report 5 grid. problem # 8 grid a plastic grid covers the open end of a cylindrical vessel...
TRANSCRIPT
Problem # 8
• Grid• A plastic grid covers the open end of
a cylindrical vessel containing water. The grid is covered and the vessel is turned upside down. What is the maximal size of holes in the grid so that the water does not flow out when the cover is removed?
112/04/18 Reporter: 知 物 達 理 2
Overview• Introduction
– Observation– Problem Analysis
• Experiment– Experimental Setup– Experiment
• Theory• Conclusions & Summary• References
112/04/18 Reporter: 知 物 達 理 3
Introduction• Observation• The water will not flow out when the holes
are small.• When a disturbance is applied to the
vessel, some water will flow out.• Vessel with larger holes are less resistant
to disturbances.
112/04/18 Reporter: 知 物 達 理 4
Introduction• Problem Analysis• Slight imbalances in pressure occurs
throughout the vessel.• The surface tension between the water
surface and the grid neutralizes the imbalances.
• When the imbalance is too great, the water surface breaks and the water flows out.
112/04/18 Reporter: 知 物 達 理 5
Experiment
• Diameter: 70mm• Thickness: 1mm• Spacing: 4mm• Hole Sizes: 4x4, 5x5, 6x6, 7x7,
7.5x7.5, 8x8, 9x9mm
112/04/18 Reporter: 知 物 達 理 7
Finding the Maximum Hole Size
Experiment
112/04/18 Reporter: 知 物 達 理 8
Finding the Maximum Hole Size
4 mm X4 mm holes 5 mmX5 mm holes
6 mmX6 mm holes 7 mmX7 mm holes
Experiment
112/04/18 Reporter: 知 物 達 理 9
Finding the Maximum Hole Size
7.5 mmX7.5 mm holes 8mmX8 mm holes
9mmX9 mm holes
Experiment
112/04/18 Reporter: 知 物 達 理 10
Finding the Maximum Hole Size
Hole Size 4x4mm 5x5mm 6x6mm
Result Success
Hole Size 7x7mm 7.5x7.5mm 8x8mm 9x9mm
Result
Successful Below
Array of 4x4
Successful Below
Array of 3x3
Successful Below
Array of 2x2
Fail
The critical hole size is between 7x7mm and 8x8mm
Theory• The Rayleigh-Taylor
instability: The instability of a dense fluid above a lower density fluid in an accelerating field.
• A small perturbation will increase the local pressure difference and therefore the displacement will keep raising until the interface break.
112/04/18 Reporter: 知 物 達 理 11
1P 2P0P
0P
2 1P P P P
Theory
The pressure difference caused by gravity can be written as
The restoring pressure caused by surface tension is
112/04/18 Reporter: 知 物 達 理 12
P g
2 0 2'P P g
1 0 1'P P g
( )x
2 1
STc
PR
Theory
Surface modes that decay by are formed
Where
The effective distance of the disturbance is
112/04/18 Reporter: 知 物 達 理 13
exp( )ky
2k
1k
1k
The total effective mass is
1 2 1 2
A Am m m
k k
Given a sinusoidal perturbation
And assume that , We get
Using
Along with
We get
Theory
112/04/18 Reporter: 知 物 達 理 14
1k
( ) exp( )x ikx
21/cR k 2
STc
P kR
P g 2m A g k
From the equations
We get
So
Theory
112/04/18 Reporter: 知 物 達 理 15
1 2 1 2
A Am m m
k k
2m A g k
2
1 2
( )k g k
exp( )ST t 2
1 2
( )ST
k g k
If is real, will be an exponential growthIf is imaginary, will be a sine wave
Theory
112/04/18 Reporter: 知 物 達 理 16
exp( )ST t
STST
( )t( )t
In critical condition,
Input the constants, we get
Using , we get
Theory
112/04/18 Reporter: 知 物 達 理 17
c
gk
1368ck m
2k
0.017c m
8.5cl mm
2 0g k
Conclusion
• As the hole size increases, it became more difficult to keep the water in the vessel
• The experimental results agree with the theoretical size of 8.5*8.5mm
112/04/18 Reporter: 知 物 達 理 18
Hole Size 7x7mm 7.5x7.5mm 8x8mm 9x9mm
Result
Successful Below
Array of 4x4
Successful Below
Array of 3x3
Successful Below
Array of 2x2
Fail