Modeling this differential equation

[ramb]

Hi,

My differential equations professor wanted us to think about this problem:

How would you go about modeling the rate at which water raises on a paper towel dipped into a puddle with fixed water level.

I figure this would be a differential equation, but I'm not sure as to the form of this. He hinted that the solution could be, say a number raised to the variable, or the variable raised to a power.

I'm having trouble with the physics. Initially, in the in class experiment, the paper towel soaked up the water very, very fast at first (within the first couple of seconds), then slowed down dramatically, Further, we found that the soaking level does not go past a certain point in the paper towel after a long time - and that when this experiment was repeated, it stopped at the same place. We figured that it had to do with the rate at which the water was evaporating was making up for the flow rate that the water was going upwards the towel.

So the differential equation should involve both evaporation, something to do with the towel's absorbtion, and gravity, right? The independent variable should be time, and the dependent variable should be the height on the towel the water has gotten to.

Thanks

 

[Eynstone]

The calculation will be simpler with the principle of least action. Consider the difference between the surface energies of water-air & water-paper . Add the gravitational potential to it & subtract the kinetic energy. Integrating over time yields the action.

 

[jambaugh]

Sounds like his hints are toward a simple functional model. I don't think evaporation is an issue in the time scale you are using.

If you are considering a function form consider the initial rate and the asymptotic behavior as t->infinity.

If you're going the physical route a la a differential equation then consider that gravity IS a factor and then consider how the paper behaves if it is bent to run horizontal at various heights above the liquid surface.

Just from measurements you should be able to map out the rates to get:
$$\frac{dx}{dt} = r(x)$$
determining r(x) from the rate for a given height of the bend.
You can then guess how it would be affected by gravity and what functional form r(x) takes. Then solve to see if it fits the function form you might have guessed earlier with its observed asymptotic form. You should be able to solve for constants using the max height.