On surface levelling in viscous liquids and gels

Summary

Surface tension at an irregular boundary of a liquid or gel gives rise to stresses within the material. In the two-dimensional case, for a Newtonian viscous liquid, the resulting flow is analysed and an expression for the decay of the irregularities is derived.

Since the stresses do not involve viscosity, they can be obtained from the viscous flow theory. Stress components obtained for gels in this way are identical with those deduced by means of the elastic theory of plane strain¹) taking Poisson’s ratio as 1/2. The maximum shearing stress in the gel, which governs the onset of flow, is obtained from the derived stress distribution.

It is assumed that the surface gradient is everywhere small. In the viscous flow case, it is additionally assumed that variations in the depth of the liquid are also relatively small; but failure to observe this restriction closely does not lead to serious error.

The effect of gravity can readily be allowed for when the fluid extends in a horizontal direction. Surface tension is the dominant factor for surface-profile components of wavelength less than about one cm.