Solitary waves in a tapered prestressed fluid-filled elastic tube

In the present work, treating the arteries as a tapered, thin walled, long and circularly conical prestressed elastic tube and using the longwave approximation, we have studied the propagation of weakly nonlinear 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 equations admit a solitary wave type of solution with variable wave speed. It is observed that, the wave speed increases with distance for positive tapering while it decreases for negative tapering.     

Weakly nonlinear waves in a linearly tapered elastic tube filled with a fluid

In the present work, treating the arteries as a tapered, thin-walled, long and circularly conical prestressed elastic tube and using the long-wave approximation, we have studied the propagation of weakly nonlinear 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 equations admits a solitary wave type of solution with variable wave speed. It is observed that the wave speed increases with the scaled time parameter τ for positive tapering while it decreases for negative tapering, as expected.     

Solitary waves in a fluid-filled thin elastic tube with variable cross-section

The present work treats the arteries as a thin walled prestressed elastic tube with variable cross-section and uses the longwave approximation to study the propagation of weakly nonlinear 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 equations admits a solitary wave type of solution with variable wave speed. It is observed that, for soft biological tissues with an exponential strain energy function the wave speed increases with distance for narrowing tubes while it decreases for expanding tubes.     

Amplitude Modulation of Nonlinear Waves in Fluid-Filled Tapered Tubes

We study the modulation of nonlinear waves in fluid-filled prestressed tapered tubes. For this, we obtain the nonlinear dynamical equations of motion of a prestressed tapered tube filled with an incompressible inviscid fluid. Assuming that the tapering angle is small and using the reductive perturbation method, we study the amplitude modulation of nonlinear waves and obtain the nonlinear Schrödinger equation with variable coefficients as the evolution equation. A traveling-wave type of solution of such a nonlinear equation with variable coefficients is obtained, and we observe that in contrast to the case of a constant tube radius, the speed of the wave is variable. Namely, the wave speed increases with distance for narrowing tubes and decreases for expanding tubes.     

Variable electrical and thermal conductivity in the theory of generalized thermoelastic diffusion

This paper deals with the problem of a thermoelastic half-space with a permeating substance in contact with the bounding plane in the context of the theory of generalized thermoelastic diffusion with one relaxation time and with variable electrical and thermal conductivity. The bounding surface of the half-space is taken to be traction free and subjected to a time dependent thermal shock. The solution is obtained in the Laplace transform domain by a direct approach. A numerical technique is employed to obtain the solution in the physical domain. It is found that there exist two coupled waves, one of which is elastic and the other is thermal, and a third wave affects diffusion mainly. A comparison is made with the results obtained in a thermoelastic medium with and without diffusion in the following cases : (a) the electrical and thermal conductivities have constant values, (b) the presence of magnetic field and (c) the generalized theory in thermoelasticity. Received: June 1, 2005     

DFT modal analysis of spectral element methods for the 2D elastic wave equation

The DFT modal analysis is a dispersion analysis technique that transforms the equations of a numerical scheme to the discrete Fourier transform domain sampled in the mesh nodes. This technique provides a natural matching of exact and approximate modes of propagation. We extend this technique to spectral element methods for the 2D isotropic elastic wave equation, by using a Rayleigh quotient approximation of the eigenvalue problem that characterizes the dispersion relation, taking full advantage of the tensor product representation of the spectral element matrices. Numerical experiments illustrate the dependence of dispersion errors on the grid resolution, polynomial degree, and discretization in time. We consider spectral element methods with Chebyshev and Legendre collocation points.     

Resonant hydraulic sloshing in a tank of variable depth

The forced resonant oscillations of a fluid in a tank of variable depth are considered within the hydraulic approximation. It is shown that for certain bottom topographies a continuous periodic output dominated by the first normal mode is possible. This contrasts with the case of a tank of constant depth, where hydraulic jumps are a feature of the motion. The amplitude and frequency of the output are connected by a cubic equation. The fluid response can act like that of a hard or soft spring, depending on the bottom topography. There is also a critical bottom topography that yields a higher order response amplitude.     

Stability of post-fertilization traveling waves

This paper studies the stability of a family of traveling wave solutions to the system proposed by Lane et al. [D.C. Lane, J.D. Murray, V.S. Manoranjan, Analysis of wave phenomena in a morphogenetic mechanochemical model and an application to post-fertilization waves on eggs, IMA J. Math. Appl. Med. Biol. 4 (4) (1987) 309-331], to model a pair of mechanochemical phenomena known as post-fertilization waves on eggs. The waves consist of an elastic deformation pulse on the egg's surface, and a free calcium concentration front. The family is indexed by a coupling parameter measuring contraction stress effects on the calcium concentration. This work establishes the spectral, linear and nonlinear orbital stability of these post-fertilization waves for small values of the coupling parameter. The usual methods for the spectral and evolution equations cannot be applied because of the presence of mixed partial derivatives in the elastic equation. Nonetheless, exponential decay of the directly constructed semigroup on the complement of the zero eigenspace is established. We show that small perturbations of the waves yield solutions to the nonlinear equations decaying exponentially to a phase-modulated traveling wave.     

Three-scale asymptotics for a diffusion problem coupled with the wave equation

In this article, we present an asymptotic analysis of waves of elastic stress in an infinite solid whose boundary is subject to a rapid thermal load. The problem under consideration couples the wave equation and the heat equation, and the asymptotic approximation of the solution requires three-scaled variables. The asymptotic approximation is supplied with a rigorous remainder estimate and is illustrated numerically.     

A non-trivial explicit solution for the 14-velocity Cabannes kinetic model

We study the shock wave problem for the Cabannes 14-velocity model of the Boltzmann equation in one space dimension (x$x$-axis) and obtain a non-trivial explicit solution which asymptotically connects two particular Maxwellian states. We prove that such a solution exists provided that a suitable condition on the differential elastic cross sections hold.     

Painlevé property,Lax pair and Darboux transformation of the variable-coefficient modified Kortweg-de Vries model in fluid-filled elastic tubes

With the consideration on the artery as a thin walled prestressed elastic tube with variable radius, a variable-coefficient modified Kortweg-de Vries (vc-mKdV) equation is obtained by the long wave approximation for the blood which is assumed as the incompressible non-viscous fluid. In the present paper, we firstly investigate the Painlevé property of the vc-mKdV equation. Furthermore, with the Ablowitz-Kaup-Newell-Segur procedure and symbolic computation, the Lax pair of the vc-mKdV equation is constructed, by virtue of which we construct the Darboux transformation and a new soliton solution. Finally, the features of the new solution are discussed to illustrate the influences of the constant and variable coefficients in the solitonic propagation.     

Waves in an Inhomogeneous Tube with a Flowing Two-Phase Fluid

Results of theoretical and mathematical justification of the problem on a pulsating flow of a two-phase barotropic bubbly fluid enclosed in an elastic semi-infinite cylindrical tube inhomogeneous along its length are presented. Linear one-dimensional equations are used. It is assumed that the tube is rigidly attached to the surrounding medium and therefore its displacement in the axial direction is absent. At infinity, the tube material is assumed to be homogeneous. To describe the pressure, flow rate, and displacement of the fluid, a pulsating pressure is given at the tube end. The problem stated is reduced to a singular Sturm-Liouville boundary-value problem, which in turn is reduced to a Volterra-type integral equation. This equation is solved by the method of successive approximations. By assuming that the corresponding potential is integrable, it is proved that these approximations converge to the exact solution of the problem. It is shown that this assumption also covers the very important practical case of piecewise inhomogeneity. For numerical realization, we consider a homogeneous tube with flowing water containing a small amount of bubbles. The effect of the volume content of bubbles on wave characteristics is revealed. In particular, it is stated that, for the oscillation regime selected, an increased bubble volume content decreases the wave velocity and considerably increases the flow speed (rate).     

Closed-form solutions, extremality and nonsmoothness criteria in a large deformation elasticity problem

The pure azimuthal shear problem for a circular cylindrical tube of nonlinearly elastic material, both isotropic and anisotropic, is examined on the basis of a complementary energy principle. For particular choices of strain-energy function, one convex and one non-convex, closed-form solutions are obtained for this mixed boundary-value problem, for which the governing differential equation can be converted into an algebraic equation. The results for the non-convex strain energy function provide an illustration of a situation in which smooth analytic solutions of a nonlinear boundary-value problem are not global minimizers of the energy in the variational statement of the problem. Both the global minimizer and the local extrema are identified and the results are illustrated for particular values of the material parameters.      

On crack propagation shapes in elastic bodies

We consider a two dimensional elastic isotropic body with a curvilinear crack. The formula for the derivative of the energy functional with respect to the crack length is discussed. It is proved that this derivative is independent of the crack path provided that we consider quite smooth crack propagation shapes. An estimate for the derivative of the energy functional being uniform with respect to the crack propagation shape is derived.     

Blow-up and global existence for a general class of nonlocal nonlinear coupled wave equations

We study the initial-value problem for a general class of nonlinear nonlocal coupled wave equations. The problem involves convolution operators with kernel functions whose Fourier transforms are nonnegative. Some well-known examples of nonlinear wave equations, such as coupled Boussinesq-type equations arising in elasticity and in quasi-continuum approximation of dense lattices, follow from the present model for suitable choices of the kernel functions. We establish local existence and sufficient conditions for finite-time blow-up and as well as global existence of solutions of the problem.     

Three-dimensional gravity waves in a channel of variable depth

We consider existence of three-dimensional gravity waves traveling along a channel of variable depth. It is well known that the long-wave small-amplitude expansion for such waves results in the stationary Korteweg–de Vries equation, coefficients of which depend on the transverse topography of the channel. This equation has a single-humped solitary wave localized in the direction of the wave propagation. We show, however, that there exists an infinite set of resonant Fourier modes that travel at the same speed as the solitary wave does. This fact suggests that the solitary wave confined in a channel of variable depth is always surrounded by small-amplitude oscillatory disturbances in the far-field profile.     

High-order one-way model nesting in dispersive non-uniform media

Global-regional model interaction is considered for two-dimensional linear time dependent waves in a dispersive non-uniform medium with a continuously varying wave speed. The setup, which is sometimes called ‘one-way nesting,’ arises in Numerical Weather Prediction (NWP) as well as in other fields concerning waves in very large domains. The Carpenter scheme for this type of problem is revisited, in the context of the dispersive wave equation with a variable wave speed. The original Carpenter scheme is based on the Sommerfeld radiation operator, and thus is associated with low-order accuracy. By replacing the Sommerfeld operator with the high-order Hagstrom-Warburton absorbing operator, a modified Carpenter open boundary condition emerges which possesses high-order accuracy. This is demonstrated via a numerical example in a wave guide with a wave speed which varies linearly in the cross section.     

Instability of solitary waves on Euler’s elastica

Stability of solitary waves in a thin inextensible and unshearable rod of infinite length is studied. Solitary-wave profile of the elastica of such a rod without torsion has the form of a planar loop and its speed depends on a tension in the rod. The linear instability of a solitary-wave profile subject to perturbations escaping from the plane of the loop is established for a certain range of solitary-wave speeds. It is done using the properties of the Evans function, an analytic function on the right complex half-plane, that has zeros if and only if there exist the unstable modes of the linearization around a solitary-wave solution. The result follows from comparison of the behaviour of the Evans function in some neighbourhood of the origin with its asymptotic at infinity. The explicit computation of the leading coefficient of the Taylor series of the Evans function near the origin is performed by means of the symbolic computer language. Received: April 6, 2004; revised: December 12, 2004     

动脉中脉搏波传播分析

将血管简化为弹性管,并考虑组织对血管壁的约束,利用力学方法建立血液流过血管的力学模型．通过理论分析对脉搏波在血管中的传播规律进行研究,同时分析了血液粘性、血管壁弹性模量、管径对波的传播的影响．通过对考虑血液粘性和不考虑血液粘性的结果比较,发现血液的粘性对脉搏波的传播的影响不能忽略,并且当弹性模量增大时,传播速度增大,血流的压力值增高;血管直径减小时,血流压力也增高,脉搏波速度增大．理论分析得到的结果也有助于利用脉搏波的信息来分析和辅助诊断一些人体疾病的病因．     

A note on the wave propagation in water of variable depth

In the present work, utilizing the two dimensional equations of an incompressible inviscid fluid and the reductive perturbation method we studied the propagation of weakly nonlinear waves in water of variable depth. For the case of slowly varying depth, the evolution equation is obtained as the variable coefficient Korteweg-de Vries (KdV) equation. Due to the difficulties for the analytical solutions, a numerical technics so called “the method of integrating factor” is used and the evolution equation is solved under a given initial condition and the bottom topography. It is observed the parameters of bottom topography causes to the changes in wave amplitude, wave profile and the wave speed.