What is the solution for the Continuity Equation at r=0?

[Clausius2]

Hi guys. I am solving the axisymmetric free jet of an incompressible fluid. But I have troubles at r=0. Continuty equation can be written in cylindrical coordinates as:

1/r*d(rv)/dr + du/dz=0

v=radial velocity (v=0 at r=0)
u=axial velocity.
hz=delta(z)
hr=delta(r)

What happens at r=0?. I have to obtain a finite difference scheme in order to integrate the problem, so I have created a grid that has "i" index for "z" coordinate and "j" index for "r" coordinate. j=1 corresponds to the symmetry axe.

for j>1 I have no problem because r>0. But in r=0 there is a singularity point. How can I deal with this?. As you can see I know v(i,1)=0

 

[mika]

1/r*d(vr)/dr+du/dz=0 =>

1/r*(v+r*dv/dr)+du/dz = v/r + dv/dr +du/dz

at r=0, v=0 ->

dv/dr + du/dz = 0

does this help?

 

[Clausius2]

Hey, thanks for your reply.

But it does not help me so much. At r=0, v=0, and it is undetermined. I have had time to investigate and it is dealed by expanding du/dr it in Taylor Series.