[ordinary]

Hello everyone,

I asked this question on Reddit but didn't get any solution.

I drew the image below to calculate volumetric dilation of a fluid element in cylindrical coordinates:
cylindery.png
Then I wrote velocity components in terms of Taylor's series expansion:
velocities.png
After that I started to calculate dilation:
dilation.png
I'm missing something but I can't figure it out. Do you have any idea?

Regards,
Emre

[FMDenaro]

The volume dilatation is measured by the divergence of the velocity. You can write that in your reference system.
Or write the integral over your small volume and then take the averaging tending to a vanishing volume.

[tas38]

Originally Posted by ordinary

Hello everyone,

I asked this question on Reddit but didn't get any solution.

I drew the image below to calculate volumetric dilation of a fluid element in cylindrical coordinates:
Attachment 78274
Then I wrote velocity components in terms of Taylor's series expansion:
Attachment 78275
After that I started to calculate dilation:
Attachment 78276
I'm missing something but I can't figure it out. Do you have any idea?

Regards,
Emre

Your expression is correction except for a missing 1/r multiplying the circumferential velocity term.


Note also when comparing your derived result with the known divergence...

[ordinary]

Originally Posted by FMDenaro

The volume dilatation is measured by the divergence of the velocity. You can write that in your reference system.
Or write the integral over your small volume and then take the averaging tending to a vanishing volume.

Thank you very much. I just wanted to prove it via different method.

[ordinary]

Originally Posted by tas38

Your expression is correction except for a missing 1/r multiplying the circumferential velocity term.


Note also when comparing your derived result with the known divergence...

Thank you very much. So that means last definition of partial derivative Vr is correct. Only something is wrong with circumferential velocity term. I will check it.