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Chapter 7: Problem 33

In a fluid mechanics laboratory experiment a tank of water, with diameter \(D,\) is drained from initial level \(h_{0}\). The smoothly rounded drain hole has diameter \(d\). Assume the mass flow rate from the tank is a function of \(h, D, d, g, \rho,\) and \(\mu,\) where \(g\) is the acceleration of gravity and \(\rho\) and \(\mu\) are fluid properties. Measured data are to be correlated in dimensionless form. Determine the number of dimensionless parameters that will result. Specify the number of repeating parameters that must be selected to determine the dimensionless parameters. Obtain the \(\Pi\) parameter that contains the viscosity.

Short Answer

Expert verified
The number of dimensionless parameters is \(3\). The number of repeating parameters is \(3\). The \(\Pi\) parameter that contains the viscosity is \(\Pi_{\mu} = \frac{\mu \cdot \sqrt{d}}{\sqrt{g}}\)

Step by step solution

01

Identify the Variables

First, let's identify the variables from the problem. The variables involved are \(h\), \(D\), \(d\), \(g\), \(\rho\), and \(\mu\). The total number of variables is \(6\).
02

Identify the Fundamental Dimensions

The fundamental dimensions involved in this problem are mass \(M\), length \(L\), and time \(T\). The dimensions of these variables in terms of \(M\), \(L\), and \(T\) are as follow: for \(D\), \(d\), and \(h\), the dimensions are \(L\); for \(g\), the dimensions are \(\frac{L}{T^{2}}\); for \(\rho\), the dimensions are \(\frac{M}{L^{3}}\); and for \(\mu\), the dimensions are \(\frac{ML}{T}\).
03

Determine the Number of Dimensionless Parameters

The number of dimensionless parameters can be determined using Buckingham's \(\Pi\) Theorem, which states that the number of dimensionless parameters is equal to \(n-m\), where \(n\) is the total number of variables and \(m\) is the number of fundamental dimensions. In this case, \(n=6\) and \(m=3\), therefore, the number of dimensionless parameters is \(6-3=3\).
04

Determine the Number of Repeating Parameters

The number of repeating parameters that must be selected to determine the dimensionless parameters is equal to the number of fundamental dimensions involved in the problem. In this case, there are \(3\) fundamental dimensions (\(M\), \(L\), \(T\)), so the number of repeating parameters is \(3\).
05

Obtain the \(\Pi\) Parameter that Contains the Viscosity

To obtain the \(\Pi\) parameter, we can choose any three repeating variables, but one of them must be \(\mu\), the viscosity. For example, if we choose \(d\), \(g\), and \(\mu\) as the repeating variables, we can form the \(\Pi\) parameter that contains the viscosity as follows: \(\Pi_{\mu} = \mu^{a} \cdot d^{b} \cdot g^{c}\). To make this parameter dimensionless, we solve for \(a\), \(b\), and \(c\) using the dimensional homogeneity, that gives \[ \Pi_{\mu} = \frac{\mu \cdot \sqrt{d}}{\sqrt{g}} \]

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