Periodic travelling interfacial hydroelastic waves with or without mass II: Multiple bifurcations and ripples

2018 ◽  
Vol 30 (04) ◽  
pp. 756-790 ◽  
Author(s):  
BENJAMIN F. AKERS ◽  
DAVID M. AMBROSE ◽  
DAVID W. SULON
Keyword(s):  
Travelling Waves ◽  
Global Bifurcation ◽  
Travelling Wave ◽  
Two Dimensional ◽  
Wave Problem ◽  
Bifurcation Theorem ◽  
One Dimensional ◽  
Spatially Periodic ◽  
Parameter Values ◽  
The Waves

In a prior work, the authors proved a global bifurcation theorem for spatially periodic interfacial hydroelastic travelling waves on infinite depth, and computed such travelling waves. The formulation of the travelling wave problem used both analytically and numerically allows for waves with multi-valued height. The global bifurcation theorem required a one-dimensional kernel in the linearization of the relevant mapping, but for some parameter values, the kernel is instead two-dimensional. In the present work, we study these cases with two-dimensional kernels, which occur in resonant and non-resonant variants. We apply an implicit function theorem argument to prove existence of travelling waves in both of these situations. We compute the waves numerically as well, in both the resonant and non-resonant cases.

Mathematical modelling of atherosclerosis as an inflammatory disease

2009 ◽  
Vol 367 (1908) ◽  
pp. 4877-4886 ◽  
Author(s):  
N. El Khatib ◽  
S. Génieys ◽  
B. Kazmierczak ◽  
V. Volpert
Keyword(s):  
Inflammatory Response ◽  
Inflammatory Disease ◽  
Travelling Waves ◽  
Reaction Diffusion ◽  
Travelling Wave ◽  
Reaction Diffusion Equations ◽  
Bistable System ◽  
Two Dimensional ◽  
Chronic Inflammatory Response ◽  
One Dimensional

Atherosclerosis is an inflammatory disease. The atherosclerosis process starts when low-density lipoproteins (LDLs) enter the intima of the blood vessel, where they are oxidized (ox-LDLs). The anti-inflammatory response triggers the recruitment of monocytes. Once in the intima, the monocytes are transformed into macrophages and foam cells, leading to the production of inflammatory cytokines and further recruitment of monocytes. This auto-amplified process leads to the formation of an atherosclerotic plaque and, possibly, to its rupture. In this paper we develop two mathematical models based on reaction–diffusion equations in order to explain the inflammatory process. The first model is one-dimensional: it does not consider the intima’s thickness and shows that low ox-LDL concentrations in the intima do not lead to a chronic inflammatory reaction. Intermediate ox-LDL concentrations correspond to a bistable system, which can lead to a travelling wave that can be initiated by certain conditions, such as infection or injury. High ox-LDL concentrations correspond to a monostable system, and even a small perturbation of the non-inflammatory case leads to travelling-wave propagation, which corresponds to a chronic inflammatory response. The second model we suggest is two-dimensional: it represents a reaction–diffusion system in a strip with nonlinear boundary conditions to describe the recruitment of monocytes as a function of the cytokines’ concentration. We prove the existence of travelling waves and confirm our previous results, which show that atherosclerosis develops as a reaction–diffusion wave. The results of the two models are confirmed by numerical simulations. The latter show that the two-dimensional model converges to the one-dimensional one if the thickness of the intima tends to zero.

TRAVELLING WAVES AND OSCILLATIONS IN SAL’NIKOV’S COMBUSTION REACTION IN A COMPRESSIBLE GAS

2015 ◽  
Vol 56 (3) ◽  
pp. 233-247 ◽  
Author(s):  
RHYS A. PAUL ◽  
LAWRENCE K. FORBES
Keyword(s):  
Asymptotic Solution ◽  
Multiple Scales ◽  
Travelling Waves ◽  
Method Of Lines ◽  
Travelling Wave ◽  
Reaction Scheme ◽  
Compressible Viscous Gas ◽  
The Method Of Lines ◽  
Temperature Waves ◽  
Parameter Values

We consider a two-step Sal’nikov reaction scheme occurring within a compressible viscous gas. The first step of the reaction may be either endothermic or exothermic, while the second step is strictly exothermic. Energy may also be lost from the system due to Newtonian cooling. An asymptotic solution for temperature perturbations of small amplitude is presented using the methods of strained coordinates and multiple scales, and a travelling wave solution with a sech-squared profile is derived. The method of lines is then used to approximate the full system with a set of ordinary differential equations, which are integrated numerically to track accurately the evolution of the reaction front. This numerical method is used to verify the asymptotic solution and investigate behaviours under different conditions. Using this method, temperature waves progressing as pulsatile fronts are detected at appropriate parameter values.

scholarly journals Snakes mimic earthworms: propulsion using rectilinear travelling waves

2013 ◽  
Vol 10 (84) ◽  
pp. 20130188 ◽  
Author(s):  
Hamidreza Marvi ◽  
Jacob Bridges ◽  
David L. Hu
Keyword(s):  
Theoretical Study ◽  
Travelling Waves ◽  
Vertical Motion ◽  
Muscular Contraction ◽  
Travelling Wave ◽  
Wave Frequency ◽  
One Dimensional ◽  
Anatomical Constraints ◽  
Local Body ◽  
Body Lifting

In rectilinear locomotion, snakes propel themselves using unidirectional travelling waves of muscular contraction, in a style similar to earthworms. In this combined experimental and theoretical study, we film rectilinear locomotion of three species of snakes, including red-tailed boa constrictors, Dumeril's boas and Gaboon vipers. The kinematics of a snake's extension–contraction travelling wave are characterized by wave frequency, amplitude and speed. We find wave frequency increases with increasing body size, an opposite trend than that for legged animals. We predict body speed with 73–97% accuracy using a mathematical model of a one-dimensional n -linked crawler that uses friction as the dominant propulsive force. We apply our model to show snakes have optimal wave frequencies: higher values increase Froude number causing the snake to slip; smaller values decrease thrust and so body speed. Other choices of kinematic variables, such as wave amplitude, are suboptimal and appear to be limited by anatomical constraints. Our model also shows that local body lifting increases a snake's speed by 31 per cent, demonstrating that rectilinear locomotion benefits from vertical motion similar to walking.

scholarly journals Nonlinear dynamics of surfactant-laden two-fluid Couette flows in the presence of inertia

2016 ◽  
Vol 802 ◽  
pp. 5-36 ◽  
Author(s):  
A. Kalogirou ◽  
D. T. Papageorgiou
Keyword(s):  
Stokes Equations ◽  
Travelling Waves ◽  
Type Theorem ◽  
Three Dimensional ◽  
Two Dimensional ◽  
One Dimensional ◽  
Wide Range ◽  
Time Periodic ◽  
Two Fluid ◽  
Permanent Form

The nonlinear stability of immiscible two-fluid Couette flows in the presence of inertia is considered. The interface between the two viscous fluids can support insoluble surfactants and the interplay between the underlying hydrodynamic instabilities and Marangoni effects is explored analytically and computationally in both two and three dimensions. Asymptotic analysis when one of the layers is thin relative to the other yields a coupled system of nonlinear equations describing the spatio-temporal evolution of the interface and its local surfactant concentration. The system is non-local and arises by appropriately matching solutions of the linearised Navier–Stokes equations in the thicker layer to the solution in the thin layer. The scaled models are used to study different physical mechanisms by varying the Reynolds number, the viscosity ratio between the two layers, the total amount of surfactant present initially and a scaled Péclet number measuring diffusion of surfactant along the interface. The linear stability of the underlying flow to two- and three-dimensional disturbances is investigated and a Squire’s type theorem is found to hold when inertia is absent. When inertia is present, three-dimensional disturbances can be more unstable than two-dimensional ones and so Squire’s theorem does not hold. The linear instabilities are followed into the nonlinear regime by solving the evolution equations numerically; this is achieved by implementing highly accurate linearly implicit schemes in time with spectral discretisations in space. Numerical experiments for finite Reynolds numbers indicate that for two-dimensional flows the solutions are mostly nonlinear travelling waves of permanent form, even though these can lose stability via Hopf bifurcations to time-periodic travelling waves. As the length of the system (that is the wavelength of periodic waves) increases, the dynamics becomes more complex and includes time-periodic, quasi-periodic as well as chaotic fluctuations. It is also found that one-dimensional interfacial travelling waves of permanent form can become unstable to spanwise perturbations for a wide range of parameters, producing three-dimensional flows with interfacial profiles that are two-dimensional and travel in the direction of the underlying shear. Nonlinear flows are also computed for parameters which predict linear instability to three-dimensional disturbances but not two-dimensional ones. These are found to have a one-dimensional interface in a rotated frame with respect to the direction of the underlying shear and travel obliquely without changing form.

One- and two-dimensional travelling wave solutions in gas-fluidized beds

1996 ◽  
Vol 306 ◽  
pp. 183-221 ◽  
Author(s):  
B. J. Glasser ◽  
I. G. Kevrekidis ◽  
S. Sundaresan
Keyword(s):  
Fluidized Beds ◽  
Equations Of Motion ◽  
High Amplitude ◽  
Travelling Wave ◽  
Numerical Continuation ◽  
Model Parameters ◽  
Travelling Wave Solutions ◽  
Two Dimensional ◽  
One Dimensional ◽  
Wave Solutions

Making use of numerical continuation techniques as well as bifurcation theory, both one- and two-dimensional travelling wave solutions of the ensemble-averaged equations of motion for gas and particles in fluidized beds have been computed. One-dimensional travelling wave solutions having only vertical structure emerge through a Hopf bifurcation of the uniform state and two-dimensional travelling wave solutions are born out of these one-dimensional waves. Fully developed two-dimensional solutions of high amplitude are reminiscent of bubbles. It is found that the qualitative features of the bifurcation diagram are not affected by changes in model parameters or the closures. An examination of the stability of one-dimensional travelling wave solutions to two-dimensional perturbations suggests that two-dimensional solutions emerge through a mechanism which is similar to the overturning instability analysed by Batchelor & Nitsche (1991).

scholarly journals Two-dimensional nonlinear travelling waves in magnetohydrodynamic channel flow

2014 ◽  
Vol 760 ◽  
pp. 387-406 ◽  
Author(s):  
Jonathan Hagan ◽  
Jānis Priede
Keyword(s):  
Magnetic Field ◽  
Linear Stability ◽  
Channel Flow ◽  
Travelling Waves ◽  
Transverse Magnetic ◽  
Travelling Wave ◽  
Linear Stability Theory ◽  
Two Dimensional ◽  
Order Of Magnitude ◽  
The Magnetic Field

AbstractThis study is concerned with the stability of a flow of viscous conducting liquid driven by a pressure gradient in the channel between two parallel walls subject to a transverse magnetic field. Although the magnetic field has a strong stabilizing effect, this flow, similarly to its hydrodynamic counterpart – plane Poiseuille flow – is known to become turbulent significantly below the threshold predicted by linear stability theory. We investigate the effect of the magnetic field on two-dimensional nonlinear travelling-wave states which are found at substantially subcritical Reynolds numbers starting from $\mathit{Re}_{n}=2939$ without the magnetic field and from $\mathit{Re}_{n}\sim 6.50\times 10^{3}\mathit{Ha}$ in a sufficiently strong magnetic field defined by the Hartmann number $\mathit{Ha}$. Although the latter value is a factor of seven lower than the linear stability threshold $\mathit{Re}_{l}\sim 4.83\times 10^{4}\mathit{Ha}$, it is still more than an order of magnitude higher than the experimentally observed value for the onset of turbulence in magnetohydrodynamic (MHD) channel flow.

scholarly journals A polymer–solvent model of biofilm growth

2010 ◽  
Vol 467 (2129) ◽  
pp. 1449-1467 ◽  
Author(s):  
H. F. Winstanley ◽  
M. Chapwanya ◽  
M. J. McGuinness ◽  
A. C. Fowler
Keyword(s):  
Numerical Solutions ◽  
Travelling Waves ◽  
Travelling Wave ◽  
Bacterial Biofilms ◽  
Travelling Wave Solutions ◽  
Biofilm Growth ◽  
One Dimensional ◽  
Solvent Model ◽  
Time Dependent Problem ◽  
Viscoelastic Rheology

We provide and analyse a model for the growth of bacterial biofilms based on the concept of an extracellular polymeric substance as a polymer solution, whose viscoelastic rheology is described by the classical Flory–Huggins theory. We show that one-dimensional solutions exist, which take the form at large times of travelling waves, and we characterize their form and speed in terms of the describing parameters of the problem. Numerical solutions of the time-dependent problem converge to the travelling wave solutions.

The growth, saturation, and scaling behaviour of one- and two-dimensional disturbances in fluidized beds

1998 ◽  
Vol 362 ◽  
pp. 83-119 ◽  
Author(s):  
M. F. GÖZ ◽  
S. SUNDARESAN
Keyword(s):  
Fluidized Bed ◽  
Growth Rates ◽  
Fluidized Beds ◽  
Secondary Growth ◽  
Two Dimensional ◽  
Small Perturbations ◽  
One Dimensional ◽  
Gas Fluidized Beds ◽  
The Waves ◽  
Primary Instability

It is well-known that fluidized beds are usually unstable to small perturbations and that this leads to the primary bifurcation of vertically travelling plane wavetrains. These one-dimensional periodic waves have been shown recently to be unstable to two-dimensional perturbations of large transverse wavelength in gas-fluidized beds. Here, this result is generalized to include liquid-fluidized beds and to compare typical beds fluidized with either air or water. It is shown that the instability mechanism remains the same but there are big differences in the ratio of the primary and secondary growth rates in the two cases. The tendency is that the secondary growth rates, scaled with the amplitude of a fully developed plane wave, are of similar magnitude for both gas- and liquid-fluidized beds, while the primary growth rate is much larger in the gas-fluidized bed. This means that the secondary instability is accordingly stronger than the primary instability in the liquid-fluidized bed, and consequently sets in at a much smaller amplitude of the primary wave. However, since the waves in the liquid-fluidized bed develop on a larger time and length scale, the primary perturbations need longer time and thereby travel farther until they reach the critical amplitude. Which patterns are more amenable to being visually recognized depends on the magnitude of the initially imposed disturbance and the dimensions of the apparatus. This difference in scale plays a key role in bringing about the differences between gas- and liquid-fluidized beds; it is produced mainly by the different values of the Froude number.

scholarly journals Localized States in Bi-Pattern Systems

2009 ◽  
Vol 2009 ◽  
pp. 1-9 ◽  
Author(s):  
U. Bortolozzo ◽  
M. G. Clerc ◽  
F. Haudin ◽  
R. G. Rojas ◽  
S. Residori
Keyword(s):  
Optical System ◽  
Nonlinear Optical ◽  
Amplitude Equation ◽  
Localized States ◽  
Two Dimensional ◽  
One Dimensional ◽  
Self Organized ◽  
Localized Structures ◽  
Spatially Periodic ◽  
Different Shapes

We present a unifying description of localized states observed in systems with coexistence of two spatially periodic states, calledbi-pattern systems. Localized states are pinned over an underlying lattice that is either a self-organized pattern spontaneously generated by the system itself, or a periodic grid created by a spatial forcing. We show that localized states are generic and require only the coexistence of two spatially periodic states. Experimentally, these states have been observed in a nonlinear optical system. At the onset of the spatial bifurcation, a forced one-dimensional amplitude equation is derived for the critical modes, which accounts for the appearance of localized states. By numerical simulations, we show that localized structures persist on two-dimensional systems and exhibit different shapes depending on the symmetry of the supporting patterns.

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