It's a relatively simple problem I'm sure, but I'm a bit confused on how exactly to go about it:
There are three parallel plates with water (viscosity of 0.8807 cp @ 30°C) between plates 1 and 2 (plate order of 1 on bottom, 2 in middle, 3 on top), and toluene (viscosity of 0.5179 cp @ 30°C) between plates 2 and 3. The distance between each plate pair is 10 cm and plate 3 (top plate) moves at 3 m/s while plate 1 (bottom plate) is at rest.
a) I need to find the velocity of plate 2 at steady-state
b) And the F/A (shear stress) on plate 3 that is needed to maintain the 3 m/s velocity
Please excuse any formatting issues you may come across. I am new to forums and still learning how to properly format questions and comments.
Shear stress = [itex]\tau[/itex]yx = -[itex]\mu[/itex]*vx/y where [itex]\mu[/itex] = viscosity and vx = velocity in the positive x direction (the same direction plate 3 is moving in)
[itex]\tau[/itex]yx = F/A
The Attempt at a Solution
I was a bit unsure initially of how to go about the problem but my attempt was to solve for the shear stress between plates 1 & 2 and plates 2 & 3, and then equate them to get the velocity of the middle plate.
So after multiplying dy by both sides and integrating (y from 0 to 10, and vx from 0 to v2 for plates 1 & 2 and from v2 to 3 for plates 2 & 3), I got the equations:
Plates 1 & 2: [itex]\tau[/itex]yx = -0.08007*v2
Plates 2 & 3: [itex]\tau[/itex]yx = -0.15937 + 0.5179*v2
Equating these gave me the velocity of plate 2 to be 1.178 m/s
For the 2nd part of the question (solving for F/A), I think you only need to solve for shear stress as that equals F/A. So using v2, I plugged it into one of the equations to get [itex]\tau[/itex]yx = F/A = -0.094322 N.
The problem I have with my methods is that I'm not too sure about equating the two shear stresses as I see no real reason that they should be equal in the first place. Also, the 2nd part of the question specifically asks about the F/A on plate 3, which really makes me think that the shear stress must be different for each plate. While my first answer seems to make sense, I don't think a negative force value for the second one fits, so I would assume that I'm doing something wrong there. Thanks in advance for any help, the hardest part I have with this stuff is that it's really confusing to me what actually needs to be done and which equations to use to get there.