Problem Set #6 - Solutions

Goal of Problem Set #6: This assignment is meant to help you understand:
  1. how inward and outward bends affect a fluid's pressure, speed, kinetic energy, and potential energy,
  2. how nozzles affect a fluid's pressure, speed, kinetic energy, and potential energy,
  3. laminar and turbulent flows in fluids,
  4. viscous and pressure drag forces in fluid flow,
  5. lift forces in fluid flow

You are on a road trip with three of your closest friends. You are sitting in the front passenger seat of a four-door sedan as it heads straight down the open road at the speed limit. All of the car's windows and its sunroof are closed. Suddenly, a truck in front of you drops a big sack of cotton puffs. The bag bursts and the puffs sail up into the air in front of your car. For a few seconds, you get to see the precise airstream around the car, highlighted by the moving cotton puffs. What you see it pretty simple: the airstream hits the front of the car and spreads out in all directions in order to flow around the vehicle. Some of this air flows around your side of the car and you pay particular attention to that portion of the airstream. As this air flows off the windshield, it bends inward toward your side window in order to follow your side of the car. The air continues to bend inward until it passes your window but manages to straighten out by the time it reaches the rear side window. The air flows straight past that window, parallel to its surface. The air then bends inward again toward the rear of the car, but doesn't get very far before it breaks up into a huge swirling air pocket behind the car.
1. The cotton is long gone and you decide to open your side window to let in some air. The moment you open the window a little, the pressure inside the car drops below normal atmospheric pressure and your ears pop. Evidently the pressure just outside your side window is lower than atmospheric pressure. Explain briefly why that should be the case.

Answer: The airstream is bending inward, toward the car's front side window. That inward bend requires that the pressure just outside the window be lower than the pressure far from the window (i.e. atmospheric pressure).

Why: Airstreams only bend because of pressure differences on their two sides (neglecting the minor influence of gravity). For the airstream to bend toward the front side window, the pressure must be lower on the window side of the airstream than on the distant non-window side. Since the pressure far from the car is atmospheric, the pressure at the side window itself must be below atmospheric.

2. You roll your window up and the air inside the car soon returns to atmospheric pressure. Now your friend in the rear seat opens the rear side window, but there is no sudden pressure drop. Why does opening the rear side window have so much less effect on the air pressure in the car than opening the front side window?

Answer: The airstream has straightened out, so it is no longer bending and has similar pressures on both sides. The air outside the rear side window is therefore approximately atmospheric pressure.

Why: Once the airstream stops bending, its pressure becomes uniform from side-to-side. The relatively straight airstream near the rear side window has about the same pressure on the window side as on the distant non-window side. Since the pressure far from the car is atmospheric, so is the pressure near the window itself.

3. The day is getting warmer and so you all open your side windows. Things are going great until you decided to toss your chewing gum out your window. You feel a good breeze drawing air out your window, so you flip your gum into that breeze and the gum goes out the window with it. Unfortunately, the gum then blows into the rear side window and startles your friend. How can air be flowing (or even accelerating) out of your side window and then flowing (or even accelerating) back into the rear side window?

Answer: Since air accelerates toward lower pressure, air inside the car accelerates toward the lower pressure outside the front side window. As air flows out the front side window, the pressure inside the car decreases and atmospheric pressure air outside the rear side window begins to accelerate into the car. A circulation develops.

Why: The pressures outside the two side windows are different and air flows from the higher pressure outside the rear side window, through the car, toward the lower pressure outside the front side window.

4. You find this funny pressure effect outside your window interesting, so you stick your hand out into the passing airstream. You expect the air to have a very low pressure. But with your palm turned to face the onrushing air, you feel a tremendous pressure on your palm, pushing your hand toward the rear of the car. Why does the air in your palm have such a high pressure?

Answer: The airstream is bending outward, away from your palm, so the pressure at your palm must be higher than far away from it. Since the pressure far from your hand is atmospheric pressure, the pressure on your hand is above atmospheric pressure.

Why: Overall, your hand is behaving like a funny shaped ball placed in a rushing airstream. Air bends outward to spread out around it, then inward to arc around its sides, and then tries to rejoin and bend outward to leave the back of your hand. In reality, it fails to rejoin behind your hand and you end up with a large air pocket behind your hand. The end result is pressure drag -- your hand is pushed backward hard by the airstream.

5. You turn the upper edge of your hand forward toward the onrushing air and now you feel your entire hand pushed upward, as well as backward. If something is pushing your hand upward, your hand must be pushing that something downward. What is being pushed downward and how does that downward push affect its motion?

Answer: The air is being pushed downward and it accelerations downward (the airstream bends downward).

Why: The only thing touching your hand is the air and since something is pushing your hand up, it must be that air. The air behaves like anything else experiencing a downward push: it accelerates downward. Since it is passing by horizontally upon arrival, it leaves having been bent downward.

6. The windows are all closed again and you decide to open the car's sunroof. Once again, the air pressure inside the car suddenly drops below atmospheric pressure when you begin to open the sunroof. Evidently the air pressure on top of the car is less than atmospheric. (A) How does that reduced pressure affect the support force that the roadway exerts on the car? (B) How does that change in support force affect the car's traction – the maximum force of static friction that the wheels can exert on the roadway?

Answer: (A) The support force is reduced. (B) The traction is reduced.

Why: The low pressure above the car leads to an unbalanced pressure around the car and an overall upward force due to that pressure imbalance. The air is partially supporting the car. With the air helping it, the pavement doesn't need to provide as large a support force on the car to support the car. The air is bearing some of the car's weight. Once the wheels and pavement aren't pushing against one another as hard, the frictional forces between them become weaker; the car loses traction.

7. To compensate for this traction problem, the car has a spoiler above the trunk and rear wheels. This wing-like surface deflects the passing airstream upward. Why does that action improve the traction for the rear wheels of the car?

Answer: The spoiler pushes the air upward, so the air pushes the spoiler downward. To balance this additional downward force on the car, the pavement must push upward harder on the car's rear wheels. The traction increases for those wheels.

Why: The spoiler makes the air push downward on the rear of the car and presses the car onto the pavement. Since the traction increases when the forces between tires and pavement increase, this airstream deflection leads to better traction.

8. The huge air pocket that forms behind the car is not good news for the car's fuel efficiency. Why not?

Answer: The air pocket is associated with pressure drag and the car must consume fuel to do work against that pressure drag force.

Why: The car does work against pressure drag. The amount of that work is equal to the drag force times the distance the car travels against it. The larger the air pocket gets, the stronger the drag force becomes (assuming a constant speed) and the worse the work load it imposes on the car.