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Hi guys. Let me just say at the outset that I know very little fluid mechanics but I keep coming back to the same issue over and over in a general relativity related problem so I figured I'd just ask the fluid mechanics question here.
In countless places the interpretation of the vorticity vector of a fluid is given as follows. If we consider any fluid element and an infinitesimal displacement vector between this and a neighboring fluid element, such that the displacement vector remains fixed to this neighboring fluid element, then the vorticity vector measures the instantaneous angular velocity of the displacement vector, say relative to inertial guidance gyroscopes comoving with the considered fluid element. Of course this makes perfect sense when the fluid has a vanishing strain tensor so that it describes a rigid motion and indeed the claim is very easy to prove mathematically in the case of rigid motion.
But does it still hold if the motion isn't rigid i.e. if the strain tensor is non-vanishing? If the displacement vector is ##\delta x^i## then the relative velocity is ## \delta\dot x^i = \sigma^{i}{}{}_j \delta x^j## in the absence of vorticity and compression, so that there is only shear. But if ##\sigma_{ij}## has off-diagonal components then certainly ##\delta \dot x^i## will have a part that represents rotational velocity, right? Only if ##\delta x^i## is aligned with the principal axes of ##\sigma_{ij}## will there be no such rotational velocity. And indeed I have searched as much as I can and I can't find any mathematical proof of the statement that vorticity measures the entirety of the instantaneous angular velocity of the infinitesimal displacement vector between neighboring fluid elements when the strain tensor is non-zero.
Thanks in advance.
In countless places the interpretation of the vorticity vector of a fluid is given as follows. If we consider any fluid element and an infinitesimal displacement vector between this and a neighboring fluid element, such that the displacement vector remains fixed to this neighboring fluid element, then the vorticity vector measures the instantaneous angular velocity of the displacement vector, say relative to inertial guidance gyroscopes comoving with the considered fluid element. Of course this makes perfect sense when the fluid has a vanishing strain tensor so that it describes a rigid motion and indeed the claim is very easy to prove mathematically in the case of rigid motion.
But does it still hold if the motion isn't rigid i.e. if the strain tensor is non-vanishing? If the displacement vector is ##\delta x^i## then the relative velocity is ## \delta\dot x^i = \sigma^{i}{}{}_j \delta x^j## in the absence of vorticity and compression, so that there is only shear. But if ##\sigma_{ij}## has off-diagonal components then certainly ##\delta \dot x^i## will have a part that represents rotational velocity, right? Only if ##\delta x^i## is aligned with the principal axes of ##\sigma_{ij}## will there be no such rotational velocity. And indeed I have searched as much as I can and I can't find any mathematical proof of the statement that vorticity measures the entirety of the instantaneous angular velocity of the infinitesimal displacement vector between neighboring fluid elements when the strain tensor is non-zero.
Thanks in advance.