Pressure distrubution inside accelerating vehicle

[rulmismo]

Homework Statement

We have an accelerating vehicle with constant a. Air inside it with mass m.
It is kind of closed box of length L and front area S.
First objetive: we want to derive the steady state pressure distribution inside the box.
Second objetive: we want to derive the transient pressure distribution inside the box.

Homework Equations

My first guess was to assimilate the problem to an atmosfere with gravitatory field of g=a and use the barometric formula for calculating the pressure distribution.
Second try is to use simplified Navier eq. but I don´t know how exactly...

The Attempt at a Solution

First try with barometric formula:
Force over the mass of air inside: Fa = m a
Reaction pressure over the rear side: Pa=Fa/S
Assuming barometric distribution inside with g=a (M molar mass of air)
P(x)=C exp( -M a x / (RT) )
with initial condition at rear side P(0)=C=Pa
This gives a very weak variation with length (i.e for 60m it just changes 0,1Pa) so I don´t trust this solution.

Second try:
Assuming 1D problem, incompressible fluid and no viscosity:
dV/dt = f - (1/ro) dP/dx
If steady state: f = (1/ro) dP/dx

update: checking dimensinally it seems that f is an acceleration so it should be a, and it is in opposite sense of x so:
dP = -ro a dx

integrating P(x)=-ro a x + C,

Now I know that in x=0, I have Pa= m a / S, so P(x)= - ro a x + m a /S

With 1m/s2 this gives a linear gradient of pressure about 1.2Pa/m length of the vehicle.
¿it makes sense this result?
I would appreciate any comment or help!

 

[mark726]

Thank you for your interesting post. The problem you have described is indeed a challenging one, and I would like to offer some suggestions for your solution attempts.

Firstly, for the steady state pressure distribution inside the box, it may be helpful to consider the ideal gas law, which relates pressure, volume, and temperature of a gas. In this case, the volume of the box is fixed, but the temperature may change due to the acceleration of the vehicle. You can use this relationship to derive the pressure distribution.

For the transient pressure distribution, it may be useful to consider the equation of motion for a fluid element in an accelerating flow. This equation takes into account both the acceleration and the change in pressure with respect to distance. By solving this equation, you can obtain the transient pressure distribution inside the box.

I would also recommend considering the boundary conditions at the front and rear of the box, as these will affect the pressure distribution inside. Additionally, it may be helpful to consider the effects of viscosity and compressibility on the pressure distribution.

I hope these suggestions help you in your solution attempts. If you have any further questions or need clarification, please feel free to ask. Good luck with your calculations!