Assume that at point r and time t a fluid particle has a property Q. As this particle moves about, this property will change with time (and space). Again, in the Lagrangian description, Q is only a function of time, i.e.

(Eq. 1)

However, from the Eulerian point of view, any property of the fluid is a function of time and space, which is also a function of time implicitly. Then

(Eq. 3)

where we have used the chain rule to account for the spatial dependence on time. Remembering some of the formulas from dynamics, we have

(Eq. 4)

upon substitution of Eq. 4 into Eq. 3, we finally obtain the material derivative for a scalar

(Eq. 5)

To obtain the material derivative for a vector field, we follow a similar procedure keeping in mind the directional nature of a vector. We illustrate this using the velocity field - keep in mind that this works for any kind of vector field. Again, in a Lagrangian reference, the velocity is only a function of time. In the Eulerian view, the velocity has the following form

(Eq. 7)

Again, noting that the partial derivative with respect to time in Eq. 7 (first term) is evaluated at a fixed position in space, the unit vectors associated with the fluid particle at that point are fixed as viewed from an Eulerian reference, therefore,

(Eq. 8)

To evaluate the remaining terms in Eq. 7, we have to first remember some equations from dynamics or vector calculus about differential changes in unit vectors in spherical coordinates. These are given by

(Eq. 12)

Voila!!!

Once Eqs. 8 through 12 are put together, one obtains the full expression for the material derivative of a vector field in spherical coordinates.

In the next post, I will present an invariant vector form for the material derivative so that we don't have to go through all the hassle of using chain rule differentiation to evaluate the material derivative. But it was worth it to see how it works using good old calculus.

Once Eqs. 8 through 12 are put together, one obtains the full expression for the material derivative of a vector field in spherical coordinates.

In the next post, I will present an invariant vector form for the material derivative so that we don't have to go through all the hassle of using chain rule differentiation to evaluate the material derivative. But it was worth it to see how it works using good old calculus.

Cite as:

Saad, T. "3. The Material Derivative in Spherical Coordinates".
Weblog entry from
Please Make A Note.
http://pleasemakeanote.blogspot.com/2008/08/derivation-of-navier-stokes-equations_18.html

thankzz alooot dude..:P

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ReplyDeleteI checked this site to compare three other derivations (including my own) with this one, and I believe either Eqn. 9 has a sign error in the second term, or the author has defined an unconventional meridional coordinate system (originating from the southern rather than northern pole). As a general critique, it may be helpful to the reader if the coordinate system were clearly defined from the start.

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