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rulmismo

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## 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!

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