Can Couette Flow with Wavy Plates?

[binbagsss]

These are probably a stupid questions.



For Couette planar flow with horizontal infinite length plates, the top one moving with a constant velocity in the horizontal direction, with a single layer, one assumes there is only velocity in the horizontal direction.



1)In the case of Couette flow with a wavy / sinusouidal bottom plate( where eddie's can occur), can the velocity have vertical components in this case?



2) In the case of two immiscible layers, the free surface between the two layers is vulnerable to instabilities, due to, e.g. differing densities and viscosities. In such a set up, can the velocities have vertical components as well?

 

[Baluncore]

1)In the case of Couette flow with a wavy / sinusouidal bottom( where eddie's can occur), can the velocity have vertical components in this case?

What is the "bottom"?
Which way is "vertical"?
Does the distance between the plates vary?
Or does the extent of the plates in 2D have a boundary?

 

[binbagsss]

What is the "bottom"?
Which way is "vertical"?
Does the distance between the plates vary?
Or does the extent of the plates in 2D have a boundary?

vertical is given since the plates are infinite length and i've specified their direction

the set-up is in horizontal alignment. the distance does not vary. one would just assume a traditional couette flow if nothing else is specified?

unsure on the last question re. a boundary, i said of infinite length

 

[Chestermiller]

These are probably a stupid questions.



For Couette planar flow with horizontal infinite length plates, the top one moving with a constant velocity in the horizontal direction, with a single layer, one assumes there is only velocity in the horizontal direction.



1)In the case of Couette flow with a wavy / sinusouidal bottom plate( where eddie's can occur), can the velocity have vertical components in this case?

Of course.

2) In the case of two immiscible layers, the free surface between the two layers is vulnerable to instabilities, due to, e.g. differing densities and viscosities. In such a set up, can the velocities have vertical components as well?

Yes.

But as long as the sinusoidal length period. is very large compared to the height of the channel, the vertical components of velocity will be small.

 

[binbagsss]

Yes.

But as long as the sinusoidal length period. is very large compared to the height of the channel, the vertical components of velocity will be small.

thank you, what about the case of amplitude similar to the height of the channel but a large period.

And what about case 2? e.g. with the case of differing densities with the heavier layer on top, wouldn't one expect some vertical movement downward to displacing the lighter under layer?

thanks

 

[Chestermiller]

thank you, what about the case of amplitude similar to the height of the channel but a large period.

In this case, you can use lubrication theory in which the plates are treated as "locally parallel", and in which. although over a period of the sinusoid, the pressure change is zero, the pressure varies locally axially, and there is a net axial flow (although it is not the value that would be obtained if the plates were perfectly parallel.

And what about case 2? e.g. with the case of differing densities with the heavier layer on top, wouldn't one expect some vertical movement downward to displacing the lighter under layer

The situation you describe here is hydrodynamically unstable, and any perturbation to the system will cause the layers to invert. This includes moving the plates.

thanks

 

[binbagsss]

In this case, you can use lubrication theory in which the plates are treated as "locally parallel", and in which. although over a period of the sinusoid, the pressure change is zero, the pressure varies locally axially, and there is a net axial flow (although it is not the value that would be obtained if the plates were perfectly parallel.

The situation you describe here is hydrodynamically unstable, and any perturbation to the system will cause the layers to invert. This includes moving the plates.

what about the case of immiscible layers?

 

[binbagsss]

In this case, you can use lubrication theory in which the plates are treated as "locally parallel", and in which. although over a period of the sinusoid, the pressure change is zero, the pressure varies locally axially, and there is a net axial flow (although it is not the value that would be obtained if the plates were perfectly parallel.

What is meant by axially sorry? thanks.

 

[Chestermiller]

What is meant by axially sorry? thanks.

In the direction parallel to the plate movement.

 

[binbagsss]

what about the case of immiscible layers?

what about immiscible layers so the layers can not invert? what do you mean by moving plates? thanks

 

[Chestermiller]

what about immiscible layers so the layers can not invert? what do you mean by moving plates? thanks

The layers can invert even if they are immiscible.

 

[binbagsss]

Yes.

But as long as the sinusoidal length period. is very large compared to the height of the channel, the vertical components of velocity will be small.

probably a very stupid question, but when eddies are occurring, there must be vertical components to the velocity?

 

[Chestermiller]

probably a very stupid question, but when eddies are occurring, there must be vertical components to the velocity?

Sure.