Nonlinear waves in a prestressed elastic tube filled with a layered fluid

In this work, we studied the propagation of weakly nonlinear waves in a prestressed thin elastic tube filled with an incompressible layered fluid, where the outer layer is assumed to be inviscid whereas the cylindrical core is considered to be viscous. Using the reductive perturbation technique, the propagation of weakly nonlinear waves in the longwave approximation is studied. The governing equation is shown to be the Korteweg-de Vries-Burgers' (KdV-B) equation. A travelling wave type of solution to this evolution equation is sought and it is observed that the formation of shock wave becomes evident with increasing core radius parameter.     

Evolution equations for nonlinear waves in a tapered elastic tube filled with a viscous fluid

In this work, employing the reductive perturbation method and treating the arteries as a tapered, thin walled, long and circularly conical prestressed elastic tube, the propagation of weakly nonlinear waves is investigated in such a fluid-filled elastic tube. By considering the blood as an incompressible viscous fluid, depending on the viscosity and perturbation parameters we obtained various evolution equations as the extended Korteweg-de Vries (KdV), extended KdV Burgers and extended perturbed KdV equations. Progressive wave solutions to these evolution equations are obtained and it is observed that the wave speeds increase with the distance for negative tapering while they decrease for positive tapering.     

Non-symmetrical waves in a prestressed elastic tube filled with an inviscid fluid

In order for a better understanding of the effect of initial stress on flow in elastic tubes, the propagation of time harmonic waves in a prestressed elastic tube filled with an inviscid fluid is studied. Although the blood is known to be a non-Newtonian fluid, for simplicity in the mathematical analysis, it is assumed to be non-viscous while the tube material is considered to be incompressible, isotropic and elastic. Utilizing the theory of small deformations superimposed on large initial static deformation, for a non-symmetrical perturbed motion, the governing differential equations are obtained in the cylindrical polar coordinates. Due to variability of the coefficients of the resulting differential equations of the solid body, the field equations are solved by a truncated power series method. Applying the boundary conditions, the dispersion relation is obtained as a function of inner pressure, axial stretch and the thickness ratio. It is observed that the wave speed of the non-symmetrical wave is large as compared to the symmetrical case. Various special cases as well as some numerical results are also discussed in the paper.     

Modulation of nonlinear waves in a viscous fluid contained in a tapered elastic tube

In the present work, treating the arteries as a tapered, thin walled, long and circularly conical prestressed elastic tube and the blood as a Newtonian fluid, we have studied the amplitude modulation of nonlinear waves in such a fluid-filled elastic tube, by use of the reductive perturbation method. The governing evolution equation is obtained as the dissipative nonlinear Schrödinger equation with variable coefficients. It is shown that this type of equations admit solitary wave solutions with variable wave amplitude and speed. It is observed that, the wave speed increases with distance for tubes of descending radius while it decreases for tubes of ascending radius. The dissipative effects cause a decay in wave amplitude and wave speed.     

Forced perturbed Korteweg-de Vries equation in an elastic tube filled with a viscous fluid

In the present work, treating the arteries as a prestressed thin walled elastic tube with a stenosis and the blood as a Newtonian fluid, we have studied the propagation of weakly nonlinear waves in such a composite medium, in the long wave approximation, by use of the reductive perturbation method [A. Jeffrey, T. Kawahara, Asymptotic Methods in Nonlinear Wave Theory, Pitman, Boston, 1981]. We obtained the forced perturbed Korteweg-de Vries equation with variable coefficients as the evolution equation. By use of the coordinate transformation, it is shown that this type of evolution equation admits a progressive wave solution with variable wave speed. As might be expected from physical consideration, the wave speed reaches its maximum value at the center of stenosis and gets smaller and smaller as we go away from the center of the stenosis. The results seem to be consistent with physical intuitions.     

Forced Korteweg-de Vries equation in an elastic tube filled with an inviscid fluid

In the present work, treating the arteries as a prestressed thin walled elastic tube with a stenosis and the blood as an inviscid fluid, we have studied the propagation of weakly nonlinear waves in such a composite medium, in the long wave approximation, by use of the reductive perturbation method [C.S. Gardner, G.K. Morikawa, Similarity in the asymptotic behavior of collision-free hydromagnetic waves and water waves, Courant Institute Math. Sci. Report, NYO-9082 (1960) 1-30, T. Taniuti, C.C. Wei, Reductive perturbation method in non-linear wave propagation I, J. Phys. Soc. Jpn., 24 (1968) 941-946]. We obtained the forced Korteweg-de Vries (FKdV) equation with variable coefficients as the evolution equation. By use of the coordinate transformation, it is shown that this type of evolution equation admits a progressive wave solution with variable wave speed. As might be expected from physical consideration, the wave speed reaches its maximum value at the center of stenosis and gets smaller and smaller as we go away from the center of the stenosis. The variations of radial displacement and the fluid pressure with the distance parameter are also examined numerically. The results seem to be consistent with Bernoulli’s law for inviscid fluid.     

Nonlinear waves in an elastic tube with variable prestretch filled with a fluid of variable viscosity

In 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.     

Elastic waves interacting with a thin, prestressed, fiber-reinforced surface film

Elastic surface waves propagating at the interface between an isotropic substrate and a thin, transversely isotropic film are analyzed. The transverse isotropy is conferred by fibers lying parallel to the interface. A rigorous leading-order model of the thin-film/substrate interface is derived from the equations of three-dimensional elasticity for prestressed, transversely isotropic films having non- uniform properties. This is used to study Love waves.     

Shear waves in a fluid saturated elastic plate

In the present context, we consider the propagation of shear waves in the transverse isotropic fluid saturated porous plate. The frequency spectrum for SH-modes in the plate has been studied. It is observed that the frequency of the propagation is damped due to the two-phase character of the porous medium. The dimensionless phase velocities of the shear waves have also been calculated and presented graphically. It is interesting to note that the frequency and phase velocity of shear waves in porous media differ significantly in comparison to that in isotropic elastic media.     

Diffraction of elastic waves by four rigid strips embedded in an infinite orthotropic medium

In the present paper, the elastodynamic response of four coplanar rigid strips embedded in an infinite orthotropic medium due to elastic waves incident normally on the strips is analyzed. The resulting mixed boundary-value problem is solved by an integral-equation method. The normal stress and the vertical displacement are derived in closed analytic form. Numerical values of stress-intensity factors at the edges of the strips and vertical displacements at point in the plane of the strips for several orthotropic materials are calculated and plotted graphically to show the effect of material orthotropy.     

Wave propagation in a thin walled fluid filled viscoelastic tube

Summary The dynamic response of a thin walled, fluid filled, viscoelastic tube, subjected to the sudden release of a uniformly distributed circumferential line loading, is analyzed. It is assumed that the fluid is incompressible and inviscid and that the behavior of the tube material is represented by the standard viscoelastic model. A simple approximate shell theory, for tethered tubes, is employed. Results, for parameters appropriate to biological applications, are obtained by numerical inversion of Fourier transforms.With 7 Figures     

Concentrated force acting on an elastic inclusion in a thick-walled tube

A two-dimensional problem is investigated on the action of a concentrated force applied to the axis of a circular cylindrical, elastic inclusion embedded in an elastic thick-walled tube. This is a generalization of an indentation problem in infinite space, previously studied by Noble and Hussain [4] and revised by Omar and Hassan [5]. The problem is solved using a fast numerical approximation technique and numerical results are presented that allow us to evaluate the angle of contact and to establish a comparison with the case of embedding in an infinite space.     

Net flow of compressible viscous liquids induced by travelling waves in porous media

The influence of ultrasonic radiation on the flow of a liquid through a porous medium is analyzed. The analysis is based on a mechanism proposed by Ganiev et al. according to which ultrasonic radiation deforms the walls of the pores in the shape of travelling transversal waves. Like in peristaltic pumping, the travelling transversal wave induces a net flow of the liquid inside the pore. In this article, the wave amplitude is related to the power output of an acoustic source, while the wave speed is expressed in terms of the shear modulus of the porous medium. The viscosity as well as the compressibility of the liquid are taken into account. The Navier–Stokes equations for an axisymmetric cylindrical pore are solved by means of a perturbation analysis, in which the ratio of the wave amplitude to the radius of the pore is the small parameter. In the second-order approximation a net flow induced by the travelling wave is found. For various values of the compressibility of the liquid, the Reynolds number and the frequency of the wave, the net flow rate is calculated. The calculations disclose that the compressibility of the liquid has a strong influence on the net flow induced. Furthermore, by a comparison with the flow induced by the pressure gradient in an oil reservoir, the net flow induced by a travelling wave can not be neglected, although it is a second-order effect.     

Boundary integral equations for the scattering of elastic waves by elastic inclusions with thin interface layers

Elastic waves are scattered by an elastic inclusion. The interface between the inclusion and the surrounding material is imperfect: the displacement and traction vectors on one side of the interface are assumed to be linearly related to both the displacement vector and the traction vector on the other side of the interface. The literature on such inclusion problems is reviewed, with special emphasis on the development of interface conditions modeling different types of interface layer. Inclusion problems are formulated mathematically, and uniqueness theorems are proved. Finally, various systems of boundary integral equations over the interface are derived.     

Motion of a viscous fluid with suspended particles in a curved tube

In this paper we have discussed the motion of a viscous fluid with suspended particles through a curved tube of small curvature ratio. The system is treated as two separate interacting continua. Solutions for axial and secondary velocities are obtained in the form of asymptotic expansions in powers of Dean Number. The streamline pattern for the particulate phase reveals many interesting features. The influence of the particulate continium on the fluid is described by the parameter τ which depends on the density ratio of the two continua. The concentration distribution of the particles in a given cross section is determined. It is noticed that the particles move closer to the wall for certain values of the concentration and the density ratio.     

Plane waves in a thermally conducting viscous liquid

The aim of this paper is to investigate plane waves in a thermally conducting viscous liquid half-space with thermal relaxation times. There exist three basic waves, namely; thermal wave, longitudinal wave and transverse wave in a thermally conducting viscous liquid half-space. Reflection of plane waves from the free surface of a thermally conducting viscous liquid half-space is studied. The results are obtained in terms of amplitude ratios and are compared with those without viscosity and thermal disturbances.     

充粘性液体管材中超声导波的应力分析

对超声纵向导波在充粘液管材中的应力分布进行了分析,并讨论了用各模式检测充粘液管材的最佳频厚积范围和检测位置。分析表明,随频厚积的增加,L(0,1)和L(0,3)模式的平面内应力在管外壁上的值由负向变为正向,而L(0,2)和L(0,4)模式面内应力的值则相反,变换方向的点恰好在各模式的下一高阶模式群速度最大时的频厚积点附近;而在管内壁上,法向应力分布曲线达到零点的位置恰好在各模式群速度最大时的频厚积点附近。在各模式群速度较大的频厚积区域内,该模式在管内外表面上的平面内应力较大,而法向应力较小,因此能量泄漏较小。故在各模式群速度较大的频厚积区域内,用该模式来检测充粘液管材较合适。     

Axial power flow distributions of ultrasonic guided waves in viscous liquid-filled pipes

The axial power flow (APF) magnitude and attenuation distributions of ultrasonic longitudinal guided waves in viscous liquid-filled elastic pipes are investigated. The optimal location, optimal mode and its frequency-thickness product (fd) for the test of pipes filled with viscous liquid are chosen according to APF and attenuation distributions. The results show that the APF magnitude distribution is an important parameter in choosing the modes and parameters. A particular mode has weak dispersion in ranges of fd values with large group velocity, while other modes with smaller group velocity in the same fd ranges have stronger dispersion. It has been observed that, within these ranges, the chosen mode has a larger APF on the pipe's wall. Therefore, in the region of fd values where a particular mode has a large group velocity, this mode will be effective to be used in testing elastic pipes filled with viscous liquid. The results obtained from both the APF analysis and attenuation distribution are consistent.