4. The Material Derivative in Vector Form

[Previous Article: 3. The Material Derivative in Spherical Coordinates]

After going over the derivation of the material derivative in Cartesian, cylindrical, and spherical coordinates and seeing all the trouble that we had to go through, it is time to present the material derivative in a vector invariant form.

An invariant vector form is independent of the coordinate system used. The form will be the same for all coordinate. Of course, vector operations (i.e. gradient and curl) are not the same for different coordinate systems [see this post] once they are expanded; but the gradient is always a gradient and the curl is always a curl - only the expansion is different.

I will follow Karamcheti's explanation for obtaining the vector form of the material derivative. We start by considering a fluid particle at R measured from the origin of the coordinate system and time t. Consider also a generic scalar fluid property Q such as the temperature, pressure, or density. The scalar restriction will be removed once we obtain the general form for the material derivative. At point R and time t, the property is defined as Q(R, t). At time (t + Δt), the fluid particle moves a distance Ds and the fluid property changes accordingly to Q(R + V Δt, t + Δt)

(Fig. 1)

The total change in Q from t to (t + Δt) is

(Eq. 1)

then, the time derivative is defined as

(Eq. 2)

using Taylor's series for Q, we have

(Eq. 3)

If we substitute Eq. 3 into Eq. 2, we get the following

(Eq. 4)

Note that all high order terms disappear as the limit in the derivative is applied. The second term in Eq. 4 can be cast in vector form because it represents the derivative of Q in the direction of the streamline, tangent to the velocity vector. This means that it can be written as the dot product of the gradient of Q and the unit vector along the streamline, i.e. parallel to the velocity. Mathematically, this can be written as

(Eq. 5)

at the outset, we recover

(Eq. 6)

Voila!

This is the expression we are looking for; Eq. 6 represents the time derivative of a transported fluid property as seen from an Eulerian point of view. This also works when Q is a vector field, call it A

(Eq. 7)

However, the form given by Eq. 7 only works for Cartesian coordinates because it not invariant under coordinate transformation. This means that it does not hold true when using curvilinear coordinates such as Cylindrical or Spherical. Fortunately, we can write it using invariant form as follows

(Eq. 8)

Voila!

Specifically, when the vector field is the velocity field, then Eq. 8 simplifies quite nicely as

(Eq. 9)

[Next Article: 5. The Reynolds Transport Theorem]

Cite as:
Saad, T. "4. The Material Derivative in Vector Form". Weblog entry from Please Make A Note. http://pleasemakeanote.blogspot.com/2008/08/derivation-of-navier-stokes-equations_20.html