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File DIMEN > contains functions for automatic dimensional analysis,
and file DIMEN DEMO contains a demonstration. Usage is of the form
NONDIMENSIONALIZE(list of physical quantities);
The returned value is a sufficient list of nondimensional products of
powers of the physical quantities. A physical relation between only
the given physical quantities must be expressible as a relation between
the nondimensional quantities. There are usually fewer nondimensional
than physical quantities, which reduces the number of experiments or
numerical computations necessary to establish the physical relation to
a specified resolution, in comparison with the number if all but one
dependent physical variable were independently varied. Also, the
absence of any given physical quantity in the output reveals that
either the quantity is irrelevant or others are necessary to describe
the relation.
The program already knows an extensive number of relations between
physical quantities, such as VELOCITY=LENGTH/TIME. (CPUTIME plays
the role of the customary MACSYMA global variable TIME.)
The user may override or supplement the prespecified
relations by typing
DIMENSION(equation or list of equations);
where each equation is of the form indeterminate=expression, where
expression is a product or quotient of powers of none or more of the
indeterminates CHARGE, TEMPERATURE, LENGTH, TIME, or MASS. To see
if a relation is already established type
GET(indeterminate, 'DIMENSION);
The result of NONDIMENSIONALIZE usually depends upon the value of the
global variable %PURE, which is set to a list of none or more of the
indeterminates ELECTRICPERMITTIVITYOFAVACUUM, BOLTZMANNSCONSTANT,
SPEEDOFLIGHT, PLANCKSCONSTANT, GRAVITYCONSTANT, corresponding to the
relation between charge and force, temperature and energy, length and
time, length and momentum, and the inversesquare law of gravitation
respectively. Each included relation is used to eliminate one of
CHARGE, TEMPERATURE, LENGTH, TIME, or MASS from the dimensional basis.
To avoid omission of a possibly relevant nondimensional grouping,
either include the relevant constant in %PURE or in the argument
of NONDIMENSIONALIZE if the corresponding physical effect is thought to
be relevant to the problem. However, the inclusion of unnecessary
constants, especially the latter three, tends to produce irrelevant or
misleading dimensionless groupings, defeating the purpose of
dimensional analysis. As an extreme example, if all five constants are
included in %PURE, all physical quantities are already dimensionless.
%PURE is initially set to '[ELECTRICPERMITTIVITYOFVACUUM,
BOLTZMANNSCONSTANT], which is best for most engineering work.
%PURE must not include any of the other 3 constants without also
including these 2.
Send problems and suggestions to STOUTE.
REFERENCES:
R. Kurth, "Dimensional Analysis and Group Theory in Astrophysics",
Perggamon Press.
H.L. Langhaar, "Dimensional Analysis and Theory of Models", John
Wiley and Sons.
D.R. Stoutemyer, "Automatic Dimensional Analysis, Using Computer
Symbolic Mathematics", report, Electrical Engineering Department,
University of Hawaii, Honolulu, Hawaii 96822.
