The Effects of Higher-Order Approximations in a Fluid-Filled Elastic Tube with Stenosis

2006 ◽  
Vol 61 (12) ◽  
pp. 641-651
Author(s):  
Hilmi Demiray
Keyword(s):  
Evolution Equation ◽  
Wave Speed ◽  
Kdv Equation ◽  
Order Term ◽  
Perturbation Expansion ◽  
Reductive Perturbation Method ◽  
Elastic Tube ◽  
Inviscid Fluid ◽  
Weakly Nonlinear ◽  
Elastic Tubes

Treating arteries as thin-walled prestressed elastic tubes with a narrowing (stenosis) and blood as an inviscid fluid, we study the propagation of weakly nonlinear waves in such a fluid-filled elastic tube by employing the reductive perturbation method in the long wave approximation. It is shown that the evolution equation of the first-order term in the perturbation expansion may be described by the conventional Korteweg-de Vries (KdV) equation. The evolution equation for the second-order term is found to be the linearized KdV equation with a nonhomogeneous term, which contains the contribution of the stenosis. A progressive wave type solution is sought for the evolution equation, and it is observed that the wave speed is variable, which results from the stenosis. We study the variation of the wave speed with the distance parameter τ for various amplitude values of the stenosis. It is observed that near the center of the stenosis the wave speed decreases with increasing stenosis amplitude. However, sufficiently far from the center of the stenosis stenosis amplitude becomes negligibly small.

Nonlinear Wave Modulation in a Fluid-Filled Elastic Tube with Stenosis

2008 ◽  
Vol 63 (1-2) ◽  
pp. 24-34 ◽  
Author(s):  
Hilmi Demiray
Keyword(s):  
Evolution Equation ◽  
Variable Coefficient ◽  
Wave Speed ◽  
Harmonic Wave ◽  
Reductive Perturbation Method ◽  
Elastic Tube ◽  
Axial Distance ◽  
Nls Equation ◽  
Wave Speeds ◽  
Straight Lines

In the present work, treating the arteries as a thin-walled and prestressed elastic tube with a stenosis and the blood as a Newtonian fluid with negligible viscosity, we have studied the amplitude modulation of nonlinear waves in such a composite system by use of the reductive perturbation method. The governing evolution equation was obtained as the variable coefficient nonlinear Schr¨odinger (NLS) equation. By setting the stenosis function equal to zero, we observed that this variable coefficient NLS equation reduces to the conventional NLS equation. After introducing a new dependent variable and a set of new independent coordinates, we reduced the evolution equation to the conventional NLS equation. By seeking a progressive wave type of solution to this evolution equation we observed, that the wave trajectories are not straight lines anymore; they are rather some curves in the (ξ ,τ ) plane. It was further observed that the wave speeds for both enveloping and harmonic waves are variable, and the speed of the enveloping wave increases with increasing axial distance, whereas the speed of the harmonic wave decreases with increasing axial coordinates. The numerical calculations indicated that the speed of the harmonic wave decreases with increasing time parameter, but the sensitivity of wave speed to this parameter is quite weak.

Interactions of Nonlinear Waves in Fluid-Filled Elastic Tubes

2007 ◽  
Vol 62 (1-2) ◽  
pp. 21-28
Author(s):  
Hilmi Demiray
Keyword(s):  
Solitary Waves ◽  
Nonlinear Waves ◽  
Elastic Tube ◽  
Wave Number ◽  
Inviscid Fluid ◽  
Leading Order ◽  
Elastic Tubes ◽  
Longwave Approximation ◽  
Thin Walled ◽  
Analytical Phase

In this work, treating an artery as a prestressed thin-walled elastic tube and the blood as an inviscid fluid, the interactions of two nonlinear waves propagating in opposite directions are studied in the longwave approximation by use of the extended PLK (Poincaré-Lighthill-Kuo) perturbation method. The results show that up to O(k3), where k is the wave number, the head-on collision of two solitary waves is elastic and the solitary waves preserve their original properties after the interaction. The leading-order analytical phase shifts and the trajectories of two solitons after the collision are derived explicitly.

Large-amplitude unsteady flow in liquid-filled elastic tubes

1967 ◽  
Vol 29 (3) ◽  
pp. 513-538 ◽  
Author(s):  
John H. Olsen ◽  
Ascher H. Shapiro
Keyword(s):  
Water Waves ◽  
Large Amplitude ◽  
Linear Equations ◽  
Perturbation Expansion ◽  
Elastic Tube ◽  
Fluid Viscosity ◽  
Elastic Tubes ◽  
Non Linear ◽  
Laminar And Turbulent Flow ◽  
Large Amplitude Motion

Unsteady, large-amplitude motion of a viscous liquid in a long elastic tube is investigated theoretically and experimentally, in the context of physiological problems of blood flow in the larger arteries. Based on the assumptions of long wavelength and longitudinal tethering, a quasi-one-dimensional model is adopted, in which the tube wall moves only radially, and in which only longitudinal pressure gradients and fluid accelerations are important. The effects of fluid viscosity are treated for both laminar and turbulent flow. The governing non-linear equations are solved analytically in closed form by a perturbation expansion in the amplitude parameter, and, for comparison, by numerical integration of the characteristic curves. The two types of solution are compared with each other and with experimental data. Non-linear effects due to large amplitude motion are found to be not as large as those found in similar problems in gasdynamics and water waves.

PROPAGATION OF WEAKLY NONLINEAR WAVES IN A FLUID-FILLED THIN ELASTIC TUBE

1999 ◽  
Vol 23 (2) ◽  
pp. 253-265
Author(s):  
H. Demiray
Keyword(s):  
Nonlinear Waves ◽  
Vries Equation ◽  
Perturbation Technique ◽  
Fluid Pressure ◽  
Static Field ◽  
Elastic Tube ◽  
Inviscid Fluid ◽  
Weakly Nonlinear ◽  
De Vries ◽  
Weakly Nonlinear Waves

In this work, we study the propagation of weakly nonlinear waves in a prestressed thin elastic tube filled with an inviscid fluid. In the analysis, considering the physiological conditions under which the arteries function, the tube is assumed to be subjected to a uniform pressure P0 and a constant axial stretch ratio λz. In the course of blood flow in arteries, it is assumed that a finite time dependent radial displacement is superimposed on this static field but, due to axial tethering, the effect of axial displacement is neglected. The governing nonlinear equation for the radial motion of the tube under the effect of fluid pressure is obtained. Using the exact nonlinear equations of an incompressible inviscid fluid and the reductive perturbation technique, the propagation of weakly nonlinear waves in a fluid-filled thin elastic tube is investigated in the longwave approximation. The governing equation for this special case is obtained as the Korteweg-de-Vries equation. It is shown that, contrary to the result of previous works on the same subject, in the present work, even for Mooney-Rivlin material, it is possible to obtain the nonlinear Korteweg-de-Vries equation.

Higher-Order Approximations for Symmetrical Regular Long Wave Equation

2006 ◽  
Vol 61 (12) ◽  
pp. 607-614
Author(s):  
Xian-Jing Lai ◽  
Jie-Fang Zhang
Keyword(s):  
Wave Equation ◽  
Schrödinger Equation ◽  
Schrodinger Equation ◽  
Order Term ◽  
Perturbation Expansion ◽  
Higher Order ◽  
Reductive Perturbation Method ◽  
Third Order ◽  
Long Wave ◽  
Tangent Method

In this work, we extend the application of “the modified reductive perturbation method” to the symmetrical regular long wave equation and try to obtain the contribution of higher-order terms in the perturbation expansion. It is shown that the lowest-order term in the expansion is governed by the nonlinear Schrödinger equation while the second- and third-order terms are governed by the linear Schrödinger equation. By employing the hyperbolic tangent method, progressive wave type solutions are obtained for the first-, second- and third-order terms in the perturbation expansion. - PACS numbers: 02.30.Jr, 42.25.Bs, 42.81.Dp, 43.35.Kp

scholarly journals Propagation of Nonlinear Pressure Waves in Blood

2013 ◽  
Vol 2013 ◽  
pp. 1-5 ◽  
Author(s):  
A. Elgarayhi ◽  
E. K. El-Shewy ◽  
Abeer A. Mahmoud ◽  
Ali A. Elhakem
Keyword(s):  
Vries Equation ◽  
Finite Amplitude ◽  
Reductive Perturbation Method ◽  
Elastic Tube ◽  
Pressure Waves ◽  
Weakly Nonlinear ◽  
Flow Problems ◽  
Inner Radius ◽  
Reductive Perturbation ◽  
Conditions Of Stability

The propagation of weakly nonlinear pressure waves in a fluid-filled elastic tube has been investigated. The reductive perturbation method has been employed to derive the Korteweg-de Vries equation for small but finite amplitude. The effect of the final inner radius of the tube on the basic properties of the soliton wave was discussed. Moreover, the conditions of stability and the soliton existence via the potential and the corresponding phase portrait were computed. The applicability of the present investigation to flow problems in arteries is discussed.

Pressure Wave Propagation in Fluid-Filled Co-Axial Elastic Tubes Part 1: Basic Theory

2003 ◽  
Vol 125 (6) ◽  
pp. 852-856 ◽  
Author(s):  
K. Berkouk ◽  
P. W. Carpenter ◽  
A. D. Lucey
Keyword(s):  
Wave Speed ◽  
Leading Edge ◽  
Governing Equations ◽  
Cross Sectional ◽  
Strong Function ◽  
Weakly Nonlinear ◽  
Weakly Nonlinear Theory ◽  
Elastic Tubes ◽  
Classic Theory ◽  
Serious Disease

Our work is motivated by ideas about the pathogenesis of syringomyelia. This is a serious disease characterized by the appearance of longitudinal cavities within the spinal cord. Its causes are unknown, but pressure propagation is probably implicated. We have developed an inviscid theory for the propagation of pressure waves in co-axial, fluid-filled, elastic tubes. This is intended as a simple model of the intraspinal cerebrospinal-fluid system. Our approach is based on the classic theory for the propagation of longitudinal waves in single, fluid-filled, elastic tubes. We show that for small-amplitude waves the governing equations reduce to the classic wave equation. The wave speed is found to be a strong function of the ratio of the tubes’ cross-sectional areas. It is found that the leading edge of a transmural pressure pulse tends to generate compressive waves with converging wave fronts. Consequently, the leading edge of the presure pulse steepens to form a shock-like elastic jump. A weakly nonlinear theory is developed for such an elastic jump.

scholarly journals Modulation of nonlinear waves in a fluid-filled elastic tube with stenosis

2004 ◽  
Vol 2004 (60) ◽  
pp. 3205-3218 ◽  
Author(s):  
Hilmi Demiray
Keyword(s):  
Evolution Equation ◽  
Amplitude Modulation ◽  
Nonlinear Waves ◽  
Variable Coefficient ◽  
Composite System ◽  
Harmonic Wave ◽  
Reductive Perturbation Method ◽  
Elastic Tube ◽  
Reductive Perturbation ◽  
Thin Walled

Treating the arteries as a thin-walled and prestressed elastic tube with a stenosis and the blood as a Newtonian fluid with negligible viscosity, we have studied the amplitude modulation of nonlinear waves in such a composite system by use of the reductive perturbation method. The governing evolution equation is obtained as the variable-coefficient nonlinear Schrödinger equation. It is observed that the speed of the harmonic wave increases with distance from the center of stenosis.

Transcritical two-layer flow over topography

1987 ◽  
Vol 178 ◽  
pp. 31-52 ◽  
Author(s):  
W. K. Melville ◽  
Karl R. Helfrich
Keyword(s):  
Steady Flow ◽  
Solitary Waves ◽  
Numerical Solutions ◽  
Kdv Equation ◽  
Single Layer ◽  
Cubic Nonlinearity ◽  
Arbitrary Length ◽  
Weakly Nonlinear ◽  
Flow Over Topography ◽  
Layer Flow

The evolution of weakly-nonlinear two-layer flow over topography is considered. The governing equations are formulated to consider the effects of quadratic and cubic nonlinearity in the transcritical regime of the internal mode. In the absence of cubic nonlinearity an inhomogeneous Korteweg-de Vries equation describes the interfacial displacement. Numerical solutions of this equation exhibit undular bores or sequences of Boussinesq solitary waves upstream in a transcritical regime. For sufficiently large supercritical Froude numbers, a locally steady flow is attained over the topography. In that regime in which both quadratic and cubic nonlinearity are comparable, the evolution of the interface is described by an inhomogeneous extended Kortewegde Vries (EKdV) equation. This equation displays undular bores upstream in a subcritical regime, but monotonic bores in a transcritical regime. The monotonic bores are solitary wave solutions of the corresponding homogeneous EKdV equation. Again, locally steady flow is attained for sufficiently large supercritical Froude numbers. The predictions of the numerical solutions are compared with laboratory experiments which show good agreement with the solutions of the forced EKdV equation for some range of parameters. It is shown that a recent result of Miles (1986), which predicts an unsteady transcritical regime for single-layer flows, may readily be extended to two-layer flows (described by the forced KdV equation) and is in agreement with the results presented here.Numerical experiments exploiting the symmetry of the homogeneous EKdV equation show that solitary waves of fixed amplitude but arbitrary length may be generated in systems described by the inhomogeneous EKdV equation.

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