1. Collapsing Bubbles Rachel Bauer Jenna Bratz Rachel

2. Introduction Bubbles have been entertaining children for centuries. Bubbles have been entertaining children for centuries. Children blew bubbles through clay pipes back in the 1700’s. Children blew bubbles through clay pipes back in the 1700’s. Today over 200 million bottles of bubbles are sold each year! Today over 200 million bottles of bubbles are sold each year! Although fun, there is a direct mathematical reason for which they appear—soap films seek to minimize their surface energy, which means minimizing surface area, making it a sphere. Although fun, there is a direct mathematical reason for which they appear—soap films seek to minimize their surface energy, which means minimizing surface area, making it a sphere.

3. Procedure We suspended a tube off of a lab stand. We suspended a tube off of a lab stand. Grid paper (1cm blocks) was set up behind the tube. Grid paper (1cm blocks) was set up behind the tube. Three different tubes were used--a capillary tube, a straw and a large plastic tube Three different tubes were used--a capillary tube, a straw and a large plastic tube Rachel dabbed the soap solution onto the tube and blew a bubble and Jenna started and stopped the high speed camera to capture the collapse of the bubble. Rachel dabbed the soap solution onto the tube and blew a bubble and Jenna started and stopped the high speed camera to capture the collapse of the bubble. The camera took 60 frames per second. The camera took 60 frames per second.

4. Procedure (cont.) The videos were stored as a sequence of pictures. The videos were stored as a sequence of pictures. Pictures were put into Matlab and a program was used to find the least squares circle to fit the data points around the bubble Pictures were put into Matlab and a program was used to find the least squares circle to fit the data points around the bubble The average radius was then calculated The average radius was then calculated

5. Data Straw Trial 3: Initial BubbleAfter.45 secAfter.7 secAfter.85 sec

6. Theory We want to begin modeling the deflation of a soap bubble through a narrow tube. We want to begin modeling the deflation of a soap bubble through a narrow tube. By Poiseuille’s equation we know that the change in the gas volume with respect to time is given by, where r is the radius, l is the length of the tube, is the viscosity of the air, and is the change in pressure. By Poiseuille’s equation we know that the change in the gas volume with respect to time is given by, where r is the radius, l is the length of the tube, is the viscosity of the air, and is the change in pressure.

7. Theory (cont.) The equation for the change in volume of the bubble is also given by where R is the radius of the bubble. The equation for the change in volume of the bubble is also given by where R is the radius of the bubble. By the Laplace-Young Law we have since there are two surfaces of the bubble. By the Laplace-Young Law we have since there are two surfaces of the bubble. Setting the two equations equal and separating variables we get the following equation Setting the two equations equal and separating variables we get the following equation with initial condition. with initial condition.

8. Theory (cont.) Solving this differential equation we find Solving this differential equation we find The radius was then calculated using a Matlab program that takes points around bubble and finds the least squares circle to fit those points. (Thanks Derek!) The radius was then calculated using a Matlab program that takes points around bubble and finds the least squares circle to fit those points. (Thanks Derek!) Next we wanted to calculate the surface tension for each of the trials. Next we wanted to calculate the surface tension for each of the trials.

9. Analysis We want to compare the actual radius we found for our data with the expected radius given by the theory. We want to compare the actual radius we found for our data with the expected radius given by the theory. Find the least squares curve that approximates our data points. Find the least squares curve that approximates our data points. t R

10. Analysis (cont.) Minimize the error between the square of the sum of the expected (theoretical) radii and the actual radii:, where Minimize the error between the square of the sum of the expected (theoretical) radii and the actual radii:, where and n is the number of data and n is the number of data points we have. points we have. Differentiating E with respect to we have: Differentiating E with respect to we have:

11. Analysis (cont.) We want to find when is equal to 0. We want to find when is equal to 0. We plotted the functions for each trial in Maple and found the x-intercept. We plotted the functions for each trial in Maple and found the x-intercept. This is the that minimizes the error. This is the that minimizes the error. Capillary Tube: Capillary Tube: Trial 1: =.0198 N/m = 19.8 dynes/cm Trial 1: =.0198 N/m = 19.8 dynes/cm Trial 2: =.0225 N/m = 22.5 dynes/cm Trial 2: =.0225 N/m = 22.5 dynes/cm Trial 3: =.022 N/m = 22 dynes/cm Trial 3: =.022 N/m = 22 dynes/cm

12. Analysis (cont.) Plastic Tube Plastic Tube Trial 1: =.00625 N/m = 6.25 dynes/cm Trial 1: =.00625 N/m = 6.25 dynes/cm Straw Straw Trial 1: =.01335 N/m = 13.35 dynes/cm Trial 1: =.01335 N/m = 13.35 dynes/cm Trial 2: =.01316 N/m = 13.16 dynes/cm Trial 2: =.01316 N/m = 13.16 dynes/cm Trial 3: =.0141 N/m = 14.1 dynes/cm Trial 3: =.0141 N/m = 14.1 dynes/cm Consistent within the same tube, inconsistent for different size tubes. Consistent within the same tube, inconsistent for different size tubes. Straw trials seems to be the best, closest to expected value (13-14 dynes/cm). Straw trials seems to be the best, closest to expected value (13-14 dynes/cm).

13. Two Bubble Theory Extend the model to one with two bubbles, one at each end of the tube: Extend the model to one with two bubbles, one at each end of the tube: Analysis will begin the same as above. We now just have two bubbles with volumes V 1 and V 2. Analysis will begin the same as above. We now just have two bubbles with volumes V 1 and V 2.

14. Two Bubbles (cont.) The change in the gas volume is: The change in the gas volume is: The change in the volume of the two bubbles is: The change in the volume of the two bubbles is:

15. Two Bubbles (cont.) The change in pressure will change. The change in pressure will change. We have. We have. Using the Laplace-Young Law we find Using the Laplace-Young Law we find Similar to before, set the two equations for dV 1 and dV 2 equal and plug in the equations for the change in pressure. Similar to before, set the two equations for dV 1 and dV 2 equal and plug in the equations for the change in pressure.

16. Two Bubbles (cont.) Find two coupled nonlinear first order differential equations: Find two coupled nonlinear first order differential equations: The total volume in this system is and therefore, The total volume in this system is and therefore, so our system has a conservation law—the volume is a constant. so our system has a conservation law—the volume is a constant.

17. Two Bubbles (cont.) Phase Plane analysis of system of equations: Phase Plane analysis of system of equations: Steady-state occurs when R 1 = R 2. Steady-state occurs when R 1 = R 2. R 1 > R 2 : R 1 > R 2 : R 1 < R 2 : R 1 < R 2 :

18. Two Bubbles (cont.) Directional Field Directional Field x=R 1, y=R 2

19. Two Bubbles (cont.) Since the volume is a constant, the equation Since the volume is a constant, the equation gives the equation for the trajectories in the phase plane: gives the equation for the trajectories in the phase plane: x=R 1, y=R 2

20. Spherical Cap Note as one bubble gets smaller, the shape changes from a sphere to a spherical cap. Note as one bubble gets smaller, the shape changes from a sphere to a spherical cap. We can modify our model to take this into account. We can modify our model to take this into account. Assume that R 1 > R 2, then R 1 will increase and R 2 will decrease as described above. Assume that R 1 > R 2, then R 1 will increase and R 2 will decrease as described above. Find equations that model the time after R 2 equals the radius of the tube. Find equations that model the time after R 2 equals the radius of the tube.

21. Spherical Cap (cont.) The equation for R 1 will stay the same, since the shape stays spherical. The equation for R 1 will stay the same, since the shape stays spherical. The volume of the spherical cap is The volume of the spherical cap is therefore the change in volume of the cap is given by therefore the change in volume of the cap is given by

22. Spherical Cap (cont.) We get the following system of equations: We get the following system of equations: This system also has a conservation of volume law., so we have This system also has a conservation of volume law., so we have

23. Spherical Cap (cont.) The volume is constant, so the equation gives the equation for a trajectory in the phase plane of this system. The volume is constant, so the equation gives the equation for a trajectory in the phase plane of this system. a = R 1, b = R 2

24. Spherical Cap (cont.) We can plot both phase planes and both trajectories to see the difference after R 2 equals the radius of the tube. We can plot both phase planes and both trajectories to see the difference after R 2 equals the radius of the tube. We start with R 1 > R 2, R 1 will increase along the black trajectory until it reaches the point where R 2 is equal to the radius of the tube, then R 1 will follow the green trajectory and will not increase as much as it would have if it followed the original trajectory. We start with R 1 > R 2, R 1 will increase along the black trajectory until it reaches the point where R 2 is equal to the radius of the tube, then R 1 will follow the green trajectory and will not increase as much as it would have if it followed the original trajectory. x=R 1, y=R 2

25. Conclusion In general, our surface tension was not consistent throughout our different trials. Possible reasons for error: In general, our surface tension was not consistent throughout our different trials. Possible reasons for error: Air hitting the bubble Air hitting the bubble Bubble not remaining steady Bubble not remaining steady Measurement error Measurement error

26. Conclusion (cont.) In the two bubble case, the theory matched small experimental results from class. In the two bubble case, the theory matched small experimental results from class. Future work: Future work: More experiments to find additional surface tension values. More experiments to find additional surface tension values. Extend two bubble case to n bubbles Extend two bubble case to n bubbles Attempt to isolate bubble from any disturbances in the lab Attempt to isolate bubble from any disturbances in the lab Complete experiments to further verify two bubble case Complete experiments to further verify two bubble case