# Mixng problem of two coaxial flows

1. Sep 13, 2009

### souviktor

Mixng problem of two coaxial flows......

1. The problem statement, all variables and given/known data

In the figure,an incompressible fluid ,velocity v is discharging from a 2D channel,height 2c into another 2D channel of height 2a. (a>c),their axes are common.pressure is uniform everywhere and v,w are uniform at the inlet.

Inner flow mixes with that of outer flow and the outer boundary of the mixing region is defined by the ordinate y0.
outside the mixing region the flow may be considered inviscid.If the velocity distribution across the channel is given by
u=U0+U/2*(1+cos(pi*y/y0)) 0<=y<=y0
u=U0 y0<=y<=a

where U0 and U are functions of x.

I need to show U0 becomes zero when
y0/a=3*[c/a*(v/w-1)+1]2/{2c/a*(v2/w2-1)+1}

please see the attached image for the figure.....

2. Relevant equations

may be the boundary layer equations.....otherwise known as blasius equations....

3. The attempt at a solution
This problem looks pretty similar to the problem of a fluid expanding through a smooth diffuser.We know in that case we can use the bernoulli's eqn (neglecting viscous losses)
to find out the drop in static pressure along the length=(v1-v2)2/(2*g) which is the head loss.

But here I dont know whether that is applicable or not.moreover at the boundary no slip b.c should be imposed while solving the b.c does that mean ...velocity at the boundary=(v-w)?
I am scratching my head for the last week and nothing comes out....I am clueless.....can any one help?

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