Suppose the pipe has very thick walls, very thin channel and flat ends, that is virtually indistinguishable from a rod. In this case, the pressure in the center of the front end of the pipe is approximately by (ro * V ^ 2 ) / 2 higher than atmospheric pressure , while at the rear end by approximately the same amount below it (ro is the air density, V is the pipe speed ) . This pressure difference causes the flow through the channel , the rate of which depends on the channel diameter , pipe length , wall roughness . In general, the pressure drop along the pipe channel and the flow rate through it depends on the pipe speed , the channel diameter , outer diameter ,pipe length , the roughness of the inner and outer walls and also on the shape of the front and rear ends of the pipe.

Unless you're moving the pipe at an angle, I would think the pressure is atmospheric all the way. If the pipe, however, is long enough that the boundary layer over the pipe's wall grow to the point that it reduces the effective area of the pipe then you would get a distribution similar to that of a nozzle.

Static pressure will be same but due to motion dynamic pressure change will be there depending upon the type of motion and angle of attack and vorticty as well if it is rotating.

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Dear Prof. Andrei-Mugur Georgescu

straight pipe is in horizontal position and moved along its axis at a zero angle with constant velocity. pipe velocity is far from turbulent regime such that air flow is completely laminar. also pipe thickness can be ignored (very thin pipe). 

Dear Prof. Alexander Shnip

the pipe moves through still air (air has zero velocity). how still air can have dynamic pressure? if the pipe is short length then boundary layers from wall will not reach to each other and hence velocity of the pipe center line(axis) remain zero through the pipe length. particle of zero velocity have zero dynamic pressure. it seems we have pressure distribution through the pipe but how it is possible for the pressure distribution to change such that satisfies zero gauge pressure at the both side? 

Dear Prof. Khaled Al-Salem

as you know if pressure atmospheric dominates all the way then due to the fact of inertia frame (frame on a constant velocity object) it is possible to reverse the motion i.e. hydrodynamic characteristics are the same if straight horizontal pipe remains stationary (V=0) and air comes through the pipe with velocity V. in this situation pressure will decreases from beginning to the end. in fact this is the pressure gradient that drive fluid through the pipe. this point makes this problem complicated.

This problem is similar to problem when pipe is stationary and air is flowing with V velocity [Relativity].Now consider the total pressure at inlet P1{static pressure + dynamic pressure},so in actual condition by including friction loss in flow outlet pressure will be P2=P1-pressure loss due to friction.This pressure difference is responsible for flow.

Now take ideal condition in which no losses take place,so P1=P2.The complication is here if there is no pressure difference then how flow will take place? The Answer to this problem lies in Newton's first low[body will retain its state till external force is applied],in this case because we are considering ideal condition means absence of any external force,fluid which is entering with V velocity will continue its motion same way without losing and pressure head.so actually in ideal case no pressure difference is required for flow to be continue only initiation is require to start the flow and then it will continue flowing for constant cross-section pipe.  

In actual condition for continuous flow pressure difference is only require to over come friction or other losses.