Re: [Phys-l] question about Bernoulli

[John Mallinckrodt <ajm@csupomona.edu>]

brian whatcott wrote:

> Hmmm..I was hoping you wouldn't talk about reduced density: the word 
> "Incompressible" is often specified in this Bernoulli connection, after all.


Well, if you increase the pressure on *any* real substance under adiabatic conditions, it's density *will* increase.  Indeed, the specification that the process is adiabatic implies that the equation of state is barotropic, so that the density is a monotonically increasing function of pressure *only*.  For gases, the density can increase a lot; for liquids, hardly at all.

Thus, it seems to me that its nice to start with the general Bernoulli eqn for compressible fluids that I presented earlier, i.e.

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

and then note that, as a substance becomes less and less compressible, two things happen--gamma becomes very large and rho becomes ~constant.  Thus, the Bernoulli eqn can be written in its more familiar form:

  (1/2)rho*v^2 + rho*g*h + p = constant

> The key idea is to consider the difference between the total pressure 
> along a stream line, and
> the total pressure across streamlines - where dynamic pressure is not 
> available?

> Dynamics does not allow us the luxury of supposing as we often can in 
> statics, that
> " pressure is the same everywhere."

I'm not sure how to completely address your objection here, but I think the pressure being referred to in Bernoulli's principle is an isotropic pressure.  Strictly speaking it is determined at each point along a streamline with the "constant" in the equations above being different for different streamlines.

John Mallinckrodt
Cal Poly Pomona