I have a long microchannel where flows some water. The reynolds number is much smaller than one. Within the structure of this microchannel there is a big defect. It looks like a bump of size approximately 20 micrometer and the microchannel's width and height are 100 micrometer.

My question is the following: On what length scale will the defect modify the Poiseuille-like structure of the flow field?

I think it's possible to show by a scaling argument that the defect's effect will be "damped' very very fast, and that the Poiseuille-like flow shape will be recover after a few micrometers past the defect.

  • $\begingroup$ I see you didn't add the turbulence tag so my point 1 is, I guess, not directed at you. But it's still a useful to think about! $\endgroup$ – tpg2114 Mar 2 '15 at 21:26

At Reynolds numbers much smaller than one, you are in the Stokes flow or creeping flow regime.

There are several questions you need to ask yourself in order to understand this problem:

  1. You tagged this question turbulence -- at what Reynolds numbers do you expect the flow to be turbulent?
  2. What determines the shape of the velocity profile (ie. what terms are important)?
  3. What terms may be neglected from the equations at very low Reynolds numbers and are they any of the terms you deemed important in the previous step?
  4. You are in the micron domain size -- what is the Knudsen number of your flow? Is continuum even a good approximation here?
  5. If you assumed a shape for the disturbance, can you show how quickly it is damped? What would a disturbed solution look like/how would you introduce it into the equations?

So those are some hints on how to think about the problem. You are right, you can in fact show how quickly/slowly disturbances will be damped. I can't tell you how because you need to work through that on your own, but everything you need is in the equations.

Assume a functional form for the disturbance, plug it in, see what happens.

  • $\begingroup$ Thank you tpg2114. I indeed did not put the turbulence tag myself. I actually wanted to put "low reynolds" but it doesn't seem to be a popular tag... $\endgroup$ – gregor02 Mar 2 '15 at 21:45
  • $\begingroup$ @gregor02 No problem, but point 1 is always a good one to think about anyway. $\endgroup$ – tpg2114 Mar 2 '15 at 21:53
  • $\begingroup$ For comment 2.), the velocity profile results from pressure difference between the two extremities of the channel. $\endgroup$ – gregor02 Mar 2 '15 at 21:57
  • $\begingroup$ @gregor02 I already know the answers to all of those :) I'm just giving you hints on how to work through the problem. Because it's a homework-and-exercises problem, I can't just outright give you the answer. But if you can derive the governing equations as they apply to this particular problem, assume a functional form for the disturbance and plug it in, then "do what you need to do" (linearize, assume minimum/maximum values, etc) to find out when they are damped, you will be done. $\endgroup$ – tpg2114 Mar 2 '15 at 22:00
  • $\begingroup$ Yes you're right concerning point 1.), it's important, but here i consider it zero... Concerning the Knudsen number, i never calculated it, but well i know it's a common problem of (micro) fluid mechanics. $\endgroup$ – gregor02 Mar 2 '15 at 22:01

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