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Idontknow84
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I've got a problem regarding tensors.
Premise: we are considering a fluid particle with a velocity [itex]\mathbf{u}[/itex] and a position vector [itex]\mathbf{x}[/itex]; [itex]S_{ij}[/itex] is the strain rate tensor, defined in this way:
[itex]\displaystyle{S_{ij}=\frac{1}{2}\left(\frac{\partial u_i}{\partial x_j} +\frac{\partial u_j}{\partial x_i}\right)}.[/itex]
OK, the problem is in this paragraph taken from Fluid Mechanics. Fifth Edition, P. K. Kundu, I. M. Cohen, D. R. Dowling, 2011, p. 78:
My question is: why is [itex]S_{ij}[/itex] independent of the frame of reference in which it is observed? Sure, it is zero in every frame in which the fluid particle translates with constant linear-velocity [itex]\mathbf{U}[/itex] and rotates with constant angular-velocity [itex]\mathbf{\Omega}[/itex], but this doesn't explain why it should be the case "even if [itex]\mathbf{U}[/itex] depends on time and the frame of reference is rotating."
[*] This is the Exercise 3.17:
[[itex]R_{ij}[/itex] is the rotation tensor: [itex]\displaystyle{\frac{\partial u_i}{\partial x_j} -\frac{\partial u_j}{\partial x_i}}[/itex].]
Premise: we are considering a fluid particle with a velocity [itex]\mathbf{u}[/itex] and a position vector [itex]\mathbf{x}[/itex]; [itex]S_{ij}[/itex] is the strain rate tensor, defined in this way:
[itex]\displaystyle{S_{ij}=\frac{1}{2}\left(\frac{\partial u_i}{\partial x_j} +\frac{\partial u_j}{\partial x_i}\right)}.[/itex]
OK, the problem is in this paragraph taken from Fluid Mechanics. Fifth Edition, P. K. Kundu, I. M. Cohen, D. R. Dowling, 2011, p. 78:
Here we also note that [itex]S_{ij}[/itex] is zero for any rigid body motion composed of translation at a spatially uniform velocity [itex]\mathbf{U}[/itex] and rotation at a constant rate [itex]\mathbf{\Omega}[/itex] (see Exercise 3.17).[*] Thus, [itex]S_{ij}[/itex] is independent of the frame of reference in which it is observed, even if [itex]\mathbf{U}[/itex] depends on time and the frame of reference is rotating.
My question is: why is [itex]S_{ij}[/itex] independent of the frame of reference in which it is observed? Sure, it is zero in every frame in which the fluid particle translates with constant linear-velocity [itex]\mathbf{U}[/itex] and rotates with constant angular-velocity [itex]\mathbf{\Omega}[/itex], but this doesn't explain why it should be the case "even if [itex]\mathbf{U}[/itex] depends on time and the frame of reference is rotating."
[*] This is the Exercise 3.17:
For the flow field [itex]\mathbf{u} = \mathbf{U} + \mathbf{\Omega} \times \mathbf{x}[/itex], where [itex]\mathbf{U}[/itex] and [itex]\mathbf{\Omega}[/itex] are constant linear- and angular-velocity vectors, use Cartesian coordinates to a) show that [itex]S_{ij}[/itex] is zero, and b) determine [itex]R_{ij}[/itex].
[[itex]R_{ij}[/itex] is the rotation tensor: [itex]\displaystyle{\frac{\partial u_i}{\partial x_j} -\frac{\partial u_j}{\partial x_i}}[/itex].]
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