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Chapter 7: Problem 9

The equation describing motion of fluid in a pipe due to an applied pressure gradient, when the flow starts from rest, is $$\frac{\partial u}{\partial t}=-\frac{1}{\rho} \frac{\partial p}{\partial x}+\nu\left(\frac{\partial^{2} u}{\partial r^{2}}+\frac{1}{r} \frac{\partial u}{\partial r}\right)$$ Use the average velocity \(\bar{V}\), pressure drop \(\Delta p\), pipe length \(L\) and diameter \(D\) to nondimensionalize this equation. Obtain the dimensionless groups that characterize this flow.

Short Answer

Expert verified
The non-dimensional equation could be given by substituting the non-dimensional variables, simplifying the equation and identifying the resulting dimensionless groups. The exact form would depend on additional specifics not given in the problem.

Step by step solution

01

Identify the suitable variables

The first step is to note the variables given that we can use to nondimensionalize our equation. These include the average velocity \(\bar{V}\), the pressure drop \(\Delta p\), the pipe length \(L\), and the diameter \(D\).
02

Define non-dimensional variables and substitute them

Define a set of non-dimensional variables using the given variables. For instance, we might define: \(u' = \frac{u}{\bar{V}}\), \(t' = \frac{t\bar{V}}{L}\), \(p' = \frac{p}{\Delta p}\), \(x' = \frac{x}{L}\), and \(r' = \frac{r}{D/2}\). Substitute these new variables into the original equation.
03

Simplify the equation

After substitution, simplify the terms and group together items that are dimensionally the same to make the equation as simple as possible.
04

Identify the dimensionless groups

The dimensionless groups in the equation are the terms that appear after the non-dimensionalization process. They are constants of proportionality that are free of dimensions and specify the character of the flow in the pipe.

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Most popular questions from this chapter

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