Chapter 2

Minimal Energy and Stability Rubrics

2.1 Abstract

This chapter describes basic themes of stability for liquids in contact with various geometries of walls. A few simple rules can lead to a well-developed intuition of the stability of liquid shapes in general circumstances. There are some configurations for which it is possible to prove mathematically what the absolute minimum energy shapes are. Obviously, minimum energy shapes are stable.

A brief summary of the rubrics (very loosely stated):

  • Spherical drops are usually minimal energy.
  • Rotational symmetry means lower energy.
  • Pressure increasing with volume means stability.
  • Pressure decreasing with volume can give rise to a “double-bubble” instability.

The first section focuses on spheres, and works through a gallery of examples where a spherical section is the minimum energy surface. Included are droplets on planes, droplets in wedges, bridges between parallel planes, and droplets outside convex bodies. It is also possible to include configurations with mobile walls. The second section describes how “symmetrization” to rotational symmetry lowers the energy. This leads to the classic rotationally symmetric surfaces of constant mean curvature: the sphere, the cylinder, the catenoid, the unduloid, and the nodoid. Sections of these surfaces often turn up as bridges between parallel substrates. The third section describes a general method of showing stability in cases where the pressure increases as volume increases. This ...

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