Bifurcation of a partially immersed plate between two parallel plates

Abstract

The three-plate system in which a vertical plate is located between two spaced parallel plates partially immersed in an infinite water bath in a downward gravity field is considered. With different contact angles and distance between the plates on both sides, the force profiles of the middle plate in this three-plate system are investigated using the Young–Laplace equation in two dimensions, and five non-trivial qualitative force profiles are found to possibly depend on the contact angles and the distance. The study is then extended to the qualitative changes of stability and behaviours in the system, and the striking properties related to the bifurcation theory come to light. Results show that, for different contact angles, there are at most eight possible bifurcation diagrams where the distance between the plates on both sides is chosen as the bifurcation parameter. By analysing the force profile of the middle plate in each of the eight bifurcation diagrams, the stabilities of the equilibria of the plate can be obtained. The number and the stabilities of equilibria will change when the bifurcation parameter passes the critical value.