Capillary phenomena. Part 14.—Approximate treatment for sessile-drop and captive-bubble meridians and properties

Abstract

New approximations are given for the first and second integrals of the Pascal–Laplace differential equations for sessile drops and captive bubbles. The first integral gives the ordinate Z(Φ) as a function of X(Φ), i.e. Z=Z(X, Φ), for meridian angle Φ= arctan (dZ/dX). The second integral is the meridian curve Z=Z(X). The first integral is obtained in the form Z=(1 – cos Φ)^½[I₀(X√2)–1]/I₁(X√2) and the second integral is given by X=X*+X_∞(Φ)[I₀(X*√2)– 1]/I₁(X*√2) with X_∞(Φ)=–{ln|tan[(Φ+ 180°)/4]|– 2 sin (Φ/2)}/√2 –C. I₀ and I₁ are modified Bessel functions of the first kind and X* is a shape factor which is a reasonably large value of X chosen at a constant meridian angle Φ=Φ*; C is a constant determined by Φ*, e.g. C= 0.37677 when Φ*= 90°.

The first of these equations gives the variation of drop or bubble height as X is increased at constant Φ; Z passes through a maximum. The three equations can be used to obtain approximate meridian curves. A drop of water surrounded by air on a solid with which it makes a contact angle of 30° would be well represented provided the contact radius exceeded ca. 5 mm. The accuracy increases to larger angles as the drop size increases. Extremely good meridian curves can be obtained by using Blaisdell's approximation for the equtorial region and the above equations for the crown region of drops.