Numerical Investigation Of Nonlinear Waves Connected To Blood Flow In An Elastic Tube With Variable Radius

Abstract

Abstract. We investigate flow of incompressible fluid in a cylindrical tube with elastic walls. The radius of the tube may change along its length. The discussed problem is connected to the fluid-structure interaction in large human arteries and especially to nonlinear effects. The long-wave approximation is applied to solve model equations. The obtained model Korteweg-deVries equation possessing a variable coefficient is reduced to a nonlinear dynamical system of three first order differential equations. The low probability of a solitary wave arising is shown. Periodic wave solutions of the model system of equations are studied and it is shown that the waves, that are consequence of the irregular heart pulsations may be modelled by a sequence of parts of such periodic wave solutions.

Nonlinear Evolution Equation for Propagation of Waves in an Artery with an Aneurysm: An Exact Solution Obtained by the Modified Method of Simplest Equation

“…For that purpose, the constitutive relation for tube material must be specified. Here, unlike [18]- [22], we assume that the arterial wall is an incompressible, anisotropic and hyperelastic material. The mechanical behaviour of such a material can be defined by the strain energy function of Fung for arteries [43]:…”

“…Using a specific perturbation method, in a long-wave approximation the authors obtained the forced Korteweg-de Vries (KdV) equation with variable coefficients [18], forced perturbed KdV equation with variable coefficients [19], and forced Korteweg-de Vries-Burgers (KdVB) equation with variable coefficients as evolution equations [20]. The same theoretical frame was used in [21], [22] to examine nonlinear wave propagation in an artery with a variable radius. Considering the artery as a long inhomogeneous prestretched thin elastic tube with an imperfection (presented at large by an unspecified function f (z)), and the blood as an incompressible inviscid fluid the authors reached again the forced KdV equation with variable coefficients.…”

“…Considering the artery as a long inhomogeneous prestretched thin elastic tube with an imperfection (presented at large by an unspecified function f (z)), and the blood as an incompressible inviscid fluid the authors reached again the forced KdV equation with variable coefficients. Apart from solitary propagation waves in such a system, in [22], possibility of periodic waves was discussed at appropriate initial conditions. In the present work we shall focus on consideration of the blood flow through an artery with a local dilatation (an aneurysm).…”

Simple Equations Method (SEsM): Algorithm, Connection with Hirota Method, Inverse Scattering Transform Method, and Several Other Methods

“…Because of this the complex systems attract much research attention in the last decades, see for examples, [ 8 , 9 , 10 , 11 , 12 , 13 , 14 , 15 , 16 , 17 , 18 , 19 , 20 , 21 , 22 ]. Most of the complex systems are nonlinear—many such examples can be found in the fluid mechanics or solid-state physics [ 23 , 24 , 25 , 26 , 27 , 28 , 29 ]. The effects connected to the nonlinearity can be studied, for example, by means of time series analysis or by means of models based on differential or difference equations (additional information about the methodology of the nonlinear time series analysis, some applications of this methodology and basic information about nonlinear differential equations, can be seen, e.g., in [ 30 , 31 , 32 , 33 , 34 , 35 , 36 , 37 , 38 , 39 , 40 , 41 , 42 , 43 , 44 , 45 ]).…”

Numerical modeling of dynamics of a population system with time delay

“…() We note especially the application of theory of stochastic processes and dynamics with delay to different fields of science—fluid mechanics, economics, social sciences population dynamics, and medicine. () The inclusion of time delay into the dynamic models leads to changes in their properties. () Below we introduce time delay in the model of Dimitrova‐Vitanov() and investigate the changes in the dynamics with increasing time delay.…”