The profile of a dew drop

Abstract

A stationary dew drop assumes a shape that minimizes the sum of its gravitational and surface energies. This shape, however, cannot be described by a mathematical function in closed form because the governing differential equations of the drop do not admit of an analytical solution but must be integrated numerically to yield the drop profile. Hence, this problem provides a good case for using the calculus of variations to obtain the governing differential equations and presents advanced students with an opportunity to gain insight and experience in numerical computations involving a real and interesting physical problem. In this paper we obtain the differential equations governing the shape of the drop in two ways: (i) by considering the balance of forces on the drop in static equilibrium; (ii) by seeking the profile which minimizes the total energy of the drop through the use of the calculus of variations. The resulting differential equations are then integrated numerically to obtain the theoretical profile. Two computational strategies, one for each of the two classes of wetting and nonwetting drops, are discussed. Further, it is shown how one may obtain the values of the surface tension and the contact angle for a liquid drop by matching the computed profile to the experimentally observed shape.

The profile of a capillary liquid bridge between solid surfaces

“…22 These phenomena have inspired the publication of several papers on the subject. [23][24][25][26][27] Theoretical descriptions of capillary bridges have been given for several cases, such as a liquid bridge between two surfaces, 20,28 or the inverse, a hole in a thin liquid film. 29,30 In most of these cases, the liquid bridge is in contact with a reservoir of the same fluid so that the pressure difference with the outer fluid is zero.…”

Reliable determination of contact angle from the height and volume of sessile drops

“…The following tables list our input data (volume and height) as well as the computed values (contact angle, footprint radius, and the apex radius of curvature) for water drops, ranging in volume from 40 μl to 10 μl, on two substrates, Teflon, and Lucite. For comparison, we give the experimental values of contact angle measured from the drop profile 33 , Water on Teflon:…”

Roughness effects on contact angle measurements

“…Some studies have focused on determining numerical 6 or analytical expressions 7 to describe the shape of the fluid interface. Experimental investigations have addressed the effect of surface chemistry on contact angle values.…”