BASIC MATH I 38103 FINAL EXAM 12 NOV 1998
COPY DOWN EACH PROBLEM, AND THE PROBLEM NUMBER. (YOU NEED NOT
COPY THE LONG SENTENCES, ABBREVIATE!) SHOW ALL WORK. WRITE
OUT ALL STEPS THAT YOU THINK OF. DON'T DO ANY IN YOUR HEAD.
WORK SLOWLY BUT SURELY AND YOU WILL FINNISH IN TIME. CIRCLE
YOUR ANSWERS. DON'T FORGET TO CHECK SOLUTIONS TO 'SOLVE FOR X'
PROBLEMS. LOOK AT THE ORIGINAL QUESTION WHEN YOU FINNISH A
PROBLEM, TO MAKE SURE YOU'VE ANSWERED THE QUESTION. QUESTIONS
ARE WORTH 4 POINTS EACH (UNLESS MARKED LIKE []) TO MAKE A TOTAL
OF 200 POINTS. THERE ARE 4 PAGES, 57 PROBLEMS.
PART I
1) WHAT IS THE SLOPE OF THE LINE: Y= -3X+2 ? [1]
2) WHAT IS THE Y-INTERCEPT OF THE LINE Y=4X+3 ? [1]
3) DOES EVERY STRAIGHT LINE HAVE A Y-INTERCEPT? [2]
4) GRAPH THE LINE Y= -3
. 3
5) GRAPH THE LINE Y= - ---X + 6
. 2
6) FIND THE VALUE OF THE SLOPE OF THE LINE BETWEEN THE POINTS:
(6,3) AND (-2,3), OR TELL IF THE SLOPE IS UNDEFINED.
7) WRITE THE EQUATION OF THE LINE WITH A SLOPE OF -3/2 THAT
PASSES THROUGH THE POINT (-4,3)
8) WRITE THE EQUATION OF THE LINE THAT PASSES THROUGH THE
POINTS: (4,-5) AND (4,3)
9) WRITE THE EQUATION OF THIS LINE:
PART II
10) DOES EVERY SYSTEM OF EQUATIONS HAVE A SOLUTION? [2]
11) CAN A SYSTEM OF LINEAR EQUATIONS HAVE EXACTLY TWO DIFFERENT
SOLUTIONS? [2]
12) MATCH THE SYSTEM OF EQUATIONS WITH THE CORRECT CONCLUSION:
[9]
A) Y=2X+1 B) X+Y=1 C) X+Y=1
. Y=X+2 2X+2Y=2 2X+2Y=4
1) NO SOLUTION 2) EXACTLY ONE 3) INFINITE NUMBER
. SOLUTION OF SOLUTIONS
. 2
13) IS THE GRAPH OF THE EQUATION Y=X A STRAIGHT LINE? [bonus]
SOLVE EACH SYSTEM OF LINEAR EQUATIONS. DON'T FORGET TO CHECK!
14) BY SUBSTITUTION: 15) BY GRAPHING: 16) BY ADDITION:
. 4X + Y = 14 Y = X - 1 X - Y = 4
. 3X - 2Y = 5 2Y + X = 4 2X - 3Y = 9
[6] [9] [6]
PART III
. -5 6
17) WHICH IS BIGGER 4 OR 3 ? [1]
. 2 2
18) TRUE OR FALSE: (X+4) = X + 16 [2]
19) WHAT IS THE AREA OF THE FLOOR OF A ROOM 10 FT. WIDE, 8 FT.
LONG AND 12 FT. HIGH?:
A) 80 FT. B) 960 SQUARE FT. B) 960 CUBIC FT. C) 80 SQUARE FT.
. 3 2 2 2 0 0
20) ( -3X Y ) *(4X Y) = 21) 4 * 3 =
. -2
22) WRITE 10 AS A DECIMAL.
SIMPLIFY, WRITING THE ANSWER WITHOUT NEGATIVE EXPONENTS.
. 2 -2 0 2 7 -4
23) 4X Y 24) 4X Y 25) (X )*(X ) =
. -------- ----------
. 2 -2 4
. (XY) 12Y X
. 3 2 2
26) (8X - 3X - 4Y) - (-4X + 8X + 2Y) =
. 2 2
27) (X-1) * ( X + X - 1) = 28) (4X-3) =
29) WRITE 0.00821 IN SCIENTIFIC NOTATION
. -5 2
30) WHICH IS GREATER? 8.98 X10 OR 1.001 X10 [2]
. 2
. -X - X
31) IF Y = -------- AND X= -2, CALCULATE THE VALUE OF Y.
. 3
. X
PART IV
32) HOW MANY DIFFERENT SOLUTIONS MIGHT A QUADRATIC EQUATION
HAVE? [2]
33) HOW MANY DIFFERENT SOLUTIONS MIGHT A 3RD DEGREE EQUATION
HAVE? [bonus]
. 2
34) SOLVE X = 4 35) FACTOR: 6X+9 [1]
36) DO THE SOLUTIONS OF QUADRATIC EQUATIONS REFER TO THE POINTS
OF INTERSECTIONS OF CURVED LINES OR STRAIGHT LINES? [bonus]
. 2 3 2
37) FACTOR: 2X - 5X - 3 38) FACTOR: X - 6X - 8X
. 2
39) IS: 3X + 1 = 4 A QUADRATIC EQUATION? SOLVE FOR X.
. 2
40) IS: 5X = X - 14 A QUADRATIC EQUATION? SOLVE FOR X.
SOLVE FOR X:
. 2
41) X +100 = 0
. 3
42) X(X+3) = 2X + 6 43) X -9X = 0
. 2
44) DIVIDE BY LONG DIVISION: (2X - 5X -3) - (X-3)
PART V
45) ARRANGE THE FOLLOWING NUMBERS IN ORDER FROM THE SMALLEST TO
LARGEST: 4 4 4 4 4 4 4
. ---, ----, ---, ---, ---, ---, -----
. 2 1 8 1 4 -1 1
. --- ---
. 2 10
. 4
46) WHERE WOULD THE FRACTION ---, GO IF YOU IMAGINE IT HAS A
VALUE? 0 [bonus]
. 2 2
47) TRUE OR FALSE: X + X X
[2] ---------- = ---
. 4 + X 4
48) TRUE OR FALSE: 6X + 3 9
. -------- = -----
[2] X(X+1) X+1
. 2
. X -3X - 28
49) SIMPLIFY --------------
. 2
. X - 49
. 2 2
. X - 64 X -8X
50) SIMPLIFY --------- - -------
. 2
. 8X + 64 8X
. 3 4
51) ADD: --- + ----- =
. X X+1
. X 2
52) ADD ------ - ------- =
. 2
. 2X+4 X +2X
. 3 4
53) SOLVE FOR X: ----- = ---
. X-2 X
. 6 3X
54) SOLVE FOR X: X + ----- = -----
[6] X-2 X-2
. 1 1
55) SIMPLIFY: X - --- 56) ------ =
. X [2] 1
. --------- ---
. X + 1 2
PART VI
CHOOSE ONE QUESTION AND WRITE ABOUT A HALF A PAGE ON IT. USE
EXAMPLES, EQUATIONS, CALCULATIONS, GRAPHS, ETC... WRITE IN
ENGLISH SENTENCES! [10]
. 0
A) WHY IS 4 =1?
. 2 3
B) WHY CAN'T I ADD X + X = X ?
C) NOW LOOK OVER QUESTIONS #11, #13, #32 AND #36 AND DESCRIBE
THE SITUATION IN DETAIL.
. 4
D) EXPLAIN WHY --- IS UNDEFINED.
. 0
Thank you. I hope you learned more than just some algebra
this semester. I enjoyed your company. Have a jolly winter!
Friday, January 7, 2011
Elementary Algebra Final Exam I Gave Once (or use it as a review!)
Thursday, January 6, 2011
How Should You Tilt a Bottle To Pour The Liquid Out Fastest?
Blackskimmmer:
engineering question: tube closed at one open at the other. 1"wide, 6 feet long. filled to the brim with water.
what angle below the hoizontal drains it the fastest?
http://sfbay.craigslist.org/forums/?ID=67718622
1)
90 degrees -- upside down would be fastest
Blackskimmmer:
are you sure? the bubbles get in the way.
Zillionaire:
nope. Depends on how stable the tube. The "stickiness" of the water and geometry of the tube determines the angle... I think.
Gotta find the optimum glug, glug frequecy, bubble size and fall rate...
Um... 45 degrees?
Zillionaire:
Start at zero go to 90
Blackskimmmer:
describe the function of angle versus time
Zillionaire:
well... If you did it really fast at the right focal point, you could generate significant cetripetal acceleration. How about infitity degrees per second?
Blackskimmmer:
that's cheating, don't make things complicated yet.
Cyclopia:
Water weighs 1 gram/cc
and is very wet.
It's a good solvent, too.
My favorite beers are about 94% water.
And I am composed mostly of water.
It's good stuff.
1.6)
Blackskimmmer:
second guess the instabilities?
tminus7:
Rayleigh-Taylor instabilities.
http://en.wikipedia.org/wiki/Rayleigh-Taylor_instability
The real reason the water falls out of the tube.
Blackskimmmer:
that article hardly made a lick'o' sense, but i like the idea that crab nebula is RT fingers. my 2nd favorite place in the universe.
well as i aint got no answers to my problem, i'll have to go out and get a tube and experiment myself.
i'll report on it.
why do you say these instablilties is why the water flows out of the tube? in the turbulent regime? or is it just what gets it initiating the flow?
tminus7:
It is what intiates the flow.
Remember the only reason we are talking about this problem at all is that the outside air pressure is pushing the water back into the closed pipe with more force than the water weight pusing out. This is the condition Of RT instability. A low density fluid, air, pushing on a high density fluid, water. The simulation picture in the wiki article, if you turn it upside down is, is exactly the OP's problem. This is for the pipe straight vertical.
One simple proof is the old playing card/ glass of water trick. Place a card over the open end of a glass of water. Hold it and invert the glass. Let go of the card and, magically, the water does not fall out! The card is a stiff solid and changes the conditions away from the RT condition. The card is not free to flow. But it shows the outside air pressure is great enough to hold the water (and card) in the glass. So gravity is insufficient to pull the water out of an inverted glass. You need RT to make it fall!
Blackskimmmer:
aha! you are assuming that the if the water leaves the tube at first without any air flowing all the way to the end first, that the water will have to leave a vacuum at the opposite end, and thus... i'm not sure i understand... ok i understand merely a bubble flowing up through the tube of water but it's flowing up because the water over it is flowing down it, bouyancy and all... i still don't understand how to make a free body diagram with the volume elements to show why things move...
argh.... physics is hard!
ok, at least with the card trick i see that if the water tries to bulge out the upsidown glass with the card, it's gotta leave a vacuum at the top.. so air pressure keeps the card horizontal..
if i replace the card with the surface of the water... (surface tension effects? argghhh).. same problem, so .. you got to get some instability hapening... wow what would a movie be of the shape of the surface of that water the first millisecond that you remove the card (without causing any eddies where it touches... yikes, how to do it clean?)
surface tension IS important. if you do the trick with a hollow glass stirir, the water doesnt come out at all! hmm... what's the cuttof diameter per given hight of column...
yikes! fluids!
2)
Zillionaire:
answer is: 0
Zillionaire:
i mean negative tan(dia/lenth)
Blackskimmmer:
curious suggestion. that's the angle of the triangle from the lip of the tube to the opposite edge.
hmmm... gonna have to try this.
InOldenDays:
Yeah, nice idea, Z.
Angular measure, though: "arctan" instead of "tan".
3)
TheRealTryanJ:
1 degree below hoizontal so it doesnt bulid vacum to prevent water from flowing.
Blackskimmmer:
that's the idea! but how do you know 3degrees isn't faster? tricky problem.
i'm gonna have to go try it later.
4)
Geoloseth:
the angle will have to change in relation to the amount of water left in the tube. In essence you tip it over and once it starts to drain you keep tipping it to the maximum angle before air gets trapped in the tube.
Blackskimmmer:
that's a second version of the question.
5)
Toober:
I'll guess. The glu-glug frequency is a function of viscosity. I'm also not convinced that the recoil due to glugging would be enough to overcome a velocity of the maximum downward angle.
BUT, assuming you want to minimize glugging. That means getting a complete layer of incoming air to the very back of the tube.
So, place the tube at zero degrees horizontal, and draw a line from the lower open lip back to the upper closed lip. (you can figure it out from the dimensions you gave).I'm guessing that that is the minimum tipping angle for non-glugging.
6)
Blackskimmmer:
Ok, here is the report on my preliminary experiments:
well, i didn't get a tube yet, so i used two bottles:
I) louza bottle 1.5" wide mouth, 12" high, 3"wide bottom walls fairly streight
horizontal: 9sec
10sec
it starts off fast and then the last bits dribble out
30deg? about 2 sec, some glugging
45deg: 2sec? some glugging
2.5sec
vertical: 1.5sec
II)beer bottle: 3/4"opening, 4" of 1" neck, 10" tall, 2" at base
horizontal: 20sec, glugging, doesn't empty all the way slows to a dribble etc..
20deg? close to making the line from upper lip to rear bottom end horizontal: 11sec
11sec
45deg: 10sec fairly periodic glugging
9 sec
vertical: 8.5sec nonperiodic glugging? the glugging caused my hand to shake
8.5sec
7.5 sec
8.5sec
dynamic: 10sec
10sec
8 sec starting vertical and going more horizontal at end
10sec
9sec
dynamic maintaining NO glugging: 16sec
18sec
dynamic vertical then horiz then final vertical spill: 9sec
9sec
none of the measurements are precise. to the nearest second because how well could i time when i released the water with my palm? i used my computers clock with seconds. windows XP and IBM thinkpad.
http://sfbay.craigslist.org/forums/?ID=67868883
Blackskimmmer:
wow, i haven't done a physics exp in a while
alot of work!
analysis: while the glugging close to vertical is pretty bad, the speed of emptying between glugs and the speed of emptying at the end (less glugs) seems to more than make up for it.
would be interesting to time precisely the vertical emptying times maybe 100 times and see what kind of distribution i get.
would be interesting to analyse the periodicity of the gluging at various angles.
curious that dynamically changing the angle i couldn't get much better timing!
this is fun. it would be interesting to get a good long cylinder and a stand with a clamp and measure the angle (arcsin(height of rear end/len) more accurately. the longer the tube, the less imprecise my timings will be due to my impreciese "shutter" opening!
Toober:
To standardize:
When I had done similar experiments with students, we used PVC pipe closed-off with a PVC valve that you can cheaply purchase (>$8?) at Home Depot. The experiments we had done were pressure vs. height vs. velocity experiments, but only at 90-degrees vertical.
please consider this response: http://forums.newyork.craigslist.org/?act=Q&ID=67751574
Someday, I'd like to get back to Newtonian experiments.
Blackskimmmer:
yes, i hope to get around to finding some tubing, should be fun.
i suppose if i had glass tubing i could do things like time how long a bubble takes to rise to the top. i recall an exhibit on davinci and one of the things he messed with was the chaotic motion of bubbles...
Toober:
He described bubbles, but only as a part of turbulence (as I recall from my readings).
Blackskimmmer:
this stuff, from:
http://www.esam.northwestern.edu/~miksis/
Computation of Moving Boundary Problems in Fluid Dynamics
Rising gas bubbles play an important role in many physical and biological processes, such as the dynamics of multiphase flows, cavitation processes, and the flow of bubbles in the bloodstream. The rise of gas bubbles and the observation of a path instability has been documented since the time of Leonardo Da Vinci, but questions related to the origin of this instability still exist. CatherineNorman thesis topic was concerned with the development of a level-set numerical method to study the dynamics of rising bubbles. She considered both bubbles rising under an inclined plane and bubble free rising. She considered both cases were there was a film of liquid between the bubble and the plane and classes where there was a three-phase contact line. Hecode allowed for adaptive meshing and she developed a full second-order method.
On the right is a numerical calculation using the level-set code of C. Normann. Here we see a three-dimensional gasbubble rising from rest. The bubble initially rising along the center-line and flattens in the direction it is moving. With increasing distance, a spiraling path instability occurs. Because the bubble is rising inside of a finite channel, the spiraling instability eventually becomes a zig-zag instability where the bubble move back and forth in a center-plane normal to two of the faces.
References
* C.Norman and M.J. Miksis, "Dynamics of a Gas Bubble in an Inclined Channel at Finite Reynolds Number", Phys. Fluids, 17(2), 022102, 2005.
* C.Norman and M.J. Miksis, "Gas Bubble with a Moving Contact Line Rising in an Inclined Channel at Finite Reynolds Number", Physica D, 209, 191-204, 2005.
* C.E. Norman, "A level-set numerical method to determine the dynamics of gas bubbles in inclined channels", Ph.D. Thesis, Northwestern University, June 2005.
another description with davinci quote:
http://ct-cr4.chem.uva.nl/single_bubble/nature.pdf
engineering question: tube closed at one open at the other. 1"wide, 6 feet long. filled to the brim with water.
what angle below the hoizontal drains it the fastest?
http://sfbay.craigslist.org/forums/?ID=67718622
1)
90 degrees -- upside down would be fastest
Blackskimmmer:
are you sure? the bubbles get in the way.
Zillionaire:
nope. Depends on how stable the tube. The "stickiness" of the water and geometry of the tube determines the angle... I think.
Gotta find the optimum glug, glug frequecy, bubble size and fall rate...
Um... 45 degrees?
Zillionaire:
Start at zero go to 90
Blackskimmmer:
describe the function of angle versus time
Zillionaire:
well... If you did it really fast at the right focal point, you could generate significant cetripetal acceleration. How about infitity degrees per second?
Blackskimmmer:
that's cheating, don't make things complicated yet.
Zillionaire:
It is complicated, I would need to know the properties of water and air.
It is complicated, I would need to know the properties of water and air.
Cyclopia:
Water weighs 1 gram/cc
and is very wet.
It's a good solvent, too.
My favorite beers are about 94% water.
And I am composed mostly of water.
It's good stuff.
1.6)
Blackskimmmer:
second guess the instabilities?
tminus7:
Rayleigh-Taylor instabilities.
http://en.wikipedia.org/wiki/Rayleigh-Taylor_instability
The real reason the water falls out of the tube.
Blackskimmmer:
that article hardly made a lick'o' sense, but i like the idea that crab nebula is RT fingers. my 2nd favorite place in the universe.
well as i aint got no answers to my problem, i'll have to go out and get a tube and experiment myself.
i'll report on it.
why do you say these instablilties is why the water flows out of the tube? in the turbulent regime? or is it just what gets it initiating the flow?
tminus7:
It is what intiates the flow.
Remember the only reason we are talking about this problem at all is that the outside air pressure is pushing the water back into the closed pipe with more force than the water weight pusing out. This is the condition Of RT instability. A low density fluid, air, pushing on a high density fluid, water. The simulation picture in the wiki article, if you turn it upside down is, is exactly the OP's problem. This is for the pipe straight vertical.
One simple proof is the old playing card/ glass of water trick. Place a card over the open end of a glass of water. Hold it and invert the glass. Let go of the card and, magically, the water does not fall out! The card is a stiff solid and changes the conditions away from the RT condition. The card is not free to flow. But it shows the outside air pressure is great enough to hold the water (and card) in the glass. So gravity is insufficient to pull the water out of an inverted glass. You need RT to make it fall!
Blackskimmmer:
aha! you are assuming that the if the water leaves the tube at first without any air flowing all the way to the end first, that the water will have to leave a vacuum at the opposite end, and thus... i'm not sure i understand... ok i understand merely a bubble flowing up through the tube of water but it's flowing up because the water over it is flowing down it, bouyancy and all... i still don't understand how to make a free body diagram with the volume elements to show why things move...
argh.... physics is hard!
ok, at least with the card trick i see that if the water tries to bulge out the upsidown glass with the card, it's gotta leave a vacuum at the top.. so air pressure keeps the card horizontal..
if i replace the card with the surface of the water... (surface tension effects? argghhh).. same problem, so .. you got to get some instability hapening... wow what would a movie be of the shape of the surface of that water the first millisecond that you remove the card (without causing any eddies where it touches... yikes, how to do it clean?)
surface tension IS important. if you do the trick with a hollow glass stirir, the water doesnt come out at all! hmm... what's the cuttof diameter per given hight of column...
yikes! fluids!
2)
Zillionaire:
answer is: 0
Zillionaire:
i mean negative tan(dia/lenth)
Blackskimmmer:
curious suggestion. that's the angle of the triangle from the lip of the tube to the opposite edge.
hmmm... gonna have to try this.
InOldenDays:
Yeah, nice idea, Z.
Angular measure, though: "arctan" instead of "tan".
3)
TheRealTryanJ:
1 degree below hoizontal so it doesnt bulid vacum to prevent water from flowing.
Blackskimmmer:
that's the idea! but how do you know 3degrees isn't faster? tricky problem.
i'm gonna have to go try it later.
4)
Geoloseth:
the angle will have to change in relation to the amount of water left in the tube. In essence you tip it over and once it starts to drain you keep tipping it to the maximum angle before air gets trapped in the tube.
Blackskimmmer:
that's a second version of the question.
5)
Toober:
I'll guess. The glu-glug frequency is a function of viscosity. I'm also not convinced that the recoil due to glugging would be enough to overcome a velocity of the maximum downward angle.
BUT, assuming you want to minimize glugging. That means getting a complete layer of incoming air to the very back of the tube.
So, place the tube at zero degrees horizontal, and draw a line from the lower open lip back to the upper closed lip. (you can figure it out from the dimensions you gave).I'm guessing that that is the minimum tipping angle for non-glugging.
6)
Blackskimmmer:
Ok, here is the report on my preliminary experiments:
well, i didn't get a tube yet, so i used two bottles:
I) louza bottle 1.5" wide mouth, 12" high, 3"wide bottom walls fairly streight
horizontal: 9sec
10sec
it starts off fast and then the last bits dribble out
30deg? about 2 sec, some glugging
45deg: 2sec? some glugging
2.5sec
vertical: 1.5sec
II)beer bottle: 3/4"opening, 4" of 1" neck, 10" tall, 2" at base
horizontal: 20sec, glugging, doesn't empty all the way slows to a dribble etc..
20deg? close to making the line from upper lip to rear bottom end horizontal: 11sec
11sec
45deg: 10sec fairly periodic glugging
9 sec
vertical: 8.5sec nonperiodic glugging? the glugging caused my hand to shake
8.5sec
7.5 sec
8.5sec
dynamic: 10sec
10sec
8 sec starting vertical and going more horizontal at end
10sec
9sec
dynamic maintaining NO glugging: 16sec
18sec
dynamic vertical then horiz then final vertical spill: 9sec
9sec
none of the measurements are precise. to the nearest second because how well could i time when i released the water with my palm? i used my computers clock with seconds. windows XP and IBM thinkpad.
http://sfbay.craigslist.org/forums/?ID=67868883
Blackskimmmer:
wow, i haven't done a physics exp in a while
alot of work!
analysis: while the glugging close to vertical is pretty bad, the speed of emptying between glugs and the speed of emptying at the end (less glugs) seems to more than make up for it.
would be interesting to time precisely the vertical emptying times maybe 100 times and see what kind of distribution i get.
would be interesting to analyse the periodicity of the gluging at various angles.
curious that dynamically changing the angle i couldn't get much better timing!
this is fun. it would be interesting to get a good long cylinder and a stand with a clamp and measure the angle (arcsin(height of rear end/len) more accurately. the longer the tube, the less imprecise my timings will be due to my impreciese "shutter" opening!
Toober:
To standardize:
When I had done similar experiments with students, we used PVC pipe closed-off with a PVC valve that you can cheaply purchase (>$8?) at Home Depot. The experiments we had done were pressure vs. height vs. velocity experiments, but only at 90-degrees vertical.
please consider this response: http://forums.newyork.craigslist.org/?act=Q&ID=67751574
Someday, I'd like to get back to Newtonian experiments.
Blackskimmmer:
yes, i hope to get around to finding some tubing, should be fun.
i suppose if i had glass tubing i could do things like time how long a bubble takes to rise to the top. i recall an exhibit on davinci and one of the things he messed with was the chaotic motion of bubbles...
Toober:
He described bubbles, but only as a part of turbulence (as I recall from my readings).
Blackskimmmer:
this stuff, from:
http://www.esam.northwestern.edu/~miksis/
Computation of Moving Boundary Problems in Fluid Dynamics
Rising gas bubbles play an important role in many physical and biological processes, such as the dynamics of multiphase flows, cavitation processes, and the flow of bubbles in the bloodstream. The rise of gas bubbles and the observation of a path instability has been documented since the time of Leonardo Da Vinci, but questions related to the origin of this instability still exist. CatherineNorman thesis topic was concerned with the development of a level-set numerical method to study the dynamics of rising bubbles. She considered both bubbles rising under an inclined plane and bubble free rising. She considered both cases were there was a film of liquid between the bubble and the plane and classes where there was a three-phase contact line. Hecode allowed for adaptive meshing and she developed a full second-order method.
On the right is a numerical calculation using the level-set code of C. Normann. Here we see a three-dimensional gasbubble rising from rest. The bubble initially rising along the center-line and flattens in the direction it is moving. With increasing distance, a spiraling path instability occurs. Because the bubble is rising inside of a finite channel, the spiraling instability eventually becomes a zig-zag instability where the bubble move back and forth in a center-plane normal to two of the faces.
References
* C.Norman and M.J. Miksis, "Dynamics of a Gas Bubble in an Inclined Channel at Finite Reynolds Number", Phys. Fluids, 17(2), 022102, 2005.
* C.Norman and M.J. Miksis, "Gas Bubble with a Moving Contact Line Rising in an Inclined Channel at Finite Reynolds Number", Physica D, 209, 191-204, 2005.
* C.E. Norman, "A level-set numerical method to determine the dynamics of gas bubbles in inclined channels", Ph.D. Thesis, Northwestern University, June 2005.
another description with davinci quote:
http://ct-cr4.chem.uva.nl/single_bubble/nature.pdf
When You Drip Milk In Coffee, Which One Splashes Up?
Preliminary discussion and experiments:
Question:
dripping drops into a pool of liquid. when you drip liquid A into a pool of liquid B, one drop at a time, you get a tiny splash upon impact.
Is the splash that you see a small amount of liquid B being flung into the air, or is it liquid A bouncing off of the surface of liquid B?
http://sfbay.craigslist.org/forums/?ID=91386638
Blackskimmmer:
lets look:
maybe the website describes the experimental setup and we could use two immiscible fluids of different colors to find out?
cool pics of splashes:
http://courses.ncssm.edu/hsi/class2000/splashes/pictures2.htm
find the equations for this!
Blackskimmmer:
more pics and an article about bubbles in champagne:
http://courses.ncssm.edu/hsi/class2005/splashes/a/Balloon_Index.htm
http://courses.ncssm.edu/hsi/class2005/splashes/a/water_index.htm
http://www.europhysicsnews.com/full/13/article3/article3.html
Toober:
Perhaps a moving frame of reference
When I learned Navier Stokes as a chemical engineer, we always resorted to solvable boundary conditions within a static frame of reference (like a pipe wall, or sphere boundary, or rectangular channel). But I don't think that stasis is required. Or, another way to solve it is to treat the drop as a Newtonian sphere yet allow for "differentiable" elastic liquid smearing at the surface of impact. That's the ticket.
2)
well, my personal observation of dripping half-n-half into a cup of black coffee, is that the splash is white.
3)
iamlucky13:
Here's an example
First of all, yes these are real. It's from a photo contest site I sometimes browse. The photographer spent hours setting them up.
The ones to check out are the milk drops into coffee. You can see some mixing in the center column, but the outer edges appear to be mostly milk, suggesting that the milk drop rebounds against the coffee.
http://www.dpchallenge.com/portfolio.php?USER_ID=52549&collection_id=20111
iamlucky13:
Oops...correction
Reading her comments in a the "Octopus" photo, it sounds like the white milk umbrella forms when a second drop impacts the rising coffee column from a previous drop.
So the splash appears to be mixed coffee and milk.
4)
Blackskimmmer:
my results: mixing
water into soy milk from 6" to a foot, white spikes a cm or so
soy milk into water from ditto white spikes a cm or so
soy milk into olive oil from a foot, really viscous white blobs (the olive oil was in a 2cm deep bottle cap
olive oil into soy milk 6" to a foot, white spikes a cm or so
well, use imagination and interpret results. some mixing or at least the matrix fluid is clinging to the surface of the dropped fluid as it bounces back out.
this is fun, i'll try more variations when i get some time
Question:
dripping drops into a pool of liquid. when you drip liquid A into a pool of liquid B, one drop at a time, you get a tiny splash upon impact.
Is the splash that you see a small amount of liquid B being flung into the air, or is it liquid A bouncing off of the surface of liquid B?
http://sfbay.craigslist.org/forums/?ID=91386638
Blackskimmmer:
lets look:
maybe the website describes the experimental setup and we could use two immiscible fluids of different colors to find out?
cool pics of splashes:
http://courses.ncssm.edu/hsi/class2000/splashes/pictures2.htm
find the equations for this!
Blackskimmmer:
more pics and an article about bubbles in champagne:
http://courses.ncssm.edu/hsi/class2005/splashes/a/Balloon_Index.htm
http://courses.ncssm.edu/hsi/class2005/splashes/a/water_index.htm
http://www.europhysicsnews.com/full/13/article3/article3.html
Toober:
Perhaps a moving frame of reference
When I learned Navier Stokes as a chemical engineer, we always resorted to solvable boundary conditions within a static frame of reference (like a pipe wall, or sphere boundary, or rectangular channel). But I don't think that stasis is required. Or, another way to solve it is to treat the drop as a Newtonian sphere yet allow for "differentiable" elastic liquid smearing at the surface of impact. That's the ticket.
2)
well, my personal observation of dripping half-n-half into a cup of black coffee, is that the splash is white.
3)
iamlucky13:
Here's an example
First of all, yes these are real. It's from a photo contest site I sometimes browse. The photographer spent hours setting them up.
The ones to check out are the milk drops into coffee. You can see some mixing in the center column, but the outer edges appear to be mostly milk, suggesting that the milk drop rebounds against the coffee.
http://www.dpchallenge.com/portfolio.php?USER_ID=52549&collection_id=20111
iamlucky13:
Oops...correction
Reading her comments in a the "Octopus" photo, it sounds like the white milk umbrella forms when a second drop impacts the rising coffee column from a previous drop.
So the splash appears to be mixed coffee and milk.
4)
Blackskimmmer:
my results: mixing
water into soy milk from 6" to a foot, white spikes a cm or so
soy milk into water from ditto white spikes a cm or so
soy milk into olive oil from a foot, really viscous white blobs (the olive oil was in a 2cm deep bottle cap
olive oil into soy milk 6" to a foot, white spikes a cm or so
well, use imagination and interpret results. some mixing or at least the matrix fluid is clinging to the surface of the dropped fluid as it bounces back out.
this is fun, i'll try more variations when i get some time
Why Does Water Stay In The Straw With Your Thumb Over It?
Preliminary discussion and experiments:
Hydrogyrophage:
Question (possibly leading)
You know that straw trick where you stick a straw in your drink, put your finger over the top of it, and pull the straw out of the drink so your straw is now full of water? And then you remove your finger, and all the liquid falls out.
As I understand it, the liquid doesn't fall out while your finger is there because to do so, it would pull a vacuum in the space between your finger and the liquid.
But if you tried this in a vacuum, doesn't that change it? If you're already in a vacuum, then you can't create more vacuum, so there should be nothing stopping the straw from allowing the vacuum to form between finger and liquid.
[...]
http://sfbay.craigslist.org/forums/?ID=158772545
1)
Blackskimmmer:
i thought it was surface tension at the bottom that holds the water in place.
do some simple experiments, try it with longer and wider straws.
after a certain width the lower surface breaks and the water spills out, no?
ditto with a longer straw? the weight of the water above the lower surface will break it?
now do i have any tubing to try this with? now i'm not so sure about a longer straw...
maybe there is also the interface between the water and the straw that is holding the water in there.
Kenguy119:
You're a genius
I was thinking about this after my initial reactive post. And was thinking the real culprit is surface tension. What I was thinking was what if the straw was 3 ft in diameter. No way it would hold. Since a park is being built near by there was some waste PVC. I tried my experiments and I couldn't get water to stay in anything with a diameter of larger than .5 in.
I liked your strait forward logical approach.
Blackskimmmer:
wait, simpler experiment, try it with soap in the water. or another fluid with less surface tension.
now where is that staw...
Blackskimmmer:
HA experiment rules! just tried it, made a staw
out of a pen, got the water in it.
touched the bottom with my finger, nothing
touched it with water, nothing.
touched it with dish soap, boom! half of it poured out till it reached half way up the straw, where there was no longer soap and a miniscus formed again
Hydrogyrophage:
I know that surface tension plays a part, but not the whole part. Otherwise, removing your finger at the top wouldn't cause the water to fall. Also, bulb pipettes would never work.
Blackskimmmer:
ok, did you see my post above? the next peice of teh puzzle is when you said the water doesn't fall cause you can't make a vacuum between the water and your thumb. but if the collumn of water is more than about 30 feet (at which the weight of the water is = to the weight of the atmosphere of similar cross section) the water DOES fall, and DOES leve a vacuum
Blackskimmmer:
ok, so further experimentation with my straw (4" long by 1/4" inside diameter) shows that when i lift my finger, surface tension WILL keep the water in the straw if the collumn is about 2 or 3mm high. that's the extent to the force due to surface tension.
it's confusing. with the finger, atmospheric pressure holds the water up, but only if the surface tension is there. once i break it, the interactions at the lower surface become complex and...
[...]
Hydrogyrophage:
Now this is EXACTLY what I was getting at!
Atmospheric pressure HAS to play a role in siphons.
Surface tension definitely works to counteract gravity to some extent (see capillary action), but it also plays a role in preventing air from moving through the liquid to the top.
I wonder if the water in your experiments isn't forming bubbles when you add the soap, which move to the top and allow the water to fall without pulling a vacuum.
Blackskimmmer:
thinking about the geometry and the forces at the surface of the water with or without soap gave me a headache. maybe i'll think about it some more.
the surface tension allows the atmospheric pressure to act on the entire surface as a whole, i.e. the lower surface essentially becomes a rigid membrane. and thus the atmospheric pressure can exert its force on the entire column.
when i shake the tube when the surface is convex out, a drop will pinch off, and the surface will reform, flat or concave in.
if i shake it some more a bubble will form and will rise a certain distance, which i suppose means that the collumn of water above it is sinking a distance of the thickness of the bubble. it then usualy stops half way up the tube and gets stuck! maybe at this point the collumn of water above it is not heavy enough to exert enough force to move it and surface tension/adhesion holds it in place?
another puzzle. even with a column of air above a column of water, the water does not spill out. it's got atm pressure below pushing up. but what's the presure in the collumn of air above? i don't have the tools with which to measure it.
even if there is just a bubble, i don't know if the volume of air forming the bubble gets expanded i the formation of the bubble thus lowering the air pressure inside the bubble.
it's all very confusing.
It's still confusing to me when you use the phrase "pulling a vacuum"
2)
Kenguy1192:
For the Straw: A vacuum makes no difference. It isn't a pressure differential that is keeping the water "up" It is the seal (and therefore closed system) between the fluid, straw, and your finger, that prevent a bigger vacuum.
Hydrogyrophage:
Now explain that one to me...
What scientific principle keeps the liquid from falling?
Can we draw a force diagram? What is acting on the liquid to counteract the force of gravity?
FWIW (probably not much), Wikipedia claims that atmospheric pressure plays a part.
Blackskimmmer:
force diagram... it's a fluid.. but, mg for the mass of liquid, down. air pressure times cross sectional area pushing up. now the complicated part: at the miniscus: it's i don't know what shape.. but it ISN'T horizontal! so that leads me to beleive that mg of the water is pushing down on it, and then there is the surface tension pulling with horizontal component towards the wall of the straw and vertical component up.
Blackskimmmer:
better observation: for my 4" water column in ~1/4 in. inner diameter plastic straw, i can make the bottom surface of the water convex, flat or concave.
that's curious. wait, that's with air between the top of the water and my thumb...
more experiments:
so i immerse the straw and my thumb in teh glass of water: pull it out and no space between top of water and my thumb. the miniscus is convex, bowing downward.
i shake it a little and a drop of water comes off, miniscus horizontal.
shake it again, and teh miniscus goes concave into the straw.
shake it again and another drop of water comes off and a BUBBLE sloooowly rises up the straw till it reaches finger, and now there's a region of AIR between the top of the water and my finger.
but the water STILL doesn't pour out! hmm what's the miniscus at the top look like?
lets see..
Blackskimmmer:
straw too cloudy at top to see, but here's something curious:
holding column of water in with finger at top.
shake it a little and drop of water comes out and bubble rises to the top of the collumn
shake it again and another bubble rises half way and gets STUCK in the column of water, so there is from top to bottom:
finger, thin layer of air, 2" column of water, 1/5" layer of air, 2" column of water!
Hydrogyrophage:
You missed it.
1) We can ignore the meniscus and surface tension forces. This is obviously not sufficient to keep the liquid suspended, as demonstrated when the finger at the top of the straw is removed.
2) My question is regarding this situation in a vacuum, so air pressure is NOT what is pushing the water upward to counteract gravity.
So what is?
Blackskimmmer:
i know, my answer is only part of your question... but wait, you at least read the results of my experiment? with the finger in place the water stays in the tube. when i destroy the surface tension with soap, the water pours out!
onward:
lets start here: if i raise the straw out of the water, mg in the column of the water is down. weight of atmosphere transmitted to the cross section of the straw at the surface pushes up.
the water does not fall while the weight of the water is less than the weight of the atmosphere above that column.
when you lift the straw up about 30 feet, finally the weight of the water IS greater and the higher you lift it, the water WILL fall, and WILL leave a vacuum below your finger.
this is why we invented steam engines to work at the bottom of mines. pumps at the surface could not pull water up when it was below 30 feet!
3) also, i failed to point out that probably with a THIN enough straw the water will probably stay in even with finger off. but now maybe adhesive forces between water and the walls plays a part? it's similar to my result of a 2-3mm tall column of water will stick in the tube with air above and below it and no finger holding top.
4) We also had discussion of why siphoning water with a tube from one bucket to a lower bucket works. It's related but adds MORE complications.
5) Important note from another discussion:
tminus7:
Rayleigh-Taylor instabilities.
http://en.wikipedia.org/wiki/Rayleigh-Taylor_instability
The real reason the water falls out of the tube.
Blackskimmmer:
why do you say these instablilties is why the water flows out of the tube? in the turbulent regime? or is it just what gets it initiating the flow?
tminus7:
It is what intiates the flow.
Remember the only reason we are talking about this problem at all is that the outside air pressure is pushing the water back into the closed pipe with more force than the water weight pusing out. This is the condition Of RT instability. A low density fluid, air, pushing on a high density fluid, water. The simulation picture in the wiki article, if you turn it upside down is, is exactly the OP's problem. This is for the pipe straight vertical.
One simple proof is the old playing card/ glass of water trick. Place a card over the open end of a glass of water. Hold it and invert the glass. Let go of the card and, magically, the water does not fall out! The card is a stiff solid and changes the conditions away from the RT condition. The card is not free to flow. But it shows the outside air pressure is great enough to hold the water (and card) in the glass. So gravity is insufficient to pull the water out of an inverted glass. You need RT to make it fall!
Hydrogyrophage:
Question (possibly leading)
You know that straw trick where you stick a straw in your drink, put your finger over the top of it, and pull the straw out of the drink so your straw is now full of water? And then you remove your finger, and all the liquid falls out.
As I understand it, the liquid doesn't fall out while your finger is there because to do so, it would pull a vacuum in the space between your finger and the liquid.
But if you tried this in a vacuum, doesn't that change it? If you're already in a vacuum, then you can't create more vacuum, so there should be nothing stopping the straw from allowing the vacuum to form between finger and liquid.
[...]
http://sfbay.craigslist.org/forums/?ID=158772545
1)
Blackskimmmer:
i thought it was surface tension at the bottom that holds the water in place.
do some simple experiments, try it with longer and wider straws.
after a certain width the lower surface breaks and the water spills out, no?
ditto with a longer straw? the weight of the water above the lower surface will break it?
now do i have any tubing to try this with? now i'm not so sure about a longer straw...
maybe there is also the interface between the water and the straw that is holding the water in there.
Kenguy119:
You're a genius
I was thinking about this after my initial reactive post. And was thinking the real culprit is surface tension. What I was thinking was what if the straw was 3 ft in diameter. No way it would hold. Since a park is being built near by there was some waste PVC. I tried my experiments and I couldn't get water to stay in anything with a diameter of larger than .5 in.
I liked your strait forward logical approach.
Blackskimmmer:
wait, simpler experiment, try it with soap in the water. or another fluid with less surface tension.
now where is that staw...
Blackskimmmer:
HA experiment rules! just tried it, made a staw
out of a pen, got the water in it.
touched the bottom with my finger, nothing
touched it with water, nothing.
touched it with dish soap, boom! half of it poured out till it reached half way up the straw, where there was no longer soap and a miniscus formed again
Hydrogyrophage:
I know that surface tension plays a part, but not the whole part. Otherwise, removing your finger at the top wouldn't cause the water to fall. Also, bulb pipettes would never work.
Blackskimmmer:
ok, did you see my post above? the next peice of teh puzzle is when you said the water doesn't fall cause you can't make a vacuum between the water and your thumb. but if the collumn of water is more than about 30 feet (at which the weight of the water is = to the weight of the atmosphere of similar cross section) the water DOES fall, and DOES leve a vacuum
Blackskimmmer:
ok, so further experimentation with my straw (4" long by 1/4" inside diameter) shows that when i lift my finger, surface tension WILL keep the water in the straw if the collumn is about 2 or 3mm high. that's the extent to the force due to surface tension.
it's confusing. with the finger, atmospheric pressure holds the water up, but only if the surface tension is there. once i break it, the interactions at the lower surface become complex and...
[...]
Hydrogyrophage:
Now this is EXACTLY what I was getting at!
Atmospheric pressure HAS to play a role in siphons.
Surface tension definitely works to counteract gravity to some extent (see capillary action), but it also plays a role in preventing air from moving through the liquid to the top.
I wonder if the water in your experiments isn't forming bubbles when you add the soap, which move to the top and allow the water to fall without pulling a vacuum.
Blackskimmmer:
thinking about the geometry and the forces at the surface of the water with or without soap gave me a headache. maybe i'll think about it some more.
the surface tension allows the atmospheric pressure to act on the entire surface as a whole, i.e. the lower surface essentially becomes a rigid membrane. and thus the atmospheric pressure can exert its force on the entire column.
when i shake the tube when the surface is convex out, a drop will pinch off, and the surface will reform, flat or concave in.
if i shake it some more a bubble will form and will rise a certain distance, which i suppose means that the collumn of water above it is sinking a distance of the thickness of the bubble. it then usualy stops half way up the tube and gets stuck! maybe at this point the collumn of water above it is not heavy enough to exert enough force to move it and surface tension/adhesion holds it in place?
another puzzle. even with a column of air above a column of water, the water does not spill out. it's got atm pressure below pushing up. but what's the presure in the collumn of air above? i don't have the tools with which to measure it.
even if there is just a bubble, i don't know if the volume of air forming the bubble gets expanded i the formation of the bubble thus lowering the air pressure inside the bubble.
it's all very confusing.
It's still confusing to me when you use the phrase "pulling a vacuum"
2)
Kenguy1192:
For the Straw: A vacuum makes no difference. It isn't a pressure differential that is keeping the water "up" It is the seal (and therefore closed system) between the fluid, straw, and your finger, that prevent a bigger vacuum.
Hydrogyrophage:
Now explain that one to me...
What scientific principle keeps the liquid from falling?
Can we draw a force diagram? What is acting on the liquid to counteract the force of gravity?
FWIW (probably not much), Wikipedia claims that atmospheric pressure plays a part.
Blackskimmmer:
force diagram... it's a fluid.. but, mg for the mass of liquid, down. air pressure times cross sectional area pushing up. now the complicated part: at the miniscus: it's i don't know what shape.. but it ISN'T horizontal! so that leads me to beleive that mg of the water is pushing down on it, and then there is the surface tension pulling with horizontal component towards the wall of the straw and vertical component up.
Blackskimmmer:
better observation: for my 4" water column in ~1/4 in. inner diameter plastic straw, i can make the bottom surface of the water convex, flat or concave.
that's curious. wait, that's with air between the top of the water and my thumb...
more experiments:
so i immerse the straw and my thumb in teh glass of water: pull it out and no space between top of water and my thumb. the miniscus is convex, bowing downward.
i shake it a little and a drop of water comes off, miniscus horizontal.
shake it again, and teh miniscus goes concave into the straw.
shake it again and another drop of water comes off and a BUBBLE sloooowly rises up the straw till it reaches finger, and now there's a region of AIR between the top of the water and my finger.
but the water STILL doesn't pour out! hmm what's the miniscus at the top look like?
lets see..
Blackskimmmer:
straw too cloudy at top to see, but here's something curious:
holding column of water in with finger at top.
shake it a little and drop of water comes out and bubble rises to the top of the collumn
shake it again and another bubble rises half way and gets STUCK in the column of water, so there is from top to bottom:
finger, thin layer of air, 2" column of water, 1/5" layer of air, 2" column of water!
Hydrogyrophage:
You missed it.
1) We can ignore the meniscus and surface tension forces. This is obviously not sufficient to keep the liquid suspended, as demonstrated when the finger at the top of the straw is removed.
2) My question is regarding this situation in a vacuum, so air pressure is NOT what is pushing the water upward to counteract gravity.
So what is?
Blackskimmmer:
i know, my answer is only part of your question... but wait, you at least read the results of my experiment? with the finger in place the water stays in the tube. when i destroy the surface tension with soap, the water pours out!
onward:
lets start here: if i raise the straw out of the water, mg in the column of the water is down. weight of atmosphere transmitted to the cross section of the straw at the surface pushes up.
the water does not fall while the weight of the water is less than the weight of the atmosphere above that column.
when you lift the straw up about 30 feet, finally the weight of the water IS greater and the higher you lift it, the water WILL fall, and WILL leave a vacuum below your finger.
this is why we invented steam engines to work at the bottom of mines. pumps at the surface could not pull water up when it was below 30 feet!
3) also, i failed to point out that probably with a THIN enough straw the water will probably stay in even with finger off. but now maybe adhesive forces between water and the walls plays a part? it's similar to my result of a 2-3mm tall column of water will stick in the tube with air above and below it and no finger holding top.
4) We also had discussion of why siphoning water with a tube from one bucket to a lower bucket works. It's related but adds MORE complications.
5) Important note from another discussion:
tminus7:
Rayleigh-Taylor instabilities.
http://en.wikipedia.org/wiki/Rayleigh-Taylor_instability
The real reason the water falls out of the tube.
Blackskimmmer:
why do you say these instablilties is why the water flows out of the tube? in the turbulent regime? or is it just what gets it initiating the flow?
tminus7:
It is what intiates the flow.
Remember the only reason we are talking about this problem at all is that the outside air pressure is pushing the water back into the closed pipe with more force than the water weight pusing out. This is the condition Of RT instability. A low density fluid, air, pushing on a high density fluid, water. The simulation picture in the wiki article, if you turn it upside down is, is exactly the OP's problem. This is for the pipe straight vertical.
One simple proof is the old playing card/ glass of water trick. Place a card over the open end of a glass of water. Hold it and invert the glass. Let go of the card and, magically, the water does not fall out! The card is a stiff solid and changes the conditions away from the RT condition. The card is not free to flow. But it shows the outside air pressure is great enough to hold the water (and card) in the glass. So gravity is insufficient to pull the water out of an inverted glass. You need RT to make it fall!
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