Solitons, solitary waves, and voidage disturbances in gas-fluidized beds

1994 ◽  
Vol 266 ◽  
pp. 243-276 ◽  
Author(s):  
S. E. Harris ◽  
D. G. Crighton
Keyword(s):  
Vries Equation ◽  
Solitary Wave Solution ◽  
Finite Amplitude ◽  
Order Equation ◽  
Leading Order ◽  
Dimensional Model ◽  
Weakly Nonlinear ◽  
One Dimensional ◽  
Weakly Nonlinear Waves ◽  
Gas Fluidized Beds

In this paper, we consider the evolution of an initially small voidage disturbance in a gas-fluidized bed. Using a one-dimensional model proposed by Needham & Merkin (1983), Crighton (1991) has shown that weakly nonlinear waves of voidage propagate according to the Korteweg–de Vries equation with perturbation terms which can be either amplifying or dissipative, depending on the sign of a coefficient. Here, we investigate the unstable side of the threshold and examine the growth of a single KdV voidage soliton, following its development through several different regimes. As the size of the soliton increases, KdV remains the leading-order equation for some time, but the perturbation terms change, thereby altering the dependence of the amplitude on time. Eventually the disturbance attains a finite amplitude and corresponds to a fully nonlinear solitary wave solution. This matches back directly onto the KdV soliton and tends exponentially to a limiting size. We interpret the series of large-amplitude localized pulses of voidage formed in this way from initial disturbances as corresponding to the ‘voidage slugs’ observed in gas fluidization in narrow tubes.

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.

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.

The growth of periodic waves in gas-fluidized beds

1996 ◽  
Vol 325 ◽  
pp. 261-282 ◽  
Author(s):  
S. E. Harris
Keyword(s):  
Nonlinear Waves ◽  
Fluidized Beds ◽  
Kdv Equation ◽  
Periodic Waves ◽  
Weakly Nonlinear ◽  
Strongly Nonlinear ◽  
The Kdv Equation ◽  
Weakly Nonlinear Waves ◽  
Gas Fluidized Beds ◽  
The One

In this paper, we analyse the development of initially small, periodic, voidage disturbances in gas-fluidized beds. The one-dimensional model was proposed by Needham & Merkin (1983), and Crighton (1991) showed that weakly nonlinear waves satisfied a perturbed Korteweg–de Vries or KdV equation. Here, we take periodic cnoidal wave solutions of the KdV equation and follow their evolution when the perturbation terms are amplifying. Initially, all such waves grow, but at a later stage a rescaling shows that shorter wavelengths are stabilized in a weakly nonlinear state. Longer wavelengths continue to develop and eventually strongly nonlinear solutions are required. Necessary conditions for periodic waves are found and matching back onto the growing cnoidal waves is possible. It is shown further that these fully nonlinear waves also reach an equilibrium state. A comparison with numerical results from Needham & Merkin (1986) and Anderson, Sundaresan & Jackson (1995) is then carried out.

scholarly journals Propagation of Nonlinear Pressure Waves in Blood

2013 ◽  
Vol 2013 ◽  
pp. 1-5 ◽  
Author(s):  
A. Elgarayhi ◽  
E. K. El-Shewy ◽  
Abeer A. Mahmoud ◽  
Ali A. Elhakem
Keyword(s):  
Vries Equation ◽  
Finite Amplitude ◽  
Reductive Perturbation Method ◽  
Elastic Tube ◽  
Pressure Waves ◽  
Weakly Nonlinear ◽  
Flow Problems ◽  
Inner Radius ◽  
Reductive Perturbation ◽  
Conditions Of Stability

The propagation of weakly nonlinear pressure waves in a fluid-filled elastic tube has been investigated. The reductive perturbation method has been employed to derive the Korteweg-de Vries equation for small but finite amplitude. The effect of the final inner radius of the tube on the basic properties of the soliton wave was discussed. Moreover, the conditions of stability and the soliton existence via the potential and the corresponding phase portrait were computed. The applicability of the present investigation to flow problems in arteries is discussed.

Modelling Techniques in Applied Glaciology: Numerical Modelling of Avalanches

1983 ◽  
Vol 4 ◽  
pp. 297-297
Author(s):  
G. Brugnot
Keyword(s):  
Finite Difference Method ◽  
Transverse Velocity ◽  
Longitudinal Velocity ◽  
Momentum Conservation ◽  
Dimensional Model ◽  
One Dimensional ◽  
Modelling Techniques ◽  
Reference Grid ◽  
The One ◽  
Near Future

We consider the paper by Brugnot and Pochat (1981), which describes a one-dimensional model applied to a snow avalanche. The main advance made here is the introduction of the second dimension in the runout zone. Indeed, in the channelled course, we still use the one-dimensional model, but, when the avalanche spreads before stopping, we apply a (x, y) grid on the ground and six equations have to be solved: (1) for the avalanche body, one equation for continuity and two equations for momentum conservation, and (2) at the front, one equation for continuity and two equations for momentum conservation. We suppose the front to be a mobile jump, with longitudinal velocity varying more rapidly than transverse velocity.We solve these equations by a finite difference method. This involves many topological problems, due to the actual position of the front, which is defined by its intersection with the reference grid (SI, YJ). In the near future our two directions of research will be testing the code on actual avalanches and improving it by trying to make it cheaper without impairing its accuracy.

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