Perturbation Methods in Applied Mathematics

Kevorkian, J.; Cole, J. D.

doi:10.1007/978-1-4757-4213-8

Cited by 981 publications

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“…Following Keverkian and Cole [3], the matching is carried out for both x and y components of velocity of fluid and dust by the following principles:…”

Section: Formulation Of the Problemmentioning

confidence: 99%

“…Solutions are obtained in terms of series expansion with respect to the small amplitude by regular perturbation method. The inner (boundary layer flow) and the outer (flow beyond the boundary layer), solutions are matched by a matching process given by Kevorkian and Cole [3]. Graphs of the velocity components, both for the outer and the inner flow for various values of mass-concentration of the dust particles are drawn.…”

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confidence: 99%

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Induced dusty flow due to normal oscillation of wavy wall

Kannan¹,

Ramamurthy²

2001

International Journal of Mathematics and Mathematical Sciences

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Abstract. A two-dimensional viscous dusty flow induced by normal oscillation of a wavy wall for moderately large Reynolds number is studied on the basis of boundary layer theory in the case where the thickness of the boundary layer is larger than the amplitude of the wavy wall. Solutions are obtained in terms of a series expansion with respect to small amplitude by a regular perturbation method. Graphs of velocity components, both for outer flow and inner flow for various values of mass concentration of dust particles are drawn. The inner and outer solutions are matched by the matching process. An interested application of present result to mechanical engineering may be the possibility of the fluid and dust transportation without an external pressure. . Tanaka studied the problem both for small and moderately large Reynolds number. While studying the problem for moderately large Reynolds numbers, he has shown that if the thickness of the boundary layer is larger than the amplitude of the wavy wall, the technique employed for small Reynolds numbers can also be applied to the case of moderately large Reynolds numbers.Recently, while studying the flow of blood through mammalian capillaries, blood is taken to be a binary system of plasma (liquid phase) and blood cells (solid phase). In order to gain some insight into the peristaltic motion of blood in capillaries, it is of interest to study the induced flow of a dusty or two-phase fluid by sinusoidal motion of a wavy wall.Tanaka [5] discussed a two-dimensional flow of an incompressible viscous fluid due to an infinite sinusoidal wavy wall which executes progressive motion with constant speed. In 1980, S. K. Nag extended the same problem for a dusty fluid, up to secondorder solution. Ramamurthy and Rao [4] studied the two-dimensional flow of a dusty fluid which is an extension of Tanaka's work. Dhar and Nandha [2] studied the motion of the two-dimensional fluid with the motion of the wall being described as a nonprogressive wave (y = a cos(2π x/L) sin ωt) up to third-order solution. In the present paper, we have studied the two-dimensional flow of an incompressible dusty fluid, by taking the motion of the wall as a nonprogressive wave (as in Dhar's work) and discussed the problem up to fourth-order solution. The effect of dust parameter on

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“…Following Keverkian and Cole [3], the matching is carried out for both x and y components of velocity of fluid and dust by the following principles:…”

Section: Formulation Of the Problemmentioning

confidence: 99%

mentioning

confidence: 99%

Induced dusty flow due to normal oscillation of wavy wall

Kannan¹,

Ramamurthy²

2001

International Journal of Mathematics and Mathematical Sciences

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“…Then there exist a sequence {e n } with e n | 0 and a sequence of open balls {S n } 9 S r , = -{f 'efi 11 " 1 ; |f 7 -fi| <rj such that (2. 3) for e x in 5,.…”

Section: Lemma 1 (Due To Nagumomentioning

confidence: 99%

“…Apply Lemma It must be remarked that L(A r )=/?i f0 =t=0 and P^JV r/ )=/ 7 2,o=t = 0-Kevorkian and Cole's suggestive example in §4.1.2. in [7] is as follows.…”

Section: Putmentioning

confidence: 99%

On Nagumo's $H^s$ Stability in Singular Perturbations

Ashino

1991

Publ. Res. Inst. Math. Sci.

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“…Examples include double-well potentials and the strongly nonlinear Hamiltonian system describing generic weakly nonlinear 1-1 primary resonance [see, for example, Guckenheimer and Holmes (1983) and Kevorkian and Cole (1981)]. Other examples are described by Neishtadt (1991) and Henrard (1993).…”

Section: Introductionmentioning

confidence: 99%

Higher-order change of the Hamiltonian (energy) for nearly homoclinic orbits

Haberman¹,

Ho²

1994

Methods and Applications of Analysis

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ABSTRACT. Conservative dynamical systems with a homoclinic orbit are analyzed under a small dissipative perturbation. The asymptotic expansions of a nearly homoclinic orbit are shown to fail as the surrounding saddle regions are approached. A weakly nonlinear analysis of the surrounding saddle regions is derived assuming the dissipation can be approximated by linear damping in the neighborhood of the saddle point. By matching (to sufficiently high order) the asymptotic expansion of the nearly homoclinic orbit to the solutions in the surrounding saddle regions, the change in the Hamiltonian (the energy dissipation) over a complete nearly homoclinic orbit is obtained. We show there is a new higher-order logarithmic correction to the well-known Melnikov integral for the change of the Hamiltonian from one saddle approach to the next. Numerical computations of this change in the Hamiltonian for a double-well potential are shown to compare favorably with the asymptotic result.

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Perturbation Methods in Applied Mathematics

Abstract: With 79 figures. AII rights reservcd.No part of this book may be translated or reproduced in any form without written permission from Springer-Verlag Berlin Heidelberg GmbH.

Cited by 981 publications

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Induced dusty flow due to normal oscillation of wavy wall

Induced dusty flow due to normal oscillation of wavy wall

On Nagumo's $H^s$ Stability in Singular Perturbations

Higher-order change of the Hamiltonian (energy) for nearly homoclinic orbits

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