Dimensional Analysis: With Case Studies in Mechanics

Thus, the governing momentum equation reduces to the non-dimensional form of Euler equations:. Once the variables are selected and the dimensional analysis is performed, the experimenter seeks to achieve similarity between the model tested and the prototype to be designed. Instead of complete similarity, the engineering literature speaks of particular types of similarity:.

Fluid Mechanics/Dimensional Analysis

A model and a prototype are geometrically similar if and only if all body dimensions in all three coordinates have the same linear-scale ratio. All angles are preserved in geometric similarity. All flow directions are preserved. The orientations of model and prototype with respect to the surroundings must be identical.

The motions of two systems are kinematically similar if homologous particles lie at homologous points at homologous times. In other words, kinematic similarity requires that the model and prototype have dependent length-scale and time-scale ratios.

This requires, for example, similarity in Re or Mach number. The result is that the velocity-scale ratio will be the same for both:. One exception , in which the length and the time-scales becomes independent is the frictionless low-speed flow, without any free-surface.

Perspective of Dimensional Analysis in Medical Science

Dynamic similarity exists when the model and the prototype have the same length scale ratio, time-scale ratio, and force-scale or mass-scale ratio. The dynamic-similarity ensure that each of these forces will be in the same ratio and have equivalent directions between model and prototype. Again geometric similarity is a first requirement. Then dynamic similarity exists, simultaneous with kinematic similarity, if the model and prototype force and pressure coefficients are identical.

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This is ensured if:. Consider the drag force on the surface of the ship shown. To have complete dynamic similarity between test model and prototype:. Only liquid which has less kinematic viscosity than water is mercury, and even with that fluid Re similarity can not be established.

Since Froude number becomes more important than Re, in the experiments scaling is done using only Fr number. The flow in the Handihaler is unsteady. Thus, visualization is preferred in order to understand the flow character within the inhaler.

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The up-scaled models in water are used to slow down the speed of the flow so that we can record streaklines with a high-speed camera. After preliminary visualization experiments, scenarios were planned. The flow region was divided into sub-regions. University of California San Diego. Introduction to the Foundations of Applied Mathematics. Texts in Applied Mathematics A fundamental and necessary extension of classical dimensional analysis theory. International Communications in Heat and Mass Transfer, 33, Journal of Applied Physiology, 1, Statistical Analysis and Data Mining, 1 6 , The role of the Womersley number in pulsatile blood flow a theoretical study of the Cassonmodel.

Journal of Biomechanics, 34 1 , The effects of low-density lipoprotein and high-density lipoprotein on blood viscosity correlate with their association with risk of atherosclerosis in humans. What exactly is the drag a function of?

These variables need to be chosen correctly, though selection of such variables depends largely on one's experience in the topic. It is known that drag depends on.

Dimensional Analysis: With Case Studies in Mechanics - Qing-Ming Tan - Google Книги

Note that the dimensions of the left side, force, must equal those on the right side. Here, we use only the three independent dimensions for the variables on the right side: M, L, and T. Write the equation in terms of dimensions only, i. The equation then becomes.

It is seen that there are three equations, but 5 unknown variables. This means that a complete solution cannot be obtained. Thus, we choose to solve a , b , and d in terms of c and e.

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These choices are based on experience. Such dimensionless groups keep reoccurring throughout Fluid Mechanics and other fields.