Capillary Stability in Tilted Square Cylinders

Abstract

The stability of a liquid mass above a capillary surface, as in the left sketch of Fig. 1, in a circular cylinder began to be examined long ago.1,2 Later, in 1968 Concus3 used a axisymmetric numerical shooting method that integrates the curvature defined by the pressure fields to solve for equilibrium interface shapes as functions of contact angle and Bond number. The limits of existence for these solutions give us the critical Bond number as a function of contact angle. Note that while Concus did not perform a stability analysis, his limits of static equilibrium solutions agree with analytical work at 90 degrees contact angle1,2 and with Surface Evolver computations.4 The stability limit is generally given in terms of a critical Bond number. Figure 2, from Collicott and Weislogel in 2004,4 shows both original and collected results for the critical Bond number for the classical case: liquid in the circular cylinder aligned with gravity (top line). This figure also presents results for a square cylinder with the axis aligned with gravity are shown as the solid black diamonds between contact angles of 45 and 135 degrees (for contact angles outside of this range, critical wetting eliminates the possibility of the free-surface configuration needed for this type of stability analysis.) See again the left side of Fig. 1 for the geometry. Note that the results are symmetric about 90 degrees contact angle. For reference, though not relevant to the present work, Fig. 2 also presents two-dimensional and rectangular channel results. Cylinders with liquid giving Bond numbers less than the critical Bond number will have a stable state with liquid above air. With B = ρgR2/σ, then for a given liquid in earth’s gravity, this means that the critical Bond number dictates a critical tube radius above which the stable solution can not exist. When gravity and the axis of circular cylinder are not aligned, as sketched on the right in Fig. 1, the equilibrium interface is no longer an axisymmetric surface and thus both analysis and numerical shooting methods are impractical in most cases. This geometry is also that of a tube in a spacecraft with a steady acceleration of general orientation. This was addressed in a January, 2009 AIAA conference paper5 for only a circular cylinder. Those results generate questions about how similar or different is the behavior of other cross-section cylinders. Thus, here the capillary stability in a tilted square cylinder is considered. Again, a critical Bond number specifies the stability limit. Unlike the circular cylinder case, the square can be tilted in different directions relative to the cross section, as sketched in Fig. 3. For example, the square cylinder can be tilted such that one edge of the square faces down (called zero cant angle in this work) or with a vertex facing down (called 45 degrees cant angle in this work). Both of these limits, and all cases between them, are