Derivations for the Velocity Expressions Later Used
-- Three Dimensional Velocity overview

The arc length parameter for the 3D case is different than the 2D case.
In the 2D case, the inlet is a line, whereas for the 3D case, the inlet is a circular surface.

To visualize the 3D inlet arc length parameters, bring up the Plot Parameters dialog box. Clear all Plot type checkboxes in the General tab. Go to the Boundary tab and check the Boundary plot checkbox. In the Expression field enter s1 and press Apply.

s1 Arc Length Parameter

Notice the gradient of the inlet port goes from blue to green or from s=0 to s=0.5 and not to 1. Enter s2 in the Expression field. Again the gradient of the inlet port only goes from blue to green and the direction of the gradient is perpendicular to the direction of the gradient for the parameter s1.The velocity expression for the inlet in the 3D case is similar to the 2D case. The velocity needs to be 0 at the boundary and maximum at the center. An expression that has these features is 1-16*((s1-0.25)^2+(s2-0.25)^2). Enter this expression into the Expression field of the Boundary tab and press Apply.It is difficult to judge whether this expression is valid due to the low contrast ratio of the color map of the inlet. Go to the General tab. Check the Element selection checkbox. Enter the inequality x<0.001 && y>0.135 && y<0.155 in the Logical expression for inclusion field.
Inlet Close-up of Arc Length Expression: 1-16*((s1-0.25)^2+(s2-0.25)^2)

The figure above shows that the expression does give the desired results, 0 at the boundaries and maximum (1) at the center. Uncheck the Element selection checkbox. Press OK to exit Plot Parameters.

The next step is to generate the final velocity expression whose mean velocity is equal to the user specified Vinlet.
The expression:

will give a mean velocity equal to Vinlet and is derived as follows.

Take

with

Substitute x and y for s1 and s2 as follows.

These substitutions give

Make another substitution

to get

Simplifying results,            

The mean velocity is defined as

The integral can be evaluated by transforming the coordinate system to polar coordinates

The integral on the right hand side evaluates to

Plugging this back into the mean velocity equation gives

or

The final velocity expression with a parabaloidal profile and a mean velocity of Vinlet is
.