Sphere falling in countercurrent liquid flow

In summary, the equation for finding the velocity of a sphere falling through a liquid without countercurrent flow is v(t) = [(p_s - p_f)*V*g/b]*[1-exp(-b*t/m)]. This information can be found at http://en.wikipedia.org/wiki/Viscous_resistance (under very low reynold's number - stokes drag). However, for countercurrent flow, the terminal velocity of the sphere can be found by subtracting the countercurrent velocity from the falling velocity. This method may not work for reynolds numbers over 1. There is also uncertainty about whether the momentum should be applied to the projected area of the sphere (Pi*r^2) or half the total
  • #1
Bakery87
11
0
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.
 
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  • #2
Bakery87 said:
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.
 

1. What is a sphere falling in countercurrent liquid flow?

A sphere falling in countercurrent liquid flow occurs when a solid object, in the shape of a sphere, is dropped into a liquid that is flowing in the opposite direction. This creates a unique scenario in which the sphere must overcome the force of gravity and the resistance of the liquid flow in order to reach its desired destination.

2. How does the size of the sphere affect its movement in countercurrent liquid flow?

The size of the sphere can greatly impact its movement in countercurrent liquid flow. Smaller spheres will typically experience more resistance from the liquid flow, making it more difficult for them to move against the current. Larger spheres, on the other hand, will have a greater mass and may be able to overcome the resistance more easily.

3. What factors affect the velocity of a sphere falling in countercurrent liquid flow?

The velocity of a sphere falling in countercurrent liquid flow is influenced by several factors. These include the size and shape of the sphere, the density and viscosity of the liquid, and the strength of the countercurrent flow. Additionally, any external forces, such as wind or turbulence, can also impact the velocity of the sphere.

4. How is the motion of a sphere in countercurrent liquid flow affected by the shape of the sphere?

The shape of the sphere can play a significant role in its motion in countercurrent liquid flow. Spheres with a smooth and streamlined shape will typically experience less resistance from the liquid flow, allowing them to move more easily against the current. Spheres with a rough or irregular shape may face more resistance and have a harder time moving against the flow.

5. What are some practical applications of studying sphere falling in countercurrent liquid flow?

Studying sphere falling in countercurrent liquid flow has various practical applications. It can be used to understand the behavior of objects in different fluid environments, which can be helpful in fields such as engineering, hydrodynamics, and oceanography. This knowledge can also be applied to the design of objects such as ships, submarines, and underwater vehicles, to ensure they can move efficiently through countercurrent liquid flows.

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