Mixng problem of two coaxial flows

[souviktor]

Mixng problem of two coaxial flows...

Homework Statement

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...

Homework Equations

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

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 don't 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 anyone help?

 

[KataKoniK]

Dear fellow scientist,

I understand your confusion and I would be happy to help you with this problem. Let's start by looking at the given velocity distribution equation. We can see that it is a combination of two parts - one for the inner flow and one for the outer flow. The inner flow has a cosine term, which indicates that there is some mixing happening between the two flows. This mixing is happening within the mixing region, defined by the ordinate y0. Outside of this region, the flow is inviscid, meaning that there is no mixing between the two flows.

Now, let's consider the boundary conditions at the mixing region. At the boundary, we need to impose the no-slip condition, which means that the velocity at the boundary will be equal to the difference between the velocities of the two flows (v-w). This is because at the boundary, the two flows are in contact and they will have the same velocity.

Now, let's look at the given equation for U0. We can see that it is a function of x, which means that it will vary along the length of the channel. In order to find U0, we need to solve the boundary layer equations, also known as the Blasius equations. These equations will give us a relationship between U0 and y0, which we can then use to find the value of U0 when y0/a=3*[c/a*(v/w-1)+1]^2/{2c/a*(v^2/w^2-1)+1}.

I hope this helps you in your solution. Let me know if you have any further questions or if you need any clarifications. Good luck!

 

[Mene]

This problem can be approached through the use of the Navier-Stokes equations and the continuity equation. The velocity distribution given in the problem can be used to find the velocity gradients at the boundary between the two flows. By setting the Navier-Stokes equations equal to each other at this boundary and using the continuity equation, we can solve for the velocity of the inner flow (U0) in terms of the velocity of the outer flow (U) and the ratio of their viscosities (v/w). This will give us an expression for U0 in terms of the other variables.

Next, we can use the given relation between y0 and a to substitute for y0 in the expression for U0. By simplifying and rearranging the resulting equation, we can show that U0 becomes zero when y0/a=3*[c/a*(v/w-1)+1]^2/{2c/a*(v^2/w^2-1)+1}. This shows that the mixing region extends to a point where the inner flow velocity becomes zero, and the flow outside this point is inviscid.

The boundary conditions for this problem should take into account the no-slip condition at the boundary between the two flows. This means that the velocity at the boundary should be equal to the difference between the velocities of the two flows (v-w). This can be used to determine the velocity distribution at the boundary and to solve for U0 as described above.

In addition, the boundary layer equations (also known as the Blasius equations) can be used to analyze the viscous effects in the mixing region. These equations can be solved numerically to find the velocity and pressure distributions in the mixing region.

Overall, this problem involves the use of various fluid mechanics concepts and equations, including the Navier-Stokes equations, the continuity equation, and the boundary layer equations. By utilizing these tools, we can analyze the mixing of two coaxial flows and determine the conditions at which the mixing process is complete.