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So I was thinking about this while driving home the other day.

I've never been quite clear on why when you drive with the windows down air rushes into your car. I thought this might be explained by Bernoulli's equation for incompressible flow, but I ran into what seems to be a contradiction. If we consider the problem from the reference of the car, the air in the car is stationary and the air outside the car has a certain velocity. Then, Bernoulli's equation implies the pressure outside the car is lower than that inside the car. However, if we take the reference frame of the road, the air in the car is moving and then the pressure in side the car is lower. Intuitively, this second situation seems to be correct since air apparently flows into the car (from high pressure to low pressure). However there seems to be a contradiction, as the pressure gradient depends on reference frame. So my question is what has gone wrong here? Is this a situation in which bernoulli's principle simply isn't applicable? Did I make some sort of mistake in my application of the principle?

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  • $\begingroup$ If it were a flow from one pressure regime to another, it would only last a very short time, since the car's volume is enclosed. After that, you get flow out equals flow in. So if you feel air blowing into the car, you can be sure it is also blowing out, but since your hand is inside the car, you feel the inflow. You don't feel the outflow. (Give poor Bernoulli a rest :) $\endgroup$ – Mike Dunlavey Nov 13 '12 at 0:34
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    $\begingroup$ Nevertheless, the conceptual question regarding reference frames and Bernoulli's principle is a good one, even if it doesn't apply so much to the car window question $\endgroup$ – kleingordon Nov 13 '12 at 1:04
  • $\begingroup$ @Mike Dunlavey; Ok the part about the air blowing in being the same as the air blowing makes sense, otherwise my car would implode/explode. But maybe this is a better question: If I'm driving with the windows closed and can instantly open them (maybe by smashing them or something), which way will the air flow initially? $\endgroup$ – PatEugene Nov 13 '12 at 6:30
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    $\begingroup$ The Bernoulli equation is frame-dependent. PDF here. The pressure drop depends on the frame of reference. $\endgroup$ – Vijay Murthy Nov 13 '12 at 9:49
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    $\begingroup$ @VijayMurthy; yes that seems to answer my question. thank you. $\endgroup$ – PatEugene Nov 13 '12 at 10:58
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Bernoulli's equation is frame-dependent as the following paper shows it in a nice way

The Bernoulli equation in a moving reference frame

The essence of the argument is to realize that in a frame where the obstacles, around which the fluid moves, are not stationary, these surfaces do non-zero work. And one must account for this work done when using the Bernoulli equation.

A better way is to look at the generalized Bernoulli equation as done here, which also covers viscous fluids.

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    $\begingroup$ Bernoulli equation is related to energy conservation. One must be careful when using the law of conservation of energy in a moving frame. Constraint forces can do non-zero work, and this must be accounted for. The correct way is to directly look at equations motion or to use the work-energy theorem, which is frame-independent. A nice situation that exemplifies this idea is here $\endgroup$ – Vijay Murthy Jan 12 '13 at 13:12
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    $\begingroup$ Hi, this second link and third link have rotted. $\endgroup$ – Mark Mar 29 '18 at 16:03

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