Fluid dynamics (2nd year undergrad) question on jet incident on a plate

[hai2410]

Homework Statement

A circular jet of water of radius r1 is propelled vertically downwards and hits
a circular plate of radius r2 (> r1 ) so that the centre of the jet is incident onto the
centre of the plate. The velocity of the jet as it hits the plate is v and is normal to
the plate.

Show that for a plate of radius r=\sqrt(2)r1 the net force on the plate due to the
change in momentum of the fluid and the differences in pressure on each side of
the plate is zero.

[You may assume that the velocity of the water changes direction from being
normal to the plate to lying in the plane of the plate as it hits the front of the plate
and that the rear of the plate is covered by a layer of the water that has been
brought to rest.]

Homework Equations

Bernoulli:
0.5p v^2 + P =constant (p being density, P being pressure, v being velocity)

The Attempt at a Solution

I can find out the thickness of the film as a function of radial distance from centre under the assumption that the pressure is equal on the surface of the jet as it was in the incident jet [and thus using Bernoulli and conservation of mass], and I can work out the force on the plate due to the change in momentum of the jet normal to the plate.

I can also work out for a particular stagnation point above the centre of the plate, that the pressure [by Bernoulli] is P0 + 0.5pv0^2 where I've called the jet initial pressure P0, and initial velocity v0.

However both of these forces would act down on the plate, so that it was blasted away from the incident jet (as you'd expect in real life). I therefore can't think of any force which would act toward the jet, and so have no idea how to solve this problem.

Thanks for any help you can give! :)
PS for this part you don't take into account the weight of the fluid, that comes later apparently

 

[ideasrule]

I can find out the thickness of the film as a function of radial distance from centre under the assumption that the pressure is equal on the surface of the jet as it was in the incident jet [and thus using Bernoulli and conservation of mass], and I can work out the force on the plate due to the change in momentum of the jet normal to the plate.

You don't need to figure out the thickness of the film. You do need the force exerted on the plate, however, but that doesn't require you to know the thickness of the film.

I can also work out for a particular stagnation point above the centre of the plate, that the pressure [by Bernoulli] is P0 + 0.5pv0^2 where I've called the jet initial pressure P0, and initial velocity v0.

It's actually P0-0.5pv0^2. Bernoulli's law says that 0.5p v^2 + P =P0, where P0 is the pressure at a place where velocity is 0, so P=P0-0.5pv^2.

However both of these forces would act down on the plate, so that it was blasted away from the incident jet (as you'd expect in real life). I therefore can't think of any force which would act toward the jet, and so have no idea how to solve this problem.

The pressure acts upwards because on the other side of the plate, the pressure is higher.