文档介绍：3. Capillary Shape and Stability
Two equivalent methods of deriving the capillary equation are considered
here. They are based on
• the balance of forces on each element of the liquid surface. Each surface
element experiences fluid-static pressure on its area and surface tension
along its perimeter. This view was promoted by Laplace [1805, 1806] and
Young [1805].
• the minimization of the total energy of the liquid under the constraint of
constant liquid volume. This aspect was promoted by Gauss [1830].
The capillary equation is also termed the Gauss–Laplace equation for this
reason.
Balance of Forces
Interest in the capillary behavior of fluids started rather early. In 1712, Taylor
reported:
The following Experiment seeming to be of use, in discovering the Proportions
of the Attractions of Fluids, I shall not forbear giving an Account of it; tho’ I
have not here Conveniences to make it in so essful a manner, as I could
wish. – I fasten’d two pieces of Glass together, as flat as I could get; so that
they were inclined in an Angle of about 2 Degrees and a half. Then I set
them in Water, with the contiguous Edges perpendicular. The upper part of
the Water, by rising between them, made this Hyperbola; which is as I copied
it from the Glass. – I have examined it as well as I can, and it seems to
approach very near to mon Hyperbola. But my Apparatus was not
nice enough to discover this exactly. – The perpendicular Assymptote was
exactly determined by the Edge of the Glass; but the Horizontal I could not
so well discover.
Taylor’s result was confirmed soon afterwards by Hauksbee [1712, 1713a,
1713b]. Hauksbee showed additionally that the hyperbolic character of the
meniscus is independent of the inclination of the wedge relative to the reser-
voir. Hauksbee repeated the experiments using “spirit of wine”.
Dieter Langbein: Capillary Surfaces, STMP 178, 41–65 (2002)
c Springer-Verlag Berlin Heidelberg 2002
42 3. Capillary Shape and Sta