# Homework Help: Pressure distrubution inside accelerating vehicle

1. Mar 24, 2012

### rulmismo

1. The problem statement, all variables and given/known data
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.

2. Relevant 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...

3. 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 lenght (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!

Last edited: Mar 24, 2012