Let's say I have a free jet of air leaving a pipe into the atmosphere. I know that the static gauge pressure at the pipe exit is equal to the atmospheric. But what about the static gauge pressure 10 meters away if the air is still traveling as a free jet? Is it still atmospheric?
Fluid Dynamics – Is Static Gauge Pressure of a Free Jet Always Atmospheric?
classical-mechanicsfluid dynamics
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May it now be that by definition a free stream has no accelerations? And as derived in the linked page, no acceleration implies no pressure gradients. That with the fact that all the streams considered are connected with the atmosphere at some points, should mean all the stream is at atmospheric pressure.
What is the reason behind suction?
The main reason behind suction in an eductor is due to a special case of Bernoulli principle called the Venturi Effect.
When fluid flows through a pipe, it has a specific rate of flow which can be identified by the equation of continuity
$$P_1v_1A_1 = P_2v_2A_2$$
Where $\boldsymbol {v_1}$ and $\boldsymbol{v_2}$ are the velocities at position $(1)$ and $(2)$ respectively. Similarly $P$ and $A$ are the pressure and area.
At position $(1)$, the area is $A_1$ and velocity is $\boldsymbol {v_1}$. As the fluid reaches position $(2)$, The area of cross-section decreases to $A_2$, which means the velocity increases to $\boldsymbol{v_2}$
In fluid dynamics, a fluid's velocity must increase as it passes through a constriction in accord with the principle of mass continuity, while its static pressure must decrease in accord with the principle of conservation of mechanical energy. - wikipedia
Now by the equation of continuity, At position $(2)$, the pressure should decrease i.e $P_2 < P_1$. Now pressure outside the suction chamber is equal to atmospheric pressure ($P_{atm}$) which is greater than $P_2$. This low pressure area inside the suction chamber forces fluid in the container to enter the chamber. This is how suction in an eductor works.
Search for Venturi Eductor to know more.
P.S: It seems that your main question is "How increase in rate of flow(higher velocity) results in decrease in pressure of that region?"
This can be explained in a number of ways. One way is to use the conservation of mechanical energy.
$$P+hρg+\frac{1}{2}ρv^2$$
This is constant through out the motion of the fluid. Hence increase in velocity results in reduce of pressure.
I suggest you to read this question from Quora : Why does pressure in a nozzle decrease as the fluid velocity increases?
Watch these videos which clearly answers how increase in velocity results in decrease in pressure:
Best Answer
I'm confused. Bernoulli's eqn. says the static pressure inside the jet should be less than atmospheric. As you go further out and the jet slows down, then it should approach atmospheric pressure. The pressure gradient between the atmosphere outside and the low pressure inside the jet leads to air getting sucked into the jet (entrainment).