On representation of expansion in fluid flow

  1. In Helmholtz original thesis On integrals of the hydrodynamical equations, which express vortex-motion, he mentioned in the first section that the change undergone by an arbitrary infinitesimal volume of water under the time dt is composed of three different motions. One of them is an expansion or contraction of the particle parallel to three main axes of dilatation. However in modern language, this part is represented by the negative gradient of a scalar function. My question is why the gradient which I think contains one direction can represent three main directions of dilatation?
  2. jcsd
  3. boneh3ad

    boneh3ad 1,889
    Science Advisor
    Gold Member

    Dilatation is usually expressed with a divergence, not a gradient, and the divergence has no direction.
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