Steady-state diffusion through a finite pore into an infinite reservoir: An exact solution

  • R. B. Kelman


An outline is given of an analysis that leads to an exact solution for the problem of steady-state diffusion through a finite thick pore into an infinite region surrounding the mouth of the pore. From this exact formula a simple expression for the flux is derived. This expression approximates the flux with a relative error of less than 3.42 per cent independently of the ratiol/a wherel is the length of the pore anda its radius. If desired, more accurate expressions for the flux can be obtained from the exact solution.


Dual Integral Equation Longitudinal Diffusion Infinite Region Infinite Reservoir Axisymmetric 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.


  1. Brown, H. T. and F. Escombe. 1900. “Static Diffusion of Gases and Liquids in Relation to the Assimilation of Carbon and Translocation in Plants.”Phil. Trans. Roy. Soc. (Lond.) B,193, 223–291.CrossRefGoogle Scholar
  2. Carslaw, H. S. and J. C. Jaeger. 1959.Conduction of Heat in Solids, 2nd ed. Oxford: Clarendon Press.Google Scholar
  3. Collins, W. D. 1960a. “On the Solution of Some Axisymmetric Boundary Value Problems by Means of Integral Equations III. Some Electrostatic and Hydrodynamic Problems For Two Spherical Caps.”Proc. Lond. Math. Soc.,10, 428–460.CrossRefGoogle Scholar
  4. — 1960b. “On the Solution of Some Axisymmetric Boundary Value Problems by Means of Integral Equations. IV. The Electrostatic Potential Due to a Spherical Cap between Two Infinite Conducting Planes.”Proc. Edin. Math. Soc.,12, 95–106.MATHCrossRefGoogle Scholar
  5. — 1962. “On the Solution of Some Axisymmetric Boundary Value Problems by Means of Integral Equations. VII. The Electrostatic Potential Due to a Spherical Cap.”Ibid.,13, 13–24.MATHCrossRefGoogle Scholar
  6. Cooke, J. C. 1956. “A Solution of Tranter's Dual Integral Equation Problem.”Quart. Jour. Mech. Appl. Math.,9, 103–110.MATHCrossRefMathSciNetGoogle Scholar
  7. — 1958. “The Coaxial Circular Disc Problem.”Z. Angew. Math. Mech.,38, 349–356.CrossRefMathSciNetGoogle Scholar
  8. Cooke, J. D. and J. C. Tranter. 1959. “Dual Fourier Bessel Series.”Quart. Jour. Mech. Appl. Math.,12, 379–386.MATHCrossRefMathSciNetGoogle Scholar
  9. Csáky, T. Z. 1963. “A Possible Link Between Active Transport of Electrolytes and Nonelectrolytes.”Fed. Proc.,22, 3–7.Google Scholar
  10. Forsythe, G. E. 1958.Numerical Analysis and Partial Differential Equations, New York: J. Wiley.MATHGoogle Scholar
  11. Gray, A., B. B. Mathews, and T. M. MacRobert. 1931.A Treatise on Bessel Functions and Their Applications to Physics. London: Macmillan and Co., Ltd.Google Scholar
  12. Hobson, E. W. 1955.The Theory of Spherical and Ellipsoidal Harmonics, reprint. New York: Chelsea Pub. Co.Google Scholar
  13. Hogben, C. A. M. 1960. “Movement of Material Across Cell Membranes.”Physiologist,3, 56–62.Google Scholar
  14. Kelman, R. B. 1963a. “Axisymmetric Potentials in Composite Geometries: Finite Cylinder and Half Space.”Contrib. Diff. Eqs.,2, 421–440.MathSciNetGoogle Scholar
  15. — 1963b. “Axisymmetric Potentials Arising From Biological Considerations.”Bull. Amer. Math. Soc.,69, 835–838.MATHCrossRefMathSciNetGoogle Scholar
  16. — 1965. “Longitudinal Diffusion Along the Nephron During Stop Flow.”Bull. Math. Biophysics,27, 53–56.CrossRefMathSciNetGoogle Scholar
  17. Knight, R. C. 1936. “The Potential of a Sphere Inside an Infinite Circular Cylinder.”Quart. Jour. Math.,7, 124–133.CrossRefGoogle Scholar
  18. Landahl, H. D. 1953. “An Approximation Method For Diffusion and Related Problems.”Bull. Math. Biophysics,15, 49–61.CrossRefMathSciNetGoogle Scholar
  19. Patlak, C. S. 1959. “A Contribution to the Study of the Diffusion of Neutral Particles Through Pores.”Bull. Math. Biophysics,21, 129–140.CrossRefGoogle Scholar
  20. Rashevsky, N. 1960.Mathematical Biophysics, Vol. 1, 3rd and Rev. Ed. New York: Dover Publications, Inc.Google Scholar
  21. Riesz, F. and B. Sz-Nagy. 1952.Leçons d'Analyse Functionelle. Budapest: Akadémia Kiado.Google Scholar
  22. Smythe, W. R. 1953a. “Current Flow in Cylinders.”Jour. Appl. Physics,24, 70–73.MATHCrossRefGoogle Scholar
  23. — 1953b. “Charged Disc in Cylindrical Box.”Ibid.,24, 773–775.CrossRefGoogle Scholar
  24. — 1960. “Charged Sphere in a Cylinder.”Ibid.,31, 553–556.CrossRefMathSciNetGoogle Scholar
  25. Stoker, J. J. 1962. “Some Observations on Continuum Mechanics with Emphasis on Elasticity.”Bull. Amer. Math. Soc.,68, 239–278.MATHCrossRefMathSciNetGoogle Scholar
  26. Strutt, J. W. (Baron Rayleigh). 1945.The Theory of Sound, reprint. New York: Dover Publications, Inc.MATHGoogle Scholar
  27. Tranter, C. J. 1959.Integral Transforms in Mathematical Physics, 2nd ed. London: Methuen.Google Scholar
  28. Williams, W. E. 1962. “The Reduction of Boundary Value Problems to Fredholm Integral Equations of the Second Kind.”Z. Angew. Math. Phys.,13, 133–152.MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© N. Rashevsky 1965

Authors and Affiliations

  • R. B. Kelman
    • 1
  1. 1.Institute for Fluid Dynamics and Applied MathematicsUniversity of MarylandCollege Park

Personalised recommendations