An Introduction to: Velocity Model Building by Ian F. Jones

By Ian F. Jones

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An additional term is generated that involves the velocity of the moving boundary. This situation arises in problems involving moving boundaries, due to phase transition, net mass addition or removal from a system, and dissolution or precipitation of a phase. In step 6 we divide through by the dimensional coefficient of one term in each differential and algebraic equation involved in the describing equations for the particular transport or reaction process. These dimensional coefficients will consist of known parameters of the process, such as the density, viscosity, thermal conductivity, and so on, as well as the unspecified scale and reference factors used to nondimensionalize the variables in the describing equations.

2-5) Step 2 involves introducing arbitrary scale factors for each dependent and independent variable. Step 3 is unnecessary in this problem since both the velocity and spatial coordinate are naturally referenced to zero. 2-5). Indeed, one could introduce a scale factor for the second derivative. If this were done, one would find that there was no dimensionless group to determine the appropriate scale factor for the velocity. However, the latter could be obtained by integrating the scale for the second derivative of the velocity, in which case one would obtain the same scale for the velocity that will be obtained here by introducing its scale factor directly.

Scaling Analysis in Modeling Transport and Reaction Processes: A Systematic Approach to Model Building and the Art of Approximation, By William B. Krantz Copyright © 2007 John Wiley & Sons, Inc. 19 20 APPLICATIONS IN FLUID DYNAMICS dynamics, specifically creeping, lubrication, boundary-layer, and quasi-steady-state flows. We then apply ◦(1) scaling to show how it can be used to justify ignoring end and sidewall effects. We also consider the application of ◦(1) scaling in simplifying more complex flows, such as those involving free surfaces, porous media, and compressible fluids.

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