Existence of traveling wave solutions for a nonlinear dissipative-dispersive equation

In this paper, we consider a dissipative-dispersive nonlinear equation appliable to many physical phenomena. Using the geometric singular perturbation method based on the theory of dynamical systems, we investigate the existence of its traveling wave solutions with the dissipative terms having sufficiently small coefficients. The results show that the traveling waves exist on a two-dimensional slow manifold in a three-dimensional system of ordinary differential equations （ODEs）. Then, we use the Melnikov method to establish the existence of a homoclinic orbit in this manifold corresponding to a solitary wave solution of the equation. Furthermore, we present some numerical computations to show the approximations of such wave orbits.     

The propagation mechanism of high speed turbulent deflagrations

The propagation regimes of combustion waves in a 30 cm by 30 cm square cross–sectioned tube with an obstacle array of staggered vertical cylindrical rods (with BR=0.41 and BR=0.19) are investigated. Mixtures of hydrogen, ethylene, propane, and methane with air at ambient conditions over a range of equivalence ratios are used. In contrast to the previous results obtained in circular cross–sectioned tubes, it is found that only the quasi–detonation regime and the slow turbulent deflagration regimes are observed for ethylene–air and for propane–air. The transition from the quasi–detonation regime to the slow turbulent deflagration regime occurs at (where D is the tube “diameter” and is the detonation cell size). When , the quasi–detonation velocities that are observed are similar to those in unobstructed smooth tubes. For hydrogen–air mixtures, it is found that there is a gradual transition from the quasi–detonation regime to the high speed turbulent deflagration regime. The high speed turbulent deflagration regime is also observed for methane–air mixtures near stoichiometric composition. This regime was previously interpreted as the “choking” regime in circular tubes with orifice plate obstacles. Presently, it is proposed that the propagation mechanism of these high speed turbulent deflagrations is similar to that of Chapman–Jouguet detonations and quasi-detonations. As well, it is observed that there exists unstable flame propagation at the lean limit where . The local velocity fluctuates significantly about an averaged velocity for hydrogen–air, ethylene–air, and propane–air mixtures. Unstable flame propagation is also observed for the entire range of high speed turbulent deflagrations in methane–air mixtures. It is proposed that these fluctuations are due to quenching of the combustion front due to turbulent mixing. Quenched pockets of unburned reactants are swept downstream, and the subsequent explosion serves to overdrive the combustion front. The present study indicates that the dependence on the propagation mechanisms on obstacle geometry can be exploited to elucidate the different complex mechanisms of supersonic combustion waves. Received 5 November 2001 / Accepted 12 June 2002 / Published online 4 November 2002 Correspondence to: J. Chao (e-mail: jenny.chao@mail.mcgill.ca) An abridged version of this paper was presented at the 18th Int. Colloquium on the Dynamics of Explosions and Reactive Systems at Seattle, USA, from July 29 to August 3, 2001.     

Transition layers for singularly perturbed delay differential equations with monotone nonlinearities

Transition layers arising from square-wave-like periodic solutions of a singularly perturbed delay differential equation are studied. Such transition layers correspond to heteroclinic orbits connecting a pair of equilibria of an associated system of transition layer equations. Assuming a monotonicity condition in the nonlinearity, we prove these transition layer equations possess a unique heteroclinic orbit, and that this orbit is monotone. The proof involves a global continuation for heteroclinic orbits.     

Mixed Convection Heat Transfer in the Annulus Between Two Concentric Vertical Cylinders Using Porous Layers

A numerical investigation of the steady-state, laminar, axi-symmetric, mixed convection heat transfer in the annulus between two concentric vertical cylinders using porous inserts is carried out. The inner cylinder is subjected to constant heat flux and the outer cylinder is insulated. A finite volume code is used to numerically solve the sets of governing equations. The Darcy–Brinkman–Forchheimer model along with Boussinesq approximation is used to solve the flow in the porous region. The Navier–Stokes equation is used to describe the flow in the clear flow region. The dependence of the average Nusselt number on several flow and geometric parameters is investigated. These include: convective parameter, λ, Darcy number, Da, thermal conductivity ratio, K _r, and porous-insert thickness to gap ratio (H/D). It is found that, in general, the heat transfer enhances by the presence of porous layers of high thermal conductivity ratios. It is also found that there is a critical thermal conductivity ratio on which if the values of Kr are higher than the critical value the average Nusselt number starts to decrease. Also, it found that at low thermal conductivity ratio (K _r ≈ 1) and for all values of λ the porous material acts as thermal insulation.     

Time-Asymptotic Limit of Solutions of a Combustion Problem

A combustion model which captures the interacting among nonlinear convection, chemical reaction and radiative heat transfer is studied. New phenomena are found with radiative heat transfer present. In particular, there is a detonation wave solution in which there is a sonic point inside the reaction zone. As a consequence, the traveling wave speed cannot be determined before the problem is solved. The shooting method is used to prove the existence of the traveling wave. The condition we shoot at is the compatibility condition at the sonic point. Furthermore, the speed of the detonation wave decreases as the heat loss coefficient increases, as expected physically. We study the time-asymptotic limit of solutions of initial value problem for the same problem. We prove that the solution exists globally and the solution converges uniformly, away from the shock, to a shifted traveling wave solution as t + for certain compact support initial data.     

Traveling Waves in Porous Media Combustion: Uniqueness of Waves for Small Thermal Diffusivity

We study traveling wave solutions arising in Sivashinsky’s model of subsonic detonation which describes combustion processes in inert porous media. Subsonic (shockless) detonation waves tend to assume the form of a reaction front propagating with a well defined speed. It is known that traveling waves exist for any value of thermal diffusivity [5]. Moreover, it has been shown that, when the thermal diffusivity is neglected, the traveling wave is unique. The question of whether the wave is unique in the presence of thermal diffusivity has remained open. For the subsonic regime, the underlying physics might suggest that the effect of small thermal diffusivity is insignificant. We analytically prove the uniqueness of the wave in the presence of non-zero diffusivity through applying geometric singular perturbation theory. Dedicated to Mr. Brunovsky in honor of his 70th birthday.     

Chaos in Acoustic Subspace Raised by the Sommerfeld–Kononenko Effect

This paper deals with vibrations of an infinite plate in contact with an acoustic medium where the plate is subjected to a point excitation by an electric motor of limited power-supply. The whole system is divided into two “exciter - foundation” and “foundation-plate-medium”. In the system “motor-foundation” three classes of steady state regimes are determined: stationary, periodic and chaotic. The vibrations of the plate and the pressure in the acoustic fluid are described for each of these regimes of excitation. For the first class they are periodic functions of time, for the second they are modulated periodic functions, in general with an infinite number of carrying frequencies, the difference between which is constant. For the last class they correspond to chaotic functions. In another mathematical model where the exciter stands directly on an infinite plate (without foundation) it was shown that chaos might occur in the system due to the feedback influence of waves in the infinite hydro-elastic subsystem in the regime of motor shaft rotation. In this case the process of rotation can be approximately described as a solution of the fourth order nonlinear differential equation and may have the same three classes of steady state regimes as the first model. That is the electric motor may generate periodic acoustic waves, modulated waves with an infinite number of frequencies or chaotic acoustic waves in a fluid.     

Traveling Waves for a Biological Reaction-Diffusion Model

We investigate the existence of traveling wave solutions for a system of reaction–diffusion equations that has been used as a model for microbial growth in a flow reactor and for the diffusive epidemic population. The existence of traveling waves was conjectured early but only has been proved recently for sufficiently small diffusion coefficient by the singular perturbation technique. In this paper we show the existence of traveling waves for an arbitrary diffusion coefficient. Our approach is a shooting method with the aid of an appropriately constructed Liapunov function.Dedicated to Professor Shui-Nee Chow on the occasion of his 60th birthday.Wenzhang Huang-Research was supported in part by NSF Grant DMS-0204676.     

Radiation Preheating can Trigger Transition from Deflagration to Detonation

In this article the possibility of radiation heat transfer to trigger transition to detonation is studied. It is assumed that the premixed deflagration front is able to emit and the unburnt mixture to absorb radiation heat effectively. Under this assumption the ability of the flame to significantly preheat the unburnt mixture and to form a promoting temperature gradient is investigated. First, we estimate the temperature rise due to the radiation preheating of the unburnt mixture, when it is traveling through deep flame wrinkles. Subsequently, we carry out numerical simulations of premixed gaseous combustion in a tube. The simulations confirm the possibility of formation of promoting temperature gradients within flame folds and initiation of the detonation waves. They demonstrate the plausibility of the proposed mechanism of the deflagration to detonation transition.     

The effect of thermal dispersion on free convection film condensation on a vertical plate with a thin porous layer

An analytical solution to the problem of condensation by natural convection over a thin porous substrate attached to a cooled impermeable surface has been conducted to determine the velocity and temperature profiles within the porous layer, the dimensionless thickness film and the local Nusselt number. In the porous region, the Darcy–Brinkman–Forchheimer (DBF) model describes the flow and the thermal dispersion is taken into account in the energy equation. The classical boundary layer equations without inertia and enthalpyterms are used in the condensate region. It is found that due to the thermal dispersion effect, the increasing of heat transfer is significant. The comparison of the DBF model and the Darcy–Brinkman (DB) one is carried out.     

Nonsimilar boundary-layer flow over a stationary permeable plate in a rotating fluid with a magnetic field

Summary  The nonsimilar boundary-layer flow and heat transfer over a stationary permeable surface in a rotating fluid in the presence of magnetic field, mass transfer and free stream velocity are studied. The parabolic partial differential equations governing the flow have been solved numerically by using a difference–differential method. For small streamwise distance, these partial differential equations are also solved by a perturbation technique with Shanks transformation. For uniform mass transfer, analytical solutions are obtained. The surface skin friction coefficients and the Nusselt number increase with the magnetic field, suction and streamwise distance from the leading edge of the plate except the skin friction coefficient in the y-direction which decreases with the increasing magnetic field. Received 4 December 2001; accepted for publication 24 September 2002     

Convective Flows in a TVZ-like Setting with a Brittle/Ductile Transition

This article presents a simple method for coupling the equations for fluid and heat flow in a porous medium with a rock mechanics model based on a simplified Mohr–Coulomb yield criterion. The aim is to investigate the effects of introducing a brittle–ductile transition zone (BDTZ) into models of the large scale hydrology of systems like the Taupo Volcanic Zone (TVZ) in New Zealand. The coupling between fluid/heat flow and rock mechanics is only partial in that it assumes that the local strain rate and lithostatic pressure are known a priori. A simple empirical relationship between the permeabilities of rock in its brittle and ductile states is also assumed, which mimics the effect of slow thermal creep in closing fluid pathways by reducing the effective permeability. This model is first applied to a number of simple situations where large scale convection of groundwater occurs above a depth of 20-km. These examples demonstrate the formation of a horizontal brittle–ductile transition zone under conductive conditions with uniform strain, and how this responds to local changes in pressure, temperature and strain rate. The presence of low permeability below the BDTZ effectively defines the greatest depth to which groundwater can steadily penetrate, providing a feedback to these models which influences the transport of heat and mass on all the length scales. The TVZ provides inspiration for the second model setting. Localised sources of water and heat, modelled on magmatic sill-like structures at a depth of 10-km, induce a BDTZ which shallows from 14-km to less than 10-km, consistent with geophysical estimates in the TVZ. A common feature of the models is that strong downflows of surface water occur in permeable regions adjacent to the heat sources, which depress the BDTZ by several kilometres between the geothermal areas. Water with near-surface temperatures can thus exist at great depths in these regions. Lastly, the models imply that the geothermal areas in the TVZ probably do not occur directly above any localised heat source, but are instead displaced towards the centre of the TVZ rift.     

Heat transfer during heat sterilization and cooling processes of canned products

In this paper, an analysis of transient heat transfer during heat sterilization and cooling processes of a cylindrical canned product is presented. In the analysis, most practical case including the boundary condition of third kind (i.e., convection boundary condition, leading to 0.1 ≤ Bi ≤ 100) was employed. A simple analytical model for determining effective heat transfer coefficients for such products is developed. For the heat sterilization process, heating coefficient is incorporated into heat transfer coefficient model. An experimental study was performed to measure the thermal center temperatures of the short-cylindrical canned products (i.e., Tuna fish) during heat sterilization at the retort medium temperatures of 115^∘C and 121^∘C, and during cooling process at 16^∘C. The effective heat transfer coefficient model used the experimental temperature data. Using these effective heat transfer coefficients the center temperature distributions were calculated and compared with the experimental temperature distributions. Agreement was found considerably high. The results of the present study indicate that the heat-transfer analysis technique and heat-transfer coefficient model are reliable, and can provide accurate results for such problems. Received on 12 November 1997     

The Geometry of the Solution Set of Nonlinear Optimal Control Problems

In an optimal control problem one seeks a time-varying input to a dynamical systems in order to stabilize a given target trajectory, such that a particular cost function is minimized. That is, for any initial condition, one tries to find a control that drives the point to this target trajectory in the cheapest way. We consider the inverted pendulum on a moving cart as an ideal example to investigate the solution structure of a nonlinear optimal control problem. Since the dimension of the pendulum system is small, it is possible to use illustrations that enhance the understanding of the geometry of the solution set. We are interested in the value function, that is, the optimal cost associated with each initial condition, as well as the control input that achieves this optimum. We consider different representations of the value function by including both globally and locally optimal solutions. Via Pontryagin’s maximum principle, we can relate the optimal control inputs to trajectories on the smooth stable manifold of a Hamiltonian system. By combining the results we can make some firm statements regarding the existence and smoothness of the solution set.     

A study on a flexible wing with up–down vibration in a pulsating flow of cooling air to improve heat transfer efficiency

We investigated a flexible wing that can function as a folding fan by vibrating smoothly on a heated surface, and the effects of this vibration on heat transfer. For flexible up–down vibrations of the wing in a pulsating flow, we propose a novel milli-scale flexible wing shape with a relatively large body and a narrow connecting leg. The shape was optimized such that its deformation became much larger at a low air flow. We performed two-way fluid–structure interaction analyses to predict performance, and an experimental validation was also conducted. The details of flow, heat transfer, and structural deformation are summarized qualitatively. Our results show that the heat transfer coefficient of a heated surface with a single flexible wing was approximately 11.3 % greater than that of a flat plate.     

不同自燃模式对压力波动形成的影响

在内燃机中,HCCI(均值混合气压燃)爆震、汽油机常规爆震、汽油机超级爆震都是由未燃混合气自燃引起的化学能突然释放,从而产生振荡燃烧,但其压升率及压力振荡幅值却截然不同。为了阐明其中机理,根据上述的带震荡的燃烧压力波变化规律,提出以实验测得的放热率为基础的“能量注入法”,建立了3种自燃模式。通过对能量方程中的热源项进行分类改变,进而对3种自燃模式进行数值模拟、对其产生的压力波动进行比较分析。模拟结果表明,不同震荡特征的燃烧压力波源于不同的自燃模式,从而导致其宏观表现的压升率以及压力波振荡幅值有极大的差异。以放热率为基础的“能量注入法”能准确、快捷地探究内燃机燃烧室中压力波的形成与传播。     

Expanding the applications of holographic interferometry to the quantitative visualization of oscillatory thermofluid processes using temperature as tracer

 In this paper we discuss a novel application of holographic interferometry in the simultaneous quantitative visualization of high-speed, oscillatory flow and temperature fields in complex flow geometries. We consider cases of (i) self-sustained oscillatory flows with main flow imposed in grooved and communicating channels as well as (ii) oscillating thermofluid processes with zero mean velocity in a thermoacoustic refrigerator model. Examples showing unsteady temperature distributions obtained by real-time holographic interferometry combined with high-speed cinematography illustrate the possibilities of the approach introduced in the paper. Our study shows that temperature distributions accurately mirror flow structures in certain types of complex, unsteady flows, thus allowing, apart from the measurement of temperature profiles and heat transfer, also the measurement of oscillatory amplitudes, frequencies, wavelengths as well as the speed of propagation of traveling waves by applying digital image processing techniques. In the grooved and communicating channels it is possible to visualize the structure of the Tollmien–Schlichting waves through isotherms by using the infinite fringe field alignment of holographic interferometry. In the thermoacoustic refrigerator model, small amplitude temperature oscillations generated by the acoustic standing wave are visualized and measured. Image processing as well as data reduction procedures used in the analysis of these flow fields are discussed in the paper. Experimental data obtained by applying the techniques introduced in the paper show good agreement with theory and results of numerical simulations. Our study suggests that using temperature as tracer offers numerous advantages in the study of certain types of complex, unsteady flows. Received: 7 January 1997/Accepted: 18 September 1997     

Bifurcation analysis in a time-delay model for prey–predator growth with stage-structure

A time-delay model for prey–predator growth with stage-structure is considered. At first, we investigate the stability and Hopf bifurcations by analyzing the distribution of the roots of associated characteristic equation. Then, an explicit formula for determining the stability and the direction of periodic solutions bifurcating from Hopf bifurcations is derived, using the normal form theory and center manifold argument. Finally, some numerical simulations are carried out for supporting the analytic results.     

On the Range of Applicability of the Reissner–Mindlin and Kirchhoff–Love Plate Bending Models

We show that the Reissner–Mindlin plate bending model has a wider range of applicability than the Kirchhoff–Love model for the approximation of clamped linearly elastic plates. Under the assumption that the body force density is constant in the transverse direction, the Reissner–Mindlin model solution converges to the three-dimensional linear elasticity solution in the relative energy norm for the full range of surface loads. However, for loads with a significant transverse shear effect, the Kirchhoff–Love model fails. This revised version was published online in June 2006 with corrections to the Cover Date.