Sphere falling in countercurrent liquid flow

[Bakery87]

I'm attempting to find the velocity of a sphere falling through a liquid with an upward countercurrent flow. I already have the velocity as a function of time in a stagnant flow regime, but do not have one with countercurrent flow.

Here is the equation without countercurrent flow:
v(t) = [(p_s - p_f)*V*g/b]*[1-exp(-b*t/m)]

where:
p_s = density of sphere
p_f = density of fluid
V = volume of sphere
g = 9.8 m/s^2
t = time
m = mass of sphere
b = 6(Pi)(fluid viscosity)(sphere radius)

The information can be found at: http://en.wikipedia.org/wiki/Viscous_resistance
(under very low reynold's number - stokes drag)

I know the stokes drag will work for reynolds numbers under 1, which may not be the case.

The addition of a countercurrent flow I assume cannot be added by simply finding the terminal velocity of a falling sphere and then subtracting the countercurrent velocity. I assumed a force balance on the system by subtracting the momentum of the fluid (before the derivation to the above) but the equation doesn't seem to properly identify the velocity. I also wasn't sure if the momentum needed to be applied to the projected area of the sphere (Pi*r^2) or half the total surface area (2*Pi*r^2).

Any help would be appreciated.

 

[Chestermiller]

I'm attempting to find the velocity of a sphere falling through a liquid with an upward countercurrent flow. I already have the velocity as a function of time in a stagnant flow regime, but do not have one with countercurrent flow.

Here is the equation without countercurrent flow:
v(t) = [(p_s - p_f)*V*g/b]*[1-exp(-b*t/m)]

where:
p_s = density of sphere
p_f = density of fluid
V = volume of sphere
g = 9.8 m/s^2
t = time
m = mass of sphere
b = 6(Pi)(fluid viscosity)(sphere radius)

The information can be found at: http://en.wikipedia.org/wiki/Viscous_resistance
(under very low reynold's number - stokes drag)

I know the stokes drag will work for reynolds numbers under 1, which may not be the case.

The addition of a countercurrent flow I assume cannot be added by simply finding the terminal velocity of a falling sphere and then subtracting the countercurrent velocity.

Wrong. That's exactly what can be done.