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.

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.

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.

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.

Existence and uniqueness of travelling waves for a neural network

1993 ◽  
Vol 123 (3) ◽  
pp. 461-478 ◽  
Author(s):  
G. Bard Ermentrout ◽  
J. Bryce McLeod
Keyword(s):  
Neural Network ◽  
Wave Front ◽  
Existence And Uniqueness ◽  
Travelling Waves ◽  
Travelling Wave ◽  
Steady States ◽  
Stable States ◽  
One Dimensional ◽  
Travelling Wave Front

SynopsisA one-dimensional scalar neural network with two stable steady states is analysed. It is shown that there exists a unique monotone travelling wave front which joins the two stable states. Some additional properties of the wave such as the direction of its velocity are discussed.

Theoretical study of the susceptibility of nearly one-dimensional VF2

1978 ◽  
Vol 27 (11) ◽  
pp. 1079-1081 ◽  
Author(s):  
S. Konishi ◽  
K. Motizuki
Keyword(s):  
Theoretical Study ◽  
One Dimensional

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.

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.

Nonlinear travelling internal waves with piecewise-linear shear profiles

2018 ◽  
Vol 856 ◽  
pp. 984-1013 ◽  
Author(s):  
K. L. Oliveras ◽  
C. W. Curtis
Keyword(s):  
Piecewise Linear ◽  
Travelling Waves ◽  
Tangential Velocity ◽  
Travelling Wave ◽  
Numerical Continuation ◽  
Travelling Wave Solutions ◽  
Water Wave Problem ◽  
Two Fluids ◽  
Wave Solutions ◽  
Linear Shear

In this work, we study the nonlinear travelling waves in density stratified fluids with piecewise-linear shear currents. Beginning with the formulation of the water-wave problem due to Ablowitz et al. (J. Fluid Mech., vol. 562, 2006, pp. 313–343), we extend the work of Ashton & Fokas (J. Fluid Mech., vol. 689, 2011, pp. 129–148) and Haut & Ablowitz (J. Fluid Mech., vol. 631, 2009, pp. 375–396) to examine the interface between two fluids of differing densities and varying linear shear. We derive a systems of equations depending only on variables at the interface, and numerically solve for periodic travelling wave solutions using numerical continuation. Here, we consider only branches which bifurcate from solutions where there is no slip in the tangential velocity at the interface for the trivial flow. The spectral stability of these solutions is then determined using a numerical Fourier–Floquet technique. We find that the strength of the linear shear in each fluid impacts the stability of the corresponding travelling wave solutions. Specifically, opposing shears may amplify or suppress instabilities.

Theoretical study of soliton dynamics of a finite one-dimensional hydrogen-bonded system

1986 ◽  
Vol 107 (2-3) ◽  
pp. 389-396 ◽  
Author(s):  
Yoshiki Kashimori ◽  
Fuchun Chien ◽  
Kichisuke Nishimoto
Keyword(s):  
Theoretical Study ◽  
One Dimensional ◽  
Soliton Dynamics ◽  
Hydrogen Bonded
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