scholarly journals Frictional hysteresis and particle deposition in granular free-surface flows

2019 ◽  
Vol 875 ◽  
pp. 1058-1095 ◽  
Author(s):  
A. N. Edwards ◽  
A. S. Russell ◽  
C. G. Johnson ◽  
J. M. N. T. Gray
Keyword(s):  
Numerical Simulations ◽  
Particle Deposition ◽  
Initial Conditions ◽  
Travelling Waves ◽  
Travelling Wave ◽  
Free Surface Flows ◽  
Uniform Flow ◽  
Glass Beads ◽  
Deposit Thickness ◽  
Frictional Hysteresis

Shallow granular avalanches on slopes close to repose exhibit hysteretic behaviour. For instance, when a steady-uniform granular flow is brought to rest it leaves a deposit of thickness $h_{stop}(\unicode[STIX]{x1D701})$ on a rough slope inclined at an angle $\unicode[STIX]{x1D701}$ to the horizontal. However, this layer will not spontaneously start to flow again until it is inclined to a higher angle $\unicode[STIX]{x1D701}_{start}$, or the thickness is increased to $h_{start}(\unicode[STIX]{x1D701})>h_{stop}(\unicode[STIX]{x1D701})$. This simple phenomenology leads to a rich variety of flows with co-existing regions of solid-like and fluid-like granular behaviour that evolve in space and time. In particular, frictional hysteresis is directly responsible for the spontaneous formation of self-channelized flows with static levees, retrogressive failures as well as erosion–deposition waves that travel through the material. This paper is motivated by the experimental observation that a travelling-wave develops, when a steady uniform flow of carborundum particles on a bed of larger glass beads, runs out to leave a deposit that is approximately equal to $h_{stop}$. Numerical simulations using the friction law originally proposed by Edwards et al. (J. Fluid Mech., vol. 823, 2017, pp. 278–315) and modified here, demonstrate that there are in fact two travelling waves. One that marks the trailing edge of the steady-uniform flow and another that rapidly deposits the particles, directly connecting the point of minimum dynamic friction (at thickness $h_{\ast }$) with the deposited layer. The first wave moves slightly faster than the second wave, and so there is a slowly expanding region between them in which the flow thins and the particles slow down. An exact inviscid solution for the second travelling wave is derived and it is shown that for a steady-uniform flow of thickness $h_{\ast }$ it produces a deposit close to $h_{stop}$ for all inclination angles. Numerical simulations show that the two-wave structure deposits layers that are approximately equal to $h_{stop}$ for all initial thicknesses. This insensitivity to the initial conditions implies that $h_{stop}$ is a universal quantity, at least for carborundum particles on a bed of larger glass beads. Numerical simulations are therefore able to capture the complete experimental staircase procedure, which is commonly used to determine the $h_{stop}$ and $h_{start}$ curves by progressively increasing the inclination of the chute. In general, however, the deposit thickness may depend on the depth of the flowing layer that generated it, so the most robust way to determine $h_{stop}$ is to measure the deposit thickness from a flow that was moving at the minimum steady-uniform velocity. Finally, some of the pathologies in earlier non-monotonic friction laws are discussed and it is explicitly shown that with these models either steadily travelling deposition waves do not form or they do not leave the correct deposit depth $h_{stop}$.

Numerical solution of Fisher's equation

1974 ◽  
Vol 11 (03) ◽  
pp. 445-457 ◽  
Author(s):  
Jenö Gazdag ◽  
José Canosa
Keyword(s):  
Water Waves ◽  
Initial Conditions ◽  
Travelling Waves ◽  
Nonlinear Partial Differential Equations ◽  
Travelling Wave ◽  
Step Function ◽  
Local Perturbation ◽  
Fisher’S Equation ◽  
Minimal Speed ◽  
Fisher's Equation

The accurate space derivative (ASD) method for the numerical treatment of nonlinear partial differential equations has been applied to the solution of Fisher's equation, a nonlinear diffusion equation describing the rate of advance of a new advantageous gene, and which is also related to certain water waves and plasma shock waves described by the Korteweg-de-Vries-Burgers equation. The numerical experiments performed indicate how from a variety of initial conditions, (including a step function, and a wave with local perturbation) the concentration of advantageous gene evolves into the travelling wave of minimal speed. For an initial superspeed wave this evolution depends on the cutting off of the right-hand tail of the wave, which is physically plausible; this condition is necessary for the convergence of the ASD method. Detailed comparisons with an analytic solution for the travelling waves illustrate the striking accuracy of the ASD method for other than very small values of the concentration.

scholarly journals ASYMPTOTICAL ANALYSIS OF SOME COUPLED NONLINEAR WAVE EQUATIONS

2011 ◽  
Vol 16 (1) ◽  
pp. 97-108 ◽  
Author(s):  
Aleksandras Krylovas ◽  
Rima Kriauzienė
Keyword(s):  
Wave Equations ◽  
Initial Conditions ◽  
Travelling Waves ◽  
Linear Equations ◽  
Travelling Wave ◽  
Nonlinear Wave Equations ◽  
Linear Wave ◽  
Resonance Case ◽  
Non Linear ◽  
Asymptotical Analysis

We consider coupled nonlinear equations modelling a family of travelling wave solutions. The goal of our work is to show that the method of internal averaging along characteristics can be used for wide classes of coupled non-linear wave equations such as Korteweg-de Vries, Klein – Gordon, Hirota – Satsuma, etc. The asymptotical analysis reduces a system of coupled non-linear equations to a system of integro – differential averaged equations. The averaged system with the periodical initial conditions disintegrates into independent equations in non-resonance case. These equations describe simple weakly non-linear travelling waves in the non-resonance case. In the resonance case the integro – differential averaged systems describe interaction of waves and give a good asymptotical approximation for exact solutions.

scholarly journals Recurrence of travelling waves in transitional pipe flow

2007 ◽  
Vol 584 ◽  
pp. 69-102 ◽  
Author(s):  
R. R. KERSWELL ◽  
O. R. TUTTY
Keyword(s):  
Shear Stress ◽  
Wall Shear Stress ◽  
Pipe Flow ◽  
Initial Conditions ◽  
Travelling Waves ◽  
Transition Process ◽  
Travelling Wave ◽  
Shear Stresses ◽  
Wall Shear ◽  
Dividing Surface

The recent theoretical discovery of families of unstable travelling-wave solutions in pipe flow at Reynolds numbers lower than the transitional range, naturally raises the question of their relevance to the turbulent transition process. Here, a series of numerical experiments are conducted in which we look for the spatial signature of these travelling waves in transitionary flows. Working within a periodic pipe of 5D (diameters) length, we find that travelling waves with low wall shear stresses (lower branch solutions) are on a surface in phase space which separates initial conditions which uneventfully relaminarize and those which lead to a turbulent evolution. This dividing surface (a separatrix if turbulence is a sustained state) is then minimally the union of the stable manifolds of all these travelling waves. Evidence for recurrent travelling-wave visits is found in both 5D and 10D long periodic pipes, but only for those travelling waves with low-to-intermediate wall shear stress and for less than about 10% of the time in turbulent flow at Re = 2400. Given this, it seems unlikely that the mean turbulent properties such as wall shear stress can be predicted as an expansion solely over the travelling waves in which their individual properties are appropriately weighted. Instead the onus is on isolating further dynamical structures such as periodic orbits and including them in any such expansion.

Algebra, analysis and probability for a coupled system of reaction-duffusion equations

1995 ◽  
Vol 350 (1692) ◽  
pp. 69-112 ◽  
Keyword(s):  
Differential Equation ◽  
Large Deviation ◽  
Initial Conditions ◽  
Travelling Waves ◽  
Reaction Diffusion ◽  
Coupled System ◽  
Travelling Wave ◽  
The Other ◽  
Reaction Diffusion Equation ◽  
Local Martingale

This paper is designed to interest analysts and probabilists in the methods of the ‘other’ field applied to a problem important in biology and in other contexts. It does not strive for generality. After § 1 a , it concentrates on the simplest case of a coupled reaction-diffusion equation. It provides a complete treatment of the existence, uniqueness, and asymptotic behaviour of monotone travelling waves to various equilibria, both by differential-equation theory and by probability theory. Each approach raises interesting questions about the other. The differential-equation treatment makes new use of the maximum principle for this type of problem. It suggests a numerical method of solution which yields computer pictures which illustrate the situation very clearly. The probabilistic treatment is careful in its proofs of martingale (as opposed to merely local-martingale) properties. A new change-of-measure technique is used to obtain the best lower bound on the speed of the monotone travelling wave with Heaviside initial conditions. Waves to different equilibria are shown to be related by Doob h -transforms. Large-deviation theory provides heuristic links between alternative descriptions of minimum wave speeds, rigorous algebraic proofs of which are provided. Since the paper was submitted, an alternative method of proving existence of monotone travelling waves has been developed by Karpelevich et al. (1993). We have extended our results in different directions from theirs (one of which is hinted at in § 1 a ), and have found the methods used here well equipped for these generalizations. See the Addendum.

scholarly journals On the generation of nonlinear travelling waves in confined geometries using electric fields

2014 ◽  
Vol 372 (2020) ◽  
pp. 20140066 ◽  
Author(s):  
R Cimpeanu ◽  
D. T Papageorgiou
Keyword(s):  
Numerical Simulations ◽  
Electric Fields ◽  
Travelling Waves ◽  
Travelling Wave ◽  
Direct Numerical Simulations ◽  
Small Scale ◽  
Interfacial Instabilities ◽  
Lower Electrode ◽  
Nonlinear Deformations ◽  
Moving Parts

We investigate electrostatically induced interfacial instabilities and subsequent generation of nonlinear coherent structures in immiscible, viscous, dielectric multi-layer stratified flows confined in small-scale channels. Vertical electric fields are imposed across the channel to produce interfacial instabilities that would normally be absent in such flows. In situations when the imposed vertical fields are constant, interfacial instabilities emerge due to the presence of electrostatic forces, and we follow the nonlinear dynamics via direct numerical simulations. We also propose and illustrate a novel pumping mechanism in microfluidic devices that does not use moving parts. This is achieved by first inducing interfacial instabilities using constant background electric fields to obtain fully nonlinear deformations. The second step involves the manipulation of the imposed voltage on the lower electrode (channel wall) to produce a spatio-temporally varying voltage there, in the form of a travelling wave with pre-determined properties. Such travelling wave dielectrophoresis methods are shown to generate intricate fluid–surface–structure interactions that can be of practical value since they produce net mass flux along the channel and thus are candidates for microfluidic pumps without moving parts. We show via extensive direct numerical simulations that this pumping phenomenon is a result of an externally induced nonlinear travelling wave that forms at the fluid–fluid interface and study the characteristics of the generated velocity field inside the channel.

scholarly journals Allee optimal control of a system in ecology

2018 ◽  
Vol 28 (09) ◽  
pp. 1665-1697 ◽  
Author(s):  
Emmanuel Trélat ◽  
Jiamin Zhu ◽  
Enrique Zuazua
Keyword(s):  
Optimal Control ◽  
Numerical Simulations ◽  
Travelling Waves ◽  
Travelling Wave ◽  
Computational Method ◽  
Diffusion Reaction ◽  
Macroscopic Model ◽  
Future Research ◽  
Optimal Controls ◽  
Open Problems

The Allee threshold of an ecological system distinguishes the sign of population growth either towards extinction or to carrying capacity. In practice, human interventions can tune the Allee threshold for instance thanks to the sterile male technique and the mating disruption. In this paper, we address various control problems for a system described by a diffusion–reaction equation regulating the Allee threshold, viewed as a real parameter determining the unstable equilibrium of the bistable nonlinear reaction term. We prove that this system is the mean field limit of an interacting system of particles in which the individual behaviour is driven by stochastic laws. Numerical simulations of the stochastic process show that the propagation of population is governed by travelling wave solutions of the macroscopic reaction–diffusion system, which model the fact that solutions, in bounded space domains, reach asymptotically an equilibrium configuration.An optimal control problem for the macroscopic model is then introduced with the objective of steering the system to a target travelling wave. Using well-known analytical results and stability properties of travelling waves, we show that well-chosen piecewise constant controls allow to reach the target approximately in sufficiently long time. We then develop a direct computational method and show its efficiency for computing such controls in various numerical simulations. Finally, we show the effectiveness of the obtained macroscopic optimal controls in the microscopic system of interacting particles and we discuss their advantage when addressing situations that are out of reach for the analytical methods. We conclude the paper with some open problems and directions for future research.

Numerical solution of Fisher's equation

1974 ◽  
Vol 11 (3) ◽  
pp. 445-457 ◽  
Author(s):  
Jenö Gazdag ◽  
José Canosa
Keyword(s):  
Water Waves ◽  
Initial Conditions ◽  
Travelling Waves ◽  
Nonlinear Partial Differential Equations ◽  
Travelling Wave ◽  
Step Function ◽  
Local Perturbation ◽  
Fisher’S Equation ◽  
Minimal Speed ◽  
Fisher's Equation

The accurate space derivative (ASD) method for the numerical treatment of nonlinear partial differential equations has been applied to the solution of Fisher's equation, a nonlinear diffusion equation describing the rate of advance of a new advantageous gene, and which is also related to certain water waves and plasma shock waves described by the Korteweg-de-Vries-Burgers equation. The numerical experiments performed indicate how from a variety of initial conditions, (including a step function, and a wave with local perturbation) the concentration of advantageous gene evolves into the travelling wave of minimal speed. For an initial superspeed wave this evolution depends on the cutting off of the right-hand tail of the wave, which is physically plausible; this condition is necessary for the convergence of the ASD method. Detailed comparisons with an analytic solution for the travelling waves illustrate the striking accuracy of the ASD method for other than very small values of the concentration.

scholarly journals Modelling the Effect of Cell Shedding on Avascular Tumour Growth

2000 ◽  
Vol 2 (3) ◽  
pp. 155-174 ◽  
Author(s):  
J. P. Ward ◽  
J. R. King
Keyword(s):  
Steady State ◽  
Tumour Growth ◽  
Numerical Solutions ◽  
Initial Conditions ◽  
Travelling Waves ◽  
Tumour Size ◽  
Travelling Wave ◽  
Live Cells ◽  
Long Time ◽  
Cell Shedding

Earlier mathematical models of the authors which describe avascular tumour growth are extended to incorporate the process of cell shedding, a feature known to affect the growth of multicell spheroids. A continuum of live cells is assumed within which, depending on the concentration of a generic nutrient, movement (described by a velocity field) occurs due to volume changes caused by cell birth and death. The necrotic material is assumed to contain a mixture of basic cellular material (assumed necessary for creating new cells) and a non-utilisable material which may inhibit mitosis. The rate of cell shedding is taken to be proportional to the mitotic rate, with constant of proportionality θ. Numerical solutions of the resulting system of partial differential equations indicate that, depending on θ and the initial conditions, the solution may either tend to the trivial state in finite time (by which we mean complete death of the tumour), or to one of two non-trivial states, namely a steady-state (indicating growth saturation) or a travelling wave (indicating continual linear growth). These long time outcomes are explored by deriving the travelling wave and steady-state limits of the model. Numerical solutions demonstrate that there are two branches of solutions, which we have termed the ′Major′ and ′Minor′ branches, consisting of both travelling waves and steady-states. The behaviour of the solutions along each branch is discussed, with those of the Major branch expected to be stable. Beyond some critical θ,where the Major and Minor branches merge, the spheroid ultimately vanishes whatever the initial tumour size due to the effects of cell shedding being too strong for it to survive. The regions of existence of the two long time outcomes are investigated in parameter space, cell shedding being shown to expand significantly the parameter ranges within which growth saturation occurs.

scholarly journals The Impact of Initial Conditions in N-Body Simulations of Debris Discs

2015 ◽  
Vol 32 ◽  
Author(s):  
E. Thilliez ◽  
S. T. Maddison
Keyword(s):  
Numerical Simulations ◽  
Initial Conditions ◽  
Parent Body ◽  
Dust Trapping ◽  
Physical Context ◽  
Disc Width ◽  
Wide Range ◽  
Belt Width ◽  
New Generation ◽  
The Impact

AbstractNumerical simulations are a crucial tool to understand the relationship between debris discs and planetary companions. As debris disc observations are now reaching unprecedented levels of precision over a wide range of wavelengths, an appropriate level of accuracy and consistency is required in numerical simulations to confidently interpret this new generation of observations. However, simulations throughout the literature have been conducted with various initial conditions often with little or no justification. In this paper, we aim to study the dependence on the initial conditions of N-body simulations modelling the interaction between a massive and eccentric planet on an exterior debris disc. To achieve this, we first classify three broad approaches used in the literature and provide some physical context for when each category should be used. We then run a series of N-body simulations, that include radiation forces acting on small grains, with varying initial conditions across the three categories. We test the influence of the initial parent body belt width, eccentricity, and alignment with the planet on the resulting debris disc structure and compare the final peak emission location, disc width and offset of synthetic disc images produced with a radiative transfer code. We also track the evolution of the forced eccentricity of the dust grains induced by the planet, as well as resonance dust trapping. We find that an initially broad parent body belt always results in a broader debris disc than an initially narrow parent body belt. While simulations with a parent body belt with low initial eccentricity (e ~ 0) and high initial eccentricity (0 < e < 0.3) resulted in similar broad discs, we find that purely secular forced initial conditions, where the initial disc eccentricity is set to the forced value and the disc is aligned with the planet, always result in a narrower disc. We conclude that broad debris discs can be modelled by using either a dynamically cold or dynamically warm parent belt, while in contrast eccentric narrow debris rings are reproduced using a secularly forced parent body belt.

Travelling Wave Control of Rotating Discs and Analysis of its Spillover Effect

1988 ◽  
Vol 202 (2) ◽  
pp. 119-127 ◽  
Author(s):  
C-S Kim ◽  
C-W Lee
Keyword(s):  
Travelling Waves ◽  
Spillover Effect ◽  
Polynomial Equation ◽  
Travelling Wave ◽  
Rotating Disc ◽  
System A ◽  
Finite Dimensional ◽  
Control Scheme ◽  
Actuator System ◽  
Wave Control

A modal control scheme for rotating disc systems is developed based upon the finite-dimensional sub-system model including a few lower backward travelling waves important to the disc response. For the single discrete sensor and actuator system, a polynomial equation, which determines the closed-loop system poles, is derived and the spillover effect is analysed, providing a sufficient condition for stability. Finally, simulation studies are performed to show the effectiveness of the travelling wave control scheme proposed.

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