# Shear Rate in a Rectangular Channel

• I
quantstr

## Main Question or Discussion Point

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?

## Answers and Replies

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Chestermiller
Mentor
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?

quantstr
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...

Chestermiller
Mentor
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?

quantstr
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.

Chestermiller
Mentor
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)

quantstr
Thank you!