Re: Water speed in a funnel in a flowing stream?

Robert,
     A very insightful question.  The answer is no; the speed of the 
water passing through the small end would not be greater, it would be
the same as that of the fluid outside of the funnel.
     This may seem counter-intuitive, so I will explain in detail:

The reason is The Law of Mass Conservation.

Let's look at the small end of the funnel.  When fluid exits the funnel
it must be traveling with the same velocity as the fluid flowing around
the funnel; otherwise, a discontinuity in the flow velocity occurs,
which (for this case) is impossible.
But common-sense tells us that the restricted area through which the flow
is traveling should act to increase the velocity.
Since we must meet the restraint at the exit of the funnel, we must
figure out what happens to the flow at the entrance of the funnel.
The same amount of mass must flow out of the funnel as flows into it
for a given period of time.  This is the Law of Mass Conservation.
The flowrate of a fluid (mass/time) is given by the following 
approximation:

(mass flowrate) = (fluid density)*(avg velocity)*(cross-sectional area)

Let's apply this equation to the entrance and exit of the funnel:

1.  State the Law of Conservation of Mass.

     (mass flowrate in) = (mass flowrate out)

2.  Substitute the appropriate equations.

     (fluid density in)*(avg velocity in)*(cross-sectional area in) =
     (fluid density out)*(avg velocity out)*(cross-sectional area out)

3.  Notice that the density of the fluid does not change through the
funnel.  (fluid density in) = (fluid density out)  We can reduce our
equation to the following.

      (avg velocity in)*(cross-sectional area in) =
      (avg velocity out)*(cross-sectional area out)

4.  Now we can use deduction to infer what happens to the flow at the
entrance to the funnel.  
If (cross-sectional area in) > (cross-sectional area out), then
(avg velocity in) < (avg velocity out)

5.  We have concluded, by deduction from the Law of Mass Conservation,
that the flow at the funnel entrance must slow down so that the flow 
going into the funnel will be able to exit at the same velocity as the 
flow moving around it.
This causes a very slight increase in pressure at the funnel entrance,
but would likely not be detectable in a slow fluid (such as a river).
The fluid will tend to "spill over" the entrance to the funnel so that 
this reduction in velocity can occur.

This is easily verified by experiment with dyes.  More simply, try to
fill a funnel at a constant flowrate.  The fluid coming out of the bottom
is flowing steadily, while the entrance to the funnel is overflilling.
Eureka!!  The Law of Mass Conservation is verified yet again!

Thank You,
Randy