Pseudo-sedimentation dialysis: an elliptic transmission problem
Author:
Ludwig C. Nitsche
Journal:
Quart. Appl. Math. 52 (1994), 83-102
MSC:
Primary 76R50; Secondary 92E20
DOI:
https://doi.org/10.1090/qam/1262321
MathSciNet review:
MR1262321
Full-text PDF Free Access
Abstract |
References |
Similar Articles |
Additional Information
Abstract: In this paper we propose and analyze a novel membrane-based fractionation process that combines conventional countercurrent dialysis with a phenomenon of “pseudo-sedimentation". An elliptic transmission problem is formulated to describe the steady-state concentration field of a given chemical species. Matching of spectral expansions at the transmission boundary yields an infinite system of linear algebraic equations for the eigenfunction expansion coefficients of the solution. Existence and uniqueness of the solution, framed in terms of a Fredholm alternative, are proven within a subset of the parameter space, along with regularity at the transmission boundary. The paper also develops a perturbation solution in order to interpret the theory in physical terms. Finally, numerically generated concentration profiles indicate the degree of selectivity that could be achieved with a separation apparatus based upon our physical concept.
N. K. Bary, A Treatise on Trigonometric Series, Vol. I (M. F. Mullins, Transl.), Pergamon Press, Oxford, 1964, pp. 82–83, 217
M. Carriero, Un problema misto e di trasmissione per gli operatori $\partial _{{x_1}}^2 + \partial _{{x_2}}^2e\partial _{{x_1}}^2 + {\cos ^2}\omega \partial _{{x_2}}^2$ in due angoli retti adiacenti con condizioni di tipo Dirichlet e Neumann—derivata obliqua regolare sulla semiretta comune, Matematiche (Catania) 31, 228–245 (1976)
S. Chandrasekhar, Stochastic problems in physics and astronomy, Rev. Modern Phys. 15, 1–89 (1943); Reprinted in Selected Papers on Noise and Stochastic Processes, N. Wax (Ed.), Dover, New York, 1954
S. C. Chikwendu and G. U. Ojiakor, Slow-zone model for longitudinal dispersion in two-dimensional shear flows, J. Fluid Mech. 152, 15–38 (1985)
D. Colton and R. Kress, Integral Equation Methods in Scattering Theory, Wiley, New York, 1983, §3.8
D. Colton and R. Kress, The construction of solutions to acoustic scattering problems in a spherically stratified medium. II, Quart. J. Mech. Appl. Math. 32, 53–62 (1979)
M. Costabel and E. Stephan, Boundary integral equations for Helmholtz transmission problems and Galerkin approximation, Z. Angew. Math. Mech. 64, T356–T358 (1984)
M. Costabel and E. Stephan, A direct boundary integral equation method for transmission problems, J. Math. Anal. Appl. 106, 367–413 (1985)
R. Courant and D. Hilbert, Methods of Mathematical Physics, Vol. I, Wiley-Interscience, New York, 1953, pp. 160–161
J. E. Fletcher, Mathematical modeling of the microcirculation, Math. Biosci. 38, 159–202 (1978)
J. C. Giddings, Field-flow fractionation, Sep. Sci. Technol. 19, 831–847 (1984–85); Cross-flow gradients in thin channels for separation by hyperlayer FFF, SPLITT cells, elutriation, and related methods, Ibid. 21, 831–843 (1986)
U. Groβmann, Existence and uniqueness of solutions of quasilinear transmission problems of both elliptic and pseudoparabolic type simulating oxygen transport in capillary and tissue, Math. Meth. Appl. Sci. 2, 34–47 (1980)
J. J. Hermans, Some aspects of counter current dialysis through hollow fiber membranes, Recueil, J. Roy. Netherlands Chem. Soc. 98, 133–136 (1979)
D. Hilbert, Grundzüge einer allgemeinen Theorie der linearen Integralgleichungen, Chelsea, New York, 1953, pp. VI-VII (For an exposition of Hilbert’s theory, see also the survey article, E. Hellinger and O. Toeplitz, Integralgleichungen und Gleichungen mit unendlichvielen Unbekannten, Encyklopädie der Mathematischen Wissenschaften, 2.3.2, pp. 1335–1597, H. Burkhardt, W. Wirtinger, R. Fricke, and E. Hilb (Eds.), B. G. Teubner, Leipzig, 1923–1927)
P. Jungers, J. Zingraff, N. K. Man, T. Drueke, and B. Tardieu, The Essentials in Hemodialysis, Martinus Nijhoff Medical Division, Boston, MA, 1978, chaps. II–IV
A. Kirsch, The denseness of the far field patterns for the transmission problem, IMA J. Appl. Math. 37, 213–225 (1986)
E. Klein, R. A. Ward, and R. E. Lacey, Membrane processes—dialysis and electrodialysis, Handbook of Separation Process Technology, Chapter 21, R. W. Rousseau (Ed.), Wiley-Interscience, New York, 1987
R. E. Kleinman and P. A. Martin, On single integral equations for the transmission problem of acoustics, SIAM J. Appl. Math. 48, 307–325 (1988)
D. L. Koch and J. F. Brady, Dispersion in fixed beds, J. Fluid Mech. 154, 399–427 (1985)
R. Kress and G. F. Roach, Transmission problems for the Helmholtz equation, J. Math. Phys. 19, 1433–1437 (1978)
O. A. Ladyzhenskaya and N. N. Ural’tseva, Linear and Quasilinear Elliptic Equations (transl. by Scripta Technica, Inc.; L. Ehrenpreis, Transl. Ed.), Academic Press, New York, 1968, chap. 3, §16
J. Lishang and L. Jin, The perturbation of the interface of the two-dimensional diffraction problem and an approximating Muskat model, J. Partial Differential Equations 3, 85–96 (1990)
E. Meister and F.-O. Speck, Diffraction problems with impedance conditions, Applicable Anal. 22, 193–211 (1986)
L. C. Nitsche, J. M. Nitsche, and H. Brenner, Existence, uniqueness and regularity of a time-periodic probability density distribution arising in a sedimentation-diffusion problem, SIAM J. Math. Anal. 19, 153–166 (1988)
I. Noda and C. C. Gryte, Multistage membrane separation processes for the continuous fractionation of solutes having similar permeabilities, AIChE J. 27, 904–912 (1981)
T. N. Phillips, Fourier series solutions to Poisson’s equation in rectangularly decomposable regions, IMA J. Numer. Anal. 9, 337–352 (1989)
G. F. Roach and B. Zhang, On transmission problems for the Schrödinger equation, SIAM. J. Math. Anal. 22, 991–1006 (1991)
J. C. Van Stone, Principles and mechanics of dialysis, Dialysis and the Treatment of Renal Insufficiency, Chapter 5, J. C. Van Stone (Ed.), Grune & Stratton, New York, 1983
T. von Petersdorff, Boundary integral equations for mixed Dirichlet, Neumann and transmission problems, Math. Methods Appl. Sci. 11, 185–213 (1989)
P. Wilde, The limiting behaviour of solutions of the acoustic transmission problems, J. Math. Anal. Appl. 126, 24–38 (1987)
N. K. Bary, A Treatise on Trigonometric Series, Vol. I (M. F. Mullins, Transl.), Pergamon Press, Oxford, 1964, pp. 82–83, 217
M. Carriero, Un problema misto e di trasmissione per gli operatori $\partial _{{x_1}}^2 + \partial _{{x_2}}^2e\partial _{{x_1}}^2 + {\cos ^2}\omega \partial _{{x_2}}^2$ in due angoli retti adiacenti con condizioni di tipo Dirichlet e Neumann—derivata obliqua regolare sulla semiretta comune, Matematiche (Catania) 31, 228–245 (1976)
S. Chandrasekhar, Stochastic problems in physics and astronomy, Rev. Modern Phys. 15, 1–89 (1943); Reprinted in Selected Papers on Noise and Stochastic Processes, N. Wax (Ed.), Dover, New York, 1954
S. C. Chikwendu and G. U. Ojiakor, Slow-zone model for longitudinal dispersion in two-dimensional shear flows, J. Fluid Mech. 152, 15–38 (1985)
D. Colton and R. Kress, Integral Equation Methods in Scattering Theory, Wiley, New York, 1983, §3.8
D. Colton and R. Kress, The construction of solutions to acoustic scattering problems in a spherically stratified medium. II, Quart. J. Mech. Appl. Math. 32, 53–62 (1979)
M. Costabel and E. Stephan, Boundary integral equations for Helmholtz transmission problems and Galerkin approximation, Z. Angew. Math. Mech. 64, T356–T358 (1984)
M. Costabel and E. Stephan, A direct boundary integral equation method for transmission problems, J. Math. Anal. Appl. 106, 367–413 (1985)
R. Courant and D. Hilbert, Methods of Mathematical Physics, Vol. I, Wiley-Interscience, New York, 1953, pp. 160–161
J. E. Fletcher, Mathematical modeling of the microcirculation, Math. Biosci. 38, 159–202 (1978)
J. C. Giddings, Field-flow fractionation, Sep. Sci. Technol. 19, 831–847 (1984–85); Cross-flow gradients in thin channels for separation by hyperlayer FFF, SPLITT cells, elutriation, and related methods, Ibid. 21, 831–843 (1986)
U. Groβmann, Existence and uniqueness of solutions of quasilinear transmission problems of both elliptic and pseudoparabolic type simulating oxygen transport in capillary and tissue, Math. Meth. Appl. Sci. 2, 34–47 (1980)
J. J. Hermans, Some aspects of counter current dialysis through hollow fiber membranes, Recueil, J. Roy. Netherlands Chem. Soc. 98, 133–136 (1979)
D. Hilbert, Grundzüge einer allgemeinen Theorie der linearen Integralgleichungen, Chelsea, New York, 1953, pp. VI-VII (For an exposition of Hilbert’s theory, see also the survey article, E. Hellinger and O. Toeplitz, Integralgleichungen und Gleichungen mit unendlichvielen Unbekannten, Encyklopädie der Mathematischen Wissenschaften, 2.3.2, pp. 1335–1597, H. Burkhardt, W. Wirtinger, R. Fricke, and E. Hilb (Eds.), B. G. Teubner, Leipzig, 1923–1927)
P. Jungers, J. Zingraff, N. K. Man, T. Drueke, and B. Tardieu, The Essentials in Hemodialysis, Martinus Nijhoff Medical Division, Boston, MA, 1978, chaps. II–IV
A. Kirsch, The denseness of the far field patterns for the transmission problem, IMA J. Appl. Math. 37, 213–225 (1986)
E. Klein, R. A. Ward, and R. E. Lacey, Membrane processes—dialysis and electrodialysis, Handbook of Separation Process Technology, Chapter 21, R. W. Rousseau (Ed.), Wiley-Interscience, New York, 1987
R. E. Kleinman and P. A. Martin, On single integral equations for the transmission problem of acoustics, SIAM J. Appl. Math. 48, 307–325 (1988)
D. L. Koch and J. F. Brady, Dispersion in fixed beds, J. Fluid Mech. 154, 399–427 (1985)
R. Kress and G. F. Roach, Transmission problems for the Helmholtz equation, J. Math. Phys. 19, 1433–1437 (1978)
O. A. Ladyzhenskaya and N. N. Ural’tseva, Linear and Quasilinear Elliptic Equations (transl. by Scripta Technica, Inc.; L. Ehrenpreis, Transl. Ed.), Academic Press, New York, 1968, chap. 3, §16
J. Lishang and L. Jin, The perturbation of the interface of the two-dimensional diffraction problem and an approximating Muskat model, J. Partial Differential Equations 3, 85–96 (1990)
E. Meister and F.-O. Speck, Diffraction problems with impedance conditions, Applicable Anal. 22, 193–211 (1986)
L. C. Nitsche, J. M. Nitsche, and H. Brenner, Existence, uniqueness and regularity of a time-periodic probability density distribution arising in a sedimentation-diffusion problem, SIAM J. Math. Anal. 19, 153–166 (1988)
I. Noda and C. C. Gryte, Multistage membrane separation processes for the continuous fractionation of solutes having similar permeabilities, AIChE J. 27, 904–912 (1981)
T. N. Phillips, Fourier series solutions to Poisson’s equation in rectangularly decomposable regions, IMA J. Numer. Anal. 9, 337–352 (1989)
G. F. Roach and B. Zhang, On transmission problems for the Schrödinger equation, SIAM. J. Math. Anal. 22, 991–1006 (1991)
J. C. Van Stone, Principles and mechanics of dialysis, Dialysis and the Treatment of Renal Insufficiency, Chapter 5, J. C. Van Stone (Ed.), Grune & Stratton, New York, 1983
T. von Petersdorff, Boundary integral equations for mixed Dirichlet, Neumann and transmission problems, Math. Methods Appl. Sci. 11, 185–213 (1989)
P. Wilde, The limiting behaviour of solutions of the acoustic transmission problems, J. Math. Anal. Appl. 126, 24–38 (1987)
Similar Articles
Retrieve articles in Quarterly of Applied Mathematics
with MSC:
76R50,
92E20
Retrieve articles in all journals
with MSC:
76R50,
92E20
Additional Information
Article copyright:
© Copyright 1994
American Mathematical Society
