Are eductors working on the Bernoulli principle? 
This is a picture of an eductor. The wikipedia article claims that it works on the Bernoulli principle.
The Bernoulli principle is famous due to the misconception about how aeroplanes fly and because it is convenient for highschools. Children are introduced to the Bernoulli principle because  doesn't need calculus to "understand" it.
Pressure difference is the obvious reason behind acceleration and that's all. Acceleration cannot diminish the pressure around a flow
My question is; What is the reason behind suction?
Edit:
From  the book "Fluid mechanics" of Frank.M.White

When a fluid flow leaves a confined internal duct and exits into an ambient
  “atmosphere,” its free surface is exposed to that atmosphere. Therefore the jet
  itself will essentially be at atmospheric pressure also. 
Only two effects could maintain a pressure difference between the atmosphere and a free exit jet. The first is surface tension which is usually negligible.
The second effect is a supersonic jet, which can separate itself from an atmosphere with expansion or compression waves . For the majority of applications, therefore, we shall set the pressure in an exit jet as atmospheric.

 A: 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:

*

*Demonstration of Venturi Effect (important)

*Bernoulli's Principle Demonstration: Bernoulli's Ball
A: Yes, the Bernoulli priciple is an integral concept in understanding the operation of this device.
As David White has observed, it is the momentum of the input stream which causes the output stream to flow out of the nozzle at a high velocity.  However, if one were to neglect the Bernoulli principle, one might be puzzled as to why the input stream interacts with the suction stream at all, rather than just spraying straight out the nozzle and leaving the suction stream stagnant.
Let us consider the above hypothetical scenario of a stagnant suction stream.  By application of the Bernoulli principle, the pressure in the output nozzle must be lower than in atmosphere outside of the eductor, since the two regions are connected via the nozzle, and the flow is much faster in the nozzle than out.
But now we see that the pressure at the end of the suction stream in the nozzle must be lower than at the start of the suction stream outside of the eductor, so the suction stream has suction and must start flowing!
This is precisely how and why the device functions.
A: Take another look at your picture and imagine that the suction port is actually in outer space...in a real vacuum. And imagine that the inlet port is hooked up to a 100psi air compressor. And the outlet port is connected to a balloon.
Is it clear that up to a certain point, you will be able to get quite a bit of air into the balloon? And if the suction port is quite big (and there's no reason it wouldn't be) there's going to be not much pressure difference between, the suction chamber and the external vacuum? So I don't think it's hard to convince yourself that the pressure at the outlet (B) can be quite a bit higher than the pressure in the suction chamber(C). If you replace the vaccuum with a supply pressure that is greater than (C) and less than (B), the device will then function as an eductor.
Of course, that condition can only persist as long as there is air still flowing straight through. Once you get to the steady state (the balloon is full), the net flow is from the air compressor to the vaccuum, and the pressure in the balloon (the outlet port) is the same as the pressure in the suction chamber. But at this stage, it's no longer functioning as an eductor.
A: The input stream is high pressure and high velocity.  When the input stream mixes with the suction stream, it slows down, and entrains the suction stream.  This means that the velocity of the output stream is higher than that of the suction stream, and lower than that of the input stream.
A: Is the output stream a mixture of air and water? If so, that air must come from somewhere, that somewhere being the inflow of the suction. On a molecular level, you might imagine that some of the air molecules will bounce against the water flow from the nozzle and thus be accelerated towards the 'output' and quickly enough exit the eductor entirely. That creates a low pressure inside the eductor which will cause air molecules to flow into the eductor via the suction inlet.
A: 
Why should an accelerated
  flow diminish pressure next
  to the flow? Yet, this device
  works.


First , I would throw some light on venturi effect with water.
As you can clearly see in image, I have represented arrows to show distance b/w water molecules.
Initially , water molecules are very close as they flow in wider pipe. When pipe becomes narrow, speed of water increases. As a result when water molecules pass towards narrow tube, they cover more distance than molecules behind them. This causes, distance b/w water molecules to increase as shown by black lines.
Further towards more narrow end of tube, distance between water molecules increases very much.
This reduces density of water. But, water does not want to change its density. To counter this change, a region of low pressure is induced. Water in pipe attracts water from vertical tube. This reduces level of water by ∆x in vertical tube.

Now, similarly for sprayers used by barbers, when air passes with huge velocity, more air is attracted from chamber by the stream. This reduces pressure in chamber. As a result water sucks in.
Edit: Sorry! I never read what an eductor is. 

Wiki quote

"It consists of a large bore straight
tube to which is attached a hose
pipe through which clean water is
pumped . The Bernoulli effect from
the flow of pumped water causes
suction at the mouth of the dredge.
Water and sediment are sucked
from the excavation site and
released from the far end of the
tube"

This time water/ air/ steam etc.(usually water) at more velocity attracts water, create suction, lift water and then debris along with it.
@Jokela 
Wiki quote:

The lifting injector uses the Venturi
  effect of a converging-diverging
  nozzle to convert the pressure
  energy of a motive fluid to velocity
  energy which creates a low pressure
  zone that draws in and entrains a
  suction fluid. After passing through
  the throat of the injector, the mixed
  fluid expands and the velocity is
  reduced which results in
  recompressing the mixed fluids by
  converting velocity energy back into
  pressure energy. The motive fluid
  may be a liquid, steam or any other
  gas. The entrained suction fluid may
  be a gas, a liquid, a slurry, or a
  dust-laden gas stream.
  The adjacent diagram depicts a
  typical modern injector. It consists
  of a motive fluid inlet nozzle and a
  converging-diverging outlet nozzle.
  Water , air , steam , or any other fluid
  at high pressure provides the
  motive force at the inlet.

A: My question is; What is the reason behind suction?
This is a conceptual misunderstanding. There is no negative pressure, and thus there is no suction. Vacuum has a zero pressure. (Absolute zero temperature analogy)
After you get this concept right, you understand, that the "suction" is nothing else than the atmospheric-pressure pushing material to the low pressure chamber. 
The nozzle produces this low pressure. But it's always the higher pressure pushing. 
This is why it works, though you can't get it working inside the nozzle by using just newton's second.
It's always "push" and newer "pull".
You can test this by closing the intake. All material inside the "suction-channel" will immediately stop moving as you prevent the "push" though there is full "pull" present. 
Edit; 
Pressure must be understood by the 
kinetic theory of gases.
It's nothing else than particles with velocity colliding to the wall in continuous rates. So there is actually no "pressure" there is only velocity distribution to all directions. When all movement is rectified and directed to only one direction, the pressure on other directions must be zero as there are no particle colliding to these directions. This is what truly happens in Bernoulli principle. 
You can also say this other vice; there is pressure only in one direction in the flow of the nozzle.
Now, as nothing is pushing here particles away from this jet, but the outside particles are pushing particles in the jet, there will be collisions and behavior according to the Newtons second in front of the nozzle, but only there. 
Edit2; 
At your own answer this statement; 

Why should an accelerated flow diminish pressure next to the flow? Yet, this device works. My intuition is that it is an effect of vorticity. But i don't know
  any sources that support my intuition.

The vorticity is not the cause, it's what follows form the mixing of two velocities. The pressure just beside the jet out of the nozzle is actually very close the atmospheric pressure. (Blue arrows in your question), only the flow losses from the "suction" is reduced. The pressure is thus "very low" only inside the  nozzle. (light grey in your picture) Google "Pelton Turbine Nozzle".
Edit 3; 
The answer of A.Goel and it's comment forced me to make this clarifying picture; 

At the left picture it's shown how the pure nozzle doesn't produce any suction. 
At the right picture it's shown how the educator would work even if the the first nozzle is just a pipe providing a high speed flow! The nozzle in the input side is NOT used to change the pressure; it's only used to decrease the flow velocity on the providing pipe-line and to reduce the flow losses there. Ofcourse this simultaniously means that the velocity is increased on the nozzle, and the pressure is thus degreased, but this doens't mean anything for the functionality of the educator. 
A: From the book Applied Gas Dynamics of Ethirajan Rathakrishnan:
"As the jet fluid travels further away from its origin, it slows down due to mixing with the stagnant ambient fluid entrained and inducted into the jet field. 
This is due to the boundary layer at the nozzle exit which develops roll up structures, or ring vortices that grow in size when they move downstream due to the entrainment of ambient fluid into the jet stream. Hence, to conserve momentum the centerline velocity decreases
...The law of conservation of mass is not valid for jet flow.The entrained mass is at a lower momentum compared to the momentum of fluid elements and they will try to come to an equilibrium"
I think that turbulence and conservation of momentum is the answer.
When two masses stick together, usually energy is not conserved( some escape to heat mostly), but momentum is always conserved.
A divergent- convergent nozzle always gives supersonic flows. In supersonic flows the fluid expands outwards all around and so the pressure is not constant across the flow. Only for supersonic flows we can assume the pressure at the exit is not equal to atmospheric pressure. 
A: I don't understand the arguments of the given answers and so i will try to half-answer my own question.
The Bernoulli equation is derived from the conservation of energy strictly along a streamline. If a fluid accelerates, then the cause must be a pressure difference.
Pressure difference is the cause and acceleration the effect. It cannot be the other way. Pressure is not lowered just because a fluid flows.
Why should an accelerated flow diminish pressure next to the flow? Yet, this device works. 
My intuition is that it is an effect of vorticity. But i don't know any sources that support my intuition.
