scholarly journals Analysis and computations of a non-local thin-film model for two-fluid shear driven flows

2019 ◽  
Vol 475 (2230) ◽  
pp. 20190367 ◽  
Author(s):  
D. T. Papageorgiou ◽  
S. Tanveer
Keyword(s):  
Thin Film ◽  
Viscous Fluid ◽  
Travelling Wave ◽  
Travelling Wave Solutions ◽  
Large Wave ◽  
Film Model ◽  
Bifurcation Points ◽  
Wave Solutions ◽  
Thin Film Model ◽  
Non Local

This paper is concerned with analysis and computations of a non-local thin-film model developed in Kalogirou & Papageorgiou ( J. Fluid Mech. 802 , 5–36, 2016) for a perturbed two-layer Couette flow when the thickness of the more viscous fluid layer next to the stationary wall is small compared to the thickness of the less viscous fluid. Travelling wave solutions and their stability are determined numerically, and secondary bifurcation points are identified in the process. We also determine regions in parameter space where bistability is observed with two branches being linearly stable at the same time. The travelling wave solutions are mathematically justified through a quasi-solution analysis in a neighbourhood of an empirically constructed approximate solution. This relies in part on precise asymptotics of integrals of Airy functions for large wave numbers. The primary bifurcation about the trivial state is shown rigorously to be supercritical, and the dependence of bifurcation points, as a function of Reynolds number R and the primary wavelength 2 πν −1/2 of the disturbance, is determined analytically.

Travelling waves for delayed reaction–diffusion equations with global response

2005 ◽  
Vol 462 (2065) ◽  
pp. 229-261 ◽  
Author(s):  
Teresa Faria ◽  
Wenzhang Huang ◽  
Jianhong Wu
Keyword(s):  
Travelling Waves ◽  
Reaction Diffusion ◽  
Travelling Wave ◽  
Reaction Diffusion Equations ◽  
Diffusion Equations ◽  
Index Formula ◽  
Delayed Response ◽  
Travelling Wave Solutions ◽  
Wave Solutions ◽  
Non Local

We develop a new approach to obtain the existence of travelling wave solutions for reaction–diffusion equations with delayed non-local response. The approach is based on an abstract formulation of the wave profile as a solution of an operational equation in a certain Banach space, coupled with an index formula of the associated Fredholm operator and some careful estimation of the nonlinear perturbation. The general result relates the existence of travelling wave solutions to the existence of heteroclinic connecting orbits of a corresponding functional differential equation, and this result is illustrated by an application to a model describing the population growth when the species has two age classes and the diffusion of the individual during the maturation process leads to an interesting non-local and delayed response for the matured population.

scholarly journals Travelling-wave analysis of a model describing tissue degradation by bacteria

2007 ◽  
Vol 18 (5) ◽  
pp. 583-605 ◽  
Author(s):  
D. HILHORST ◽  
J. R. KING ◽  
M. RÖGER
Keyword(s):  
Reaction Diffusion ◽  
Travelling Wave ◽  
Reaction Diffusion System ◽  
Diffusion System ◽  
Travelling Wave Solutions ◽  
Large Wave ◽  
Minimal Speed ◽  
Wave Solutions ◽  
Degradation Rates ◽  
Tissue Degradation

We study travelling-wave solutions for a reaction-diffusion system arising as a model for host-tissue degradation by bacteria. This system consists of a parabolic equation coupled with an ordinary differential equation. For large values of the ‘degradation-rate parameter’ solutions are well approximated by solutions of a Stefan-like free boundary problem, for which travelling-wave solutions can be found explicitly. Our aim is to prove the existence of travelling waves for all sufficiently large wave speeds for the original reaction-diffusion system and to determine the minimal speed. We prove that for all sufficiently large degradation rates, the minimal speed is identical to the minimal speed of the limit problem. In particular, in this parameter range,non-linearselection of the minimal speed occurs.

scholarly journals Mathematical study of a system of multi-dimensional non-local evolution equations describing surfactant-laden two-fluid shear flows

2021 ◽  
Vol 477 (2252) ◽  
pp. 20210307
Author(s):  
Demetrios T. Papageorgiou ◽  
Saleh Tanveer
Keyword(s):  
Global Existence ◽  
Evolution Equations ◽  
Coupled System ◽  
Travelling Wave ◽  
Two Dimensions ◽  
Travelling Wave Solutions ◽  
Rigorous Results ◽  
Wave Solutions ◽  
Non Local ◽  
Two Fluid

This article studies a coupled system of model multi-dimensional partial differential equations (PDEs) that arise in the nonlinear dynamics of two-fluid Couette flow when insoluble surfactants are present on the interface. The equations have been derived previously, but a rigorous study of local and global existence of their solutions, or indeed solutions of analogous systems, has not been considered previously. The evolution PDEs are two-dimensional in space and contain novel pseudo-differential terms that emerge from asymptotic analysis and matching in the multi-scale problem at hand. The one-dimensional surfactant-free case was studied previously, where travelling wave solutions were constructed numerically and their stability investigated; in addition, the travelling wave solutions were justified mathematically. The present study is concerned with some rigorous results of the multi-dimensional surfactant system, including local well posedness and smoothing results when there is full coupling between surfactant dynamics and interfacial motion, and global existence results when such coupling is absent. As far as we know such results are new for non-local thin film equations in either one or two dimensions.

scholarly journals Travelling wave solutions for a non-local evolutionary-epidemic system

2019 ◽  
Vol 267 (2) ◽  
pp. 1467-1509 ◽  
Author(s):  
L. Abi Rizk ◽  
J.-B. Burie ◽  
A. Ducrot
Keyword(s):  
Travelling Wave ◽  
Travelling Wave Solutions ◽  
Wave Solutions ◽  
Non Local

scholarly journals Monostable-Type Travelling Wave Solutions of the Diffusive FitzHugh-Nagumo-Type System in

2012 ◽  
Vol 2012 ◽  
pp. 1-9
Author(s):  
Chih-Chiang Huang
Keyword(s):  
Diffusion Coefficients ◽  
Type System ◽  
Single Equation ◽  
Travelling Wave ◽  
Travelling Wave Solutions ◽  
Local Equation ◽  
Wave Solutions ◽  
Non Local

This paper is concerned with monostable-type travelling wave solutions of the diffusive FitzHugh-Nagumo-type system (FHN) in for the two components and . By solving in terms of , this system can be reduced to a nonlocal single equation for . When the diffusion coefficients in the system are equal, we construct travelling wave solutions for the non-local equation by the method of super- and subsolutions developed by Morita and Ninomiya (2008) Moreover, we propose a condition for , which is similar to the condition Reinecke and Sweers (1999) used to transform (FHN) into a quasimonotone system.

scholarly journals The instability of periodic surface gravity waves

2011 ◽  
Vol 675 ◽  
pp. 141-167 ◽  
Author(s):  
BERNARD DECONINCK ◽  
KATIE OLIVERAS
Keyword(s):  
Shallow Water ◽  
Gravity Waves ◽  
Travelling Wave ◽  
Coordinate Frame ◽  
Travelling Wave Solutions ◽  
Water Wave Problem ◽  
Full Spectrum ◽  
Wave Solutions ◽  
Periodic Travelling Wave ◽  
Non Local

Euler's equations describe the dynamics of gravity waves on the surface of an ideal fluid with arbitrary depth. In this paper, we discuss the stability of periodic travelling wave solutions to the full set of nonlinear equations via a non-local formulation of the water wave problem, modified from that of Ablowitz, Fokas & Musslimani (J. Fluid Mech., vol. 562, 2006, p. 313), restricted to a one-dimensional surface. Transforming the non-local formulation to a travelling coordinate frame, we obtain a new formulation for the stationary solutions in the travelling reference frame as a single equation for the surface in physical coordinates. We demonstrate that this equation can be used to numerically determine non-trivial travelling wave solutions by exploiting the bifurcation structure of this new equation. Specifically, we use the continuous dependence of the amplitude of the solutions on their propagation speed. Finally, we numerically examine the spectral stability of the periodic travelling wave solutions by extending Fourier–Floquet analysis to apply to the associated linear non-local problem. In addition to presenting the full spectrum of this linear stability problem, we recover past well-known results such as the Benjamin–Feir instability for waves in deep water. In shallow water, we find different instabilities. These shallow water instabilities are critically related to the wavelength of the perturbation and are difficult to find numerically. To address this problem, we propose a strategy to estimate a priori the location in the complex plane of the eigenvalues associated with the instability.

Finite-amplitude gravity waves in the atmosphere: travelling wave solutions

2017 ◽  
Vol 826 ◽  
pp. 1034-1065 ◽  
Author(s):  
Mark Schlutow ◽  
R. Klein ◽  
U. Achatz
Keyword(s):  
Wave Packet ◽  
Wave Packets ◽  
Vertical Component ◽  
Travelling Wave ◽  
Moving Frame ◽  
Travelling Wave Solutions ◽  
Mean Flow ◽  
Large Wave ◽  
Modulation Equations ◽  
Wave Solutions

Wentzel–Kramers–Brillouin theory was employed by Grimshaw (Geophys. Fluid Dyn., vol. 6, 1974, pp. 131–148) and Achatz et al. (J. Fluid Mech., vol. 210, 2010, pp. 120–147) to derive modulation equations for non-hydrostatic internal gravity wave packets in the atmosphere. This theory allows for wave packet envelopes with vertical extent comparable to the pressure scale height and for large wave amplitudes with wave-induced mean-flow speeds comparable to the local fluctuation velocities. Two classes of exact travelling wave solutions to these nonlinear modulation equations are derived here. The first class involves horizontally propagating wave packets superimposed over rather general background states. In a co-moving frame of reference, examples from this class have a structure akin to stationary mountain lee waves. Numerical simulations corroborate the existence of nearby travelling wave solutions under the pseudo-incompressible model and reveal better than expected convergence with respect to the asymptotic expansion parameter. Travelling wave solutions of the second class also feature a vertical component of their group velocity but exist under isothermal background stratification only. These waves include an interesting nonlinear wave–mean-flow interaction process: a horizontally periodic wave packet propagates vertically while draining energy from the mean wind aloft. In the process it decelerates the lower-level wind. It is shown that the modulation equations apply equally to hydrostatic waves in the limit of large horizontal wavelengths. Aside from these results of direct physical interest, the new nonlinear travelling wave solutions provide a firm basis for subsequent studies of nonlinear internal wave instability and for the design of subtle test cases for numerical flow solvers.

Solitary and Travelling Wave Solutions of Certain Nonlinear Diffusion-Reaction Equations

2020 ◽  
Author(s):  
Miftachul Hadi
Keyword(s):  
Nonlinear Diffusion ◽  
Travelling Wave ◽  
Diffusion Reaction ◽  
Travelling Wave Solutions ◽  
Wave Solutions ◽  
Reaction Equations

We review the work of Ranjit Kumar, R S Kaushal, Awadhesh Prasad. The work is still in progress.

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