Report 5 Grid

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

Introduction

112/04/18 Reporter: 知 物 達 理 6

Fig.1 The grid apparatus

Plastic Cover

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