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? 2015/4/15 Reporter: 知 物 達 理 1

Overview Introduction – Observation – Problem Analysis Experiment – Experimental Setup – Experiment Theory Conclusions & Summary References 2015/4/15 Reporter: 知 物 達 理 2

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. 2015/4/15 Reporter: 知 物 達 理 3

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. 2015/4/15 Reporter: 知 物 達 理 4

Introduction 2015/4/15 Reporter: 知 物 達 理 5 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 2015/4/15 Reporter: 知 物 達 理 6 Finding the Maximum Hole Size

Experiment 2015/4/15 Reporter: 知 物 達 理 7 Finding the Maximum Hole Size 4 mm X4 mm holes 5 mmX5 mm holes 6 mmX6 mm holes 7 mmX7 mm holes

Experiment 2015/4/15 Reporter: 知 物 達 理 8 Finding the Maximum Hole Size 7.5 mmX7.5 mm holes 8mmX8 mm holes 9mmX9 mm holes

Experiment 2015/4/15 Reporter: 知 物 達 理 9 Finding the Maximum Hole Size Hole Size4x4mm5x5mm6x6mm ResultSuccess Hole Size7x7mm7.5x7.5mm8x8mm9x9mm 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. 2015/4/15 Reporter: 知 物 達 理 10

Theory The pressure difference caused by gravity can be written as The restoring pressure caused by surface tension is 2015/4/15 Reporter: 知 物 達 理 11

Theory Surface modes that decay by are formed Where The effective distance of the disturbance is 2015/4/15 Reporter: 知 物 達 理 12 The total effective mass is

Given a sinusoidal perturbation And assume that, We get Using Along with We get Theory 2015/4/15 Reporter: 知 物 達 理 13

From the equations We get So Theory 2015/4/15 Reporter: 知 物 達 理 14

Ifis real, will be an exponential growth Ifis imaginary, will be a sine wave Theory 2015/4/15 Reporter: 知 物 達 理 15

In critical condition, Input the constants, we get Using, we get Theory 2015/4/15 Reporter: 知 物 達 理 16

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 2015/4/15 Reporter: 知 物 達 理 17 Hole Size7x7mm7.5x7.5mm8x8mm9x9mm Result Successful Below Array of 4x4 Successful Below Array of 3x3 Successful Below Array of 2x2 Fail