Computational geology 24: Patterns, dimensions and viscosity

Journal of Geoscience Education, Mar 2003 by Vacher, H L

Keywords: Education - geoscience; education undergraduate; miscellaneous and mathematical geology.

Topics this issue

Mathematics: Dimensional analysis; orders of magnitude.

Geology: Lava flows, rising magmas; Stokes Law; Darcy's Law.

INTRODUCTION: THE SCIENCE OF PATTERNS

Keith Devlin and Lynn Steen are two of the important authors for the Mathematics Association of America, the mathematics society that focuses on teaching mathematics at the college level. Devlin, author of more than twenty books, writes Devlin's Angle, a monthly column posted on the MAA Website (www.maa.org/ news/devangle.html ). Steen, a former president of the MAA, is the intellectual force behind the national quantitative literacy movement (see Mathematics and Democracy, www.woodrow.org/nced/mathematics democracy.html). They both have characterized mathematics as "the science of patterns" - Devlin in a coffee-table book published by Scientific American (Devlin, 1997), Steen in an essay published in Science on the 100th birthday of the American Mathematical Society (Steen, 1988).

What does it mean to say mathematics is the science of patterns? According to Steen (1988, p. 616):

Mathematics is often defined as the science of space and number, as the discipline rooted in geometry and arithmetic. Although the diversity of modern mathematics has always exceeded this definition, it was not until the recent resonance of computers and mathematics that a more apt definition became fully evident.

Mathematics is the science of patterns. The mathematician seeks patterns in number, in space, in science, in computers, and in imagination. Mathematical theories explain the relations among patterns; functions and maps, operators and morphisms bind one type of pattern to another to yield lasting mathematical structures. Applications of mathematics use these patterns to 'explain' and predict natural phenomena that fit the patterns. Patterns suggest other patterns, often yielding patterns of patterns. In this way mathematics follows its own logic, beginning with patterns from science and completing the portrait by adding all patterns that derive from the initial ones.

In the spirit of mathematics as the science of patterns, this column discusses a pattern evident in equations relating flow rate to viscosity in undergraduate geology courses. Three such equations are listed in Table 1. The pattern of interest is that all of these equations are dimensionally the same. The dimensional variables combine in the same way. The purpose of this column is to explain why that is.

CONCLUDING REMARKS

It may seem that there is a bewildering variety of mathematical equations sprinkled through geological textbooks. Actually, the variety is not so great; there are recurring kinds of equations. Each kind of equation has its own pattern, defined by the underlying principles being applied and the dimensional variables called for by those principles. In the examples discussed in this column, the underlying principle is the balance of forces in the context of laminar flow of a viscous fluid. The dimensional variables are velocity, specific weight, viscosity, and a length. Because of dimensional homogeneity, the velocity is proportional to specific weight over viscosity times the length variable squared.

Just as one learns to make sense of graphs by looking first at the axes and their scale (arithmetic vs. logarithmic), one can make great strides in understanding equations derived from first principles by looking first at the dimensions of the variables.

REFERENCES CITED

Blatt, H., Middleton, G., and Murray, R., 1980, Origin of sedimentary rocks, Englewood Cliffs, Prentice-Hall, 782 pp.

Devlin, IK., 1997, Mathematics: the science of patterns, New York, Scientific American Library, 216 pp.

Macdonald, G.A., 1954, Activity of Hawaiian volcanoes during the years of 1940-1950, Bulletin Volcanologique, ser. 2, v. 15, p. 120-179.

Philpotts, A.R., 1990, Principles of Igneous and Metamorphic Petrology, Upper Saddle River, Prentice-Hall, 498 pp.

Resnick, R., Halliday, . and Krane, K.S., 1992, Physics, New York, Wiley, 592 pp.

Steen, L.A., The science of patterns, Science, v. 240, no. 4852, p. 611-616.

Turcotte, D.L. and Schubert, G., 1982, Geodynamics: Applications of continuum physics to geological problems, New York, Wiley, 450 pp.

Walker, G.P.L., 1967, Thickness and viscosity of Etnean lavas, Nature, v. 213, n. 5075, p. 484-485.

H. L. Vacher

Department of Geology, University of South Florida, SCA 528

4202 E. Fowler Ave., Tampa, FL 33620-5201, vacher@chuma.cas.usf.edu