Shear Rate in a Rectangular Channel

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Discussion Overview

The discussion centers on calculating the shear rate at the bottom of a rectangular channel given a known volume flow rate. Participants explore the implications of laminar flow and the differences between finite and infinite channel widths, particularly in relation to shear rate distribution.

Discussion Character

  • Technical explanation
  • Conceptual clarification
  • Debate/contested

Main Points Raised

  • Some participants inquire about the calculation of shear rate in a rectangular channel, noting the lack of resources specific to this geometry compared to circular channels.
  • One participant expresses a need for material that elaborates on the topic, indicating that their situation may not fit typical Couette flow scenarios.
  • Another participant suggests that if the channel width were infinite, the shear rate would be uniform across the bottom surface, raising questions about the implications for finite width cases.
  • There is discussion about the potential limitations of the finite width case and how shear rate might vary as a function of distance from the channel center or walls.
  • One participant proposes that a ratio, such as w/h, could be relevant in understanding the relationship between finite and infinite cases, suggesting that this ratio may fall between 20-30 in their specific context.
  • A later reply mentions that the shear rate at the wall would be nearly constant for a ratio of 20-30, except near the edges of the channel.
  • References to external literature are provided for further exploration of the topic, specifically regarding average shear rates in rectangular ducts.

Areas of Agreement / Disagreement

Participants generally agree on the focus of the discussion being on laminar flow and the challenges posed by finite channel width. However, multiple competing views remain regarding the specifics of shear rate calculations and the implications of channel geometry.

Contextual Notes

Limitations include the dependence on the assumptions of laminar flow and the specific geometry of the channel, as well as unresolved mathematical steps related to shear rate calculations in finite width scenarios.

quantstr
Say you have a rectangular channel with a width, w (m), and a height, h (m) and an infinite length. The channel itself is fixed and none of the sides can move with respect to one another. If you know the volume flow rate, V (m3/s) of fluid through the channel, how do you calculate the shear rate at the bottom of the channel?

I've tried looking for material that can elaborate on this, but it's mostly just for circular channels. Any ideas?
 
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quantstr said:
Say you have a rectangular channel with a width, w (m), and a height, h (m) and an infinite length. The channel itself is fixed and none of the sides can move with respect to one another. If you know the volume flow rate, V (m3/s) of fluid through the channel, how do you calculate the shear rate at the bottom of the channel?

I've tried looking for material that can elaborate on this, but it's mostly just for circular channels. Any ideas?
Hi. What is the big picture here? What is your motivation for wanting to know the wall shear rate for this channel?
 
I am working on a research project and this is one of the key elements of it. I don't need to be baby-stepped through the problem if it's not very difficult, but I would appreciate some material that I could go through that elaborates on the topic with respect to rectangular channels. The situation I have is not quite Couette flow, because none of the walls are sliding with respect to each other, but it may be a special case of symmetry? Not sure...
 
quantstr said:
I am working on a research project and this is one of the key elements of it. I don't need to be baby-stepped through the problem if it's not very difficult, but I would appreciate some material that I could go through that elaborates on the topic with respect to rectangular channels. The situation I have is not quite Couette flow, because none of the walls are sliding with respect to each other, but it may be a special case of symmetry? Not sure...
You still haven't answered my question regarding what this is all about. It would help to limit the scope.
I assume you are interested exclusively in laminar flow, correct? If the width w were infinite and you knew the flow rate per unit width, would you then be able to solve the problem? (This would be pressure-driven flow between parallel plates).

Are you aware that the wall shear rate varies with position around the circumference of the rectangle?
 
Yes, exclusively laminar flow. I can solve the problem if the width is infinite and, if this were the case, the shear rate would be the same everywhere on the bottom surface. I guess I'd like to know what kinds of limitations exist for the finite width case. Namely, whether or not it can be solved and what the shear rate may look like as a function of distance from the center of the channel (or one of the walls, etc.). Also, maybe there is some kind of ratio, like w/h or something similar, which could be related to how well the finite width case agrees with the infinite case near the center of the channel? Something like that would be quite useful, as this ratio may be between, say 20-30 for what we are doing, but I'd have to check exactly what it is.

Apologies for not being clear about all of this earlier. I've never really had a formal background into this kind of stuff and am learning it as I go for the most part.
 
quantstr said:
Yes, exclusively laminar flow. I can solve the problem if the width is infinite and, if this were the case, the shear rate would be the same everywhere on the bottom surface. I guess I'd like to know what kinds of limitations exist for the finite width case. Namely, whether or not it can be solved and what the shear rate may look like as a function of distance from the center of the channel (or one of the walls, etc.). Also, maybe there is some kind of ratio, like w/h or something similar, which could be related to how well the finite width case agrees with the infinite case near the center of the channel? Something like that would be quite useful, as this ratio may be between, say 20-30 for what we are doing, but I'd have to check exactly what it is.

Apologies for not being clear about all of this earlier. I've never really had a formal background into this kind of stuff and am learning it as I go for the most part.
If you want to see the solution to this problem, you can Google something like "Laminar flow in a duct of rectangular cross section."

For a ratio of 20-30, the shear rate at the wall is going to be virtually constant, except for a region on the order of about h or 2h from the two edges. If you would like to calculate the average shear rate around the perimeter of the rectangle, see the following reference:

Miller, C., Predicting Non-Newtonian Flow Behavior in ducts of Unusual Cross Section, I&EC Fundamentals, 11, 524-528 (1972)
 
Thank you!
 

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