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Fluid Flow and the Continuity Equation

Fluids, by definition can flow, but are essentially incompressible. This provides some very useful information about how fluids behave when they flow through a pipe, or a hose. Consider a hose whose diameter decreases along its length, as shown in the Figure below. The ``continuity equation'' is a direct consequence of the rather trivial fact that what goes into the hose must come out. The volume of water flowing through the hose per unit time (i.e. the flow rate at the left must be equal to the flow rate at the right or in fact anywhere along the hose. Moreover, the flow rate at and point in the hose is equal to the area of the hose at that point times the speed with which the fluid is moving:

\fbox{\parbox{4.5in}{\vspace*{7pt}flow rate = (area) x (velocity)\vspace*{7pt}}} 

You can easily verify that (area)x(velocity) has units m3/t which is correct for volume per unit time.

 

Figure 7.1: Fluid flow in a hose of variable size
\begin{figure} \begin{center} \leavevmode \epsfysize=5.5 cm \epsfbox{figs/fluids-1.eps} \end{center} \end{figure}

These considerations lead us directly to the continuity equation, which states that

\fbox{\parbox{4.5in}{\vspace*{7pt} Area x velocity = constant \vspace*{7pt}}} 

everywhere along the hose. This has the important consequence that as the area of the hose decreases, the velocity of the fluid must increase, in order to keep the flow rate constant. Anyone who has pinched one end of a garden hose has experienced this effect: the smaller you pinch the end of the hose, the faster the water comes out.