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Chapter 7: Problem 19

Consider a typical situation involving the flow of a fluid that you encounter almost every day. List what you think are the important physical variables involved in this flow and determine an appropriate set of pi terms for this situation.

Short Answer

Expert verified
The important physical variables are fluid velocity (V), fluid density (ρ), fluid viscosity (μ), pressure difference (ΔP), and pipe diameter (D). The appropriate set of Pi terms in this situation can be \(\Pi_1 = \frac{\Delta P D^{2}}{\rho V^{2}}\) (analogous to the Euler number), and \(\Pi_2 = \frac{\rho V D}{\mu}\) (the Reynolds number).

Step by step solution

01

Identify Important Physical Variables

Consider a situation of fluid flowing through a pipe. It's possible to identify the following physical variables: fluid velocity (V), fluid density (ρ), fluid viscosity (μ), pressure difference (ΔP), and pipe diameter (D).
02

Determine Dimensions of Variables

The next step is to determine the dimensions of these variables: fluid velocity (V) has dimensions of [LT^-1], fluid density (ρ) has dimensions [ML^-3], fluid viscosity (μ) has dimensions [ML^-1T^-1], pressure difference (ΔP) has dimensions [ML^-1T^-2], pipe diameter (D) has dimensions [L].
03

Identify Reference Variables

Choose three reference variables from which others can be formed. With the Buckingham π theorem in mind, the variables should contain all the dimensions (M, L, and T) present in the problem. Let's choose fluid velocity (V), fluid density (ρ), and pipe diameter (D).
04

Formulate Pi Terms

Using the Buckingham Pi theorem, combine the remaining variables with the reference variables to create dimensionless pi terms. Here are the possible pi terms: Pi term 1: \(\Pi_1 = \frac{\Delta P D^{2}}{\rho V^{2}}\) (similar to the Euler number), Pi term 2: \(\Pi_2 = \frac{\rho V D}{\mu}\) (the Reynolds number)

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