Capillary phenomena. Part 11.—Approximate treatment of the shape and properties of fluid interfaces of infinite extent meeting solids in a gravitational field

Abstract

The approximate expression involving modified Bessel functions, Z=(1 + cos Φ)^½K₀(X√2)/K₁(X√2), has been obtained as the first integral of Laplace's equation of capillarity, relating the meridian ordinate Z to the abscissa X and the meridian angle Φ= arctan (dZ/dX), for holm meridians. Holm meridians are those having a vertical axis of symmetry and which begin at the asymptotic limit X→∞ when Z= 0 and extend to small X values, e.g., they represent a rod dipped in a large vessel containing a liquid.

Many of the features previously predicted, e.g., applied force maxima, for solid rods and cones dipping into a liquid can be obtained by the approximate treatment given, without resort to numerical computation.

Comparison with a treatment by Ferguson (1912) gives a useful mathematical approximation between zeroth- and first-order modified Bessel functions, K₀(z)/K₁(z)≈[1 +z^–1]^–½, which is accurate to 3.3 % for z= 0.5, 0.27 % for z= 2 and 0.01 % for z= 10. Comparison has also been made with the first-integral capillary approximations of Ferguson, Tsivinskii (1962) and James (1974).

The treatment has been extended to give second-integral approximations for holm meridians Z=Z(X). The Z ordinate is given above and X is given by evaluation of X=X_min+X_∞(Φ) K₀(X_min√2)/K₁(X_min√2) where X_∞(Φ)={ln|tan [(Φ+ 180°)/4]|–2 sin (Φ/2)}/√2+ 0.376 77 and X_min, which is a shape factor, is the value of X at the waist of a holm. Agreement with computed meridians (numerical integration) is excellent for X_min > 1.