Gas pressure along the axis in effusion

I’m trying to approximate the pressure P along the axis x, i.e. P(x), for the following setup. At one end of a cylinder a gas is inserted (at some pressure, let’s call it P0). At the other end of the cylinder is a small hole, through which the gas exits (effusion) into vacuum. I assume that hole is small enough such that not all the gas exits steadily but some of it accumulates near the hole. Closer to the hole, higher the pressure of the gas.

I think the problem is very similar to the atmospheric air subject to gravity (closer the air to the ground – larger the pressure) and found the P(x) dependence for this case. However, I have troubles to link between the gravity in barometric case to pressure gradient in my case. If I use the presure-gradient force I get back to the barometric formula since it’s how the latter is derived.

Any help is appreciated. Thank you.