(Fluids) Material and local velocity and acceleration fields.

[Lavabug]

Homework Statement

The streamlines of a fluid are as follows:
x = (x0) + 3(y0)t^2
y = (y0)/(1 + 2t)
z = (z0) + 5(x0)t

Find the velocity and acceleration fields in the Eulerian description (local).

Homework Equations

Total/material acceleration: Dv/Dt = dv/dt + v.grad(v)

The Attempt at a Solution

I am given the trajectories of the fluid in Lagrangian form, meaning a function of the initial positions and time. If I set t=0 I get x=x0, y=y0, z=z0 as the initial positions. I then solve for the initial positions as a function of x,y,z and t.

Next I take the time derivative of streamlines I am given to get the velocity field in Lagrangian(material) form, after which I substitute x0, y0, z0, which gives me the velocity field in local (Eulerian) form.

How do I calculate the acceleration? I can take the time derivative of the material velocity (2nd time derivative of the streamlines given) and then substitute x0,y0,z0. I can also use the formula I suggested, using it on the velocity field in local form (no initial positions in it).

My prof insists that doing it either way is equivalent but I've gone through the math several times and it simply isn't. Can someone please clarify these concepts/definitions for me once and for all?

 

[olivermsun]

Can you show the two procedures you are describing?

 

[Lavabug]

Here it is: http://img862.imageshack.us/img862/592/p1000992.jpg