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Yayın The boundary layer approximation and nonlinear waves in elastic tubes(Pergamon-Elsevier Science, 2000-09) Antar, Nalan; Demiray, HilmiIn the present work, employing the nonlinear equations of an incompressible, isotropic and elastic thin tube and approximate equations of an incompressible viscous fluid, the propagation of weakly nonlinear waves is examined. In order to include the geometrical and structural dispersion into analysis, the wall's inertial and shear deformation are taken into account in determining the inner pressure-inner cross sectional area relation. Using the reductive perturbation technique, the propagation of weakly nonlinear waves, in the long-wave approximation, are shown to be governed by the Korteweg-de Vries (KdV) and the Korteweg-de Vries-Burgers (KdVB), depending on the balance between the nonlinearity, dispersion and/or dissipation. In the case of small viscosity (or large Reynolds number), the behaviour of viscous fluid is quite close to that ideal fluid and viscous effects are confined to a very thin layer near the boundary. In this case, using the boundary layer approximation we obtain the viscous-Korteweg-de Vries and viscous-Burgers equations.Yayın Head-on-collision of nonlinear waves in a fluid of variable viscosity contained in an elastic tube(Pergamon-Elsevier Science Ltd, 2009-08-30) Demiray, HilmiIn this work, treating the arteries as a thin walled, prestressed elastic tube and the blood as an incompressible viscous fluid of variable viscosity, we have studied the interactions of two nonlinear waves, in the long wave approximation, through the use of extended PLK perturbation method, and the evolution equations are shown to be the Korteweg-deVries-Burgers equation. The results show that, Up to O(is an element of(3/2)), the head-on-collision of two nonlinear progressive waves is elastic and the nonlinear progressive waves preserve their original properties after the collision. The phase functions for each wave are derived explicitly and it is shown that they are not straight lines anymore, they are rather some curves.Yayın Nonlinear waves in a viscous fluid contained in a viscoelastic tube(Birkhauser Verlag, 2001-11) Demiray, HilmiIn the present work the propagation of weakly nonlinear waves in a prestressed viscoelastic thin tube filled with a viscous fluid is studied. Using the reductive perturbation technique in analyzing the nonlinear equations of a viscoelastic tube and the approximate equations of a viscous fluid, the propagation of weakly nonlinear waves in the longwave approximation is studied. Depending on the order of viscous effects, various evolution equations like, Burgers', Korteweg-de Vries, Korteweg-de Vries-Burgers' equations and their perturbed forms are obtained. Ravelling wave type of solutions to some of these evolution equations are sought. Finally, utilizing the finite difference scheme, a numerical solution is presentede for the perturbed KdVB equation and the result is discussed.Yayın Modulation of nonlinear waves in a thin elastic tube filled with a viscous fluid(Pergamon-Elsevier Science Ltd, 1999-11) Demiray, HilmiIn the present work, utilizing the nonlinear equations of a prestressed thin elastic tube filled with an incompressible viscous fluid the propagation of weakly nonlinear waves in such a medium is studied. Considering that the arteries are initially subjected to a large static transmural pressure P-0 and an axial stretch lambda(z) and, in the course of blood flow, a finite time dependent displacement is added to this initial field, the nonlinear equations governing the motion of the tube in the radial direction is obtained. Utilizing the reductive perturbation technique the amplitude modulation of weakly nonlinear and dissipative but strongly dispersive waves is examined. The localized travelling wave solution to the evolution equation is given and the stability condition is discussed.Yayın Amplitude modulation of nonlinear waves in a fluid-filled tapered elastic tube(Elsevier Science Inc, 2004-07-15) Bakırtaş, İlkay; Demiray, HilmiIn the present work, treating the arteries as a tapered, thin walled, long and circularly conical prestressed elastic tube and using the reductive perturbation method, we have studied the amplitude modulation of nonlinear waves in such a fluid-filled elastic tube. By considering the blood as an incompressible inviscid fluid the evolution equation is obtained as the nonlinear Schrodinger equation with variable coefficients. It is shown that this type of equations admit a solitary wave type of solution with variable wave speed. It is observed that, the wave speed decreases with distance for tubes with descending radius while it increases for tubes with ascending radius.Yayın On head-on collision between two solitary waves in shallow water: the use of the extended PLK method(Springer, 2015-10) Özden, Ali Erinç; Demiray, HilmiIn the present work, we examined the head-on collision of solitary waves in shallow water theory, through the use of extended Poincare–Lighthill–Kuo (PLK) method based on the combination of reductive perturbation method with strained coordinates. Motivated with the result obtained by Ozden and Demiray (Int J Nonlinear Mech 69:66–70, 2015), we introduced a set of stretched coordinates that include some unknown functions which are to be determined so as to remove secularities that might occur in the solution. By expanding these unknown functions and the field variables into power series in the smallness parameter ?, introducing them into the field equations and imposing the conditions to remove the secularities, we obtained some evolution equations. By seeking a progressive wave solution to these evolution equations, we determined the speed correction terms and the phase-shift functions. The result obtained here is exactly the same with found by Ozden and Demiray (Int J Nonlinear Mech 69:66–70, 2015), wherein the analysis employed by Su and Mirie (J Fluid Mech 98:509–525, 1980) is utilized.Yayın Weakly nonlinear waves in a prestressed thin elastic tube containing a viscous fluid(Pergamon-Elsevier Science Ltd, 1999-11) Antar, Nalan; Demiray, HilmiIn this work, we studied the propagation of weakly nonlinear waves in a prestressed thin elastic tube filled with an incompressible viscous fluid. In order to include the geometrical and structural dispersion into analysis, the wall's inertial and sheer deformation are taken into account in determining the inner pressure-inner cross sectional area relation. Using the reductive perturbation technique, the propagation of weakly nonlinear waves, in the long-wave approximation, is shown to be governed by the Korteweg-de Vries-Burgers (KdVB) equation. Due to dependence of coefficients of the governing equation on the initial deformation, the material and viscosity parameters, the profile of the travelling wave solution to the KdVB equation changes with these parameters. These variations are calculated numerically for some elastic materials and the effects of initial deformation and the viscosity parameter on the propagation characteristics are discussed.Yayın Nonlinear waves in an elastic tube with variable prestretch filled with a fluid of variable viscosity(Pergamon-Elsevier Science Ltd, 2008-10) Demiray, HilmiIn the present work, by employing the reductive perturbation method to the nonlinear equations of an incompressible, prestressed, homogeneous and isotropic thin elastic tube and to the exact equations of an incompressible Newtonian fluid of variable viscosity, we have studied weakly nonlinear waves in such a medium and obtained the variable coefficient Korteweg-deVries-Burgers (KdV-B) equation as the evolution equation. For this purpose, we treated the artery as an incompressible, homogeneous and isotropic elastic material subjected to variable stretches both in the axial and circumferential directions initially, and the blood as an incompressible Newtonian fluid whose viscosity changes with the radial coordinate. By seeking a travelling wave solution to this evolution equation, we observed that the wave front is not a plane anymore, it is rather a curved surface. This is the result of the variable radius of the tube. The numerical calculations indicate that the wave speed is variable in the axial coordinate and it decreases for increasing circumferential stretch (or radius). Such a result seems to be plausible from physical considerations, like Bernoulli's law. We further observed that, the amplitude of the Burgers shock gets smaller and smaller with increasing time parameter along the tube axis. This is again due to the variable radius, but the effect of it is quite small.Yayın Weakly nonlinear waves in a viscous fluid contained in a viscoelastic tube with variable cross-section(Gauthier-Villars/Editions Elsevier, 2005-03) Demiray, HilmiIn the present work, treating the arteries as a thin walled prestressed viscoelastic tube with variable cross-section, and using the longwave approximation, we have studied the propagation of weakly nonlinear waves in such a fluid-filled viscoelastic tube by employing the reductive perturbation method. By considering the blood as an incompressible viscous fluid, depending on the order of various physical entities, various evolution equations with variable coefficients are obtained and progressive wave solutions to these evolution equations are given whenever possible. It is shown that this type of equations admit solitary wave type of solutions with variable wave speeds.Yayın Weakly non-linear waves in a tapered elastic tube filled with an inviscid fluid(Pergamon-Elsevier Science Ltd, 2005-07) Bakırtaş, İlkay; Demiray, HilmiIn the present work, treating the artery as a tapered, thin walled, long and circularly conical prestressed elastic tube and using the longwave approximation, we have studied the propagation of weakly non-linear waves in such a fluid-filled elastic tube by employing the reductive perturbation method. By considering the blood as an incompressible inviscid fluid, the evolution equation is obtained as the Korteweg-de Vries equation with a variable coefficient. It is shown that this type of equation admits a solitary wave-type solution with variable wave speed. It is observed that, the wave speed decreases with distance for positive tapering while it increases for negative tapering. It is further observed that, the progressive wave profile for expanding tubes (a > 0) becomes more steepened whereas for narrowing tubes (a < 0) it becomes more flattened.
