Zhang & Zhou 2020). This implies that the shape of the ECT is closely related to meniscus stability.…”

“…Inspired by the exotic container, Wente (2011) extended the 'exotic' property to a capillary tube and proposed a non-uniform circular tube, called the exotic capillary tube (ECT), vertically positioned in an infinite liquid which permits an entire continuum of equilibrium menisci. Following Wente (2011), Zhang & Zhou (2020) investigated the interior, exterior and planar cases of ECTs together (collectively called the exotic cylinder), and then examined their stabilities by the method of Slobozhanin & Alexander (2003). Results showed that each of the menisci for the exotic cylinder has the smallest eigenvalue λ 01 = 0 to axisymmetric perturbations and the menisci are stable, in contrast to the case of the exotic container.…”

“…Then, we discuss why the axisymmetric menisci in the exotic containers are unstable to non-axisymmetric perturbations while the axisymmetric menisci in the exotic cylinders are stable. The 'exotic' property indicates that the axisymmetric menisci have the smallest eigenvalue λ 01 = 0 to axisymmetric perturbations for the exotic containers and the exotic cylinders (see Wente 1999;Zhang & Zhou 2020). Myshkis et al (1987) have shown that for an unconstrained axisymmetric problem one has λ 01 < λ 11 , where λ 11 is the smallest eigenvalue to non-axisymmetric perturbations, which means that the unconstrained axisymmetric meniscus to axisymmetric perturbations is less stable than to non-axisymmetric perturbations.…”