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I would like to know why exactly is only tangential velocity considered in Euler's turbomachinery formula, specifically for torque. I understand that you can use rotational velocity to find torque but why is this not the case for this circumstance?enter image description here

More specifically, in this case provided by my textbook, what does it mean when the position vector r is purely radial so only the tangential velocity component counts? Thank you

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    $\begingroup$ If you're going to reproduce a bit of the textbook like this, you should cite it, The typographic style looks like something published by Pearson, perhaps Fluid mechanics by Douglas et al.. $\endgroup$ Oct 28 '21 at 17:17
  • $\begingroup$ May I ask what book is this? $\endgroup$
    – Algo
    Oct 29 '21 at 4:35
  • $\begingroup$ Fluid Mechanics 7th Edition by Frank M. White $\endgroup$ Nov 1 '21 at 19:29
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"The position vector is purely radial" means that the position vector is aligned along the direction from the origin (or rather, the $z$ axis) to the location of interest.

The cross product of position vector and velocity appears here because it is (by definition) the specific angular momentum.

The size of the cross product of two vectors is equal to the size of the first vector multiplied by the component of the second vector perpendicular to the first vector. In this case, that means the size of the position vector multiplied by the component of the velocity perpendicular to the position vector; i.e. the size of the position vector multiplied by the component of the velocity perpendicular to the radial direction.

"The tangential component" is, by definition, the (in-plane) component perpendicular to the radial direction.

Hence, the size of the cross product of the two vectors is equal to the size of the position vector multiplied by the tangential component of the velocity.

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