Minimum area surfaces

We have just seen that a liquid evolves spontaneously so as to minimize its surface area, and we illustrate this property in Figure 1.9. At equilibrium, minimum area surfaces satisfy Laplace s equation. 

The scientific study of liquid surfaces, which has led to our present knowledge of soap films and soap bubbles, is thought to date from the time of Leonardo da Vinci - a man of science and art. Since the fifteenth century researchers have carried out investigations in two distinct camps. In one camp there are the physical, chemical and biological scientists who have studied the macroscopic and molecular properties of surfaces with mutual benefit. The other camp contains mathematicians who have been concerned with problems that require the minimization of the surface area contained by a fixed boundary and related problems. A simple example of such a problem is the minimum area surface contained by a circle of wire. The solution to this problem is well known to be the disc contained by the wire. 

What have been the significant historical developments in the mathematics of minimum area surfaces John Bernoulli and his student Leonhard Euler were amongst the earliest workers to apply the methods of the calculus to the solution of these problems, thus laying the foundations for a new branch of the calculus, the Calculus of Variations. In a comprehensive work published in 1744 Euler derived his well known equation for the determination of minimum area surfaces and other variational problems that require the examination of a sequence of varied surfaces. The equation, in its simplest one dimensional form, is... 

The derivation of this equation was based on a geometric-analytic method. Euler successfully applied his equation to the celebrated problem of the determination of the minimum area surface contained by two parallel coaxial rings, arranged perpendicular to their common axis. This was found to be a catenary of revolution, or catenoid, providing the rings are sufficiently close together. 

Joseph Louis Lagrange was attracted by Euler s work and reformulated it using purely analytic methods. Equation (1.2) has since become known as the Euler-Lagrange equation. The solution to the minimum area surface contained by two coaxial rings remains one of the few analytic solutions available in this field. 

It was Joseph Plateau s experiments with soap films that provided mathematicians with renewed motivation to investigate the problems of minimum area surfaces. Some of these beautiful analogue solutions are examined in Chapter 4 with coloured plate illustrations. Although substantial efforts were made to obtain analytic solutions to these problems it was not until the 1930 s that significant progress was made by mathematicians such as Jesse Douglas, who obtained some general solutions, and Tibor Rado. " ... 

Planar minimum area surfaces bounded by a circle. 

The methods of Euler and Lagrange consider a continuous sequence of surfaces, contained by the boundary, which deviate from the extremum surface. For the soap film problem the extremum surface is the minimum area surface. A differential equation is then derived for the extremum surface. The solution to the differential equation will give the required surface. 

By rupturing different sections of the minimum surface one can obtain the subset of minimum surfaces associated with a number of the edges of the cube. This number is less than, or equal, to twelve. For example, by breaking the square at the centre with a dry rod, or finger, the minimum surface shown in Fig. 4.14 is obtained. It is a minimum area surface bounded by all twelve edges of the cube. It is not obvious that this surface has a larger, or smaller, surface area than the former surface with a central square and it is difficult to determine the magnitudes of these surface areas. 

Lindelbf and Moigno again dealt with, in 1861, the problem of the catenoid they find, by a shorter method, several of the results of Goldschmidt. They arrive, moreover, at this other result for an arc of catenary whose end A is taken arbitrarily, it is the minimum area surface of revolution only up until the second end B, moving away along to the curve, reaches a position such that the two tangents from A and B meet... 

Core area of air terminal device That parr of an air terminal device located within a convex closed surface of the minimum area required to include all of the air terminal device openings inside the sur face. 

Tension in the free surface of a liquid is the cause of the tendency of a liquid surface to assume the form having a minimum area, as manifested in the shape of a bubble or a drop of liquid.25 The tendency to contract is a special case of the general principle that potential energy tends toward a minimum value. 

G. Mitri and co-workers calculated the minimum area of hydrocarbon lakes which would be necessary to preserve the relative methane humidity in the lower regions of the atmosphere. The result was surprising the calculations indicated that only between 0.002 and 0.2% of the total surface area of Titan would be required (Mitri et al., 2007). 

Second, these methods do not give any information about the shape and thereby the surface area of the bubble. Although Quigley et al. (Ql) have found from photographic studies that the area evaluated assuming sphericity of bubbles is not very different from that actually measured, this evidently cannot be true for large and highly distorted bubbles since a sphere has the minimum area and any distortion in shape tends to increase it. 

Values of the critical micelle concentration (cmc), minimum area per molecules " cnic effectiveness of surface tension reduc-... 

In addition to particle size, the particle shape contributes to the surface area of the powder. Of all geometric forms, a sphere exhibits the minimum area-to-volume ratio while a chain of atoms, bonded only along the chain axis, will give the maximum area-to-volume ratio. All particulate matter... 

Different molecular areas of Langmuir monolayers can be determined. They can be defined in three ways Ao is the area per molecule extrapolated to zero differential surface tension, Ac is the minimum area per molecule at the collapse point, at the point in the tt - A isotherms where the pressure is the maximum reversible pressure (or collapse pressure ttc) and Am is the area at the midpoint pressure rrm = 0.5 TTC. 

The kinetics of the ammonia oxidation reaction are limited by the mass transfer of the chemical species (NH3, 02, NO and H20) to and from the vicinity of the catalyst surface. If the catalyst area available to reactants is higher than the minimum area permitting the reaction, the mass transfer limits the kinetics. 

It is a geometrical fact that surfaces for which the relation (1) holds are surfaces of minimum area. 

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