The Effect of Preferential Diffusion on the Soot Initiation Process in Ethylene Diffusion Flames

The influence of differential diffusion of chemical species in the soot initiation process in turbulent flows is investigated through Direct Numerical Simulations coupled to a compact global chemical mechanisms for ethylene (C₂H₄) flame combustion (Løvås et al., Combust Sci Tech 182(11):1945?C1960, 2010) featuring the important reaction steps for acetylene production. Our focus is on the formation of acetylene (C₂H₂) which is one of the most important species indicative of soot formation layers, especially in relation to the location of the H and H₂ layers. The effect of preferential diffusion is assessed by comparison of results from unity and non-unity Lewis number simulations. The results indicate that under moderate turbulent conditions, where preferential diffusion effects become prominent, and with the global scheme used preferential diffusion greatly enhances the spread of the radical H whose peak value in mass fraction is reduced by a factor of about two; the spread of H₂ is also enhanced though to a lesser extent. Importantly, the H and H₂ spread into a range of mixture fraction Z between 0.2 and 0.3 which contains the soot formation range, supporting the hypothesis that soot formation is enhanced by preferential diffusion. Nevertheless, the acetylene formation layers themselves show little adjustment in the presence of non-unity Lewis numbers suggesting that the acetylene formation is dominated under the current conditions by the direct thermal decomposition of ethylene to acetylene in the global chemistry used. The specific F _i factors that appear in flamelet models are explicitly computed; only F _H, F _H2 and F _CO show appreciable differences on the fuel lean range of mixture fraction due to non-unity Lewis numbers, suggesting that the effects of non-unity Lewis numbers could be incorporated by a selective inclusion of only a few of the F _i factors in order to save computational time.     

Symmetry and secondary bifurcation of elastic structures

We consider non-linear bifurcation problems for elastic structures modeled by the operator equation F[w;α]=0 where F:X×R^k→Y,X,Y are Banach spaces and XY. We focus attention on problems whose bifurcation equations are of the form

f_i(α₁,α₂;λ,μ)=(a_iμ+b_iλ)α_i+p_iα_i³+q_iα_i∑_j=1,j≠i^kα_j+1²+α_ih_i(λ,μ;α₁,α₂,…α_k) i=1,2,…k

which emanates from bifurcation problems for which the linearization of F is Fredholm operators of index 0. Under the assumption of F being odd we prove an important theorem of existence of secondary bifurcation. Under this same assumption we prove a symmetry condition for the reduced equations and consequently we got an existence result for secondary bifurcation. We also include a stability analysis of the bifurcating solutions.     

Transport coefficients for a partially ionized two-temperature plasma with ions and neutral particles having different masses

The viscosity, heat-conduction, and diffusion coefficients are calculated for a two-temperature three-component plasma composed of ions, neutral particles and electrons when the masses of the ions m_i and neutral particles m_a are different. Similar transport coefficients for m_i=m_a were calculated in [1–3]. The numerical values obtained for the transport coefficients are compared with the values calculated from the formulas of [1–3]. Numerical calculations are carried out for helium with a cesium additive (m_i>m_a) and for krypton with a lithium additive (m_ia).Finally, the author is grateful to V. V. Gogosov, under whose direction this work was carried out.     

Conservation laws of multidimensional diffusion--convection equations

All possible linearly independent local conservation laws for n-dimensional diffusion–convection equations u _t=(A(u)) _ii +(B ⁱ(u)) _i were constructed using the direct method and the composite variational principle. Application of the method of classification of conservation laws with respect to the group of point transformations [R.O.~Popovych, N.M. Ivanova, J. Math. Phys. 46, 2005, 043502 (math-ph/0407008)] allows us to formulate the result in explicit closed form. Action of the symmetry groups on the conservation laws of diffusion equations is investigated and generating sets of conservation laws are constructed.     

Multicomponent turbulent boundary layer on a chemically active surface

When a gaseous mixture flows past chemically active surfaces the boundary layer formed on the wetted body may contain a large number of components with different diffusion properties. This leads to the necessity for studying the diffusion of the components in the multicomponent boundary layer.The use of thebinary boundary layer concept in the general case cannot yield satisfactory results, since replacement of the mutual diffusion coefficients D_ij of the various pairs of components by a single diffusion coefficient D in many cases is a rough approximation.In the general case the number of different diffusion coefficients is equal to N(N–1)/2 (N is the number of components). Usually it is possible to identify groups of components with similar molecular weights. Then the number of different diffusion coefficients may be reduced without large error. However, even in the comparatively simple case when it is possible to divide all the components into two groups with similar molecular weights we must take account of three different diffusion coefficients (one diffusion coefficient in each group and also the diffusion coefficient for the components of one group relative to the components of the other group). Only in particular cases when the gaseous mixture consists of only two components with arbitrary molecular weights, or if all the components of the gaseous mixture have similar molecular weights, can we with justification introduce a single diffusion coefficient (if in this case there are no limitations on the direction of the diffusion).Studies have been published covering the laminar multicomponent boundary layer. An analytic method for solving the equations of the laminar multicomponent boundary layer was developed by Tirskii [1]. There are also studies in which concrete results were obtained by numerical methods with the use of computers (for example, [2, 3]).As far as the author knows, for turbulent flow there are studies (for example, [4, 5]) covering flow with chemical reactions only in the case when all the diffusion coefficients are equal (D_ij=D).The present paper presents a method for calculating the turbulent multicomponent boundary layer with account for several different diffusion coefficients.Notation x, y coordinates - u, v velocity components - density - T temperature - h heat content - H enthalpy - c_i mass concentration of the i-th component - c ₁ ⁽¹⁾ element concentrations in solid body - J_i diffusion flux of the i-th component - m molecular weight - dynamic viscosity coefficient - kinematic viscosity coefficient - heat conduction coefficient - c_p specific heat - adiabatic index - D_ij binary diffusion coefficients - P Prandtl number - S_ij Schmidt number - S_t Stanton number - M Mach number - friction - q radiant thermal flux - boundary layer thickness - D rate of displacement of gas-solid interface - degree of gasification - r_ij weight fraction of element i in component j - _ij stoichiometric coefficients - K_i reaction equilibrium constants - l number of components for which I_i0 Indices i, j component number - w quantities for y=0 - * quantities on the edge of the laminar sublayer - ⁽¹⁾ quantities at the solid body - quantities at the outer edge of the boundary layer - molar transport coefficients     

On the dispersion of a solute in oscillating flow of a non-Newtonian fluid in a channel

The paper presents an exact analysis of the dispersion of an immiscible solute in a non-Newtonian fluid (known as an incompressible second-order fluid which shows viscoelastic behaviour) flowing slowly in a parallel plate channel in the presence of a periodic pressure gradient. Using a generalized dispersion model which is valid for all times after the solute injection, the diffusion coefficients K _i (τ)(i=1,2,3,…) are obtained as functions of time τ in the case when the initial solute distribution is in the form of a slug of finite extent. The analysis leads to the novel result that K ₂(τ) (which is a measure of the longitudinal dispersion coefficient of the solute) has a steady part S in addition to a fluctuating part D ₂(τ) due to the pulsatility of the flow. It is found that S decreases with increase in the viscoelastic parameter M for given values of the amplitude λ and frequency ω of the pressure pulsation. On the other hand, it is found that at a fixed instant τ, the amplitude of D ₂(τ) increases with increase in M for given values of λ and ω. Further it is shown that at a given instant τ, the amplitude of D ₂(τ) decreases with increase in ω for given λ and M and the profile for D ₂(τ) becomes progressively flatter with increase in ω. Finally the axial distribution of the average concentration θ _m of the solute over the channel cross-section is determined at different instants after the solute injection for several values of M, λ and ω. The present study is likely to have important bearing on the problem of dispersion of tracers in blood flow through arteries.     

Numerical investigation of injection-induced electro-convection in a dielectric liquid between two eccentric cylinders

Numerical analysis of the 2D radial and azimuth electro-convection (EC) flow of dielectric liquid between two eccentric cylindrical electrodes driven by unipolar injection of ions is presented. The finite volume method is used to resolve the spatiotemporal distributions of the flow field, electric field, and charge density. The flow instability is studied in various scenarios where the radius ratio Γ = R_i/R_o ranges between 0.1 and 0.7 and the eccentricity η between 0.1 and 0.5. The bifurcation of the flow patterns depends on the electric Rayleigh number T, a ratio of the electric force to viscous force, and the two geometric parameters Γ and η. For an increasing T, the EC system develops from a weak steady convective state to chaos via different intermediate states experiencing pitchfork and Hopf bifurcations. The influence of Γ and η on the bifurcation behavior is also investigated. When Γ lies between 0.1 and 0.3, a novel periodic oscillation of the flow patterns has been observed.     

Bogdanov–Takens bifurcation in a tri-neuron BAM neural network model with multiple delays

In this paper, a tri-neuron BAM neural network model with multiple delays is considered. We show that the connection topology of the network plays a fundamental role in classifying the rich dynamics and bifurcation phenomena. There is a wide range of different dynamical behaviors which can be produced by varying the coupling strength. By choosing the connected weights c ₂₁ and c ₃₁ (the connection weights through the neurons from J-layer to I-layer) as bifurcation parameters, the critical values where a Bogdanov–Takens bifurcation occurs are derived. Then, by computing the normal forms for the system, the bifurcation diagrams are obtained. Furthermore, some interesting phenomena, such as saddle-node bifurcation, pitchfork bifurcation, homoclinic bifurcation, heteroclinic bifurcation and double limit cycle bifurcation are found by choosing the different connection strengths. Some numerical simulations are given to support the analytic results.     

Singularly continuous spectrum of singularly perturbed operators

We propose a construction of a singularly perturbed self-adjoint operator with a given compact set in its singularly continuous spectrum. In particular, the set can be a fractal of prescribed type. We use the construction of a singularly perturbed operator à for a given self-adjoint operator A in a Hilbert space $\mathcal{H}$ that solves the eigenvalue problem Ãψ _i = λ _iψ_i for a countable set Λ = {λ _i} _i=1 ^∞ of real numbers λ _i ∈ ?¹, |λ _i| < ∞, and an orthonormal system of vectors {ψ _i}, i = 1, 2 …, under certain additional general conditions.     

Couette flow of a multicomponent ionized gas

We consider equilibrium flow of a multicomponent ionized gas between two catalytic plates of infinite length, one of which moves parallel to the other with constant velocity. The results of [1] are generalized for ionized gaseous mixtures which are in local thermodynamic equilibrium. Formulas are presented for calculating the thermal flux and the effective thermal conductivity for ambipolar diffusion.Then a special ionization case is discussed.Notation A_i chemical symbol of the i-th component - W_i projection of the molar diffusive flux vector of the i-th component on the y-axis - x_i molar concentration - H_i enthalpy - m_i molecular weight - Q_s heat of the s-th reaction - K_ps(T) equilibrium constant of the s-th reaction - W_i mass formation rate of the i-th component per unit volume - Z_i charge number - e unit charge (electron charge) - E electric field intensity - distance between the plates - N number of components - v _sl stoichiometric coefficients - density - T temperature - p pressure - u projection of average velocity on y-axis - viscosity - thermal conductivity - D_ij binary diffusion coefficient - R universal gas constant - k Boltzmann constant In conclusion, the author wishes to thank G. A. Tirskii for proposing the study and for suggestions made in the course of the investigation.     

In the Chapman-Enskog Expansion¶the Burnett Coefficients Satisfy¶the Universal Relation ω3+ω4+θ3= 0

The Chapman–Enskog expansion when applied to a gas of spherical molecules yields formal expressions for the stress deviator P and energy-flux vector q, P=μP ⁽¹⁾+μ² P ⁽²⁾+…, q=μq ⁽¹⁾+μ² q ⁽²⁾+…. The Burnett terms P ⁽²⁾, q ⁽²⁾ depend on 11 coefficients ω _i , 1≦i≦6, θ&; _i , 1≦i≦ 5. This paper shows that ω₃+ω₄+θ₃= 0.     

Localized bulging in an inflated cylindrical tube of arbitrary thickness – the effect of bending stiffness

We study localized bulging of a cylindrical hyperelastic tube of arbitrary thickness when it is subjected to the combined action of inflation and axial extension. It is shown that with the internal pressure P and resultant axial force F viewed as functions of the azimuthal stretch on the inner surface and the axial stretch, the bifurcation condition for the initiation of a localized bulge is that the Jacobian of the vector function $(P,F)$ should vanish. This is established using the dynamical systems theory by first computing the eigenvalues of a certain eigenvalue problem governing incremental deformations, and then deriving the bifurcation condition explicitly. The bifurcation condition is valid for all loading conditions, and in the special case of fixed resultant axial force it gives the expected result that the initiation pressure for localized bulging is precisely the maximum pressure in uniform inflation. It is shown that even if localized bulging cannot take place when the axial force is fixed, it is still possible if the axial stretch is fixed instead. The explicit bifurcation condition also provides a means to quantify precisely the effect of bending stiffness on the initiation pressure. It is shown that the (approximate) membrane theory gives good predictions for the initiation pressure, with a relative error less than 5%, for thickness/radius ratios up to 0.67. A two-term asymptotic bifurcation condition for localized bulging that incorporates the effect of bending stiffness is proposed, and is shown to be capable of giving extremely accurate predictions for the initiation pressure for thickness/radius ratios up to as large as 1.2.     

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.     

On the dispersion of a solute in oscillating flow through a channel

The paper presents an exact analysis of the dispersion of a solute in an incompressible viscous fluid flowing slowly in a parallel plate channel under the influence of a periodic pressure gradient. Using a generalised dispersion model which is valid for all times after the solute injection, the diffusion coefficientsK _i (τ)(i=1,2,3,…) are determined as functions of timeτ when the initial distribution of the solute is in the form of a slug of finite extent. The second coefficientK ₂(τ) gives a measure of the longitudinal dispersion of the solute due to the combined influence of molecular diffusion and nonuniform velocity across the channel cross-section. The analysis leads to the novel result thatK ₂(τ) consists of a steady partS and a fluctuating partD ₂(τ) due to the pulsatility of the flow. It is shown thatS increases with increase inλ (the amplitude of pressure pulsation) for small values ofω (the frequency of the pulsation). But for largeω, S decreases with increase inλ. It is also found that for fixedλ, there is very little fluctuation inD ₂(τ) forω=1, butD ₂(τ) shows fluctuation with large amplitude whenω slightly exceeds unity. The amplitude ofD ₂(τ) then decreases with further increase inω. Thus the variation of bothS andD ₂(τ) withω is non-monotonic. Finally,? _m , the average concentration of the solute over the channel cross-section is determined for various values ofλ andω.     

Measurement of the concentration diffusion coefficient for Ne-Ar,Ne-Xe,Ne-H2, Xe-H2, H2-N2 and H2-O2 gas systems

The concentration diffusion coefficient, D ₁₂, is measured for the equimolar mixtures of Ne-Ar, Ne-Xe, Ne-H₂, Xe-H₂, H₂-N₂ and H₂-O₂ binary gas systems in a two-bulb metal apparatus in the temperature range 0 C to 100 C. These values are compared with the existing data on these systems and with the predictions of the kinetic theory in conjunction with the modified Buckingham exp-six potential. Unlike the thermal diffusion coefficient, with the simple theory it is possible to predict D ₁₂ within a few percent even for systems involving polyatomic gases. The smoothed experimental D ₁₂ values are also used to obtain data for the coefficients of viscosity and thermal conductivity at round temperatures and compositions for these systems.     

[0, k i]1 m -Factorizations Orthogonal to a Subgraph

Let G be a graph,k₁,…,k_m be positive integers. If the edges of graph G can be decomposed into some edge disjoint. [0,k₁]-factor. F₁,…,[0,k_m]-factor F_m, then we can say F={F₁,…,F_m, is a [0,k_i]_m¹-factorization of G. If H is a subgraph with m edges in graph G and |E(H)∩E(F_i)|=1 for all 1≤i≤m, then we can call that F is orthogonal to H. It is proved that if G is a..[0,k₁+…+k_m-m+1]-graph, H is a subgraph with m edges in G, then graph G has a. [0,k_i]₁^m-factorization orthogonal to H.     

Non-linear normal modes and their bifurcation of a two DOF system with quadratic and cubic non-linearity

The non-linear normal modes (NNMs) and their bifurcation of a complex two DOF system are investigated systematically in this paper. The coupling and ground springs have both quadratic and cubic non-linearity simultaneously. The cases of ω₁:ω₂=1:1, 1:2 and 1:3 are discussed, respectively, as well as the case of no internal resonance. Approximate solutions for NNMs are computed by applying the method of multiple scales, which ensures that NNM solutions can asymtote to linear normal modes as the non-linearity disappears. According to the procedure, NNMs can be classified into coupled and uncoupled modes. It is found that coupled NNMs exist for systems with any kind of internal resonance, but uncoupled modes may appear or not appear, depending on the type of internal resonance. For systems with 1:1 internal resonance, uncoupled NNMs exist only when coefficients of cubic non-linear terms describing the ground springs are identical. For systems with 1:2 or 1:3 internal resonance, in additional to one uncoupled NNM, there exists one more uncoupled NNM when the coefficients of quadratic or cubic non-linear terms describing the ground springs are identical. The results for the case of internal resonance are consistent with ones for no internal resonance. For the case of 1:2 internal resonance, the bifurcation of the coupled NNM is not only affected by cubic but also by quadratic non-linearity besides detuning parameter although for the cases of 1:1 and 1:3 internal resonance, only cubic non-linearity operate. As a check of the analytical results, direct numerical integrations of the equations of motion are carried out.     

Concentration-dependent diffusion in a composite slab with and without chemical reactions

Concentration-dependent diffusion of solute in a composite slab is investigated. The complex diffusion problem can be described by a set of nonlinear diffusion equations which is coupled to each other through the nonlinear interfacial boundary conditions. A two-layer diffusion is illustrated and the coupled nonlinear diffusion equations are conveniently solved by the orthogonal collocation method. Numerical simulation of the example reveals many interesting diffusion characteristics which are quite different from those in a single slab diffusion system.Nomenclature a _j expansion coefficient - A _i,j element of collocation matrix - B _i,j element of collocation matrix - C _a , C _b surface concentration - C _i concentration in the ith layer - D _i diffusion coefficient in the ith layer - D _i0 diffusion coefficient at very low concentration - k _i reaction rate in the ith layer - K _i dimensionless reaction rate, k _i l _i ² c _a ^m–1 /D ₁₀ - l _i thickness of the ith layer - m order of chemical reaction - n order of the orthogonal polynomial approximation - P _j–1(x _i ) orthogonal polynomial of order j - t time - x _i coordinate of the ith layer - X _i dimensionless coordinate of the ith layer, x _i/l _i - ratio of diffusion coefficient at low concentration, D ₂₀/D ₁₀ - ratio of thicknesses of layer, l ₁/l ₂ - _i dimensionless parameter in the concentration-dependent function of the ith layer - ratio of surface concentration, C _b /C _a - dimensionless time, tD ₁₀/l ₁ ² - _i dimensionless concentration in the ith layer, C _i /C _a      

Melting of snow with a diffusion-controlled analytical model

An analytical study is reported of melting of a snow layer in an aqueous solution. A diffusion-controlled analytical model was proposed to the melting under an ideal condition that an aqueous solution was instantaneously filled up by a snow layer at the same temperature as the solution. The analytical results gave a qualitative prediction of the experimental results of the melting of snow layers suddenly immersed in a calcium chloride aqueous solution. The temperature in a melting system decreased rapidly during the melting process. The melting was complete within a few seconds, which denoted a thermodynamic equilibrium. When the initial temperatureT _i and the initial porosity of snow ? _i were the same, the initial concentrationC _mi in the solution strongly affected both the decrease in temperature in the melting system and the melting mass per unit volume of snowM. WhenC _mi andT _i were the same, the maximum melting mass per unit volume of snowM _max was not largely affected by snow particle diameters. A figure was presented for the relationM _max?T _i,C _mi, and ? _i , and also a relationship was presented to easily predict the non-dimensional maximum melting massM _max ^* .