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Old Apr 7th 2017, 11:11 AM   #1
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Rotation and deformation in fluid mechanics

We can decompose the dUi/dXj to rotation (Rij) and deformation (Sij) matrices. The question is for a solid-body rotation in which U(theta)=Omega*r/2 and U(r)=0 how come Sij=0 but Rij=Rotation! I do not understand how mathematically it is possible.
Rij=d(Ui)/d(Xj)-d(Uj)/d(Xi)
Sij=d(Ui)/d(Xj)+d(Uj)/d(Xi)
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Old Apr 7th 2017, 11:54 AM   #2
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Have you tried inserting the velocity field into the rateof deformation tensor and spin tensor matrices?

If so what do you get?
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Old Apr 7th 2017, 12:06 PM   #3
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That's the thing!
When i do plug the U(theta) and U(r) into the Tau matrix in R-Theta coordinate system i'll get non-zero elements for Tau(1,2) which is d(Utheta)/d(R). Subsequently, i decomposed it to R(ij) and S(ij). Consequently, i got R(1,2), R(2,1) and S(1,2) and S(2,1). The other elements are zero. All in all i got to the point that solid body vortex does rotates rotationally as well as it deforms!
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