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

For every 2D flow cell model, the flow channel inlet is just a straight line.

Channel Inlet and Outlet for the 2D Case

In 2D COMSOL models, all geometry lines are described by a parameter s. s goes from 0 at the start point of the line to 1 at the end point of the line. This parameter allows the user to specify the distance along any geometry line which can be used in expressions to define properties along a geometry line.

To visually see what values s assumes along the inlet, go to Postprocessing->Plot Parameters…
In the General tab, uncheck all Plot type checkboxes except for the Boundary and Geometry edges checkboxes.

Plot Parameters Options

Go to the Boundary tab.
In the Expression field, enter s.
Press Apply.

Boundary Plot, Arc Length Parameter

Notice how all geometry lines change color.
Press OK to exit the Plot Parameters options.
Zoom in on the inlet.

Arc Length Parameter at the Inlet

The figure above shows that s goes from zero at the left end of the inlet to 1 at the right end of the inlet.
This arc length parameter gives the user a way to reference the inlet boundary and will be used to specify the inlet velocity along the inlet boundary.

If the inlet velocity profile is constant, the expression for the inlet velocity is Vinlet, where Vinlet is a constant.
If the inlet velocity profile is linear and goes from 0 to Vinlet, the inlet velocity expression is Vinlet*s.
The expression for the inlet velocity profile has the form V(s) = Vinlet*v(s) where v(s) is a function of s.

Possible Velocity Profile Expressions

The first requirement of the inlet velocity profile is that it needs to be a parabolic.
The second requirement is that the average velocity of this profile should equal a velocity specified by the user. This requires the mean velocity of the velocity profile to equal the inlet velocity, , where A is the area of the inlet and Vinlet is the velocity specified by the user.

A parabolic expression that will satisfy the first condition is , where C1 is a constant and s is the arc length parameter.
The actual expression should be of the form  , but since this is just the 2D case, accuracy is not that important.
The velocity expression above fulfills the first requirement but not the second requirement.
For the mean velocity to equal Vinlet, C1 should equal 6*Vinlet from the following derivation.

A = D since the geometry is 2D and the area of the inlet is the length of the inlet.
Substituting s = x/D into the velocity expression,

The mean velocity is then equal to
 
Integrating,

So,

and

The expression is thus settled upon.