Hey, I have this thing on my mind and I just cannot find any decent answer in books or the web. Maybe one of you guys can help me out.
It is about the accelerations terms in for instance the Navier Stokes equation. It is where the material derivative shows up as the sum of the Eulerian and convective acceleration: Du^i/dt=du^i+u^ju^i_:j, where u is the velocity in the superscript direction, a colon denotes contravariant differentiation and summation is implied.
As an example, in polar coordinates (radial r, azimuthal theta) it reads: du^r+u^r d(u^r)/dr+u^theta (d(u^r)/dtheta –r u^theta)
However I know from relativity the Eulerian should be: du^r – r u^theta u^theta, where the factor (-r) is from the Christoffel symbol. The convective is u^r d(u^r)/dr+u^theta (d(u^r)/dtheta –r u^theta), where the same contravariant diferentiation is used.
As you can easily see, this does not add up (because then it should be -2r u^theta u^theta). So my guess it the extra term in the Eulerian acceleration shouldn’t be there, but that would be against the geodesic motion as I remember it from some physics courses. So, am I making a mistake or is it correct to leave this one term out? If so, any suggestions how to interpret geodesic motion? Btw, I hope the equations are understandable.