Theory - Flow between two cylindrical plates

bageshwar · May 14, 2012, 09:42

I have to theoretically calculate the flow between two cylindrical plates.
This is the spec:
Two plates of 15cm radius separated between them by a 1cm gap
flow between those plates to be studied.
And its laminar flow
*I know how the flow between rectangular parallel plates of infinite length works, but not how this works, Can anyone give any link to a theory or any suggestions to solve this?

chlp · May 22, 2012, 09:44

"Cylindrical plates"? So it cylinder or plate?

bageshwar · May 22, 2012, 11:12

It is two plates parallel to each other. the places are cylindrical in shape with 0.15 m radius and 1mm height each. So i have a flow of air passing between these plates. I want to know if there is a theory for such a system, or how could it be derived

chlp · May 22, 2012, 13:06

Can you draw the pictures, eg. in mspaint?

bageshwar · May 22, 2012, 13:13

I have attached the file with this post

chlp · May 22, 2012, 13:48

How to draw this model? I do not understand...

bageshwar · May 22, 2012, 14:06

try this image

chlp · May 22, 2012, 14:24

so, no?

bageshwar · May 22, 2012, 14:29

yes the flow is through the thin region not the cross sectional area.
The flow is betwwen the plates

chlp · May 22, 2012, 14:34

Seems i have never seen before theoretical solution, simulation can be help?

bageshwar · May 22, 2012, 14:35

I have the results for the simulation, but need a theory to validate it. Thats what I am currently struggling upon.

michujo · May 23, 2012, 12:20

Hi Bageshwar, if I have understood correctly, your problem is basically a Poiseuille flow between two concentric cylinders, is that correct?

The solution is pretty straightforward, just take the Navier-Stokes equations in cylindrical coordinates and start killing terms!

For instance:
- Stationary flow: kill $\frac{\partial}{\partial t}$ terms

- No variation along z coordinate (axis of the cylinders): kill $\frac{\partial}{\partial z}$ terms, except for pressure!

- No variation along circumferential coordinate: kill $\frac{\partial}{\partial \phi}$ terms

Considering incompressible flow you do not need the continuity nor the energy equations so you end up with something like this:

$-\frac{\partial P}{\partial z} +\mu \left[ \frac{1}{r}\frac{\partial}{\partial r} \left ( r\frac{\partial u_z}{\partial r} \right )\right ]=0$

, where P is pressure, r is the radius, z is the direction along the axis and u is the velocity.

Solving for

you get:

$u_z(r)=\frac{\partial P /\partial z}{\mu }\frac{r^2}{4} +A ln(r)+B$

Now find the values of A and B imposing boundary conditions at the inner and outer cylinder walls, u=0 at R1 and R2 respectively:

$A=-\frac{\frac{\partial P/ \partial z}{\mu } \frac{R_2^2-R_1^2}{4}}{ln \left ( \frac{R_2}{R_1 }\right ) }$

and

$B=-\frac{\partial P/\partial z}{\mu }\frac{R_1^2}{4} -A \cdot ln R_1$

So that's the velocity profile in the radial direction. Then you will just have to substitute for the numerical data of your problem.

Is this what you were looking for? (please say yes!

).

Cheers.

PGodon · May 23, 2012, 12:24

I suppose the disks are not rotating, these are just two circular (not cylindrical) plates. The flow between them will depend on the viscosity and velocity, or if you prefer on the Reynolds number. You will have two boundary layers forming, one on each plate. If the Reynolds number is small, these two boundary layers might actually join together, which would likely happen far from the circular edges, towards the center. There the flow velocity will be minimal (depending on the viscosity the velocity could even be zero in the center of the circular plates). I expect the velocity to be maximal (but still smaller than the "outside" bulk velocity) at the edges, especially the edges where the flow is tangential to the circular boundary. If the Reynolds number is large, I would expect microscopic boundary layers and the flow might be almost unaffected between the plates. On the overall I expect (guts feeling) the flow velocity to be minimal towards the center (maybe within a given a radius it could have a constant value which would decrease with decreasing Reynolds number). That's my 5 cents comments. I hope it helps.

To me it seems it might be a good way to maybe use this configuration to possibly create almost a stationary flow between the plates in the center, maybe even this could be use for a two phase flow to separate or isolate one of the two components (say particles). Am I guessing right!?

michujo · May 23, 2012, 12:37

Ok, now I see that I did not understand the configuration, damn!

Could you please explain where is the inlet and where is the outlet? I do not clearly see it.

Do you inject it through a hole in the center? or the flow enters radially towards the center and goes out through a hole there?

Cheers.

bageshwar · May 24, 2012, 02:58

Hi all, the disks are not rotating. It is a compressible flow, and I do not want a stationary solution, but i need a time dependent solution. Moreover, Its a flow between two thin cylinders let us assume. the thin cylinders are parallel to each other, the flow just passes between the gaps as I had mentioned in the previous images.
Please let me know.

ndabir · April 23, 2016, 01:19

I know this is an old post but my answer might help someone in future.

I guess you need to look for Analytical Solution of "Squeezing Flow" between circular plates. Squeezing Flow is a good keyword to search this topic.

May 14, 2012, 09:42	Theory - Flow between two cylindrical plates	#1
bageshwar New Member Bhageshvar Join Date: Oct 2011 Posts: 12 Rep Power: 14	I have to theoretically calculate the flow between two cylindrical plates. This is the spec: Two plates of 15cm radius separated between them by a 1cm gap flow between those plates to be studied. And its laminar flow *I know how the flow between rectangular parallel plates of infinite length works, but not how this works, Can anyone give any link to a theory or any suggestions to solve this?

May 23, 2012, 12:24	just trying to understand	#13
PGodon New Member Patrick Godon Join Date: Apr 2010 Posts: 19 Rep Power: 17	I suppose the disks are not rotating, these are just two circular (not cylindrical) plates. The flow between them will depend on the viscosity and velocity, or if you prefer on the Reynolds number. You will have two boundary layers forming, one on each plate. If the Reynolds number is small, these two boundary layers might actually join together, which would likely happen far from the circular edges, towards the center. There the flow velocity will be minimal (depending on the viscosity the velocity could even be zero in the center of the circular plates). I expect the velocity to be maximal (but still smaller than the "outside" bulk velocity) at the edges, especially the edges where the flow is tangential to the circular boundary. If the Reynolds number is large, I would expect microscopic boundary layers and the flow might be almost unaffected between the plates. On the overall I expect (guts feeling) the flow velocity to be minimal towards the center (maybe within a given a radius it could have a constant value which would decrease with decreasing Reynolds number). That's my 5 cents comments. I hope it helps. To me it seems it might be a good way to maybe use this configuration to possibly create almost a stationary flow between the plates in the center, maybe even this could be use for a two phase flow to separate or isolate one of the two components (say particles). Am I guessing right!?

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May 22, 2012, 09:44		#2
chlp New Member Eugeny Join Date: Feb 2011 Location: Ukraine Posts: 9 Rep Power: 15	"Cylindrical plates"? So it cylinder or plate?

May 22, 2012, 11:12		#3
bageshwar New Member Bhageshvar Join Date: Oct 2011 Posts: 12 Rep Power: 14	It is two plates parallel to each other. the places are cylindrical in shape with 0.15 m radius and 1mm height each. So i have a flow of air passing between these plates. I want to know if there is a theory for such a system, or how could it be derived

May 22, 2012, 13:06		#4
chlp New Member Eugeny Join Date: Feb 2011 Location: Ukraine Posts: 9 Rep Power: 15	Can you draw the pictures, eg. in mspaint?

May 22, 2012, 13:48		#6
chlp New Member Eugeny Join Date: Feb 2011 Location: Ukraine Posts: 9 Rep Power: 15	How to draw this model? I do not understand...

May 22, 2012, 14:24		#8
chlp New Member Eugeny Join Date: Feb 2011 Location: Ukraine Posts: 9 Rep Power: 15	so, no?

May 22, 2012, 14:29		#9
bageshwar New Member Bhageshvar Join Date: Oct 2011 Posts: 12 Rep Power: 14	yes the flow is through the thin region not the cross sectional area. The flow is betwwen the plates

May 22, 2012, 14:34		#10
chlp New Member Eugeny Join Date: Feb 2011 Location: Ukraine Posts: 9 Rep Power: 15	Seems i have never seen before theoretical solution, simulation can be help?

May 22, 2012, 14:35		#11
bageshwar New Member Bhageshvar Join Date: Oct 2011 Posts: 12 Rep Power: 14	I have the results for the simulation, but need a theory to validate it. Thats what I am currently struggling upon.

May 23, 2012, 12:20		#12
michujo Senior Member Join Date: Dec 2011 Location: Madrid, Spain Posts: 134 Rep Power: 15	Hi Bageshwar, if I have understood correctly, your problem is basically a Poiseuille flow between two concentric cylinders, is that correct? The solution is pretty straightforward, just take the Navier-Stokes equations in cylindrical coordinates and start killing terms! For instance: - Stationary flow: kill $\frac{\partial}{\partial t}$ terms - No variation along z coordinate (axis of the cylinders): kill $\frac{\partial}{\partial z}$ terms, except for pressure! - No variation along circumferential coordinate: kill $\frac{\partial}{\partial \phi}$ terms Considering incompressible flow you do not need the continuity nor the energy equations so you end up with something like this: $-\frac{\partial P}{\partial z} +\mu \left[ \frac{1}{r}\frac{\partial}{\partial r} \left ( r\frac{\partial u_z}{\partial r} \right )\right ]=0$ , where P is pressure, r is the radius, z is the direction along the axis and u is the velocity. Solving for you get: $u_z(r)=\frac{\partial P /\partial z}{\mu }\frac{r^2}{4} +A ln(r)+B$ Now find the values of A and B imposing boundary conditions at the inner and outer cylinder walls, u=0 at R1 and R2 respectively: $A=-\frac{\frac{\partial P/ \partial z}{\mu } \frac{R_2^2-R_1^2}{4}}{ln \left ( \frac{R_2}{R_1 }\right ) }$ and $B=-\frac{\partial P/\partial z}{\mu }\frac{R_1^2}{4} -A \cdot ln R_1$ So that's the velocity profile in the radial direction. Then you will just have to substitute for the numerical data of your problem. Is this what you were looking for? (please say yes! ). Cheers.

May 23, 2012, 12:37		#14
michujo Senior Member Join Date: Dec 2011 Location: Madrid, Spain Posts: 134 Rep Power: 15	Ok, now I see that I did not understand the configuration, damn! Could you please explain where is the inlet and where is the outlet? I do not clearly see it. Do you inject it through a hole in the center? or the flow enters radially towards the center and goes out through a hole there? Cheers.

May 24, 2012, 02:58		#15
bageshwar New Member Bhageshvar Join Date: Oct 2011 Posts: 12 Rep Power: 14	Hi all, the disks are not rotating. It is a compressible flow, and I do not want a stationary solution, but i need a time dependent solution. Moreover, Its a flow between two thin cylinders let us assume. the thin cylinders are parallel to each other, the flow just passes between the gaps as I had mentioned in the previous images. Please let me know.

April 23, 2016, 01:19		#16
ndabir Senior Member navid Join Date: Jan 2010 Posts: 110 Rep Power: 16	I know this is an old post but my answer might help someone in future. I guess you need to look for Analytical Solution of "Squeezing Flow" between circular plates. Squeezing Flow is a good keyword to search this topic.