ODE for Water traveling up a paper towel

[RagincajunLA]

Hey guys, I am in dire need of help. In physics today, my professor gave us an experiment. It involved a strip of paper towel with dots along it from a sharp. it was hung from a metal ring stand and the edge of the paper towel was immersed into a dish of water. Water traveled up the paper towel hitting the dots of ink at certain time intervals. after the data was taken, the prof plotted a graph on his computer of height of the water as a function of time. it looks like a natural log function as it has a high slope at the beginning but the slope decreases over time until it levels off. My job is to determine the function that creates this model. any help would be greatly appreciated, i don't know where to start at all for this assignment. the only piece of information that we were given was the data.

 

[tiny-tim]

Hey RagincajunLA!

… after the data was taken, the prof plotted a graph on his computer of height of the water as a function of time. it looks like a natural log function as it has a high slope at the beginning but the slope decreases over time until it levels off. My job is to determine the function that creates this model.

Have you tried plotting it against the log of time, to see if that gives a straight line?

 

[Andy Resnick]

the prof plotted a graph on his computer of height of the water as a function of time. it looks like a natural log function as it has a high slope at the beginning but the slope decreases over time until it levels off. My job is to determine the function that creates this model.

Lots of functions display this behavior besides ln(t): 1-exp(-t), tanh(t) are two.

My point is that the appropriate function does more than simply fit the data, it *models* the underlying physics- in this case, capillary rise. Sections 1.2-1.4 of this may be particularly informative:

www.t2f.nu/s2p2/s2p2_ss_1.pdf[/URL]

 

[RagincajunLA]

Tiny-tim,
Im sorry but what would plotting it against a log of time and giving a straight line provide me? I am very confused on this problem...

 

[tiny-tim]

Hi RagincajunLA!

(just got up :zzz: …)

… what would plotting it against a log of time and giving a straight line provide me?

Well, you started by saying …

… it looks like a natural log function …

… and if it is, then a log plot would give a straight line

(and a straight line is very easy to check, unlike any other sort of curve! )

 

[RagincajunLA]

tiny tim, i did what u said and found that the log of time does provide a straight line function. but when i try to reference it back to my original function, it is not accurate at all. I am stuck. Would it help if i were to post the data that i am trying to find a function for?

 

[tiny-tim]

Hi RagincajunLA!

tiny tim, i did what u said and found that the log of time does provide a straight line function. but when i try to reference it back to my original function, it is not accurate at all. I am stuck. Would it help if i were to post the data that i am trying to find a function for?

hmm … I'm not keen on wading through a lot of data

if the graph is a straight line, how can it not fit the function?

and what do you mean by "reference it back" to the original function?

 

[RagincajunLA]

actually, the graph looks more like a y=x^.5 graph, but levels off at a certain height. it has a high slope at the beginning and the slope decreases as time goes on. So here is my method that doesn't seem to work...

i am guessing that my function is y=Cx^p with p<1
I then take the natural log of both sides to get ln(y)=Cpln(x).
So then i can natural log all my y values (heights) and all my x values(times) and i should find a straight line. I think Cp will be the slope of the line... I am very lost right now

my prof said we should start out with an autonomous ODE like y'(t)=f(t). so f(t) will be the Cx^p function... but idk where to go after that. he also said we should plot the derivatives and stuff. This is all going over my head and have been thinking about it forever

 

[RagincajunLA]

its only 10 points of data. its ok if u don't have time. I understand everyone else is pretty busy during this time of year

 

[tiny-tim]

i am guessing that my function is y=Cx^p with p<1
I then take the natural log of both sides to get ln(y)=Cpln(x).

Nooo … ln(y) = pln(x) + C

Does that put it right? ​

 

[RagincajunLA]

hmmm ill have to try that out.
but isn't is ln(y)=ln(c)+pln(x)?

 

[tiny-tim]

oops!

 

[RagincajunLA]

Ok here we go. I have my notes with me this time. we were given a set of data like this:
(0,0)(1.32,5)(2.5,.75)(6.22,1)(14.25,1.25)(24.94,1.5)(224.72,2.5)(1023.68,3.5)(4330.4)

The x-values are the time values in seconds and the y values are the heights in inches.
My professor said the starting point is that the ODE is suggested to be autonomous in the form u'=f(u). By looking at the data, it has a very high slope in the beginning and curves off to be level as time goes on. Now I believe if we plot a graph of u and f(u) on the y-axis, then f(u) should start at a high point, and then curve down and level off at 0, suggesting that the function f(u) is Cu^p with p<0. SO if i am doing this correctly, we so far have u'=Cu^p with p<0. There is a problem with this because this function suggests that there is an infinite rate at u=0 which isn't possible and also that the function never goes completely to 0, so it never would reach equilibrium. So I am majorly confused on that part =(
Another thing, if i natural log both sides of the function, I'll get Ln(u')=Ln(C) + pLn(u) which should graph to be a straight line if i plot Ln(u') and Ln(u). But where can I get the values of Ln(u') from?

One last slight problem, If i do go ahead and integrate u'=Cu^p, I get
u(t)=((p+1)t)^(p+1), which never converges to a number.
As you see, I have done some work on this tonight, but am still not coming with a solution and it's driving me crazy. Studying engineering is tough work =(