Homework Help: Velocity in microchannel with temporal temperature variation

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1. Oct 30, 2017

big dream

a microchannel of length 2L and width h in the thermal cycling region. the temperature profile .....(1)
the cyclic temperature profile leads to a time dependent density ....(2)
using the mass conservation equation i.e. ...(3)
and momentum balance equation i.e. ....(4)
we have to find the exact solution of u, v & p

The attempt at a solution
equ(1) →
equ(2) →
length scales are normalised as
so equ(2) becomes

no slip & no penetration boundary condition at the walls (y=0,h) ; and constant pressure at channel entrance(x=-L) and exit(x=L). for small reynolds number eq(3) becomes
this equation normalised as
as
and from
we can get
& is it the right process?

2. Oct 31, 2017

Staff: Mentor

What is the exact statement of the problem?

3. Nov 10, 2017

big dream

4. Nov 10, 2017

Staff: Mentor

I'm a bit confused over the problem statement. It says, "There are no spatial temperature gradients present along the length of the micro channel," and yet the temperature equation they provide indicates otherwise. I'm having trouble articulating what is happening physically in this problem. Are you supposed to assume that the length scale of the periodic temperature variations (lambda) is small compared to the distance 2L?

Please articulate what you interpret is happening in the problem. Is it that the density variations are causing fluid flow pumping in the x direction?

5. Nov 15, 2017

big dream

sorry this is the real physical problem

6. Nov 15, 2017

Staff: Mentor

The two fluid flow problems you have presented here are extremely different, and I would approach them entirely differently. I am assuming that you really want to solve the problem involving the moving temperature wave, indicated in the first post. I would assume that the spatial wavelength $\lambda$ is small compared to the length of the tube 2L, so that, for all intents and purposes, the tube could be assumed infinitely long. I would then change the frame of reference of the observer such that he is moving along with the temperature wave at the speed c. In that case, he would observe a density variation that is a function only of x. This then would reduce the problem to a steady state flow problem.

I'm prepared to reveal more hints on how to implement the approach I described above involving a moving observer if what I said is not sufficiently clear.

Last edited: Nov 16, 2017
7. Nov 16, 2017

big dream

Sir, for the second problem is my process right? Or I have to solve in another way. More preciously if this process is wrong, how I can get the exact solution of velocities?

8. Nov 16, 2017

Staff: Mentor

As I said, in my judgment, because of the combined x and t variation, you are not going to get it solved the way you are doing it. But, the method I was explaining will work.

From the frame of reference of an observer moving to the left at velocity c (i.e., moving coordinate system), you have $$T=T_0+\Delta T \cos{2\pi \frac{X}{\lambda}}$$ where X now replaces $x+ct$. So now the density is a function only of X. As reckoned from this moving coordinate system, the continuity equation becomes: $$\frac{\partial (\rho U)}{\partial X}+\frac{\partial (\rho V)}{\partial y}=0$$where U and V are the velocity components reckoned from the frame of reference of the moving observer.

Boundary conditions on the flow are now: $U=+c$ and V= 0 at y=h and y=0.

Questions??

Last edited: Nov 18, 2017
9. Nov 19, 2017

Staff: Mentor

It is too bad that @big dream has not returned. I find this a very interesting problem. Is there anyone else out there who is interested in pursuing the solution to this problem?

10. Nov 20, 2017

big dream

sir,
I was sick. so now should I put stream function? or I have to normalise continuity equation first.

11. Nov 20, 2017

Staff: Mentor

Before doing anything else, I need you to prove to yourself that, if we make the following transformation of variables,

t = T
y = Y
x = X - ct
v = V(X,Y)
u = U(X,Y) - c
p = P(X)

our equations reduce to:
$$\rho=\rho(X)$$
$$\frac{\partial (\rho U)}{\partial X}+\frac{\partial (\rho V)}{\partial Y}=0\tag{1}$$
$$\frac{\partial P}{\partial X}=\mu \frac{\partial ^2U}{\partial Y^2}\tag{2}$$
subject to the boundary conditions $U=c$ and $V=0$ at $Y=0,h$. These are equivalent to the equations that an observer who is traveling at a constant speed c in the negative x direction would write. This observer would conclude that, as reckoned from his frame of reference, the system is at steady state, with all parameters functions of X and Y only.

Last edited: Nov 20, 2017
12. Nov 20, 2017

big dream

Yes, sir I got it. Now ?

13. Nov 20, 2017

Staff: Mentor

OK. Starting with Eqn. 2, in post #11, solve for U in terms of dP/dX using the boundary conditions at y = 0, h on U. What do you obtain?

14. Nov 21, 2017

big dream

boundary condition: U= c at Y= 0,h

15. Nov 21, 2017

Staff: Mentor

Close, but this doesn't satisfy the boundary condition at Y=h. Please try once more.

16. Nov 21, 2017

big dream

17. Nov 21, 2017

Staff: Mentor

Much better. Now please take the integrated average of this between 0 and h to get the average axial velocity $\bar{U}(X)$

18. Nov 21, 2017

big dream

19. Nov 21, 2017

Staff: Mentor

OK. Now divide by h to get Ubar.

After that, please integrate the continuity equation (Eqn. 1 in post #11) between Y=0 and Y = h. What do you get?

Unfortunately, I have to leave for a few hours now, but I'll be back later.

20. Nov 21, 2017

big dream

Should I put the value of U and then evaluate?

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