Flow of fluid out of vertical tube

[swmmr1928]

Homework Statement

A liquid initially fills the inside of a vertical tube of length L and inside diameter D. The tube is capped at both ends. Then the caps are suddenly removed, and the liquid flows out of the bottom of the tube as a continuous stream until the tube is nearly empty. Assume that the flow inside the tube is laminar and that the force of the surrounding air on the stream is negligible. Gravity provides the driving force for the flow. What forces provide resistance to flow? List the parameters you think affect the rate of flow of the liquid from the tube. Give the dimensions of each parameter.

Homework Equations

na

The Attempt at a Solution

Gravity (L/t^2), Tube diameter (L), liquid density (m/L^3), liquid viscosity (m/L*t),

The forces the provide resistance to flow are inertia and viscosity(friction).

My questions to viewers: Does viscosity take care of friction at the boundary layer? Does the velocity depend on the tube length? I did not include it because it was not included in a similar example problem.

 

[Chestermiller]

Homework Statement

A liquid initially fills the inside of a vertical tube of length L and inside diameter D. The tube is capped at both ends. Then the caps are suddenly removed, and the liquid flows out of the bottom of the tube as a continuous stream until the tube is nearly empty. Assume that the flow inside the tube is laminar and that the force of the surrounding air on the stream is negligible. Gravity provides the driving force for the flow. What forces provide resistance to flow? List the parameters you think affect the rate of flow of the liquid from the tube. Give the dimensions of each parameter.

Homework Equations

na

The Attempt at a Solution

Gravity (L/t^2), Tube diameter (L), liquid density (m/L^3), liquid viscosity (m/L*t),

The forces the provide resistance to flow are inertia and viscosity(friction).

My questions to viewers: Does viscosity take care of friction at the boundary layer? Does the velocity depend on the tube length? I did not include it because it was not included in a similar example problem.

This is a bit of a tricky problem. Initially, the velocity of the fluid will be zero, so there will be no viscous drag, and inertia will dominate. As the velocity increases, you will begin to develop a (laminar) boundary layer at the wall, and the velocity profile across the tube will remain flat, except within the boundary layer. Throughout the problem, viscosity is the source of friction at the wall. It is not clear at the outset whether the length of the tube is a parameter. My intuition tells me that it is not, but I would have to set up the equations to be sure. So tentatively, I would have to say that the length of the tube is a parameter.