Re: [Phys-l] question about Bernoulli

[John Mallinckrodt <ajm@csupomona.edu>]

This probably doesn't completely address Bill Robertson's initial request for a molecular understanding of why higher velocities are associated with lower pressures, but I think that considering the case of a compressible fluid undergoing an adiabatic process as I do here may at least provide some new insight.


Bernoulli's equation, which is usually derived for the steady adiabatic laminar flow of incompressible fluids, can easily be extended to the case of compressible fluids.  The result is 

   v^2/2 + gh + [gamma/(gamma-1)](p/rho) = constant

where gamma is the ratio of specific heats.  (See http://en.wikipedia.org/wiki/Bernoulli's_principle). 

For an ideal gas this becomes

   v^2/2 + gh + [gamma/(gamma-1)](kT/m) = constant

where m is the molecular mass.  Thus, ignoring the gravitational term, we see that the temperature of the gas should be reduced within constrictions where the speed is high.  

Further, since p is proportional to rho^gamma (because it's an adiabatic process), it's easy to show that *both* the pressure and the density are reduced within the constriction.  (The pressure is reduced by a somewhat larger percentage than the density.)

As an example, I find that, if slow moving air at standard conditions is accelerated to 100 m/s in a constriction, we should see a density reduction of about 4%, a pressure reduction of about 6%, and a temperature reduction of about 2%.

I think that understanding all that helps pave the way for a better understanding of what goes on in the Bernoulli flow of an "incompressible" fluid.  Again, the reduction in pressure is accompanied by an (all but imperceptible, in this case) reduction of the density and temperature.  (Think about the relationship between pressure and density for water, for instance.)

John Mallinckrodt
Cal Poly Pomona

P.S.  I still commend my earlier question to your attention if you haven't yet thought about it.  I suspect many  people might have a hard time finding the flaw in the reasoning.  It was:
> The Bernoulli equation (for incompressible fluids) says that the sum 
> of the kinetic energy density, the gravitational potential energy density, 
> and the pressure is constant along a streamline so that if the speed 
> DECREASES from point A to point B along a horizontal streamline, 
> the pressure must be HIGHER at point B than at point A.
>
> But in the rest frame of the fluid at point A, the speed is zero at point A 
> and, thus, necessarily is higher at point B so that, in THAT frame, the 
> pressure must be LOWER at point B than at point A.
>
> What's up with that?