Fluid flow acceleration in a channel

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1. Sep 7, 2015

Proletario

MODERATOR NOTE: THIS HOMEWORK WAS SUBMITTED TO A NON-HOMEWORK FORUM SO THERE IS NO TEMPLATE

Hi everyone,

I'm stuck on a fluid mechanics problem and maybe you could help me.

As shown in figure, there is a two-dimensional converging channel, in which cross-section area varies linearly with x. If Q is the discharge, determine the flow acceleration along the x-axis (A1, A2 and L are known).

I got for area: A(x)=(A2-A1/L).x+A1

I'm not sure what to do about Q, should Q=A.v be used? Any hint about how to get the velocity field?

Last edited by a moderator: Sep 7, 2015
2. Sep 7, 2015

You don't have any other indication about what they want you to use for discharge? That's an ambiguous term.

3. Sep 7, 2015

Staff: Mentor

At steady state, the acceleration is vdv/dx. Are you familiar with this term in the momentum balance equations?

Chet

4. Sep 7, 2015

Proletario

Do you mean Navier-Stokes eq.? I still don't know how to get the velocity field...

5. Sep 7, 2015

Proletario

No, that's all the information. I think discharge here means volume per time. That's why I tried to use Q=A.v

6. Sep 7, 2015

Staff: Mentor

What about getting v(x) from the two equations in your original post:
A(x)=(A2-A1/L).x+A1
and
Q=A.v

Chet

7. Sep 7, 2015

Proletario

I thought the same, but isn't v the mean velocity? If I can use v as the v-field, then the acceleration is quite simple via material derivative.

8. Sep 7, 2015

Staff: Mentor

It's pretty safe to say that this is what they expected you to do. They just forgot to mention that the channel is converging very gradually and that locally you can assume a flat velocity profile.

Chet

9. Sep 7, 2015

Proletario

Wow, so obvious. Thanks Chet.