1. Collapsing Bubble Project By: Qiang Chen Stacey Altrichter By: Qiang Chen Stacey Altrichter

2. Introduction Remain in form due the surface tension of the soap solution. Soap decreases surface tension enabling bubbles to form. Large bubbles have a small layer of soap film hence making the surface tension of a bubble large. If the volume of air inside becomes too large, the bubble's surface tension will increase until the bubble pops. A bubble will automatically form so that the surface area is minimized for a given volume. For a closed bubble, this surface is a sphere. Can be distorted due to forces like gravity and wind. Remain in form due the surface tension of the soap solution. Soap decreases surface tension enabling bubbles to form. Large bubbles have a small layer of soap film hence making the surface tension of a bubble large. If the volume of air inside becomes too large, the bubble's surface tension will increase until the bubble pops. A bubble will automatically form so that the surface area is minimized for a given volume. For a closed bubble, this surface is a sphere. Can be distorted due to forces like gravity and wind.

3. Procedure Attached 1 cm x 1 cm graph paper to a flat plate and hung it vertically. In front of the plate, we suspend a tube. Soap solution add a small amount of fluid to the base of the tube and blow into the tube to produce a bubble. Capillary tube, a long glass tube, a red straw Three times with each tube. High speed camera to capture the evolution of the bubble over time Used MATLAB, to find the radius over time and experimentally calculate the surface tension. Attached 1 cm x 1 cm graph paper to a flat plate and hung it vertically. In front of the plate, we suspend a tube. Soap solution add a small amount of fluid to the base of the tube and blow into the tube to produce a bubble. Capillary tube, a long glass tube, a red straw Three times with each tube. High speed camera to capture the evolution of the bubble over time Used MATLAB, to find the radius over time and experimentally calculate the surface tension.

4. Pictures and Data 1. Capillary Tube 2. Glass Tube 3. Red Straw The viscosity of air is  = 1.82 x 10 -5 Pa s. Conversion for pixels to meters is: 52.75 Pixels =.01 m. 1. Capillary Tube 2. Glass Tube 3. Red Straw The viscosity of air is  = 1.82 x 10 -5 Pa s. Conversion for pixels to meters is: 52.75 Pixels =.01 m.

5. Capillary Tube Inner Radius =.00065 m Length =.1 m Inner Radius =.00065 m Length =.1 m

6. Glass Tube Inner Radius =.00155 m Length =.31 m Inner Radius =.00155 m Length =.31 m

7. Red Straw Inner Radius =.00345 m Length =.21 m Inner Radius =.00345 m Length =.21 m

8. Finding the Surface Tension Parameters and Variables V - volume of air within bubble t - time l - length of tube  p - change in pressure r - radius of tube  - viscosity of air Poiseuille's equation describes the change in volume of the bubble over time as: Parameters and Variables V - volume of air within bubble t - time l - length of tube  p - change in pressure r - radius of tube  - viscosity of air Poiseuille's equation describes the change in volume of the bubble over time as:

9. Finding the Surface Tension Change in volume where R is the radius of the bubble: Laplace’s Equation: Combine to find the following equation: Change in volume where R is the radius of the bubble: Laplace’s Equation: Combine to find the following equation:

10. Finding the Surface Tension Using the condition R(0)=R max, we find: Least squares method: R found above x i measured data for radius Took 10 data for each trial Goal: minimize error! Using the condition R(0)=R max, we find: Least squares method: R found above x i measured data for radius Took 10 data for each trial Goal: minimize error!

11. Finding the Surface Tension To minimize… Using Maple, we graphed  Error/  to find when it crosses the horizontal axis. This is the  that best suits our data found AND the formula derived. To minimize… Using Maple, we graphed  Error/  to find when it crosses the horizontal axis. This is the  that best suits our data found AND the formula derived.

12. Surface Tension Data (Nm -1 ) Trial 1Trial 2Trial 3 Capillary Tube.020.023.022 Glass Tube.028.029.028 Straw.021.022

13. Analysis of Surface Tension For every tube, the data within trials is consistent. The data is also somewhat consistent between the different tubes. The actual value of the surface tension of our soap solution is 13-14 dynes/cm which translates to 13*10 -3 N/m and 14*10 -3 N/m. Our data is ranges from being 6*10 3 N/m to 15*10 3 N/m off the expected values. For every tube, the data within trials is consistent. The data is also somewhat consistent between the different tubes. The actual value of the surface tension of our soap solution is 13-14 dynes/cm which translates to 13*10 -3 N/m and 14*10 -3 N/m. Our data is ranges from being 6*10 3 N/m to 15*10 3 N/m off the expected values.

14. Theory - 3 Bubble System 1. Spherical Bubbles 2. Spherical Caplets Bubbles 1. Spherical Bubbles 2. Spherical Caplets Bubbles

15. 1) Spherical Bubbles The system will innately want to minimize the surface area based upon on a constant volume of air constraint. Take R 1, R 2, and R 3 to be the radii of the three bubbles. The system will innately want to minimize the surface area based upon on a constant volume of air constraint. Take R 1, R 2, and R 3 to be the radii of the three bubbles.

16. 1) Spherical Bubbles Surface Area: Volume Constraint: Lagrange Multiplier,, to minimize SA: Surface Area: Volume Constraint: Lagrange Multiplier,, to minimize SA:

17. 1) Spherical Bubbles Steady States of S * : or 8 Cases Total 1 case: R i =0 for i=1, 2, 3. BUT we want V > 0! 3 cases: R i =2/ for one i but R i =0 otherwise. 3 cases: R i =0 for one i but R i =2/ otherwise. 1 case: R i =2/ for i=1, 2, 3. Steady States of S * : or 8 Cases Total 1 case: R i =0 for i=1, 2, 3. BUT we want V > 0! 3 cases: R i =2/ for one i but R i =0 otherwise. 3 cases: R i =0 for one i but R i =2/ otherwise. 1 case: R i =2/ for i=1, 2, 3.

18. 1) Spherical Bubbles If R i =2/ for one i but R i =0 otherwise, then

19. 1) Spherical Bubbles If R i =0 for one i but R i =2/ otherwise, then

20. 1) Spherical Bubbles If R i =2/ for i=1, 2, 3, then

21. 1) Spherical Bubbles The system will tend to an equilibrium that minimizes surface energy. The most stable steady-state occurs for the cases with only one bubble. The less stable case occurs for cases with two bubbles. Hence, for three bubbles we have a highly unstable surface. The system will tend to an equilibrium that minimizes surface energy. The most stable steady-state occurs for the cases with only one bubble. The less stable case occurs for cases with two bubbles. Hence, for three bubbles we have a highly unstable surface.

22. 2) Spherical Caps Consider the bubbles to be spherical caps instead of spheres Variables and Parameters R i is the radius of the full sphere a i is the height of the spherical cap θ i is the angle depicted below Consider the bubbles to be spherical caps instead of spheres Variables and Parameters R i is the radius of the full sphere a i is the height of the spherical cap θ i is the angle depicted below θiθi aiai RiRi

23. 2) Spherical Caps Here we assume that a i is smaller than R i. Otherwise, our bubbles would approach spheres rather than caps. BUT we still have the same approach as with spheres… Here we assume that a i is smaller than R i. Otherwise, our bubbles would approach spheres rather than caps. BUT we still have the same approach as with spheres…

24. 2) Spherical Caps Volume: for i=1, 2, 3 Surface Area: for i=1, 2, 3 Volume: for i=1, 2, 3 Surface Area: for i=1, 2, 3

25. 2) Spherical Caps Lagrange Multiplier: V total volume Steady States or Lagrange Multiplier: V total volume Steady States or

26. 2) Spherical Caps When R i =0, a i =0. When R i =2/λ, a i =2/λ. This is the same stuff we got from the spherical cases!! By the same procedure done previously, we find that the stable equilibria occur when there is only one bubble. Unstable ones occur for multiple bubbles. When R i =0, a i =0. When R i =2/λ, a i =2/λ. This is the same stuff we got from the spherical cases!! By the same procedure done previously, we find that the stable equilibria occur when there is only one bubble. Unstable ones occur for multiple bubbles.

27. Conclusion Possible Errors: Calculating the radius Stability of the bubble apparatus Air movement If we had more time... Vary the radii of each tube for the three bubble case Expand to n bubbles Possible Errors: Calculating the radius Stability of the bubble apparatus Air movement If we had more time... Vary the radii of each tube for the three bubble case Expand to n bubbles