23

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?

Qmechanic
  • 220,844
PatEugene
  • 391

2 Answers2

17

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.

Vijay Murthy
  • 2,376
0

The Bernoulli equation is derived from the Navier-Stokes equations under several strong simplifying assumptions: steady, irrotational, incompressible flow. It relates the variation in certain quantities (velocity, pressure, elevation) along any streamline within the flow. The Navier-Stokes equations are Galilean-invariant, meaning that they are valid in any inertial frame of reference. As a result, the Bernoulli equation must also be Galilean-invariant. However, this does not mean that the Bernoulli equation is valid in all reference frames -- some reference frames violate the underlying assumptions (in particular, that the flow must be steady).

For example, Mungan's paper (cited in previous answers) considers laminar, irrotational, incompressible flow through a pipe with a contraction. Bernoulli's equation is valid in the frame of the pipe, where the flow is steady. Bernoulli's equation is not valid in any other frame, because there is no other frame in which the flow is steady (a shift in any direction will introduce moving boundaries). One can modify/extend Bernoulli's equation to account for these effects by considering the work done on the flow by the moving boundaries, as Mungan suggests, but that is tricky, ad hoc, and not what most people mean by "Bernoulli's equation".

In the moving-car example, the Bernoulli equation cannot be used for several reasons:

  1. In both frames, the flow is turbulent (thus unsteady).

  2. In the frame of the road, the problem is unsteady.

  3. There is no steady streamline that passes from outside the car to inside the car.

As a result, Bernoulli's equation is not valid in either frame because the flow is unsteady (from 1). And, even if the flow were laminar and hence steady in the frame of the car, Bernoulli's equation would not be valid in the frame of the road (from 2). And, even for steady laminar flow in the frame of the car, it is not straightforward to use Bernoulli's equation to relate the conditions outside the car to the conditions inside the car because there is no such streamline (from 2).

Chris MacMinn
  • 1