Coping with complex boundaries

  • Avner Friedman
  • Jack F. Douglas
Part of the The IMA Volumes in Mathematics and its Applications book series (IMA, volume 67)


There are many applications in materials science involving boundary value problems in which the boundaries have complicated shapes. Examples include the computation of the capacity and translational friction coefficients of objects having general shape, the discharge of Poiseuille flow through a pipe of arbitrary cross section, and the calculation of “virial coefficients” which describe the leading order concentration dependence of the effective material properties of suspensions or composites containing a small concentration of complex shaped particles. Moreover, the scattering of small objects by electromagnetic (radar, visible light) and pressure (acoustic) waves involves mathematical problems identical to the virial coefficient calculations which have many practical applications corresponding to cases where the scattering particles have elaborate structure (e.g., snow flakes). Direct analytical approaches based on traditional differential equations methods are often not effective in dealing with this class of problems. On April 15, 1994 Jack F. Douglas from National Institute for Standards and Technology (NIST) presented a different approach based on recasting this kind of boundary value problem in terms of an averaging over random walk paths. He showed through a sequence of examples how this point of view leads to viable numerical and analytical treatment of complex boundaries. He described recent work and presented open problems.


Random Walk Effective Property Brownian Particle Poiseuille Flow Complex Boundary 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    J.B. Hubbard and J.F. Douglas, Hydrodynamic friction of arbitrarily shaped Brownian particles, Physical Review, 47 (1993), 2983–2986.CrossRefGoogle Scholar
  2. [2]
    G. Pôlya and G. Szegö, Isoperimetric Inequalities in Mathematical Physics, Annals of Math. Studies, no. 27, Princeton University Press, Princeton (1951).Google Scholar
  3. [3]
    L.E. Payne, Isoperimetric inequalities and their applications, SIAM Review, 9 (1967), 453–488.MathSciNetzbMATHCrossRefGoogle Scholar
  4. [4]
    I. Stakgold, Green’s Functions and Boundary Value Problems, WileyInterscience, New York (1979).Google Scholar
  5. [5]
    N.S. Lankof, foundation of Modern Potential Theory, Springer-Verlag, New York (1972).Google Scholar
  6. [6]
    J.F. Douglas, H.-X. Zhou and J.B. Hubbard, Hydrodynamic friction of the capacitance of arbitrarily shaped objects, Phys. Review E, 49 (1994), 5319–5331.CrossRefGoogle Scholar
  7. [7]
    M. Kac, Probability and Related Topics in Physical Sciences, Inter-science Publishers, New York (1957).Google Scholar
  8. [8]
    H.X. Zhou, A. Szabo, J.F. Douglas and J.B. Hubbard, A Brownian dynamics algorithm for calculating the hydrodynamic friction and the electrostatic capacitance of an arbitrarily shaped object, J. Chem. Phys., 100 (1994), 3821–3826.CrossRefGoogle Scholar
  9. [9]
    B.A. Luty, J.A. McCammon and H.-X. Zhou, Diffusive reaction rates from Brownian dynamics simulations: Replacing the outer cutoff surface by an analytical treatment, J. Chem. Phys., 97 (1992), 5682–5686.CrossRefGoogle Scholar
  10. [10]
    J.F. Douglas and E.J. Garboczi, Intrinsic viscosity and polarizability of particles having a wide range of shapes, submitted.Google Scholar
  11. [11]
    F. Spitzer, Electrostatic capacity, heat flow, and Brownian motion, Zeitschrift für Wahrscheinlichkeitstheorie, 3 (1964), 110–121.MathSciNetzbMATHCrossRefGoogle Scholar
  12. [12]
    J.R. Lebenhaft and R. Kapral, Diffusion-controlled processes among partially absorbing stationary sinks, J. Stat. Phys., 20 (1979), 25–56.MathSciNetCrossRefGoogle Scholar
  13. [13]
    O.D. Kellogg, Foundation of Potential Theory, Dover, New York (1953).Google Scholar
  14. [14]
    A. Dvoretsky, P. Erdös and S. Kakutani, Double points of paths of Brownian motion in n-space, Acta Sci. Math. (Szeged), 12 (1950), 75–81.MathSciNetGoogle Scholar
  15. [15]
    S.J. Taylor, The a-dimensional measure of the graph and set of zeros of a Brownian path, Proc. Comb. Phil. Soc., 51, Part 2 (1955), 265–274.CrossRefGoogle Scholar
  16. [16]
    M. Delbrück, Knotting problems in biology, Fourteenth Symp. on Appl. Math., SIAM, Philadelphia (1962), pp. 55–68.Google Scholar
  17. [17]
    F. Dean A. Stasiak, T. Koller, and N.R. Cozzarelli, Duplex DNA knots produced by Escherichia coli Topoisomerase I, Journal of Biological Chemistry, 260 (1985), 4975–4983.Google Scholar
  18. [18]
    A.G. Greenhill, On the flow of viscous fluids in a pipe or channel, Proc. Lond. Math. Soc., 13 (1881), 43–46.CrossRefGoogle Scholar
  19. [19]
    H.L. Weissberg, End correction for slow viscous flow through long tubes, Phys. Fluids, 5 (1962), 1033–1036.MathSciNetzbMATHCrossRefGoogle Scholar
  20. [20]
    J. Happel and H. Brenner, Low Reynolds Number Hydrodynamics, Prentice-Hall, Englewood Cliffs, N.J. (1965).Google Scholar
  21. [21]
    F. Hunt, J. Douglas, and J. Bernal, A path integration method for calculating Poiseuille slow velocity fields, in preparation.Google Scholar
  22. [22]
    G.I. Taylor and A.A. Griffith, The use of a soap film in solving torsion problems in Taylor’s Scientific Papers, Vol. 1, ed. G.K. Batchelor, Cambridge University Press, Cambridge (1958), pp. 1–23Google Scholar
  23. [23]
    J. Leavit and P. Ungar, Circle supports the largest sandpile, Comm. Pure Appl. Math., 14 (1962), 35–37.CrossRefGoogle Scholar
  24. [24]
    J.L. Synge, Flow of viscous fluid through pipes and channels, Proc. Symp. Appl. Math. IV, AMS, McGraw-Hill, 1953, pp. 141–165.Google Scholar
  25. [25]
    A. Pleijel, Remarks on Courant’s nodal line theorem, Comm. Pure Appl. Math., 9 (1956), 543–550.MathSciNetzbMATHCrossRefGoogle Scholar
  26. [26]
    A. Friedman, The asymptotic behavior of the first real eigenvalue of a second order elliptic operator with a small parameter in the highest derivative, Indiana Univ. Math. J., 22 (1973), 1005–1015.zbMATHCrossRefGoogle Scholar
  27. [27]
    F. Marchetti, Asymptotic exponentiality of the exit time, Statist. & Probab. Letters, 1 (1983), 167–170.MathSciNetzbMATHCrossRefGoogle Scholar
  28. [28]
    J.W. Strutt, The Theory of Second, Vol. 1, Dover (New York).Google Scholar
  29. [29]
    A.M. Skvortsov and A.A. Gorbunov, Adsorption effects in chromotography of polymers, J. Chromotography, 358 (1986), 77–83.CrossRefGoogle Scholar
  30. [30]
    S.G. Entelis, V.V. Evreinor and A.V. Gorshkov, Functionality and molecular weight distribution of telechetic polymers, Adv. in Polymer Sci., 76 (1986), 129–175.Google Scholar
  31. [31]
    J.R. Kuttler, V.G. Sigillito, Eigenvalues of the Laplacian in two dimensions, Siam Review, 26 (1984), 163–193.MathSciNetzbMATHCrossRefGoogle Scholar
  32. [32]
    P. Hsu, Probabilistic approach to the Neumann problem, Comm. Pure Appl. Math., 38 (1985), 445–472.MathSciNetzbMATHCrossRefGoogle Scholar
  33. [33]
    M.D. Donsker and M. Kac, A Sampling Method for Determining the Lowest Eigen value and the Principal Eigen function of Schrödinger’s Equation, J. Res. Nat. Bureau of Standards, 44 (1950), 551–557.MathSciNetGoogle Scholar
  34. [34]
    A. Korzeniowski, J.L. Fry, D.E. Orr and N.G. Fazleev, Feynman—Kac path integral calculation of the ground state energies of atoms, Phys. Rev. Lett., 69 (1992), 893–896.CrossRefGoogle Scholar
  35. [35]
    D.L. Cullen, M.S. Zawojski and A.L. Holbrook, Heat-Dissipation Problems Solved with Composites, 44 (1988), Publication of the Society of Plastic Engineers, Brookfield, CT, pp. 37–39.Google Scholar
  36. [36]
    J.C. Maxwell, A Treatise on Electricity and Magnetism, Dover, New York (1954).zbMATHGoogle Scholar
  37. [37]
    G. Dassios and R.E. Kleinman, On Kelvin inversion and low-frequency scattering, SIAM Review, 31 (1989), 565–585.MathSciNetzbMATHCrossRefGoogle Scholar
  38. [38]
    M. Schiffer and G. Szegö, Virtual mass and polarization, Trans. Amer. Math. Soc., 67 (1949), 130–205.MathSciNetzbMATHCrossRefGoogle Scholar
  39. [39]
    J.B. Keller, R.E. Kleinman, and T.B.A. Senior, Dipole moments in Rayleigh scattering, J. Inst. Math. Appl., 9 (1972), 14–22.MathSciNetzbMATHCrossRefGoogle Scholar
  40. [40]
    T.B.A. Senior, Low frequency scattering by a dielectric body, Radio Sci., 11 (1976), 477–482.MathSciNetCrossRefGoogle Scholar
  41. [41]
    R.E. Kleinman and T.B.A. Senior, Rayleigh scattering cross sections, Radio Science, 7 (1972), 937–942.MathSciNetCrossRefGoogle Scholar
  42. [42]
    J. Van Bladel, Low-frequency scattering by hard and soft bodies, J. Acoust. Soc., 44 (1968), 1069–1073.CrossRefGoogle Scholar
  43. [43]
    T.S. Angel and R.E. Kleinman, Polarizability tensors in low-frequency inverse scattering, Radio Science, 22 (1982), 1120–1073.CrossRefGoogle Scholar
  44. [44]
    A. Einstein, Eine Neue Besstimmung der Moleküldimensionen, Ann. Phys., 19 (1906), 289–306.CrossRefGoogle Scholar
  45. [45]
    G.K. Batchelor and J.T. Green, The determination of the bulk stress is suspension of spherical particles to order c 2, J. Fluid Mech., 56 (1972), 401–427.zbMATHCrossRefGoogle Scholar
  46. [46]
    J.M. Rallison, Suspension of rigid particles of arbitrary shape, J. Fluid Mech., 84 (1978), 237–263.MathSciNetzbMATHCrossRefGoogle Scholar
  47. [47]
    S. Haber and H. Brenner, Rheological properties of dilute suspensions of centrally symmetric Brownian particles at small shear rates, J. Coll. Inter. Sci., 97 (1984), 496–514.CrossRefGoogle Scholar
  48. [48]
    J. Kirkwood and J. Riseman, The intrinsic viscosities and diffusion constants of flexible macromolecules in solution, J. Chem. Phys., 16 (1948), 565–573.CrossRefGoogle Scholar
  49. [49]
    P. Debye and A.M. Bueche, Intrinsic viscosity, diffusion and sedimentation rate of polymers in solution, J. Chem. Phys., 16 (1948), 573–579.CrossRefGoogle Scholar
  50. [50]
    G. Pólya, A minimum problem about the motion of a solid through a fluid, Proc. Nat. Acad. Sci., 33 (1947), 218–221.MathSciNetzbMATHCrossRefGoogle Scholar
  51. [51]
    H.P. McKean, A probabilistic interpretation of equilibrium charge distributions, J. Math. Kyoto Univ., 4 (1965), 617–625.MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer-Verlag New York, Inc. 1995

Authors and Affiliations

  • Avner Friedman
    • 1
  • Jack F. Douglas
    • 2
  1. 1.Institute for Mathematics and its ApplicationsUniversity of MinnesotaMinneapolisUSA
  2. 2.Polymers DivisionNational Institute for Standards and Technology (NIST)GaithersburgUSA

Personalised recommendations