Numerical Study of Turbulent Channel Flow over Surfaces with Variable Spanwise Heterogeneities: Topographically-driven Secondary Flows Affect Outer-layer Similarity of Turbulent Length Scales

Wall-bounded turbulent flows over surfaces with spanwise heterogeneous surface roughness – that is, spanwise-adjacent patches of relatively high and low roughness – exhibit mean flow phenomena entirely different to what would otherwise exist in the absence of spanwise heterogeneity. In the outer layer, mean counter-rotating rolls occupy the depth of the flow, and are positioned such that “upwelling” and “downwelling” occurs above the low and high roughness, respectively. It has been comprehensively shown that these secondary flows are Prandtl’s secondary flow of the second kind (Anderson et al., J. Fluid Mech. 768, 316–347 2015). This behaviour indicates that spanwise spacing, s _y , between adjacent patches of high and low roughness is, itself, a problem parameter; in this study, we have systematically assessed how s _y affects turbulence structure in high Reynolds number channel flows via two-point correlations. “High roughness” is imposed with streamwise-aligned pyramid elements with height, h, selected to be ≈ 5% of the channel half height, H. For \(s_{y}/H \gtrsim 1\), we find that the aforementioned domain-scale mean circulations exist and the surface may be regarded as a topography. For s _y /H ? 0.2, turbulence statistics show characteristics very similar to a homogeneous roughness and thus the surface may be regarded as a roughness. For 0.2 ? s _y /H ? 2, the spatial extent of the counter-rotating rolls is controlled by proximity to adjacent rows, and we define such surfaces as being intermediate. We refer to such surfaces as intermediate state.     

A Comprehensive Study of Improved Delayed Detached Eddy Simulation with Wall Functions

The mechanisms for nonlinear saturation of a bluff-body stabilised turbulent premixed flame are investigated using LES with the transported flame surface density (TFSD) approach to combustion modelling. The numerical simulation is based on a previous detailed experimental investigation. Results for both the unforced non-reacting and reacting flows are validated against experiment, demonstrating that the fundamental flow features and predicted flame structure are well captured. Key terms in the FSD transport equation are then used to describe the generation and destruction of flame surface area for the unforced reacting flow. In order to investigate the non-linear response of the unsteady heat release rate to acoustic forcing, four harmonically forced flames are considered having the same forcing frequency (160 Hz) but different amplitudes of 10 %, 25 %, 50 % and 64 % of the mean inlet velocity. The flame response is characterised using the Flame Describing Function (FDF). Accurate prediction of the FDF is obtained using the current approach. The computed forced flame structure matches well with the experiment, where effects of shear layer rollup and growth of the vortices on the flame can be clearly observed. Transition to nonlinearity is also observed in the computed FDF. The mechanisms leading to the saturation of the flame response in the higher amplitude case are characterised by inspecting the terms in the FSD transport equation at conditions when the integrated heat release is at its maximum and minimum, respectively. Pinch-off and flame rollup can be seen in snapshots taken at the phase angle of maximum integrated heat release. Conversely, intense vortex shedding and flame-sheet collapse around the shear-layer, as well as small-scale destruction of flame elements in the wake, can be seen in snapshots taken at the phase angle of minimum integrated heat release. The pivotal role of FSD destruction on nonlinear saturation of the flame response is confirmed through the analysis of phase-averaged terms in the FSD transport equation taken at different locations. The phase-averaged subgrid curvature term is found to concentrate in the cusps and downstream regions where flame annihilation is dominant.     

Effect of Inflow Turbulence on an Airfoil Flow with Laminar Separation Bubble: An LES Study

The present paper is concerned with numerical investigations on the effect of inflow turbulence on the flow around a SD7003 airfoil. At a Reynolds number Re_c =?60,000, an angle of attack α =?4^° and a low or zero turbulence intensity of the oncoming flow, the flow past the airfoil is known to be dominated by early separation, subsequent transition and reattachment leading to a laminar separation bubble with a distinctive pressure plateau. The objective of the study is to investigate the effect of inflow turbulence on the flow behavior. For this purpose, a numerical methodology relying on a wall-resolved large-eddy simulation, a synthetic turbulence inflow generator and a specific source term concept for introducing the turbulence fluctuations within the computational domain is used. The numerical technique applied allows the variation of the free-stream turbulence intensity (TI) in a wide range. In order to analyze the influence of TI on the arising instantaneous and time-averaged flow field past the airfoil, the present study evaluates the range 0% ≤ TI ≤?11.2%, which covers typical values found in atmospheric boundary layers. In accordance with experimental studies it is shown that the laminar separation bubble first shrinks and finally completely vanishes for increasing inflow turbulence. Consequently, the aerodynamic performance in terms of the lift-to-drag ratio increases. Furthermore, the effect of the time and length scales of the isotropic inflow turbulence on the development of the flow field around the airfoil is analyzed and a perceptible influence is found. Within the range of inflow scales studied decreasing scales augment the receptivity of the boundary layer promoting an earlier transition.     

Turbulent Drag Reduction by a Near Wall Surface Tension Active Interface

In this work we study the turbulence modulation in a viscosity-stratified two-phase flow using Direct Numerical Simulation (DNS) of turbulence and the Phase Field Method (PFM) to simulate the interfacial phenomena. Specifically we consider the case of two immiscible fluid layers driven in a closed rectangular channel by an imposed mean pressure gradient. The present problem, which may mimic the behaviour of an oil flowing under a thin layer of different oil, thickness ratio h₂/h₁ =?9, is described by three main flow parameters: the shear Reynolds number Re _τ (which quantifies the importance of inertia compared to viscous effects), the Weber number We (which quantifies surface tension effects) and the viscosity ratio λ = ν₁/ν₂ between the two fluids. For this first study, the density ratio of the two fluid layers is the same (ρ₂ = ρ₁), we keep Re _τ and We constant, but we consider three different values for the viscosity ratio: λ =?1, λ =?0.875 and λ =?0.75. Compared to a single phase flow at the same shear Reynolds number (Re _τ =?100), in the two phase flow case we observe a decrease of the wall-shear stress and a strong turbulence modulation in particular in the proximity of the interface. Interestingly, we observe that the modulation of turbulence by the liquid-liquid interface extends up to the top wall (i.e. the closest to the interface) and produces local shear stress inversions and flow recirculation regions. The observed results depend primarily on the interface deformability and on the viscosity ratio between the two fluids (λ).     

Operator method for solving the plane elasticity problem for a strip with periodic cuts

In the present paper, we use the conformal mapping z/c = ζ?2a sin ζ (a, c?const, ζ = u + iv) of the strip {|v| ≤ v ₀, |u| < ∞} onto the domain D, which is a strip with symmetric periodic cuts. For the domain D, in the orthogonal system of isometric coordinates u, v, we solve the plane elasticity problem. We seek the biharmonic function in the form F = C ψ ₀ + S ψ*₀ + x(C ψ ₁ ? S ψ ₂) + y(C ψ ₂ + S ψ ₁), where C(v) and S(v) are the operator functions described in [1] and ψ ₀(u), …, ψ ₂(u) are the desired functions. The boundary conditions for the function F posed for v = ±v ₀ are equivalent to two operator equations for ψ ₁(u) and ψ ₂(u) and to two ordinary differential equations of first order for ψ ₀(u) and ψ*₀(u) [2]. By finding the functions ψ _j (u) in the form of trigonometric series with indeterminate coefficients and by solving the operator equations, we obtain infinite systems of linear equations for the unknown coefficients. We present an efficient method for solving these systems, which is based on studying stable recursive relations. In the present paper, we give an example of analysis of a specific strip (a = 1/4, v ₀ = 1) loaded on the boundary v = v ₀ by a normal load of intensity p. We find the particular solutions corresponding to the extension of the strip by the longitudinal force X ^∞ and to the transverse and pure bending of the strip due to the transverse force Y ^∞ and the constant moment M ^∞, respectively. We also present the graphs of normal and tangential stresses in the transverse cross-section x = 0 and study the stress concentration effect near the cut bottom.     

An iterative updating method for undamped structural systems

Finite element model updating is a procedure to minimize the differences between analytical and experimental results and can be mathematically reduced to solving the following problem. Problem P: Let M _a ∈SR ^n×n and K _a ∈SR ^n×n be the analytical mass and stiffness matrices and Λ=diag{λ ₁,…,λ _p }∈R ^p×p and X=[x ₁,…,x _p ]∈R ^n×p be the measured eigenvalue and eigenvector matrices, respectively. Find \((\hat{M}, \hat{K}) \in \mathcal{S}_{MK}\) such that \(\| \hat{M}-M_{a} \|^{2}+\| \hat{K}-K_{a}\|^{2}= \min_{(M,K) \in {\mathcal{S}}_{MK}} (\| M-M_{a} \|^{2}+\|K-K_{a}\|^{2})\), where \(\mathcal{S}_{MK}=\{(M,K)| X^{T}MX=I_{p}, MX \varLambda=K X \}\) and ∥?∥ is the Frobenius norm. This paper presents an iterative method to solve Problem P. By the method, the optimal approximation solution \((\hat{M}, \hat{K})\) of Problem P can be obtained within finite iteration steps in the absence of roundoff errors by choosing a special kind of initial matrix pair. A numerical example shows that the introduced iterative algorithm is quite efficient.     

The Effect of Grain Size Distribution on Nonlinear Flow Behavior in Sandy Porous Media

The current study provides new experimental data on nonlinear flow behavior in various uniformly graded granular materials (20 samples) ranging from medium sands (\(d_{50 }>0.39\) mm) to gravel (\(d_{50}=6.3\) mm). Generally, theoretical equations relate the Forchheimer parameters a and b to the porosity, as well as the characteristic pore length, which is assumed to be the median grain size \((d_{50})\) of the porous medium. However, numerical and experimental studies show that flow resistance in porous media is largely determined by the geometry of the pore structure. In this study, the effect of the grain size distribution was analyzed using subangular-subrounded sands and approximately equal compaction grades. We have used a reference dataset of 11 uniformly graded filter sands. Mixtures of filter sands were used to obtain a slightly more well-graded composite sand (increased \(C_{u}\) values by a factor of 1.19 up to 2.32) with respect to its associated reference sand at equal median grain size \((d_{50})\) and porosity. For all composite sands, the observed flow resistance was higher than in the corresponding reference sand at equal \(d_{50}\), resulting in increased a coefficients by factors up to 1.68, as well as increased b coefficients by factors up to 1.44. A modified Ergun relationship with Ergun constants of 139.1 for A and 2.2 for B, as well as the use of \(d_{m}-\sigma \) as characteristic pore length predicted the coefficients a and b accurately.     

Linearized Stability for Semiflows Generated by a Class of Neutral Equations,with Applications to State-Dependent Delays

We prove a principle of linearized stability for semiflows generated by neutral functional differential equations of the form x′(t) = g(? x _t , x _t ). The state space is a closed subset in a manifold of C ²-functions. Applications include equations with state-dependent delay, as for example x′(t) = a x′(t + d(x(t))) + f (x(t + r(x(t)))) with \({a\in\mathbb{R}, d:\mathbb{R}\to(-h,0), f:\mathbb{R}\to\mathbb{R}, r:\mathbb{R}\to[-h,0]}\).     

Experimental Control of Turbulent Boundary Layers with In-plane Travelling Waves

The experimental control of turbulent boundary layers using streamwise travelling waves of spanwise wall velocity, produced using a novel active surface, is outlined in this paper. The innovative surface comprises a pneumatically actuated compliant structure based on the kagome lattice geometry, supporting a pre-tensioned membrane skin. Careful design of the structure enables waves of variable length and speed to be produced in the flat surface in a robust and repeatable way, at frequencies and amplitudes known to have a favourable influence on the boundary layer. Two surfaces were developed, a preliminary module extending 152 mm in the streamwise direction, and a longer one with a fetch of 2.9 m so that the boundary layer can adjust to the new surface condition imposed by the forcing. With a shorter, 1.5 m portion of the surface actuated, generating an upstream-travelling wave, a drag reduction of 21.5% was recorded in the boundary layer with Re _τ =?1125. At the same flow conditions, a downstream-travelling produced a much smaller drag reduction of 2.6%, agreeing with the observed trends in current simulations. The drag reduction was determined with constant temperature hot-wire measurements of the mean velocity gradient in the viscous sublayer, while simultaneous laser Doppler vibrometer measurements of the surface recorded the wall motion. Despite the mechanics of the dynamic surface resulting in some out-of-plane motion (which is small in comparison to the in-plane streamwise movement), the positive drag reduction results are encouraging for future investigations at higher Reynolds numbers.     

Simulation of Receptivity and Induced Transition From Discrete Roughness Elements

Simulations have been carried out to predict the receptivity and growth of crossflow vortices created by Discrete Roughness Elements (DREs) The final transition to turbulence has also been examined, including the effect of DRE spacing and freestream turbulence. Measurements by Hunt and Saric (2011) of perturbation mode shape at various locations were used to validate the code in particular for the receptivity region. The WALE sub-grid stress (SGS) model was adopted for application to transitional flows, since it allows the SGS viscosity to vanish in laminar regions and in the innermost region of the boundary layer when transition begins. Simulations were carried out for two spanwise wavelengths: λ= 12mm (critical) and λ= 6mm (control) and for roughness heights (k) from 12 μm to 42 μm. The base flow considered was an ASU (67)-0315 aerofoil with 45 ⁰ sweep at -2.9 ⁰ incidence and with onset flow at a chord-based Reynolds number Re _c= 2.4x10 ⁶. For λ= 12mm results showed, in accord with the experimental data, that the disturbance amplitude growth rate was linear for k = 12 μm and 24 μm, but the growth rate was decreased for k = 36 μm Receptivity to λ= 6mm roughness showed equally good agreement with experiments, indicating that this mode disappeared after a short distance to be replaced by a critical wavelength mode. Analysis of the development of modal disturbance amplitudes with downstream distance showed regions of linear, non-linear, saturation, and secondary instability behaviour. Examination of breakdown to turbulence revealed two possible routes: the first was 2D-like transition (probably Tollmien-Schlichting waves even in the presence of crossflow vortices) when transition occurred beyond the pressure minimum; the second was a classical crossflow vortex secondary instability, leading to the formation of a turbulent wedge.     

Regularity and Singularities of Optimal Convex Shapes in the Plane

We focus here on the analysis of the regularity or singularity of solutions Ω ₀ to shape optimization problems among convex planar sets, namely:

$J(\Omega_{0})={\rm min} \{J(\Omega), \Omega \quad {\rm convex},\Omega \in \mathcal{S}_{\rm ad}\},$

where \({\mathcal{S}_{\rm ad}}\) is a set of 2-dimensional admissible shapes and \({J:\mathcal{S}_{\rm ad}\rightarrow\mathbb{R}}\) is a shape functional. Our main goal is to obtain qualitative properties of these optimal shapes by using first and second order optimality conditions, including the infinite dimensional Lagrange multiplier due to the convexity constraint. We prove two types of results:

1. i)
   under a suitable convexity property of the functional J, we prove that Ω ₀ is a W ^2,p -set, \({p\in[1, \infty]}\). This result applies, for instance, with p = ∞ when the shape functional can be written as J(Ω) = R(Ω) + P(Ω), where R(Ω) = F(|Ω|, E _f (Ω), λ₁(Ω)) involves the area |Ω|, the Dirichlet energy E _f (Ω) or the first eigenvalue of the Laplace–Dirichlet operator λ₁(Ω), and P(Ω) is the perimeter of Ω;
    

1. ii)
   under a suitable concavity assumption on the functional J, we prove that Ω ₀ is a polygon. This result applies, for instance, when the functional is now written as J(Ω) = R(Ω) ? P(Ω), with the same notations as above.
    

     

Decay of Almost Periodic Solutions¶of Conservation Laws

We consider the asymptotic behavior of solutions of systems of inviscid or viscous conservation laws in one or several space variables, which are almost periodic in the space variables in a generalized sense introduced by Stepanoff and Wiener, which extends the original one of H. Bohr. We prove that if u(x,t) is such a solution whose inclusion intervals at time t, with respect to ?>0, satisfy l _epsiv;(t)/t→0 as t→∞, and such that the scaling sequence u ^T (x,t)=u(T x,T t) is pre-compact as t→∞ in L _loc ¹(? ^{d +1} ₊, then u(x,t) decays to its mean value \(\), which is independent of t, as t→∞. The decay considered here is in L ¹ _loc of the variable ξ≡x/t, which implies, as we show, that \(\) as t→∞, where M _x denotes taking the mean value with respect to x. In many cases we show that, if the initial data are almost periodic in the generalized sense, then so also are the solutions. We also show, in these cases, how to reduce the condition on the growth of the inclusion intervals l _?(t) with t, as t→∞, for fixed ? > 0, to a condition on the growth of l _?(0) with ?, as ?→ 0, which amounts to imposing restrictions only on the initial data. We show with a simple example the existence of almost periodic (non-periodic) functions whose inclusion intervals satisfy any prescribed growth condition as ?→ 0. The applications given here include inviscid and viscous scalar conservation laws in several space variables, some inviscid systems in chromatography and isentropic gas dynamics, as well as many viscous 2 × 2 systems such as those of nonlinear elasticity and Eulerian isentropic gas dynamics, with artificial viscosity, among others. In the case of the inviscid scalar equations and chromatography systems, the class of initial data for which decay results are proved includes, in particular, the L ^∞ generalized limit periodic functions. Our procedures can be easily adapted to provide similar results for semilinear and kinetic relaxations of systems of conservation laws.     

Miscible Thermo-Viscous Fingering Instability in Porous Media. Part 1: Linear Stability Analysis

The development of the thermo-viscous fingering instability of miscible displacements in homogeneous porous media is examined. In this first part of the study dealing with stability analysis, the basic equations and the parameters governing the problem in a rectilinear geometry are developed. An exponential dependence of viscosity on temperature and concentration is represented by two parameters, thermal mobility ratio β _T and a solutal mobility ratio β _C , respectively. Other parameters involved are the Lewis number Le and a thermal-lag coefficient λ. The governing equations are linearized and solved to obtain instability characteristics using either a quasi-steady-state approximation (QSSA) or initial value calculations (IVC). Exact analytical solutions are also obtained for very weakly diffusing systems. Using the QSSA approach, it was found that an increase in thermal mobility ratio β _T is seen to enhance the instability for fixed β _C , Le and λ. For fixed β _C and β _T , a decrease in the thermal-lag coefficient and/or an increase in the Lewis number always decrease the instability. Moreover, strong thermal diffusion at large Le as well as enhanced redistribution of heat between the solid and fluid phases at small λ is seen to alleviate the destabilizing effects of positive β _T . Consequently, the instability gets strictly dominated by the solutal front. The linear stability analysis using IVC approach leads to conclusions similar to the QSSA approach except for the case of large Le and unity λ flow where the instability is seen to get even less pronounced than in the case of a reference isothermal flow of the same β _C , but β _T  = 0. At practically, small value of λ, however, the instability ultimately approaches that due to β _C only.     

Effects of Fuel Lewis Number on Localised Forced Ignition of Globally Stoichiometric Stratified Mixtures: a Numerical Investigation

The influences of fuel Lewis number Le_F on localised forced ignition of globally stoichiometric stratified mixtures have been analysed using three-dimensional compressible Direct Numerical Simulations (DNS) for cases with Le_F ranging from 0.8 to 1.2. The globally stoichiometric stratified mixtures with different values of root-mean-square (rms) equivalence ratio fluctuation (i.e. ?^′= 0.2, 0.4 and 0.6) and the Taylor micro-scale l_? of equivalence ratio ? variation (i.e. l_?/l_f= 2.1, 5.5 and 8.3 with l_f being the Zel’dovich flame thickness of the stoichiometric laminar premixed flame) have been considered for different initial rms values of turbulent velocity u^′. A pseudo-spectral method is used to initialise the equivalence ratio variation following a presumed bi-modal distribution for prescribed values of ?^′ and l_?/l_f for global mean equivalence ratio 〈?〉=1.0. The localised ignition is accounted for by a source term in the energy transport equation that deposits energy for a stipulated time interval. It has been observed that the maximum values of temperature and the fuel reaction rate magnitude increase with decreasing Le_F during the period of external energy deposition. The initial values of Le_F, u^′/S_b(?=1), ?^′ and l_?/l_f have been found to have significant effects on the extent of burning of the stratified mixtures following localised ignition. For a given value of u^′/S_b(?=1), the extent of burning decreases with increasing Le_F. An increase in u^′ leads to a monotonic reduction in the burned gas mass for all values of Le_F in all stratified mixture cases but an opposite trend is observed for the Le_F=0.8 homogeneous mixture. It has been found that an increase in ?^′ has adverse effects on the burned gas mass, whereas the effects of l_?/l_f on the extent of burning are non-monotonic and dependent on ?^′ and Le_F. Detailed physical explanations have been provided for the observed Le_F, u^′/S_b(?=1), ?^′ and l_?/l_f dependences.     

Viscoelastic behavior of PMMA/grafted PBA nanoparticle systems in the molten state

The viscoelastic properties of poly(methyl methacrylate) (PMMA)/grafted poly(butyl acrylate) (PBA) nanoparticle systems were investigated. The rubber particles consist of PBA core (60 nm in diameter) and grafted PMMA shell. The grafting degree, defined as the weight ratio of grafted PMMA to PBA particles, ranges from 0.8 to 1.5. Two series of samples, A series with 7.5 wt.% of PBA content and B series with 12 wt.% of PBA content, were used. The systems exhibited fast and slow relaxation process, the former reflecting the relaxation of the matrix PMMA chains and the latter being attributed to grafted PBA particles. For A series samples, time-temperature superposition (TTS) was held well over the frequency (ω) and temperature (T) ranges measured. However, for B series samples, TTS was not satisfied at low ω due to the particle-particle interaction of grafted PBA particles, although the samples obeyed TTS at high ω associated with the relaxation with entanglement of matrix PMMA. At high T and low ω region, the B series samples showed a sol-gel transition at elevating T and the critical gel behavior characterized with a power-law relationship, G′ = G″/tan(nπ/2) ∝ ωⁿ, was observed. This behavior suggested formation of a self-similar, fractal structure of grafted PBA particles. The critical gel temperature (T _gel) and the critical exponent (n) were determined for the B series samples. TEM observations revealed that as-prepared A and B samples had well-dispersed particles but the B samples after viscoelastic measurements had fragmented networks of the PBA particles, confirming that the sol-gel transition occurred for the PMMA/grafted PBA systems at elevating T.     

From linear viscoelasticity to the architecture of highly branched polyethylene

In this work, the linear viscoelastic behavior of some low-density polyethylene in the melt is used to obtain their architecture. In this way, the number of branches per molecule and long chain branching (LCB) content is determined. For this purpose, a method based on the molecular dynamics of simple star-shaped molecules is presented. It allows one to infer the topology of an average molecule through a set of 2N _c parameters {C ⁿ _i , the number concentration of a level i} and {M _bi , the mass of a segment of level i} representing an irregular Cayley tree with N _c levels. The inverse problem uses the complex shear modulus as a function of the frequency data along with a minimization algorithm. Results from the present method are compared with NMR and SEC measurements of the level of branching. It appears that SEC and rheology leads to similar results on the determination of LCB while NMR overestimate the number of branch points per molecule. Moreover, rheology allows one to go further than the basic evaluation of LCB content and shows a picture of the structure of the molecules that is in agreement with the kinetics of free radical polymerization of polyethylene.     

Effects of Fuel Lewis Number on Localised Forced Ignition of Turbulent Mixing Layers

The influences of fuel Lewis number Le _F on localised forced ignition of inhomogeneous mixtures are analysed using three-dimensional compressible Direct Numerical Simulations (DNS) of turbulent mixing layers for Le _F  = 0.8, 1.0 and 1.2 and a range of different root-mean-square turbulent velocity fluctuation u′ values. For all Le _F cases a tribrachial flame has been observed in case of successful ignition. However, the lean premixed branch tends to merge with the diffusion flame on the stoichiometric mixture fraction isosurface at later stages of the flame evolution. It has been observed that the maximum values of temperature and reaction rate increase with decreasing Le _F during the period of external energy addition. Moreover, Le _F is found to have a significant effect on the behaviours of mean temperature and fuel reaction rate magnitude conditional on mixture fraction values. It is also found that reaction rate and mixture fraction gradient magnitude \(\vert \nabla \xi \vert \) are negatively correlated at the most reactive region for all values of Le _F explored. The probability of finding high values of \(\vert \nabla \xi \vert \) increases with increasing Le _F . For a given value of u′, the extent of burning decreases with increasing Le _F . A moderate increase in u′ gives rise to an increase in the extent of burning for Le _F  = 0.8 and 1.0, which starts to decrease with further increases in u′. For Le _F  = 1.2, the extent of burning decreases monotonically with increasing u′. The extent of edge flame propagation on the stoichiometric mixture fraction ξ = ξ _st isosurface is characterised by the probability of finding burned gas on this isosurface, which decreases with increasing u′ and Le _F . It has been found that it is easier to obtain self-sustained combustion following localised forced ignition in case of inhomogeneous mixtures than that in the case of homogeneous mixtures with the same energy input, energy deposition duration when the ignition centre is placed at the stoichiometric mixture. The difficultly to sustain combustion unaided by external energy addition in homogeneous mixture is particularly prevalent in the case of Le _F  = 1.2.     

Bistable Boundary Reactions in Two Dimensions

In a bounded domain \({\Omega \subset \mathbb R^2}\) with smooth boundary we consider the problem

$\Delta u = 0 \quad {\rm{in }}\, \Omega, \qquad \frac{\partial u}{\partial \nu} = \frac1\varepsilon f(u) \quad {\rm{on }}\,\partial\Omega,$

where ν is the unit normal exterior vector, ε > 0 is a small parameter and f is a bistable nonlinearity such as f(u) = sin(π u) or f(u) = (1 ? u ²)u. We construct solutions that develop multiple transitions from ?1 to 1 and vice-versa along a connected component of the boundary ?Ω. We also construct an explicit solution when Ω is a disk and f(u) = sin(π u).

     

Families of rational solutions of the <Emphasis Type="Italic">y</Emphasis>-nonlocal Davey–Stewartson II equation

The y-nonlocal Davey–Stewartson II equation is an extension of the usual DS II equation involving a partially parity-time-symmetric potential only with respect to the spatial variable y. By using the Hirota bilinear method, families of n-order rational solutions are obtained, which include lumps in the (x, y)-plane and the (y, t)-plane, growing-and-decaying line waves in the (x, t)-plane, and hybrid solutions of interacting line rogue waves and lumps in the (x, y)-plane.