Rate of Rate of Flow In = Accumulation  + Rate of Flow Out

First, let’s get some basics laid out. The velocity field will be described as

V = vrêr +vϴêϴ + vzêz

I always prefer to use u, v, and w instead of ur, utheta, and uz to save on subscripts, although the latter nomenclature is a bit more descriptive… we’ll get used to it. Now, by construction, the volume of the differential control volume is

while the mass of fluid in the control volume is

The rate of change of mass or accumulation in the control volume is then

For the net flow through the control volume, we deal with it one face at a time. Starting with the r faces, the net inflow is

while the outflow in the r direction is

So that the net flow in the r direction is

Being O(dr^2), the last term in this equation can be dropped so that the net flow on the r faces is

The net flow in the theta direction is slightly easier to compute since the areas of the inflow and outflow faces are the same. At the outset, the net flow in the theta direction is

We now turn our attention to the z direction. The face area is that of a sector of angle d\theta:

then, the inflow at the lower z face is

while the outflow at the upper z face is

Finally, the net flow in the z direction is

Now we can put things together to obtain the continuity equation

dividing by dV and rearranging the r components of the velocity