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.

Amplification and decay of long nonlinear waves

1973 ◽  
Vol 58 (3) ◽  
pp. 481-493 ◽  
Author(s):  
S. Leibovich ◽  
J. D. Randall
Keyword(s):  
Shallow Water ◽  
Nonlinear Waves ◽  
Water Waves ◽  
Vries Equation ◽  
Variable Coefficients ◽  
Finite Amplitude ◽  
Rotating Fluids ◽  
Shallow Water Waves ◽  
Weakly Nonlinear ◽  
Weakly Nonlinear Waves

The interaction of weakly nonlinear waves with slowly varying boundaries is considered. Special emphasis is given to rotating fluids, but the analysis applies with minor modifications to waves in stratified fluids and shallow-water aves. An asymptotic solution of a variant of the Korteweg–de Vries equation with variable coefficients is developed that produces a ‘Green's law’ for the amplification of waves of finite amplitude. For shallow-water waves in water of variable depth, the result predicts wave growth proportional to the $-\frac{1}{3}$ power of the depth.

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.

scholarly journals Weakly nonlinear waves in magnetized plasma with a slightly non-Maxwellian electron distribution. Part 2. Stability of cnoidal waves

2007 ◽  
Vol 73 (6) ◽  
pp. 933-946
Author(s):  
S. PHIBANCHON ◽  
M. A. ALLEN ◽  
G. ROWLANDS
Keyword(s):  
Growth Rate ◽  
Nonlinear Waves ◽  
Electron Distribution ◽  
Magnetized Plasma ◽  
Weakly Nonlinear ◽  
Long Wavelength ◽  
First Order ◽  
Wave Solutions ◽  
Linear Waves ◽  
Weakly Nonlinear Waves

AbstractWe determine the growth rate of linear instabilities resulting from long-wavelength transverse perturbations applied to periodic nonlinear wave solutions to the Schamel–Korteweg–de Vries–Zakharov–Kuznetsov (SKdVZK) equation which governs weakly nonlinear waves in a strongly magnetized cold-ion plasma whose electron distribution is given by two Maxwellians at slightly different temperatures. To obtain the growth rate it is necessary to evaluate non-trivial integrals whose number is kept to a minimum by using recursion relations. It is shown that a key instance of one such relation cannot be used for classes of solution whose minimum value is zero, and an additional integral must be evaluated explicitly instead. The SKdVZK equation contains two nonlinear terms whose ratio b increases as the electron distribution becomes increasingly flat-topped. As b and hence the deviation from electron isothermality increases, it is found that for cnoidal wave solutions that travel faster than long-wavelength linear waves, there is a more pronounced variation of the growth rate with the angle θ at which the perturbation is applied. Solutions whose minimum values are zero and which travel slower than long-wavelength linear waves are found, at first order, to be stable to perpendicular perturbations and have a relatively narrow range of θ for which the first-order growth rate is not zero.

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