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from Part VIII - Fundaments
Published online by Cambridge University Press: 26 October 2017
The fundaments of kinematics presented include:
• B.1: an introduction to dimensional analysis;
• B.2: an introduction to the international system of units; and
• B.3: a table of parameters relevant to waves and flows on Earth and their estimated magnitudes.
Dimensional Analysis
This fundament provides an orientation to the process of dimensional analysis, whereby a problem is simplified by analyzing the dimensions of its variables and parameters. The theoretical basis for this process is provided by the Buckingham Pi theorem. An essential part of the process is the formation of a minimal number of dimensionless parameters, typically denoted by ∏, each of which is an algebraic combination of the dimensional parameters in the problem. One advantage of dimensional analysis is that we do not need a set of governing equations (though that is, of course, quite helpful). All that is needed is a “grocery list” of relevant parameters.
A related procedure is non-dimensionalization, whereby a set of governing equations and the parameters within it are made dimensionless. Often the equations are significantly simplified by this procedure, making them more amenable to analysis and solution.
Dimensional analysis employs the seven base dimensions (length, mass, time, temperature, electric current, amount of substance and luminous intensity) of the SI system. Mechanical systems involve only the first three (length, mass and time), denoted by L, M and T:
L = length M= mass T = time.
Commonly the dimensions of a variable are identified using square brackets; for example, [g] = L·T−2 means that the dimensions of acceleration are length divided by time squared.
In addition to the seven base dimensions, the SI system contains two supplemental units: plane and solid angle. These do not play a direct role in dimensional analysis and non-dimensionalization, but they do affect the numerical relationships among the dimensionless variables. The number of dimensionless parameters involved in a physical problem can easily be determined by use of the Buckingham Pi theorem which states that
N∏ = Np −Nd,
where N∏ is the number of dimensionless parameters, Np is the number of dimensional
parameters and Nd is the number of base dimensions contained in all parameters.
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