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# fluidsCh4 - Chapter 4 Dimensionless expressions...

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Chapter 4 Dimensionless expressions Dimensionless numbers occur in several contexts. Without the need for dy- namical equations, one can draw a list (real or tentative) of physically rel- evant parameters, and use the Vaschy-Buckingham theorem to construct a shorter dimensionless list. Dimensionless expressions are the required tool to compare data from different experiments (e.g. parachute data in water), leading to the recommendation that all data should be plotted in dimen- sionless form. This is generally covered at the undergraduate level, and a few points of interpretation are added here. The same dimensionless expres- sions are obtained from dynamical equations, when available: the meaningful dimensionless numbers are ratios of terms in various equations, measuring their relative importance. This can be used to approximate the equations rationally, by dropping small (dynamically inactive) terms. One notable ex- ception is when the small parameter is the coefficient of the highest-order derivative in the equation... 4.1 Dimensional analysis This material is assumed known from undergraduate courses: fill in any gaps (and practice) by consulting undergraduate textbooks from the library re- serve. We will review the procedure (same for all problems) with one familiar example as illustration. Occasional features not shown in the example will be mentioned for reference. 1. The list of parameters: 103

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104 CHAPTER 4. DIMENSIONLESS EXPRESSIONS (a) This step determines the eventual solution, and several attempts may be necessary to identify the list that makes sense of the data. The list should be sufficiently complete to account for the physics of the flow, but not to the point of introducing unnecessary com- plications: trial-and-error, from the simpler to the more elaborate, is sensible. The list is a reflection of individual insight and of the profession’s expertise. (b) Example: We will work with fully developed flow in a circular pipe. The list of parameters includes: flow parameters (pressure drop per unit length dp/dx , average speed V ), pipe configuration (diameter D or cross-sectional area A , not independent of course), and material properties (fluid density ρ and viscosity μ ). δp/L V D ρ μ not included: surface roughness; etc. (c) If some relations are known (e.g. between velocity, cross-sectional area and volume flow rate), the corresponding parameters are not independent, and one of them can be eliminated from the list for each such relation. Also, in this problem, we start from the pres- sure drop per unit length of pipe, rather than pressure drop and overall length as separate variables: the assumed proportionality between them is a valuable insight (try solving without it and note the differences!) 2. Primary dimensions: (a) ‘Dimensions’ are more general than ‘units’; e.g. meter, foot and mile and micron are units relevant to the dimension of length.
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