Research


I am interested in using methods from the calculus of variations, applied analysis, and asymptotic analysis to answer questions about ordinary and partial differential equations (ODEs and PDEs) that arises in physical and life sciences, particularly fluid dynamics. For my thesis, I investigate the role of surface tension on the fluid free surface on the natural sloshing frequencies of an incompressible, inviscid fluid in rigid containers.

Projects

Measuring fluid surface tension in a Hele-Shaw cell using Faraday waves
with C. Hohenegger, E. Kanso and B. Osting.
We propose to measure the fluid surface tension from Faraday waves generated in a narrow-width rectangular container, called a Hele-Shaw cell. The cell is placed above a speaker and made to oscillate vertically. The excitation frequency is slowly varied until two-dimensional standing Faraday waves are generated and wavelength measurements are recorded. The surface tension $\sigma$ is measured by plotting measured wavelength versus the excitation frequency and fitted the curve against the dispersion relation, by adjusting the only unknown experimental parameter, $\sigma$.

An isoperimetric sloshing problem with surface tension in a shallow container
with C. Hohenegger and B. Osting.
B. A. Troesch (1965) studied the isoperimetric sloshing problem (ISP) of determining the cross-section of a symmetric shallow canal with a given width and cross-sectional area, and the cross-section of a radially-symmetric shallow container with a given rim radius and volume, that maximises the fundamental sloshing frequency. Applying the shallow water theory and exploiting the symmetry of the containers, (ISP) is then formulated as a one-dimensional spectral optimisation problem with an area constraint, in the sense that the free surface is fixed and only the wetted bottom of the container is allowed to vary while preserving the given cross- sectional area. By means of a variational argument, Troesch showed that solving (ISP) is equivalent to solving a certain first-order singular ODEs with an integral constraint and he managed to solve for the isoperimetric shallow containers for certain cases, where the latter refers to containers that attains the isoperimetric inequality. We extend Troesch's result by including surface tension effects on the fluid free surface, restricting ourselves to a pinned contact line and a flat equilibrium free surface. Assuming finite surface tension, we found that the isoperimetric shallow containers are nonlinear perturbation of Troesch's isoperimetric shallow containers.

On the two-dimensional ice fishing problem with surface tension
with N. Willis, C. Hohenegger and B. Osting.
The ice-fishing problem (IFP) is the linear sloshing problem in a half-space bounded above by an infinite rigid plane that contains a circular or strip-like aperture. For the two-dimensional problem, the half-space is bounded above by the $x$-axis and the free surface is an interval on the $x$-axis. We consider the two-dimensional IFP with surface tension, coupled with a pinned contact line boundary condition. The motivation is that the sloshing frequencies for this problem furnish universal upper bounds for sloshing frequencies of arbitrary containers sharing the same free surface and same contact line boundary condition. We reformulate the problem as a constrained Fredholm integro-differential eigenvalue problem for the sloshing height $\xi$ on the free surface $[-1, 1]$ and the square of sloshing frequencies $\omega^2$, where the solution is constrained to satisfy the zero mean condition over $[-1, 1]$ and Dirichlet boundary condition at $x = \pm 1$. Writing the problem in weak form and expanding solutions in suitably chosen Fourier basis leads to a generalised matrix eigenvalue problem of the form $\mathbf{B\xi} = \omega^2\mathbf{C\xi}$. The entries of $\mathbf{C}$ are integrals involving logarithmic kernel and we use two Gauss quadrature rules to evaluate these integrals.

A variational characterization of fluid sloshing with surface tension
with C. Hohenegger and B. Osting.
We consider the sloshing problem for an incompressible, inviscid, irrotational fluid in an open container, including effects due to surface tension on the free surface. We restrict ourselves to a constant contact angle and seek time-harmonic solutions of the linearized problem, which describes the time-evolution of the fluid due to a small initial disturbance of the surface at rest. As opposed to the zero surface tension case, where the problem reduces to a partial differential equation for the velocity potential, we obtain a coupled system for the velocity potential and the free surface displacement. We derive a new variational formulation of the coupled problem and establish the existence of solutions using the direct method from the calculus of variations. We prove a domain monotonicity result for the fundamental sloshing eigenvalue. In the limit of zero surface tension, we recover the variational formulation of the mixed Steklov–Neumann eigenvalue problem and give the first-order perturbation formula for a simple eigenvalue.

Publication and Preprints

2. C. H. Tan, C. Hohenegger and B. Osting, An isoperimetric sloshing problem in a shallow container with pinned contact line, in preparation.

1. C. H. Tan, C. Hohenegger and B. Osting, A variational characterization of fluid sloshing with surface tension, SIAM Journal of Applied Mathematics, 77 (2017), no. 3, 995-1019. [arXiv]




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