Statistical Analysis of Scalar Dissipation Rate Transport in Turbulent Partially Premixed Flames: A Direct Numerical Simulation Study

Statistically planar turbulent premixed and partially premixed flames for different initial turbulence intensities are simulated for global equivalence ratios ??>?=?0.7 and ??>?=?1.0 using three-dimensional Direct Numerical Simulations (DNS) with simplified chemistry. For the simulations of partially premixed flames, a random distribution of equivalence ratio following a bimodal distribution of equivalence ratio is introduced in the unburned reactants ahead of the flame. The simulation parameters in all of the cases were chosen such that the combustion situation belongs to the thin reaction zones regime. The DNS data has been used to analyse the behaviour of the dissipation rate transports of both active and passive scalars (i.e. the fuel mass fraction Y _F and the mixture fraction ξ) in the context of Reynolds Averaged Navier–Stokes (RANS) simulations. The behaviours of the unclosed terms of the Favre averaged scalar dissipation rates of fuel mass fraction and mixture fraction (i.e. \(\widetilde {\varepsilon }_Y =\overline {\rho D\nabla Y_F^{\prime \prime } \cdot \nabla Y_F^{\prime \prime } } /\overline{\rho }\) and \(\widetilde {\varepsilon }_\xi =\overline {\rho D\nabla \xi ^{\prime \prime }\cdot \nabla \xi ^{\prime \prime }} /\overline {\rho })\) transport equations have been analysed in detail. In the case of the \(\widetilde {\varepsilon }_Y \) transport, it has been observed that the turbulent transport term of scalar dissipation rate remains small throughout the flame brush whereas the terms due to density variation, scalar–turbulence interaction, reaction rate and molecular dissipation remain the leading order contributors. The term arising due to density variation remains positive throughout the flame brush and the combined contribution of the reaction and molecular dissipation to the \(\widetilde {\varepsilon }_Y \) transport remains negative throughout the flame brush in all cases. However, the behaviour of scalar–turbulence interaction term of the \(\widetilde {\varepsilon }_Y \) transport equation is significantly affected by the relative strengths of turbulent straining and the straining due to chemical heat release. In the case of the \(\widetilde {\varepsilon }_\xi \) transport, the turbulent transport term remains small throughout the flame brush and the density variation term is found to be negligible in all cases, whilst the reaction rate term is exactly zero. The scalar–turbulence interaction term and molecular dissipation term remain the leading order contributors to the \(\widetilde {\varepsilon }_\xi \) transport throughout the flame brush in all cases that have been analysed in the present study. Performances of existing models for the unclosed terms of the transport equations of \(\widetilde {\varepsilon }_Y \) and \(\widetilde {\varepsilon }_\xi \) are assessed with respect to the corresponding quantities obtained from DNS data. Based on this exercise either suitable models have been identified or new models have been proposed for the accurate closure of the unclosed terms of both \(\widetilde {\varepsilon }_Y \) and \(\widetilde {\varepsilon }_\xi \) transport equations in the context of Reynolds Averaged Navier–Stokes (RANS) simulations.     

Detonation propagation in hydrogen–air mixtures with transverse concentration gradients

The influence of transverse concentration gradients on detonation propagation in \(\hbox {H}_2\)–air mixtures is investigated experimentally in a wide parameter range. Detonation fronts are characterized by means of high-speed shadowgraphy, OH* imaging, pressure measurements, and soot foils. Steep concentration gradients at low average \(\hbox {H}_2\) concentrations lead to single-headed detonations. A maximum velocity deficit compared to the Chapman–Jouguet velocity of 9 % is observed. Significant amounts of mixture seem to be consumed by turbulent deflagration behind the leading detonation. Wall pressure measurements show high local pressure peaks due to strong transverse waves caused by the concentration gradients. Higher average \(\hbox {H}_2\) concentrations or weaker gradients allow for multi-headed detonation propagation.     

Damköhler-Shelkin Paradox in the Theory of Turbulent Flame Propagation,and a Concept of the Premixed Flame at the Intermediate Asymptotic Stage

We formulated a paradox in the theory of turbulent premixed flame in the flamelet regime: discrepancy between the Damköhler (1940) and Shelkin (1943) estimate of the turbulence flame speed \(U_{t} \sim {u}^{\prime }\) in the case of strong turbulence (\({u}^{\prime }>>S_{L} \)) and numerous experiments that show a strong dependence of U_t on the speed of the instantaneous flame S_L. We name this discrepancy the Damköhler-Shelkin paradox. The first aim of the research is to validate and clarify this estimate, which is based on intuitive considerations, as the paradox must be a statement that seems contradictory to observations but is actually true. We analysed the turbulent flame in the context of the original hyperbolic combustion equation that directly describes the leading edge of the flame, which is a locus of the Zel’dovich “leading points” controlling the speed of the turbulent flame. Analysis of the corresponding characteristic equations results in the expression for speed on the steady-state turbulent flame \(U_{t} ={u}^{\prime }\sqrt {1+(S_{L} /{u}^{\prime })^{2}} \), which is the case when \({u}^{\prime }>>S_{L} \) becomes \(U_{t} \cong {u}^{\prime }\). This result confirms and improves the Damköhler-Shelkin estimate \(U_{t} \sim {u}^{\prime }\). The second aim is to resolve the Damköhler-Shelkin paradox. We explain the discrepancy with observations by the fact that turbulent flames are transient due to insufficient residence time in the real burners to reach statistical equilibrium of wrinkle structures of the random flame surface. We consider the transient flame in the intermediate asymptotic stage when the small-scales wrinkles are in statistical equilibrium, while at the same time the large-scale wrinkles are far from equilibrium. The expressions for the flame speed and width, which we deduce using the dimensional analysis and general properties of the ransom surface, \(U_{t} \sim ({u}^{\prime }S_{L})^{1/2}\) and \(\delta _{t} \sim ({u}^{\prime }Lt)^{1/2}\), show that this transient flame is in fact a turbulent mixing layer travelling with constant speed U_t depending on S_L, the intermediate steady propagation (ISP) flame. Qualitative estimations of the times required for the small-scale and large-scale wrinkles to reach statistical equilibrium show that the turbulent Bunsen- and V-flames correspond to the intermediated asymptotic stage, and the turbulent flames with a complete equilibrium structure of the wrinkled flamelet surface are not attainable under laboratory conditions. We present the results of numerical simulations of the impingent flames, which count in favour of the belief that these flames are also transient.     

On the Similarity of Pulsating and Accelerating Turbulent Pipe Flows

The near-wall region of an unsteady turbulent pipe flow has been investigated experimentally using hot-film anemometry and two-component particle image velocimetry. The imposed unsteadiness has been pulsating, i.e., when a non-zero mean turbulent flow is perturbed by sinusoidal oscillations, and near-uniformly accelerating in which the mean flow ramped monotonically between two turbulent states. Previous studies of accelerating flows have shown that the time evolution between the two turbulent states occurs in three stages. The first stage is associated with a minimal response of the Reynolds shear stress and the ensemble-averaged mean flow evolves essentially akin to a laminar flow undergoing the same change in flow rate. During the second stage, the turbulence responds rapidly to the new flow conditions set by the acceleration and the laminar-like behavior rapidly disappears. During the final stage, the flow adapts to the conditions set by the final Reynolds number. In here, it is shown that the time-development of the ensemble-averaged wall shear stress and turbulence during the accelerating phase of a pulsating flow bears marked similarity to the first two stages of time-development exhibited by a near-uniformly accelerating flow. The stage-like time-development is observed even for a very low forcing frequency; \(\omega ^{+}=\omega \nu /{\overline {u}}_{\tau }^{2}=0.00073\) (or equivalently, \({l}_{s}^{+}=\sqrt {2/\omega ^{+}}=52\)), at an amplitude of pulsation of 0.5. Some previous studies have considered the flow to be quasi-steady at \({l}_{s}^{+}=52\); however, the forcing amplitude has been smaller in those studies. The importance of the forcing amplitude is reinforced by the time-development of the ensemble-averaged turbulence field. For, the near-wall response of the Reynolds stresses showed a dependence on the amplitude of pulsation. Thus, it appears to exist a need to seek alternative similarity parameters, taking the amplitude of pulsation into account, if the response of different flow quantities in a pulsating flow are to be classified correctly.     

Measurements of Transitional and Turbulent Flow in a Randomly Packed Bed of Spheres with Particle Image Velocimetry

Particle image velocimetry (PIV) has been used to investigate transitional and turbulent flow in a randomly packed bed of mono-sized transparent spheres at particle Reynolds number, \(20<{{ Re}}_{\mathrm{p}}< 3220\). The refractive index of the liquid is matched with the spheres to provide optical access to the flow within the bed without distortions. Integrated pressure drop data yield that Darcy law is valid at \({{ Re}}_{\mathrm{p}} \approx 80\). The PIV measurements show that the velocity fluctuations increase and that the time-averaged velocity distribution start to change at lower \({{ Re}}_{\mathrm{p}}\). The probability for relatively low and high velocities decreases with \({{ Re}}_{\mathrm{p}}\) and recirculation zones that appear in inertia dominated flows are suppressed by the turbulent flow at higher \({{ Re}}_{\mathrm{p}}\). Hence there is a maximum of recirculation at about \({{ Re}}_{\mathrm{p}} \approx 400\). Finally, statistical analysis of the spatial distribution of time-averaged velocities shows that the velocity distribution is clearly and weakly self-similar with respect to \({{ Re}}_{\mathrm{p}}\) for turbulent and laminar flow, respectively.     

A practical synchronization approach for fractional-order chaotic systems

A practical synchronization approach is proposed for a class of fractional-order chaotic systems to realize perfect \(\delta \)-synchronization, and the nonlinear functions in the fractional-order chaotic systems are all polynomials. The \(\delta \)-synchronization scheme in this paper means that the origin in synchronization error system is stable. The reliability of \(\delta \)-synchronization has been confirmed on a class of fractional-order chaotic systems with detailed theoretical proof and discussion. Furthermore, the \(\delta \)-synchronization scheme for the fractional-order Lorenz chaotic system and the fractional-order Chua circuit is presented to demonstrate the effectiveness of the proposed method.     

Experimental study on flow control of the turbulent boundary layer with micro-bubbles

The effect of micro-bubbles on the turbulent boundary layer in the channel flow with Reynolds numbers (Re) ranging from \(0.87\times 10 ^{5}\) to \(1.23\times 10^{5}\) is experimentally studied by using particle image velocimetry (PIV) measurements. The micro-bubbles are produced by water electrolysis. The velocity profiles, Reynolds stress and instantaneous structures of the boundary layer, with and without micro-bubbles, are measured and analyzed. The presence of micro-bubbles changes the streamwise mean velocity of the fluid and increases the wall shear stress. The results show that micro-bubbles have two effects, buoyancy and extrusion, which dominate the flow behavior of the mixed fluid in the turbulent boundary layer. The buoyancy effect leads to upward motion that drives the fluid motion in the same direction and, therefore, enhances the turbulence intense of the boundary layer. While for the extrusion effect, the presence of accumulated micro-bubbles pushes the flow structures in the turbulent boundary layer away from the near-wall region. The interaction between these two effects causes the vorticity structures and turbulence activity to be in the region far away from the wall. The buoyancy effect is dominant when the Re is relatively small, while the extrusion effect plays a more important role when Re rises.     

Initial and Boundary Blow-U Problem for $$$$-Lalacian Parabolic Equation with General Absortion

In this article, we investigate the initial and boundary blow-up problem for the \(p\)-Laplacian parabolic equation \(u_t-\Delta _p u=-b(x,t)f(u)\) over a smooth bounded domain \(\Omega \) of \(\mathbb {R}^N\) with \(N\ge 2\), where \(\Delta _pu=\mathrm{div}(|\nabla u|^{p-2}\nabla u)\) with \(p>1\), and \(f(u)\) is a function of regular variation at infinity. We study the existence and uniqueness of positive solutions, and their asymptotic behaviors near the parabolic boundary.     

Direct Numerical Simulation of Head-On Quenching of Statistically Planar Turbulent Premixed Methane-Air Flames Using a Detailed Chemical Mechanism

A three-dimensional compressible Direct Numerical Simulation (DNS) analysis has been carried out for head-on quenching of a statistically planar stoichiometric methane-air flame by an isothermal inert wall. A multi-step chemical mechanism for methane-air combustion is used for the purpose of detailed chemistry DNS. For head-on quenching of stoichiometric methane-air flames, the mass fractions of major reactant species such as methane and oxygen tend to vanish at the wall during flame quenching. The absence of \(\text {OH}\) at the wall gives rise to accumulation of carbon monoxide during flame quenching because \(\text {CO}\) cannot be oxidised anymore. Furthermore, it has been found that low-temperature reactions give rise to accumulation of \(\text {HO}_{2}\) and \(\mathrm {H}_{2}\mathrm {O}_{2}\) at the wall during flame quenching. Moreover, these low temperature reactions are responsible for non-zero heat release rate at the wall during flame-wall interaction. In order to perform an in-depth comparison between simple and detailed chemistry DNS results, a corresponding simulation has been carried out for the same turbulence parameters for a representative single-step Arrhenius type irreversible chemical mechanism. In the corresponding simple chemistry simulation, heat release rate vanishes once the flame reaches a threshold distance from the wall. The distributions of reaction progress variable c and non-dimensional temperature T are found to be identical to each other away from the wall for the simple chemistry simulation but this equality does not hold during head-on quenching. The inequality between c (defined based on \(\text {CH}_{4}\) mass fraction) and T holds both away from and close to the wall for the detailed chemistry simulation but it becomes particularly prominent in the near-wall region. The temporal evolutions of wall heat flux and wall Peclet number (i.e. normalised wall-normal distance of \(T = 0.9\) isosurface) for both simple and detailed chemistry laminar and turbulent cases have been found to be qualitatively similar. However, small differences have been observed in the numerical values of the maximum normalised wall heat flux magnitude \(\left ({\Phi }_{\max } \right )_{\mathrm {L}}\) and the minimum Peclet number \((Pe_{\min })_{\mathrm {L}}\) obtained from simple and detailed chemistry based laminar head-on quenching calculations. Detailed explanations have been provided for the observed differences in behaviours of \(\left ({\Phi }_{\max }\right )_{\mathrm {L}}\) and \((Pe_{\min })_{\mathrm {L}}\). The usual Flame Surface Density (FSD) and scalar dissipation rate (SDR) based reaction rate closures do not adequately predict the mean reaction rate of reaction progress variable in the near-wall region for both simple and detailed chemistry simulations. It has been found that recently proposed FSD and SDR based reaction rate closures based on a-priori DNS analysis of simple chemistry data perform satisfactorily also for the detailed chemistry case both away from and close to the wall without any adjustment to the model parameters.     

Passive Boundary Layer Flow Control Using Porous Lamination

The flow over a porous laminated flat plate is investigated from a flow control perspective through experiments and computations. A square array of circular cylinders is used to model the porous lamination. We determine the velocities at the fluid–porous interface by solving the two-dimensional Navier–Stokes and the continuity equations using a staggered flow solver and using LDV in experiments. The control parameters for the porous region are porosity, \(\phi \) and Reynolds number, Re, based on the diameter of the circular cylinders used to model the porous lamination. Computations are conducted for \(0.4< \phi < 0.9\) and \(25< Re < 1000\), and the experiments are conducted for \(\phi = 0.65\) and 0.8 at \(Re \approx 391,\ 497\) and 803. The permeability of the porous lamination is observed to induce a slip velocity at the interface, effectively making it a slip wall. The slip velocity is seen to be increasing functions of \(\phi \) and Re. For higher porosities at higher Re, the slip velocity shows non-uniform and unsteady behavior and a breakdown Reynolds number is defined based on this characteristic. A map demarcating the two regimes of flow is drawn from the computational and experimental data. We observe that the boundary layer over the porous lamination is thinner than the Blasius boundary layer and the shear stress is higher at locations over the porous lamination. We note that the porous lamination helps maintain a favorable pressure gradient at the interface which delays separation. The suitable range of porosities for effective passive separation control is deduced from the results.     

Influence of hydrodynamic slip on convective transport in flow past a circular cylinder

The presence of a finite tangential velocity on a hydrodynamically slipping surface is known to reduce vorticity production in bluff body flows substantially while at the same time enhancing its convection downstream and into the wake. Here, we investigate the effect of hydrodynamic slippage on the convective heat transfer (scalar transport) from a heated isothermal circular cylinder placed in a uniform cross-flow of an incompressible fluid through analytical and simulation techniques. At low Reynolds (\({\textit{Re}}\ll 1\)) and high Péclet (\({\textit{Pe}}\gg 1\)) numbers, our theoretical analysis based on Oseen and thermal boundary layer equations allows for an explicit determination of the dependence of the thermal transport on the non-dimensional slip length \(l_s\). In this case, the surface-averaged Nusselt number, Nu transitions gradually between the asymptotic limits of \(Nu \sim {\textit{Pe}}^{1/3}\) and \(Nu \sim {\textit{Pe}}^{1/2}\) for no-slip (\(l_s \rightarrow 0\)) and shear-free (\(l_s \rightarrow \infty \)) boundaries, respectively. Boundary layer analysis also shows that the scaling \(Nu \sim {\textit{Pe}}^{1/2}\) holds for a shear-free cylinder surface in the asymptotic limit of \({\textit{Re}}\gg 1\) so that the corresponding heat transfer rate becomes independent of the fluid viscosity. At finite \({\textit{Re}}\), results from our two-dimensional simulations confirm the scaling \(Nu \sim {\textit{Pe}}^{1/2}\) for a shear-free boundary over the range \(0.1 \le {\textit{Re}}\le 10^3\) and \(0.1\le {\textit{Pr}}\le 10\). A gradual transition from the lower asymptotic limit corresponding to a no-slip surface, to the upper limit for a shear-free boundary, with \(l_s\), is observed in both the maximum slip velocity and the Nu. The local time-averaged Nusselt number \(Nu_{\theta }\) for a shear-free surface exceeds the one for a no-slip surface all along the cylinder boundary except over the downstream portion where unsteady separation and flow reversal lead to an appreciable rise in the local heat transfer rates, especially at high \({\textit{Re}}\) and Pr. At a Reynolds number of \(10^3\), the formation of secondary recirculating eddy pairs results in appearance of additional local maxima in \(Nu_{\theta }\) at locations that are in close proximity to the mean secondary stagnation points. As a consequence, Nu exhibits a non-monotonic variation with \(l_s\) increasing initially from its lowermost value for a no-slip surface and then decreasing before rising gradually toward the upper asymptotic limit for a shear-free cylinder. A non-monotonic dependence of the spanwise-averaged Nu on \(l_s\) is observed in three dimensions as well with the three-dimensional wake instabilities that appear at sufficiently low \(l_s\), strongly influencing the convective thermal transport from the cylinder. The analogy between heat transfer and single-component mass transfer implies that our results can directly be applied to determine the dependency of convective mass transfer of a single solute on hydrodynamic slip length in similar configurations through straightforward replacement of Nu and \({\textit{Pr}}\) with Sherwood and Schmidt numbers, respectively.     

The stability of two-dimensional spatial solitons in cubic–quintic–septimal nonlinear media with different diffractions and $${\varvec{\mathcal {}}}$$-symmetric potentials

A (2+1)-dimensional nonlinear Schrödinger equation in cubic–quintic–septimal nonlinear media with different diffractions and \({\mathcal {PT}}\)-symmetric potentials is studied, and (2+1)-dimensional spatial solitons are derived. The stable region of analytical spatial solitons is discussed by means of the eigenvalue method. The direct numerical simulation indicates that analytical spatial soliton solutions stably evolve within stable region in the media of focusing septimal and focusing or defocusing cubic nonlinearities with disappearing quintic nonlinearity under the 2D extended Scarf II potential. However, under the extended \({\mathcal {PT}}\)-symmetric potential with \(p=2\) and \(p=3\), analytical spatial soliton solutions stably evolve within stable region in the media of focusing quintic and septimal nonlinearities with defocusing cubic nonlinearity. In other cases, analytical spatial soliton solutions cannot sustain their original shapes, and they are distorted and broken up and finally decay into noise.     

The Effect of Wall Normal Actuation on a Turbulent Boundary Layer

In this work, a series of direct numerical simulations are conducted to study the effect of wall normal spanwise homogeneous wall actuation on a turbulent boundary layer. The moving boundary is represented by a boundary data immersion technique. A parametric study was performed, varying the actuator length, the wall normal actuation amplitude and the actuation frequency. It was found that localized actuation, relying only on wall motion instead of requiring a plenum in the case of synthetic jets, generated a net momentum flux jet affecting the flow not only in the immediate vicinity of the actuator but also for a significant distance downstream. The cases with an actuator velocity of \( u^{+}_{act}=?20.1 \) showed a particularly pronounced effect on the boundary layer and resulted in a recirculation region.     

Spreading in a Shifting Environment Modeled by the Diffusive Logistic Equation with a Free Boundary

We investigate the influence of a shifting environment on the spreading of an invasive species through a model given by the diffusive logistic equation with a free boundary. When the environment is homogeneous and favourable, this model was first studied in Du and Lin (SIAM J Math Anal 42:377–405, 2010), where a spreading–vanishing dichotomy was established for the long-time dynamics of the species, and when spreading happens, it was shown that the species invades the new territory at some uniquely determined asymptotic speed \(c_0>0\). Here we consider the situation that part of such an environment becomes unfavourable, and the unfavourable range of the environment moves into the favourable part with speed \(c>0\). We prove that when \(c\ge c_0\), the species always dies out in the long-run, but when \(0<c<c_0\), the long-time behavior of the species is determined by a trichotomy described by (a) vanishing, (b) borderline spreading, or (c) spreading. If the initial population is written in the form \(u_0(x)=\sigma \phi (x)\) with \(\phi \) fixed and \(\sigma >0\) a parameter, then there exists \(\sigma _0>0\) such that vanishing happens when \(\sigma \in (0,\sigma _0)\), borderline spreading happens when \(\sigma =\sigma _0\), and spreading happens when \(\sigma >\sigma _0\).     

Does Density Ratio Significantly Affect Turbulent Flame Speed?

In order to experimentally study whether or not the density ratio σ substantially affects flame displacement speed at low and moderate turbulent intensities, two stoichiometric methane/oxygen/nitrogen mixtures characterized by the same laminar flame speed S _L = 0.36 m/s, but substantially different σ were designed using (i) preheating from T _u = 298 to 423 K in order to increase S _L , but to decrease σ, and (ii) dilution with nitrogen in order to further decrease σ and to reduce S _L back to the initial value. As a result, the density ratio was reduced from 7.52 to 4.95. In both reference and preheated/diluted cases, direct images of statistically spherical laminar and turbulent flames that expanded after spark ignition in the center of a large 3D cruciform burner were recorded and processed in order to evaluate the mean flame radius \(\bar {R}_{f}\left (t \right )\) and flame displacement speed \(S_{t}=\sigma ^{-1}{d\bar {R}_{f}} \left / \right . {dt}\) with respect to unburned gas. The use of two counter-rotating fans and perforated plates for near-isotropic turbulence generation allowed us to vary the rms turbulent velocity \(u^{\prime }\) by changing the fan frequency. In this study, \(u^{\prime }\) was varied from 0.14 to 1.39 m/s. For each set of initial conditions (two different mixture compositions, two different temperatures T _u , and six different \(u^{\prime })\), five (respectively, three) statistically equivalent runs were performed in turbulent (respectively, laminar) environment. The obtained experimental data do not show any significant effect of the density ratio on S _t . Moreover, the flame displacement speeds measured at u′/S _L = 0.4 are close to the laminar flame speeds in all investigated cases. These results imply, in particular, a minor effect of the density ratio on flame displacement speed in spark ignition engines and support simulations of the engine combustion using models that (i) do not allow for effects of the density ratio on S _t and (ii) have been validated against experimental data obtained under the room conditions, i.e. at higher σ.     

A Reaction–Diffusion–Advection Equation with Mixed and Free Boundary Conditions

We investigate a reaction–diffusion–advection equation of the form \(u_t-u_{xx}+\beta u_x=f(u)\) \((t>0,\,0<x<h(t))\) with mixed boundary condition at \(x=0\) and Stefan free boundary condition at \(x=h(t)\). Such a model may be applied to describe the dynamical process of a new or invasive species adopting a combination of random movement and advection upward or downward along the resource gradient, with the free boundary representing the expanding front. The goal of this paper is to understand the effect of advection environment and no flux across the left boundary on the dynamics of this species. For the case \(|\beta |<c_0\), we first derive the spreading–vanishing dichotomy and sharp threshold for spreading and vanishing, and then provide a much sharper estimate for the spreading speed of h(t) and the uniform convergence of u(t, x) when spreading happens. For the case \(|\beta |\ge c_0\), some results concerning virtual spreading, vanishing and virtual vanishing are obtained. Here \(c_0\) is the minimal speed of traveling waves of the differential equation.     

Persistence and Critical Domain Size for Diffusing Populations with Two Sexes and Short Reproductive Season

A model is considered for a spatially distributed population of male and female individuals that mate and reproduce only once in their life during a very short reproductive season. Between birth and mating, females and males move by diffusion on a bounded domain \(\Omega \) under Dirichlet boundary conditions. Mating and reproduction are described by a (positively) homogeneous function (of degree one). We identify a basic reproduction number \({\mathcal {R}}_0\) that acts as a threshold between extinction and persistence. If \({\mathcal {R}}_0 <1\), the population dies out while it persists (uniformly weakly) if \({\mathcal {R}}_0 > 1\). \({\mathcal {R}}_0\) is the cone spectral radius of a bounded homogeneous map.     

Liouville Results for <Emphasis Type="Italic">m</Emphasis>-Laplace Equations in Half-Space and Strips with Mixed Boundary Value Conditions and Finite Morse Index

We consider sign-changing solutions of the equation \(-\Delta _m u= |u|^{p-1}u\) where \(m\ge 2\) and \(p>1\) in half-space and strips with nonlinear mixed boundary value conditions. We prove Liouville type theorems for stable solutions or for solutions which are stable outside a compact set. The main methods used are the integral estimates, the Pohozaev-type identity and the monotonicity formula.     

A Note on the Relative Equilibria Bifurcations in the $$$$-Body Problem

Consider the planar Newtonian \((2N+1)\)-body problem, \(N\ge 1,\) with \(2N\) bodies of unit mass and one body of mass \(m\). Using the discrete symmetry due to the equal masses and reducing by the rotational symmetry, we show that solutions with the \(2N\) unit mass points at the vertices of two concentric regular \(N\)-gons and \(m\) at the centre at all times form invariant manifold. We study the regular \(2N\)-gon with central mass \(m\) relative equilibria within the dynamics on the invariant manifold described above. As \(m\) varies, we identify the bifurcations, relate our results to previous work and provide the spectral picture of the linearization at the relative equilibria.