# Fluid Dynamics and Its Biological and Medical Applications

## Flow Rate and Its Relation to Velocity


### Learning Objectives

By the end of this section, you will be able to:
1. Calculate flow rate.
2. Define units of volume.
3. Describe incompressible fluids.
4. Explain the consequences of the equation of continuity.
Flow rate  is defined to be the volume of fluid passing by some location through an area during a period of time, as seen in . In symbols, this can be written as
where  is the volume and  is the elapsed time.
The SI unit for flow rate is , but a number of other units for  are in common use. For example, the heart of a resting adult pumps blood at a rate of 5.00 liters per minute (L/min). Note that a liter (L) is 1/1000 of a cubic meter or 1000 cubic centimeters ( or ). In this text we shall use whatever metric units are most convenient for a given situation.
Flow rate and velocity are related, but quite different, physical quantities. To make the distinction clear, think about the flow rate of a river. The greater the velocity of the water, the greater the flow rate of the river. But flow rate also depends on the size of the river. A rapid mountain stream carries far less water than the Amazon River in Brazil, for example. The precise relationship between flow rate  and velocity  is
where  is the cross-sectional area and  is the average velocity. This equation seems logical enough. The relationship tells us that flow rate is directly proportional to both the magnitude of the average velocity (hereafter referred to as the speed) and the size of a river, pipe, or other conduit. The larger the conduit, the greater its cross-sectional area.  illustrates how this relationship is obtained. The shaded cylinder has a volume
which flows past the point  in a time . Dividing both sides of this relationship by  gives
We note that  and the average speed is

. Thus the equation becomes .
 shows an incompressible fluid flowing along a pipe of decreasing radius. Because the fluid is incompressible, the same amount of fluid must flow past any point in the tube in a given time to ensure continuity of flow. In this case, because the cross-sectional area of the pipe decreases, the velocity must necessarily increase. This logic can be extended to say that the flow rate must be the same at all points along the pipe. In particular, for points 1 and 2,
This is called the equation of continuity and is valid for any incompressible fluid. The consequences of the equation of continuity can be observed when water flows from a hose into a narrow spray nozzle: it emerges with a large speed—that is the purpose of the nozzle. Conversely, when a river empties into one end of a reservoir, the water slows considerably, perhaps picking up speed again when it leaves the other end of the reservoir. In other words, speed increases when cross-sectional area decreases, and speed decreases when cross-sectional area increases.
Since liquids are essentially incompressible, the equation of continuity is valid for all liquids. However, gases are compressible, and so the equation must be applied with caution to gases if they are subjected to compression or expansion.
The solution to the last part of the example shows that speed is inversely proportional to the square of the radius of the tube, making for large effects when radius varies. We can blow out a candle at quite a distance, for example, by pursing our lips, whereas blowing on a candle with our mouth wide open is quite ineffective.
In many situations, including in the cardiovascular system, branching of the flow occurs. The blood is pumped from the heart into arteries that subdivide into smaller arteries (arterioles) which branch into very fine vessels called capillaries. In this situation, continuity of flow is maintained but it is the sum of the flow rates in each of the branches in any portion along the tube that is maintained. The equation of continuity in a more general form becomes
where  and  are the number of branches in each of the sections along the tube.

### Test Prep for AP Courses


### Section Summary

1. Flow rate  is defined to be the volume  flowing past a point in time

, or

 where
 is volume and

 is time.
2. The SI unit of volume is
.
3. Another common unit is the liter (L), which is .
4. Flow rate and velocity are related by
 where  is the cross-sectional area of the flow and

 is its average velocity.
5. For incompressible fluids, flow rate at various points is constant. That is,

### Conceptual Questions


### Problems & Exercises
