Maximal tipping angles of nonempty bottles

ESSIM 2012, DresdenGroup 12NAMES

Outline

• Problem– Restrictions

• Creating bottle– Calculations

• Calculating the liquid mass centre– Monte Carlo method– Mesh method

• Results• Conclusion

Problem

• Determine the maximal inclination angle and the corresponding fill quantity for various existing bottles.

• Figure sources:• http://www.4thringroad.com/wp-content/uploads/2009/08/coca-cola-main-design.jpg• http://s3.amazonaws.com/static.fab.com/inspiration/154695-612x612-1.png

Modelling ideas

• The bottle will fall when the system’s mass centre passes the tipping point.

• First, the problem was solved for totally full or totally empty cylindrical bottle, because it is easy to solve analytically.

• Only 2-dimensional case was considered because of the radial symmetry.

Problem restrictions

• Assumptions made:– Bottle density is homogeneous– Liquid density is homogeneous– Bottle has to be radial symmetric– Tilting point is fixed during inclination

Creating bottle

• For creating the bottle, coordinates of one edge are given

• Bottle mass is measured• Next You will see the bottles we used!

Water bottle

Coke bottle

Cylinder

Fat wine bottle

Calculating the bottle mass centre

• Take the mass centre of each line between given coordinates

• Length of the line• Mass centre of system of lines is

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R =1

Mmiri

i

∑

Calculating the mass centre of liquidMonte Carlo

Calculating the mass centre of liquidMesh

• Calculate a triangular mesh• Find the water level by minimizing the V-V(h)• Use coarse grid, but refine in the water level• Calculate the mass centre of every triangle

Will the bottle fall?

• Add mass centres of the whole system• Has the mass centre passed the tipping point?

• Picture of a bottle on the edge of falling

Results Monte Carlo method

ResultsMesh model

Conclusion

• Monte Carlo method is quite slow to use• 3D would have been more accurate• For cylindrical bottles, the maximum tipping

angle is easily calculated