Does Bernoulli's Equation Apply to Ideal Gases in a Piston-Driven System?

[mathperson]

consider a hypodermic needle like structure with a piston in the larger tube and a piston in the smaller tube as well.
let the cavity be filled with an ideal incompressible fluid.
if i understand the Bernoulli theorem correctly, if a pressure is applied to the larger piston, causing the fluid to move with respect to the structure, then the pressure on the smaller piston will be less than that pressure, in accordance with Bernoulli's equation. (i'm ignoring gravity in this. ) notice that the cavity shape is changed, but the volume is unchanged.

if i replace the liquid with an ideal gas, and keep the volume unchanged as the larger piston is pushed, does the same equation apply?
if so, is there a reference where Bernoulli's theorem is proved using the kinetic theory of gasses?

 

[mathperson]

it's been a while since i posed the question, but no response. is the question too easy or too hard?
in my physics textbook, ideal gasses are treated separately from ideal liquids.
yet, when an airplane wing is discussed, the air is treated as a liquid.

 

[wongdaisiu]

I am not sure if you can prove Bernoulli's Equation via the kinetic theory of gases. Bernoulli's Equation takes a "macroscopic" view, whereas kinetic theory of gases takes a more "microscopic" view and uses statistics to extrapolate the macroscopic view.

With that being said, Bernoulli's equation is applicable for incompressible fluids, which includes gases at low velocities. However, another assumption that must be valid in order to use Bernoulli's equation is that that that particle travels in a streamline (aka it must be moving). The scenario that you describe is a quasi-static situation where fluid statics is appropriate rather than fluid dynamics.

I hope this helps.

 

[Aero51]

You can use Bernoulli's equation to approximate the pressure in each piston. Bernoulli's equation is a simplified conservation of energy formula; it should work for both the fluid and the gas provided your fluid isn't something crazy. The ideal gas law may be easier to apply though.
For a fluid like water I do not think the ideal gas law is applicable because it assumes that the gas molecules are non-interacting hard spheres. This is not the case for a fluid like water which is polar, has surface tension and a considerably higher viscosity. An approximation may exist but I have not heard of it.

 

[mathperson]

thanks for responding.
i'm still (pardon me) up in the air on this.
if i understand what quasi-static means, then the speed of the pistons doesn't matter for a gas: the pressure is the same throughout.
on the other hand, the gas is treated as a liquid, and hence the Bernoulli equation applies.
in short, why can the gas be considered a liquid?

 

[Aero51]

Personally, I would interpret "Quasi-Static" as the piston is moving very slowly/not at all. This would translate to using hydrostatics to solve the problem. As far as the second half of your question is concerned, the gas is not being considered a liquid. The basic Bernoulli equation is valid for inviscid, incompressible, irrotational flow. This can be a gas or a liquid at the expense of the model's ability to predict real world flows.

 

[mathperson]

perhaps i should have asked why a gas can be seen as having a "flow".
does that mean the substance has streamlines?
am i the only one confused by this?

 

[Aero51]

If a gas is "flowing" in the sense that it is moving with a velocity then yes it does have streamlines. However, this is only true for the conditions assumed by potential flow, which can be modeled by Bernoulli's equation. Also, I forgot to mention that Bernoulli equation is valid for compressible flow too.

 

[mathperson]

let's think of a simpler situation:a (horizontal) tube with 2 pistons,the cylindrical volume filled with an ideal liquid. keep the pistons stationary with horizontal pressure as necessary.
case#1:tube stationary.
case#2:tube accelerating (to the left, say).
case#3:tube moving at a constant velocity.
repeat with an ideal gas in place of the liquid.
there seem to be no streamlines in any of these situations.
how have the pressures on the pistons changed, if they have?

 

[Aero51]

Perhaps it would be better if you had a visual diagram to show me. I understand the concept of a horizontal tube with 1 piston but what do you mean with 2 pistons?

 

[Sagar_C]

Flow equations for composite gases
Author: J M Burgers
Publisher: New York, Academic Press, 1969.
Series: Applied mathematics and mechanics, v. 11.

Have a look at first few pages of this book where you will know how Euler's fluid equation are derived from kinetic theory of gases. You can see what approximation goes in in deriving that. Once you have Euler's equation, you know how to get Bernoulli equation.

 

[mathperson]

Aero51,
i think of a piston as a disk with a rod attached, as in a hypodermic needle or a bicycle pump. in the (constant diameter) tube, the left hand piston has its rod pointing to the left, then there is a space for the liquid or gas, and then there is the right hand piston with its rod pointing to the right. thus a cylinder of liquid or gas is created.
(i'm not that proficient with computers: when i tried on another occasion to send a diagram, it got botched. sorry.)
Sagar C,
thanks for the reference. I've ordered it through the library link system.

 

[Aero51]

Ok. If that is the case then the answer depends on whether or not you have a fixed volume of fluid. For instance, the piston could be compressed but not eject fluid out of the syringe. If this is the case then the pressure will increase inside the container. If fluid is being ejected, then you will have streamlines and the pressure will be about equivalent to the pressure of the surroundings provided that the piston is not accelerated too greatly. Hope this helps.

 

[mathperson]

alas, i still haven't made myself clear.
in the hypodermic syringe structure, the piston in the smaller tube is moving faster than the piston in the larger tube. they are moving in the same direction at speeds that keep the VOLUME CONSTANT.

in an ideal liquid there is a smooth velocity vector field.
on the other hand, in an ideal gas, if two particles are close to each other, the usually have completely different velocity vectors.
hence, it is not obvious that a gas can be a fluid.
indeed, if the two pistons are moving at different speeds, yet the pressures on the pistons are equal, this contradicts bernoulli.
from the Burgers' book cited above: "A gas without collisions...should not be considered...an 'ideal fluid'."
of course, I'm considering the gas as ideal particles making elastic collisions with the boundary structure.
in my physics book, the gas section does not discuss changing the shape, but not the volume, of a container.

thanks for your patience.

 

[Somenath]

Bernoulli's Theorem tells that pressure on the curved surface of the thin tube will be less than the pressure on the curved surface of the thick tube. It does not tell anything about the pressure exerted on the piston.
It explains that a person on railway platform standing too close to a speeding train can be pulled towards the train. But it canot tell about the pressure on the front face of the train due to air.

 

[mathperson]

Somenath,
i've looked a a few textbooks on fluid dynamics. they give the same argument:
a fixed volume goes faster in a narrower tube region than in a wider.
the acceleration required comes from a force along the axis of the pipe.
this is why, in my example of the constant diameter tube with 2 pistons, i think the pressure on the curved surface is the same as on the pistons, whether the tube is moving or not. at least for a liquid. i don't know about a gas.

but the main problem i have is: why is a gas considered a fluid even though it's defined quite differently from a liquid?

 

[boneh3ad]

Liquid and fluid are not equivalent terms. A fluid is a material that continuously deforms under an applied shear stress. Gases, liquids, plasmadynamics and some plastic solids fit this description.

The Bernoulli equation isn't really valid in the situation you describe for most fluids as it would not be valid to treat the fluid as inviscid. However, assuming it was, it still makes sense. Bernoulli's equation is effective energy conservation in terms of pressures. It states, in essence, that the total pressure is constant along a streamline or in an invscid fluid. By applying a force to the plunger you are increasing the total pressure, so that constant is changing.