Computation of term (v*gra)v in spherical coordinates

In order to solve this problem I think it is necessary to go back to the derivation of the (v dot del) v operator also given in VAPS The gradient, divergence and laplacian are also well known for general curvilinear coordinates. It is clear that the entire v dot del operator acts on v, so it seems to be a simple matter of working out v dot del. It is clear that Eqs. (9) – (11) of Note 356(4) follow from the usual divergence, Eqs. (6) to (8). So this looks like a problem in fundamental vector algebra.

To: Emyrone@aol.com
Sent: 30/08/2016 12:24:47 GMT Daylight Time
Subj: Computation of term (v*gra)v in spherical coordinates

It is not clear to me how the term (v*grad)v is to be transformed to
spherical coordinates. You used the comparison to the div operator, I
simply multiplied v and del in spherical coordinates and applied to v.
In case of the given Coulomb field, the results for the Kambe velocity
field v are different:

Method 1 (analogous to div v):
v_r is purely imaginary (I analyzed the result and tried a plot)

Method 2 (scalar product v*del):
v_r is proportional to 1/sqrt(r) in the simplest case, depending on
integration constant, see Figure.

According to VAPS, the expressions grad and div are derived from the
tangent vectors and surface flow in the corresponding coordinates. To be
sure, we would have to analyze what the geometrical meaning of the
operator (v*grad) is and derive this for the corresponding coordinate
system.
There is not much info on this operator in the internet. I found the site
https://en.wikipedia.org/wiki/Del_in_cylindrical_and_spherical_coordinates
There the operator is denoted “material derivative” and is given. It is
much more complicated than both above methods 1 and 2. I will try this
operator and see what results.

Horst


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