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

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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=Xmin+X∞(Φ) K0(Xmin√ text-decoration:overline2)/K1(Xmin√ text-decoration:overline2) where X∞(Φ)={ln|tan [(Φ+ 180°)/4]|–2 sin (Φ/2)}/√ text-decoration:overline2+ 0.376 77 and Xmin, 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 Xmin > 1.

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