drift velocity of gas needed to get into a pressurised vessel Would a gas entering a narrow tubular  opening of an initially empty vessel at a given velocity, fill up the vessel to a definite pressure?
If so, how is the velocity of the gas related to the final pressure of the gas in the vessel?
If now, the gas is let out of the opening, what would be the initial velocity of the gas stream?
 A: Preliminary:
Pressure is the driving (or source) term in Naiver-Stokes equation that governs fluid flows in continuum region,  if we know pressure at inlet and outlet we can find the velocity for simple low speed flows. 
This process is isentopic if there is no heat flow. That problem can be solved by Bernoulli equation if the flow is irrotational, and the  equation is (this can be used for this flow). Here i'm neglecting surface tension because I'm not sure about the length to diameter of the tube you want to use.
$$ p_1 +\rho \frac{v_1^2}{2} =p_2 +\rho \frac{v_2^2}{2}$$
If there is any diverting in that passage you can consider continuity equation
$$ A_1v_1=A_2v_2$$
Here
$p_1$, $v_1$ are inlet velocity and pressure respectively. For big vessel or chamber  $v_1$ =0 (approx.)
$A_1$ and $A_2$ are cross sectional area normal to the flow at inlet and exist respectively.
I'm not aware of any analytical method to solve transient equation. For that only way is solving NS equation using numerical methods or some Computational Fluid Dynamics packages.
