Gradient operator in Natural Curvilinear Coordinates

[anmol1980]

Hi All,

I have been trying to understand some fluid mechanics in a research paper and have been wrestling with the mathematics for quite some time now without success.
I want to derive gradient operator with following coordinate system in R^3 space

Let and arbitrary curve C be locus of centroid of a cylindrical (non circular) jet in space.

Let sigma is the arclength along this curve. let rho be perpendicular distance from the locus of centroids. Let theta be the angle between the line drawn from a particular point in jet to the locus of the centroids and the normal to C at that point.

What would be the gradient operator in the cordinate system in terms of

tau (torsion)
K (curvature)
sigma, rho, theta (coordinates)
isigma, irho, itheta (unit vectors along the direction of increasing sigma, rho and theta respectively)


If you could forward me to any manterial or help me derive the gradient operator, I would really appreciate.

Thanks

 

[Andy Resnick]

AFAIK, these are standard results from differential geometry. The surface gradient operator can be found in Neumann's papers related the the generalized theory of capillarity (http://jcp.aip.org/resource/1/jcpsa6/v66/i12/p5464_s1 is a classic paper).

Struik's Dover book also has these results explicitly written out, IIRC.

It's messy, that's why I didn't barf up a bunch of LaTeX. Read the appendices to Neumann's paper and see if that helps.