1. SURFACE TENSION SPS Lectures January 2006 Wayne Lawton Department of Mathematics National University of Singapore http://math.nus.edu.sg/~matwml matwml@nus.edu.sg

2. ABSTRACT Surface tension is the phenomena whereby a body of water ‘pulls together’ so as to reduce its surface area. It gives soap bubbles, lava lamps, capillary flow, the shape of atomic nuclei and living things. We explain this miracle using geometric and probabilistic concepts, such as mean curvature of fluid boundaries and thermodynamic entropy, presented in an intuitive tension-free style.

3. Surface tension T has units of HISTORICAL “In 1751 Segner introduced the concept of the surface tension of liquids and made an unsuccessful attempt to give a mathematical description of capillary action.” http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Segner.html http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Segner.html

4. Problem: compute differential pressure SPHERICAL SOAP BUBBLES Approach: in equilibrium the work done by the air pressure to expand = energy added through increased surface area

5. We repeat these computations for a cylinder with fixed length L >> R CYLINDRICAL SOAP BUBBLES

6. of a surface S at a point p is computed with x,y,z orthonormal coordinates such that x(p)=y(p)=z(p) = 0 and near p S is the graph of a function MEAN CURVATURE Exercise: Show that above is independent of rotation of x,y coordinates

7. LAPLACE’S FORMULA Theorem For a surface deformation Proof Formula (4-4), page 183 in [1] Corollary If then Proof Laplace’s formula (61.3), page 239 in [2].

8. 1. Write out a detailed derivation that the mean curvature of a sphere with radius R at every point p equals 1/R. 2. Compute the mean curvature of a cylinder and graph of TUTORIAL PROBLEMS 4. Show that the surface of a soap bubble with no inside has minimal area among all surfaces with its edge/edges. 3. Derive corollary 1 on previous page. 5. Research&discuss minimal surfaces.

9. 1.Carry out experiments described in http://www.funsci.com/fun3_en/exper2/exper2.htm#bubbles RESEARCH PROJECTS 2.Carry out experiments described in http://www.funsci.com/fun3_en/exper2/exper2.htm#bubbles http://www.sciencebuddies.org/mentoring/project_ideas/Phys_p 010.shtml?from=Home 3.Derive and numerically simulate the shape of a water surface that supports an water bug 4.Investigate role of T in sufactant chemistry 5.Investigate role of T in protein folding http://www3.interscience.wiley.com/cgi- bin/abstract/107610849/ABSTRACT?CRETRY=1&SRETRY=0 6.Investigate role of T in nuclear physics http://prola.aps.org/abstract/PRC/v62/i2/e025804

10. [1] Lectures in Classical Differential Geometry by Dirk J. Struik, Dover 1950 REFERENCES [2] Fluid Mechanics by L. D. Landau and E. M. Lifshitz, Pergamon, 1987 [3] Numerical gauge methods for variable density and multi-phase flows, PhD thesis submitted by Jia Shuo, NUS, 2005