Chapter Review
15-7 - 15-8 Bernoulli's Equation
In general the same concepts that we use to describe the dynamics of particles also apply to fluid dynamics. With fluids it is often more convenient to express these concepts in terms of density and pressure rather than mass and force. One such example comes from the application of the work-energy theorem to fluids resulting in a mathematical relation known as Bernoulli's equation.
With a fluid we can replace the concept of a particle with a small region of the fluid called a
fluid element of density
moving at speed v while sweeping out a volume
DV under the action of a differential fluid pressure
DP. Bernoulli's equation can be obtained by applying the work-energy theorem (W
net =
DK) to this fluid element. The work done on a fluid element due to a change in the fluid pressure is
DW
pressure = (P
1 - P
2)(
DV). The work done by gravity as the fluid element changes vertical level from a height y
1 to y
2 is
. The change in the kinetic energy of the fluid element can be written as
. These three quantities combine to give Bernoulli's equation
.
This expression holds in the absence of frictional losses.
Another way of stating Bernoulli's equation is
.
This form of the equation helps make the physical consequences a little more clear because you can see, for example, that for a fluid flowing at a constant vertical level an increase in the speed of flow must be accompanied by a decrease in pressure (and vice versa). This effect, often called Bernoulli's principle, is important in understanding the consequence of air flow in many applications.
You should also notice that Bernoulli's equation is consistent with the dependence of pressure on depth that was discussed above. Examining this dependence leads to Torricelli's law for the speed of a fluid flowing from an aperture in a container placed a depth h below the surface of the fluid. If, for example, both the surface of the fluid and the aperture are open to the air then the pressure at both locations is atmospheric pressure so that P1 = P2 in Bernoulli's equation. Assuming that the fluid is essentially static at the surface (v1 = 0) leads to
,
where h = y1 - y2.
Physlet Illustration: Blood Flow in Arteries
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| Blood flows in an artery with a partial blockage from left to right in the animation (position is in millimeters and times is in seconds). A blood platelet is shown moving through the artery. How does the size of the constriction (variable from 2 to 10 mm from each wall) affect the speed of the blood flow? Positions measured in mm, time in seconds, and pressure in mm of Hg. Start |
Hints:
- Measure the speed of the platelet (in mm/s) before or after the constriction by pausing and clicking on the platelet to determine its position at two different times.
- Measure the speed of the platelet while it is in the constricted area using the same technique. Does the speed increase or decrease?
- Adjust the size of the constriction to see if the speed responds appropriately.
- How does the constriction affect the blood pressure? Move the detector to different positions and note the pressure.
- Does the blood pressure obey Bernoulli's equation? Note: To verify this quantitatively, you will need to convert the pressure to Pascals.
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Example 15.7 Water Flow at Constant Pressure: Water flows in a horizontal segment of pipe at a pressure of 85 kPa with a speed of 2.6 m/s. The pipe widens such that its area becomes larger by 35%. If the flow is to be at constant pressure, how far above the initial horizontal level should the pipe divert the water?
Picture the Problem The picture shows a pipe carrying water first horizontally, then uphill, and horizontally again.
Strategy Bernoulli's equation relates the pressure to the speed of flow and the change in vertical level. However, the changing width of the pipe can be handled by the equation of continuity, so both expressions should be used.
Solution
| 1. Using the fact that the pressure is constant, P1 = P2, Bernoulli's equation becomes: |  |
| 2. Solving for the difference in horizontal level: |  |
| 3. Use the continuity equation to solve for v2 in terms of v1: |  |
| 4. Subsitute v2 into the expression for the change in vertical level: |  |
| 5. Obtain the numerical result: |  |
Insight As this example illustrates, you should keep in mind that it is often useful to use both Bernoulli's equation and the equation of continuity to perform a complete analysis.
15-9* Viscosity and Surface Tension
(A) Viscosity
When a particle moves it usually experiences some sort of frictional resistance to its motion. The same is true for fluid flow, with fluids this resistance is called viscosity. All of the previous discussion assumed an ideal fluid which, in part, means that we assumed that there was no viscosity. In this section we will take a glimpse at what the effect of including this unavoidable phenomenon is.
For a fluid flowing through a tube of length L and cross sectional area A, the flow results from a pressure differential across the length of the tube, P
1 - P
2. What are some of the physical quantities related to this pressure difference? It works out that the pressure difference is directly proportional to the speed, v, at which the fluid flows (it requires a greater pressure difference to get faster flow). We also find that P
1 - P
2 is directly proportional to the length of the tube (it requires a greater pressure difference to sustain the flow of the larger amount of fluid contained in a longer tube). Finally, the pressure difference is inversely proportional to the cross sectional are of the tube (a wider tube provides more room for the fluid to flow more freely requiring less pressure). Thus, we have that (P
1 - P
2)
vL/A. The constant of proportionality turns out to be 8
ph, where
h us called the
coefficient of viscosity. Therefore, we have the following relation
.
We can see from the above equation that larger values of
h means the fluid is more viscous because more pressure is needed for the fluid to flow at a certain speed. The SI units of
h is
. However, a commonly used unit of
h is the
poise which is defined as
.
As mentioned previously, fluid flow is often characterized by the volume flow rate DV/Dt = Av. When the above expression for viscous flow is re-written in terms of the volume flow rate for a tube of circular cross-section (A = pr2), we get Poiseuille's equation
,
which shows that the volume flow rate increases as the fourth power of the radius.
Surface Tension
The surfaces of fluids, especially liquids, are observed to behave in a way similar to that of an elastic membrane when objects are placed on the surface. This effect is due to the surface tensionof the fluid. Surface tension results from internal forces between the molecules of a fluid that collectively resist the deformation of the fluid's surface from its equilibrium configuration. This effect is analogous to what happens in a spring and is responsible for the ability of insects to walk on water.
Exercise 15.8 Viscosity: Using the result of Example 15.3 above, estimate the volume flow rate of blood from the head to the feet of the six-foot-tall person. Assume an effective radius of 23 cm.)
Solution: We know the following information:
Given: P
1 - P
2 = 18.8 kPa, L = 1.83 m, r = 0.23 m,
h = 0.0027
Find:
DV/
Dt
Using the above given information, together with the viscosity of blood (from Table 15-2 in the text), we can directly use Poiseuille's equation
Notice that three significant figures were used for the pressure difference. That is done because the final result of Example 15.3 is now just an intermediate result for this example.
.
,
.
.
