Abstract
A general noniterative method of solution of nonlinear boundary value problems involving a parameter is presented. Through a parametric differentiation of the original system, the boundary value problem is converted into an initial value problem with the parameter as the independent variable and differentiation with respect to the original variable is completely eliminated. A step by step integration with respect to the parameter of the newly obtained initial value system yields a large family of solutions which may be of practical interest when the parameter chosen is of physical significance.
The method is applied to study the effect of uniform wall suction and injection on the laminar flow in an annulus. The integration is carried out with respect to the transpiration Reynolds number and the deviation from the Poiseuille flow can be seen at each step of integration.
Résumé
Une méthode générale non itérative est développée pour résoudre des problèmes aux limites non linéaires contenant un paramètre. La dérivation du système différentiel par rapport au paramètre transforme le problème en un problème aux valeurs initiales, où la variable indépendante est le paramètre et où la dérivation par rapport à la variable d'origine est complètement éliminée. L'intégration du nouveau système par rapport au paramètre est menée pas à pas et conduit à une famille de solutions qui peuvent être d'intérêt pratique si le paramètre a une signification physique.
La méthode est appliquée à l'étude de l'écoulement laminaire d'un fluide dans un tube annulaire à parois uniformément perméables. La variable d'intégration est le nombre de Reynolds de transpiration et la déviation par rapport à l'écoulement de Poiseuille peut être déterminée à chaque pas d'intégration.
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References
R. Bellman, R. Kalaba, andG. M. Wing,Invariant Imbedding and the Reduction of Two-Point Boundary Value Problems to Initial Value Problems, Proc. Nat. Acad. Sci.46, 1646–1649 (1960).
H. B. Keller,Shooting and Embedding for Two-Point Boundary Value Problems, J. Math. Anal. and Appl.36, 598–609 (1971).
P. E. Rubbert andM. T. Landahl,Solution of Nonlinear Flow Problems Through Parametric Differentiation, Phys. Fluids10, 831–835 (1967).
C. L. Narayana andP. Ramamoorthy,Compressible Boundary-Layer Equations Solved by the Method of Parameter Differentiation, AIAA J.10, 1085–1086 (1972).
C. W. Tan andR. DiBiano,A Study of the Falkner-Skan Problem with Mass Transfer, AIAA J.10, 923–925 (1972).
T. Y. Na,Comment on “Compressible Boundary-Layer Equations Solved by the Method of Parameter Differentiation,” AIAA J.11, 1790–1791 (1973).
T. Y. Na andC. E. Turski,Solution of the Nonlinear Differential Equations for Finite Bending of a Thin-Walled Tube by Parameter Differentiation, Aero Quart. pp. 14–18 (1974).
T. Y. Na andI. S. Habib,Non-iterative Solution of a Boundary Value Problem by Parameter Differentiation, Chem. Eng. Sci.29, 1669–1670 (1974).
J. Casti, H. Kagiwada, R. Kalaba, andS. Ueno,Reduction of Fredholm Integral Equations to Cauchy Systems, Univ. of S. California, Dept. of Electrical Eng., Techn. Rep. No. 70-19 (1970).
H. B. Keller,Numerical Methods for Two-Point Boundary Value Problems. Blaisdell Publ. Co., Waltham, Mass., 54–69 (1968).
M. Kubicek andV. Hlavacek,Solution of Non-linear Boundary Value Problems, a Novel Method, Chem. Eng. Sci.27, 743–750 (1972).
J. Morel,Similarity Solutions of Flows in Annuli with Rotating Permeable Walls, Ph.D. Thesis, Illinois Institute of Technology, Chicago, Ill. (1973)
A. S. Berman,Laminar Flow in Channels with Porous Walls, J. Appl. Phys.24, 1232–1235 (1953).
R. M. Terrill Flow through a Porous Annulus, Appl. Sci. Res.17, 204–222 (1967).
System/360 Scientific Subroutine Package, (360A-CM-03X) Version II, Programmer's Manual, H20-0205-1, IBM Co., pp. 122–126 (1967).
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Morel, J., Bernstein, B. A method of parameter differentiation applied to flow in porous annuli. Journal of Applied Mathematics and Physics (ZAMP) 27, 289–302 (1976). https://doi.org/10.1007/BF01590503
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DOI: https://doi.org/10.1007/BF01590503