Gradient operator in Natural Curvilinear Coordinates

In summary, the speaker has been struggling to understand fluid mechanics in a research paper and is specifically trying to derive the gradient operator in a coordinate system involving tau, K, sigma, rho, theta, isigma, irho, and itheta. They are looking for resources or assistance in deriving this operator and mention Neumann's papers and Struik's Dover book as potential sources.
  • #1
anmol1980
1
0
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
 
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  • #2
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.
 
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FAQ: Gradient operator in Natural Curvilinear Coordinates

What is the gradient operator in natural curvilinear coordinates?

The gradient operator in natural curvilinear coordinates is a mathematical tool used to determine the rate of change of a scalar function in a curved space. It takes into account the variations in the coordinate system and calculates the magnitude and direction of the steepest increase of the function at a given point.

How is the gradient operator expressed in natural curvilinear coordinates?

The gradient operator in natural curvilinear coordinates is expressed as a linear combination of the basis vectors of the coordinate system. This allows for the calculation of the gradient in any direction without needing to change the coordinate system.

What is the relationship between the gradient operator and the directional derivative?

The gradient operator is closely related to the directional derivative, as it represents the direction of the steepest increase of a scalar function. The directional derivative is the rate of change of the function in a specific direction, and it can be calculated using the gradient operator and the dot product.

How is the gradient operator used in physics and engineering?

The gradient operator is used extensively in physics and engineering to solve problems in curved spaces, such as fluid dynamics, electromagnetism, and general relativity. It allows for the calculation of important physical quantities, such as velocity, acceleration, and electric and magnetic fields, in non-Cartesian coordinate systems.

Are there any limitations to using the gradient operator in natural curvilinear coordinates?

While the gradient operator is a powerful tool, it does have some limitations. It cannot be used for vector functions, as it only operates on scalar functions. Additionally, in very complex coordinate systems, the calculation of the gradient can become very cumbersome and may require advanced mathematical techniques.

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