Bernoulli's Equation

If you watch a lit cigarette in still air, you will see the smoke carried by an air flow which starts out flowing smoothly, and then becomes turbulent. Turbulent flow is extremely complicated, we will not deal with it here.

The smooth flow (laminar flow) is what Bernoulli analyzed. Two other conditions: no energy loss due to drag forces, and no change of density.

Consider a horizontal flow first, in which the fluid is speeding up due to a drop in pressure in the direction of flow. (Consider a small portion of the fluid. If there is a smaller pressure on the right side than the left, there will be a net force to the right causing it to accelerate to the right.  Net force is -DPA, pressure difference times area, so work done is -DPAx. The minus sign is a convention: if fluid is accelerating to the right, there must be a drop in pressure as you move to the right.) The work done on the gas equals the change of kinetic energy, and if you divide both sides by volume (Ax is volume, and A1x1=A2x2 if the flow area changes), you find that

-D P = D (1/2 r v2), where r is density, mass/volume.

If the fluid is traveling uphill or downhill as well, we need to put in a change of gravitational potential energy mgy, which becomes
rgy when divided by volume. Then we have
-D P = D (1/2 r v2) + D (r gy)

Signs: If it is speeding up and going uphill, both terms on the right are positive. The left side is
- (P2 - P1) = P1 - P2 with the flow from point 1 to point 2.

If we rearrange our equation we find
P + 1/2 rv2 + rgy = constant, and we see from this that at the same elevation, higher velocity means lower P. Or for two points 1 and 2 in the flow,
P1 + 1/2rv12 + rgy1 = P2 + 1/2rv22 + rgy2

Applications:
1. Here is a squeeze-bulb device with a liquid below. When the bulb is squeezed, the fast-moving air above the top of the vertical tube is at a low pressure, causing the liquid to move up in the tube. When it gets into the air stream it forms little droplets and comes out in a spray. Some cologne dispensers use this principle.

 

2. Prairie dogs dig tunnels to live in, and they put one entrance higher than the other.  Faster air at the higher opening means that P is lower there, so air circulates through the tunnel from the low end to the high end. They need the fresh air.
3. Similarly, when there is wind, chimneys produce a draft even with no fire.
4. Vertical axis wind turbines (VAWT) Savonius wind machine, the Darrieus machine, and efficiency of the ideal wind turbine. Here is a Savonius machine, viewed from above, a tilde shape:  ~
5. The lift on an airplane wing is often used as an example, but obviously there is more involved because airplanes can fly upside down.  I think the Bernoulli effect is significant at low speeds.
6.  Curve balls (baseball, tennis, golf…).  Again there is more to it, but Bernoulli is part of it.  It is thought that the more important part is the fact that where the speed of air relative to the ball’s surface is greater, “separation” occurs sooner. No matter what the cause, if air causes a force to the right, air must be pushed to the left.
Ball spinning clockwise:

Air pushed up, so F on ball down:

The Bernoulli equation is for constant r . I did the algebra for the case of a variable r , which you probably won't find in your text, and I found:
P/r + 1/2v2 + gy = constant, which looks like the same thing as above, but it is not. (Just dividing both sides of the equation in the previous paragraph by r makes the left side the same as this one but the right side would be constant/r , and if r is not constant then the right side is not constant. No, you need to go back to the beginning of the derivation and note that mass in = mass out, so rAx is the constant to divide by.) For textbook problems, ignore this paragraph and use the constant r version. Here is a thought: the ideal gas law can be written
PM = r RT, so P/r is RT/M, and thus we cut the number of variables. (M is molecular mass, a constant for a given material, and variables P and r are out and variable T is in. So RT/M + 1/2v2 + gy = constant.) This shows how the temperature of a gas varies with velocity and altitude, but remember that it must be non-turbulent, and it must be in the same flow. Also please note that while the variation in P can give you some noticeable effects, T varies very little in your everyday flow situations.

The theory in the above paragraph could be tested easily with a sensitive (millidegree)
D T thermometer.

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