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Chapter 1: Problem 13

The volume rate of flow, \(Q,\) through a pipe containing a slowly moving liquid is given by the equation \\[ Q=\frac{\pi R^{4} \Delta p}{8 \mu \ell} \\] where \(R\) is the pipe radius, \(\Delta p\) the pressure drop along the pipe, \(\mu\) a fluid property called viscosity \(\left(F L^{-2} T\right),\) and \(\ell\) the length of pipe. What are the dimensions of the constant \(\pi / 8 ?\) Would you classify this equation as a general homogeneous equation? Explain.

Short Answer

Expert verified
The dimensions of the constant \(\pi / 8\) are dimensionless. The given equation can be classified as a general homogeneous equation because the dimensions match on both sides of the equation.

Step by step solution

01

Identify dimensions of each parameter

First, identify the dimensions of each parameter in the equation. Here, \(R\) is radius (Length, \(L\)), \(\Delta p\) is pressure drop (Force per unit Area, \(F/L^{2}\)), \(\mu\) is viscosity (Force \(\cdot\) Time per unit area, \(F \cdot T / L^{2}\)), and \(\ell\) is length of pipe (Length, \(L\)). The volume rate of flow, \(Q\), is volume per unit time (Length cubed over time, \(L^{3}/T\)).
02

Verify dimensions of the equation

Next, verify the dimensions on each side of the equation and identify what dimensions \(\pi / 8\) would need to be for the equation to hold true. The RHS of the equation has dimensions \(L^4 \cdot F / (F \cdot T / L^2) \cdot L = L^{7}/T\), so to match the dimensions of LHS, \(\pi / 8\) must be dimensionless (or have dimensions \(\frac{1}{L^{4}}\)).
03

Verify if this is a homogeneous equation

A general homogeneous equation has matching dimensions on both the LHS and RHS. Verify this using the information from steps 1 and 2. Since both LHS and RHS have the same dimensions (\(L^3/T\)), the given equation can be classified as a homogeneous equation.

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