Self-similar solutions for unsteady flow behind an exponential shock in an axisymmetric rotating dusty gas

Similarity solutions are obtained for one-dimensional unsteady isothermal and adiabatic flows behind a strong exponential cylindrical shock wave propagating in a rotational axisymmetric dusty gas, which has variable azimuthal and axial fluid velocities. The shock wave is driven by a piston moving with time according to an exponential law. Similarity solutions exist only when the surrounding medium is of constant density. The azimuthal and axial components of the fluid velocity in the ambient medium are assumed to obey exponential laws. The dusty gas is assumed to be a mixture of small solid particles and a perfect gas. To obtain some essential features of the shock propagation, small solid particles are considered as a pseudo-fluid; it is assumed that the equilibrium flow conditions are maintained in the flow field, and that the viscous stresses and heat conduction in the mixture are negligible. Solutions are obtained for the cases when the flow between the shock and the piston is either isothermal or adiabatic, by taking into account the components of the vorticity vector. It is found that the assumption of zero temperature gradient results in a profound change in the density distribution as compared to that for the adiabatic case. The effects of the variation of the mass concentration of solid particles in the mixture \(K_p\) , and the ratio of the density of solid particles to the initial density of the gas \(G_a\) are investigated. A comparison between the solutions for the isothermal and adiabatic cases is also made.     

Interaction of heat transfer by conduction and radiation in a nongray planar medium

This paper considers the problem of the steady energy transfer due to combined effects of conduction and radiation in a medium with frequency dependent properties. The particular problem studied is the one-dimensional energy transfer in an absorbing, emitting, and conducting medium capable of generating heat which is bounded by two black plates. The analysis is restricted to an absorption coefficient x_v of theMilne-Eddington type, i.e., x_v(T)=αv β(T), and the frequency dependence of α(v) is approximated by a rectangular model. Temperature distribution and heat fluxes are reported in the paper for two models of spectral absorption coefficient and are compared with those for the gray case.     

A self- similar solution of a shock propagation in a mixture of a non-ideal gas and small solid particles

Similarity solutions are obtained for unsteady, one-dimensional self-similar flow behind a strong shock wave, driven by a moving piston, in a dusty gas. The dusty gas is assumed to consist of a mixture of small solid particles and a non-ideal gas, in which solid particles are continuously distributed. It is assumed that the equilibrium flow-condition is maintained and variable energy input is continuously supplied by the piston. Solutions are obtained under both the isothermal and adiabatic conditions of the flow-field. The spherical case is worked out in detail to investigate to what extent the flow-field behind the shock is influenced by the non-idealness of the gas in the mixture as well as by the mass concentration of the solid particles, by the ratio of density of the solid particles to the initial density of the mixture and by the energy input due to moving piston. A comparison is also made between isothermal and adiabatic cases.     

Calculation of convective heat flux in the vicinity of the stagnation point for partially ionized gas flow past a body

From numerical solutions of the boundary layer equations for a four-component gas mixture (E, N⁺, N₂, and N) with gas injection, approximate formulas for the heat flux as a function of the variation of λρ/c_p and h^* across the boundary layer and the magnitude of the objection are obtained (λ is the thermal conductivity of the mixture,ρ is density, c_p is the specific heat, and h^* is the enthalpy of the ideal gas state of the mixture). An effective ambipolar diffusion coefficient D^(a)(i) is introduced, making possible finite formulas for the convective heat fluxes in the “frozen” boundary layer. We study the behavior of these coefficients within the boundary layer. A formula is obtained for convective heat flux to the wall from partially ionized air for a nine-component mixture (E, O⁺, N⁺, NO⁺, O, N, NO, O₂ N₂). Even for simpler four-component gas model three effective ambipolar diffusion coefficients are necessary: $$\begin{gathered} D^{(a)} (A) = D (A, M) D^{(a)} (I) = 2D (A, M), \hfill \\ D^{(a)} (M) = [ 1 + c_e (I)] D(A, M). \hfill \\ \end{gathered} $$ Here D(A, M) is the binary diffusion coefficient of the atoms into molecules, and c_e(I) is the ion concentration at the outer edge of the boundary layer. The assumption of an infinitely large charge-exchange cross section and the other simplifying assumptions used in [1] lead to overestimation of the magnitude of the dimensionless heat flux by 7–15% for the “frozen” boundary layer case.     

Thermal convection for a thermodependent herschel-bulkley fluid in an annular duct

This article presents a numerical and experimental investigation of the thermal convection for a thermodependent Herschel-Bulkley fluid in an annular duct under the conditions of a uniform heat flux density on the outer wall and an insulated inner wall. In the numerical analysis, it is assumed that: (i) the rheological behavior of the fluid can be expressed through the Herschel-Bulkley law: $\tau = \tau _s + K\dot \gamma ^n $ ; (ii) the flow is fully developed at the inlet; (iii) all fluid properties except consistency indexK are constant. TheK?T relation used isK=K ₀exp(?bT). The results obtained enable us to characterize completely the dynamical and thermal fields. The numerical solution is in good agreement with the experimental data, showing the reasonableness of the computed results.     

Self-similar solution of cylindrical shock wave propagation in a rotational axisymmetric mixture of a non-ideal gas and small solid particles

Similarity solutions are obtained for one- dimensional isothermal and adiabatic unsteady flow behind a strong cylindrical shock wave propagating in a rotational axisymmetric dusty gas, which has a variable azimuthal fluid velocity together with a variable axial fluid velocity. The shock is assumed to be driven out by a moving piston and the dusty gas to be a mixture of non-ideal (or perfect) gas and small solid particles, in which solid particles are continuously distributed. It is assumed that the equilibrium flow-condition is maintained and variable energy input is continuously supplied by the piston. The shock Mach number is not infinite, but has a finite value. The azimuthal and axial component of the fluid velocity in the ambient medium are assumed to be vary and obey power laws, and the density of the ambient medium is taken to be constant. In order to obtain the similarity solutions the angular velocity of the ambient medium is assumed to be decreasing as the distance from the axis increases. Effects of the variation of the parameter of non-idealness of the gas in the mixture, the mass concentration of solid particles and the ratio of the density of solid particles to the initial density of the gas are investigated.     

Self-similar motion of a gas heated by a nonequilibrium continuous radiation spectrum

In this paper we discuss the motion of the vapor formed during the evaporation of a solid by a continuous radiation spectrum. The vapor is assumed to be heated by this radiation to a temperature T much higher than the phase-transition temperature T_v and much higher than the temperature T_i at which significant ionization of the vapor begins. in the case, T_v and T_i can be neglected (as can the heat of evaporation Q_v and the energy Q_i expended on ionization). As a result of this motion, the vapor has a density ? much lower than the densityρ ₀ of the solid. It can therefore be assumed that the heating wave moves through an absolutely cold and infinitely dense gas. At the same time, the vapor temperature is assumed low enough that reradiation can be neglected. The radiation-absorption coefficient η for the ionized vapor can be described by a power-law dependence on T and ? for certain ranges of T, ε, and the photon energy ε. In this case, the motion of the gas is a self-similar problem. The spectrum and angular distribution of the incident radiation [φ (ε, θ)] and the η and ε dependences can be arbitrary. A system of ordinary differential equations is found and solved. Intense radiation incident on a solid surface will evaporate the solid. If the absorption coefficient η of the vapor and the flux density q of the radiation are high enough, the escaping vapor will be heated to a high temperature in a relatively short time. This temperature will not only be much higher than the evaporation temperature T_v, but it will also be higher than the “ionization temperature” T_i. If the internal energy per unit mass of the vapor is much higher than the heat of evaporation Q_v and the energy Q_i expended on ionization, and if the vapor density ? is much lower than the initial densityρ ₀ as a result of its escape, then the problem of the motion and heating of the vapor can be simplified through the assumptions. (0.1) $$T_v = T_i = Q_v = Q_i = 0,\rho _0 = \infty (v_0 = 1/\rho _0 = 0)$$ (here and below, v is the specific volume). We can therefore assume that the heating wave moves through an infinitely dense and absolutely cold gas. In the region of multiple and complete ionization, the ionized-vapor absorption coefficient η, associated with free-free electron transitions in the field of ions, and bound-free transitions from the higherlying states of atoms and ions, has an approximately power-law dependence on T and ? [1], or on p and ? (p is the pressure): Here k and K are numerical coefficients which depend on the substance and on the ranges of T, ?, and ε in which (0.2) is used. For a completely ionized gas, we have α=3/2, β=1,a=?5/2, b=?3/2, and \( - \bar 1/2\) when ε?T; or α=3/2, β=1,a=3/2, and b=?1/2 when when ε ? T. We assume that (0.2) holds for any T, for approximation (0.1). We assume the ratio of specific heats γ to be constant for a certain temperature range in the range of multiple and complete ionization. With these simplifying approximations, the problem of the planar, transient flow of a gas heated by a beam of monochromatic radiation is a self-similar problem. It has been studied in [2,3]. It is shown below that the analogous problem of the motion of a gas heated by a nonequilibrium continuous radiation spectrum is also self-similar. For a partially ionized gas, approximation (0.2) is usualy satisfied only for the long-wavelength part of the incident spectrum. For the short-wavelength part of the spectrum (that is, for photons whose energy is close to or greater than the ionization potential characteristics of the ions for the given temperature range, and which are capable of direct photoionization of these ions from the ground or first excited status), the absorption coefficient is usually much smaller (by several orders of magnitude). This “hard” radiation penetrates a short distance into the solid, causing intense heating of a thin surface layer of small mass. An afterionization wave propagates through the substance, moving under the influence of the radiation flux in the hard part of the spectrum; if the temperature of the surface layer is close to the source temperature T_e, and reradiation becomes important, there will also be a thermal wave [1]. Since the energy expended in heating is large in these waves, their propagation velocity is small (in comparison with that of the wave of evaporation, initial ionization, and heating of the plasma by the long-wavelength part of the spectrum), even if the hard and soft parts of the incidence spectrum have comparable energies (E_h and E_s). Also, the intense reradiation by the thermal wave in the hard part of the spectrum increases its propagation velocity. Finally, the energy in the short-wavelength part of the spectrum may in general be small because of self-adsorption in the source itself (for example, adsorption of the short-wavelength radiation in the cold working gas ahed of a shock wave front in an explosive source [4]). Accordingly, the heating waves for the various parts of the source spectrum may propagate differently Since the mass of the surface layer heated by the short-wave-length part of the spectrum is small, the pressure produced as a result of of the disintegration of the surface layer is small when E_h is of the order order of E_s or, especially, when E_h?E_s; that is, the hydrodynamic effects of the heating and surface-layer disintegration on the motion and and heating of the deep layers heated by the “basic” part of the spectrum can also be neglecred. The high temperature and low density of this layer only facilitate the penetration of the long-wavelength part of the spectrum into the deeper layers; however, because of the small mass of this layer, even this phenomenon has little effect on the hydrodynamic processes in the deeper layers. Accordingly, Eq. (0.2) can frequently be assumed valid for the basic part of the spectrum in the case of a partially ionized gas, also; the rest of the spectrum may simply be neglected. These restrictions on the applicability of the self-similar problem are generally removed in the case of a completely ionized gas. A state close to that of complete ionization arises when two ionization potentials typical of a given temperature range are greatly different (this occurs, for example, in the case of the alkaline metals, and also when one atomic shell has been essentially ionized, while another has not yet started to be ionized; e. g., the L- and K-shells or the M- and L-shells). We consider here the case in which the heating is caused by nonequilibrium radiation, that is, radiation such that the intrinisic radiation of the vapor may be neglected. This is a valid assumption when the vapor temperature is considerably below the source temperature T_e, or, more accurately, when the following condition holds (for a Planckian source spectrum): (0.3) $$W\sigma T_e^4 \chi \left( {\frac{{\varepsilon _1 }}{{T_e }},\frac{{\varepsilon _2 }}{{T_e }}} \right) \gg \sigma \Upsilon ^4 \chi \left( {\frac{{\varepsilon _1 }}{T},\frac{{\varepsilon _2 }}{T}} \right)$$ Here W is the source-radiation dilution coefficient due to geometric factors, σ is the Stefan-Boltzmann constant, ?₁ and ?₂ are the boundaries of the “basic part” of the spectrum, and χ is the fraction of the spectral energy of a Planckian source with a temperature T_e or T for photous with energies ?₁≤?≤?₂. We note that the boundaries ?₁ and ?₂ for the source and vapor-radiation spectra are sometimes slightly differnt, but condition (0.3) can be easily modified for this situation or for a non-Planckian source spectrum. For our problem, the radiation intensity J=J (m, t, ε, θ) is a function of four variables: the time t, the Lagrangian mass coordinate m, the photon energy ε, and the angle θ between the direction of motion and the beam direction. The intensity J_o=J (o, t, ε, θ) of the radiation incident on the boundary m=0 is assumed to be a given function. In the self-similar problem, J can be represented as (0.4) $$J = t^\lambda J(mt^{ - n} ,\varepsilon ,\theta )$$ . This can be done (when conditions (0.1)–(0.3) are satisfied) when J_o can be represented by (0.5) $$J_0 = t^\lambda \psi (\varepsilon ,\theta )(\varepsilon _1 \leqslant \varepsilon \leqslant \varepsilon _2 ,\theta _1 \leqslant \theta \leqslant \theta _2 )$$ If the source spectrum is Planckian, condition (0.5) requires that T_e=const. In this case, the power-law time dependence of the intensity J_o may reflect, for example, motion of the radiation source toward the irradiated surface; in this case, however, the limiting angle θ₂ of the incident radiation also changes (usually, θ₁=0). As before, the problem is self-similar if these angles θ₂(t) are always small; that is, if the radiation is almost completely unidirectional. The arbitrary nature of the function ψ(ε, ч), which shows the spectrum and angular distribution of thesource radiation, and the arbitrary nature of the function φ (ε), which shows the dependence of the absorption coefficient on the photon energy, permit us to analyze the effects of these functions on the heating and motion of the substance for the case of the self-similar solution.     

Natural convection heat transfer in enclosures with microemulsion phase change material slurry

This paper has dealt with the natural convection heat transfer characteristics of microemulsion slurry composed of water, fine particles of phase change material (PCM) in rectangular enclosures. The microemulsion slurry exhibited non-Newtonian pseudoplastic fluid behavior, and the phase changing process can show dramatically variations in both thermophysical and rheological properties with temperature. The experiments have been carried out separately in three subdivided regions in which the state of PCM in microemulsion is in only solid phase, two phases (coexistence of solid and liquid phases) or only liquid phase. The complicated heat transfer characteristics of natural convection have appeared in the phase changing region. The phase change phenomenon of the PCM enhanced the heat transfer in natural convection, and the Nusselt number was generalized by introducing a modified Stefan number. However, the Nusselt number did not show a linear output with the height of the enclosure, since a top conduction lid or stagnant layer was induced over a certain height of the enclosure. The Nusselt number increased with a decrease in aspect ratio (width/height of the rectangular enclosure) even including the side-wall effect. However, the microemulsion was more viscous while the PCM was in the solid phase, the side-wall effect on heat transfer was greater for the PCM in the solid region than that for the PCM in the liquid region. The correlation generalized for the PCM in a single phase is $ Nu = 1/3(1 - C_1 )Ra^{{1 \over {3.5n + 1}}} , $ where C ₁ = e ^–0.09AR for the PCM in solid phase and C ₁ = e ^–0.33AR for the PCM in liquid phase. For the PCM in the phase changing region, the correlation can be expressed as $ Nu = CRa^{{1 \over {7n + 2}}} Ste^{ - (1.9 - 1.65n)} , $ where C = 1.22 – 0.035AR for AR > 10 and C = 0.55 – 16.4e ^–1.1AR for AR < 10. The enclosure height used in the present experiments was varied from H = 5.5 [mm] to 30.4 [mm] at the fixed width W = 120 [mm] and depth D = 120 [mm]. The experiments were done in the range of modified Rayleigh number 7.0 × 10² ≤ Ra ≤ 3.0 × 10⁶, while the enclosure aspect ratio AR varied from 3.9 to 21.8.     

The Cattaneo type space-time fractional heat conduction equation

The classical heat conduction equation is generalized using a generalized heat conduction law. In particular, we use the space-time Cattaneo heat conduction law that contains the Caputo symmetrized fractional derivative instead of gradient ${{\partial_x}}$ and fractional time derivative instead of the first order partial time derivative ${{\partial_t}}$ . The existence of the unique solution to the initial-boundary value problem corresponding to the generalized model is established in the space of distributions. We also obtain explicit form of the solution and compare it numerically with some limiting cases.     

Mixed convection film condensation from downward flowing vapors onto a sphere with uniform wall heat flux

A model is developed for the study of mixed convection film condensation from downward flowing vapors onto a sphere with uniform wall heat flux. The model combined natural convection dominated and forced convection dominated film condensation, including effects of pressure gradient and interfacial vapor shear drag has been investigated and solved numerically. The separation angle of the condensate film layer, φ _s is also obtained for various pressure gradient parameters, P ^* and their corresponding dimensionless Grashof?'s parameters, Gr ^*. Besides, the effect of P ^* on the dimensionless mean heat transfer, will remain almost uniform with increasing P ^* until for various corresponding available values of Gr ^*. Meanwhile, the dimensionless mean heat transfer, is increasing significantly with Gr ^* for its corresponding available values of P ^*. For pure natural-convection film condensation, is obtained.     

The Problems of the Turbulent Burning Velocity

The paper reviews the practical problems in measuring a turbulent burning velocity that gives the mass rate of burning. These largely centre on identifying an appropriate flame surface to associate with the turbulent burning velocity, u _t , and the density of the unburned mixture. Such a flame surface has been identified, in terms of the mean reaction progress variable, $\bar {c}$ , for explosive flame propagation in a fan-stirred bomb. Measurement of $\bar {c}$ makes possible an estimation of the flame surface density, ??, from the relationship ${\it \Sigma} =k\bar {c}\left( {1-\bar {c}} \right)$ . It is shown that in such explosions, mass rates of burning derived from the measured total flame surface area agreed well with those found from the measured turbulent burning velocity. Flamelet considerations identify appropriate dimensionless correlating parameters for u _t . As a result, correlations of turbulent burning velocity divided by the effective rms turbulent velocity, are plotted against the turbulent Karlovitz stretch factor, K, for different values of the Markstein number for flame strain rate, Ma_sr. These plots cover a wide range of variables, including pressure and fuels, and are indicative of different regimes of turbulent combustion. At the lower values of K, there is some evidence of increases in u _t and k due to high-frequency flame surface wrinkling arising from flame instabilities. These increase as Ma_sr becomes more negative. It is found from the developed value of the mean flame surface density throughout the flame brush that, to a first approximation, an increase in u _t for a given mixture is accompanied by a proportional increase in the volume of the brush. The analysis shows that the volume fraction of the turbulent flame brush that is reacting is quite small.     

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

Statistically planar turbulent partially premixed flames for different initial intensities of decaying turbulence have been simulated for global equivalence ratios <????> = 0.7 and <????> = 1.0 using three-dimensional simplified chemistry based Direct Numerical Simulations (DNS). The simulation parameters are chosen such that the combustion situation belongs to the thin reaction zones regime and a random bi-modal distribution of equivalence ratio ?? is introduced in the unburned gas ahead of the flame to account for mixture inhomogeneity. The DNS data has been used to analyse the statistical behaviour of the transport of the cross-scalar dissipation rate based on the fuel mass fraction Y _F and the mixture fraction ?? fluctuations $\,\tilde{\varepsilon}_{Y\xi}={\overline{\rho D\nabla Y_{F}^{\prime\prime}.\nabla \xi^{\prime\prime}} } \big/ {\bar {\rho }}$ (where $\bar{q}$ , $\tilde{q}={\overline{\rho q} } \big/ {\bar {\rho }}$ and $q^{\prime\prime} =q-\tilde {q}$ are Reynolds average, Favre mean and Favre fluctuation of a general quantity q) in the context of Reynolds Averaged Navier?CStokes simulations where ?? is the gas density and D is the gas diffusivity. The statistical behaviours of the unclosed terms in the $\tilde{\varepsilon }_{Y\xi } $ transport equation originating from turbulent transport T ₁, density variation T ₂, scalar?Cturbulence interaction T ₃, chemical reaction rate T ₄ and the molecular dissipation rate D ₂ have been analysed in detail. It has been observed that the contributions of T ₂, T ₃, T ₄ and D ₂ play important roles in the $\tilde{\varepsilon }_{Y\xi } $ transport for the globally stoichiometric cases, but in the globally fuel-lean cases the contributions of T ₂ and T ₄ become relatively weaker in comparison to the contributions of T ₃ and D ₂. The term T ₁ remains small in comparison to the leading order contributions of T ₃ and D ₂ for all cases, but the contribution of T ₁ plays a more important role in the low Damköhler combustion cases. The term T ₂ behaves as a sink term towards the unburned gas side but becomes a source term towards the burned gas side. The scalar?Cturbulence interaction term T ₃ has been found to be generally positive throughout the flame brush, but in globally stoichiometric cases the contribution of T ₃ becomes negative in regions of intense heat release. The combined contribution of (T ₄ ?C D ₂) remains mostly as a sink in all cases studied here. Models are proposed for the unclosed terms of the $\tilde{\varepsilon }_{Y\xi } $ transport equation in the context of Reynolds Averaged Navier?CStokes simulations, which are shown to satisfactorily predict the corresponding quantities extracted from the DNS data for all cases.     

Radiative heat transfer in absorbing, emitting, and anisotropically scattering boundary-layer flows with reflecting boundary

The heat transfer in absorbing, emitting, and anisotropically scattering boundary-layer flows with reflecting boundary over a flat plate, over a 90-deg wedge, and in stagnation flow is solved by application of the Galerkin method with the particular solution boundary condition I _p (τ₀,ξ,?μ) of the equation of radiative transfer for an inhomogeneous term and the Box method. The exact integral expressions for the radiation part of this problem are developed. The coupling between convective and radiative heat transfer in boundary-layer flows is described by a set of nonlinear simultaneous equations including differential equations and integrodifferential equations. The Galerkin method and the particular solution boundary condition I _p (τ₀,ξ,?μ) are used to analyze the radiation part of the problem. The nonsimilar boundary-layer equations are solved by the Box method. The present numerical procedure solutions are compared in tables with the other exact treating results, the P-3, and P-1 approximation methods for the case of isotropically scattering boundary-layer flows. The effects of linearly anistropically scattering and reflecting surface are taken into account. It is found that the present method is a reliable and efficient numerical procedure and scattering leads to a reduction in the total heat flux. The influence of the forward-backward scattering parameter on the total heat flux decreases with the increase of the surface reflectivity.     

Temperature solutions due to time-dependent moving-line-heat sources

A closed-form model for the computation of temperature distribution in an infinitely extended isotropic body with a time-dependent moving-line-heat sources is discussed. The temperature solutions are presented for the sources of the forms: (i) $\dot Q_1 (t) = \dot Q_0 \exp ( - \lambda t)$ , (ii) $\dot Q_2 (t) = \dot Q_0 (t/t^ \star )\exp ( - \lambda t)$ , and $\dot Q_3 (t) = \dot Q_0 [1 + a\cos (\omega t)]$ , whereλ andω are real parameters andt? characterizes the limiting time. The reduced (or dimensionless) temperature solutions are presented in terms of the generalized representation of an incomplete gamma function Γ (α,x;b) and its decompositionsC _Γ andS _Γ. It is also demonstrated that the present analysis covers the classical temperature solution of a constant strength source under quasi-steady-state situations.     

The critical tube diameter in a two reaction steps detonation: the H2/NO2 mixture case

We present experimental results on the detonability of the H₂/NO₂ mixture whose detonation exhibits a single cellular structure (λ₁) for the lean mixtures and a double cellular structure (fine cells of size λ₁ inside larger cells of size λ₂) for stoichiometric and rich mixtures. Whatever the equivalence ratio ${\phi}$ , the chemical energy is released in two successive exothermic steps of heat of reaction Q ₁ and Q ₂ (Q ₁ + Q ₂ = Q, the total heat release) and characterised (for ${\phi > 1}$ ) by two chemical lengths. The detonability is evaluated on the basis of critical conditions of self-sustained detonations transmission from a cylindrical tube of i.d. d to free space. Results show that for the critical tube diameter relationship d/λ₁ = k, with respect to the equivalence ratio ${\phi}$ ranging from 0.5 to 1.3 at ambient temperature, k is higher than the classical value 13 and its variation is rather complex. Indeed, d/λ₁ increases with ${\phi}$ from 17–18 for ${\phi = 0.5}$ to 45–50 for ${\phi = 1}$ and to 90–100 for ${\phi = 1.3}$ . The highest detonability obtained for ${\phi = 0.6}$ is explained on the basis of the highest relative contribution of the first exothermic step to the total energy Q. We conclude that, as d/λ₁ drops with Q ₂ decreasing, it should tend to 13 with the vanishing second exothermic reaction.     

Advective Transport in Heterogeneous Formations: The Impact of Spatial Anisotropy on the Breakthrough Curve

Water flow and solute transport take place in formations of spatially variable conductivity K. The logconductivity Y?= ln K is modeled as a random stationary space function, of normal univariate pdf (of mean In K _G and variance ${\sigma_{Y}^{2}}$ ) and of axisymmetric autocorrelation of integral scales I _h,I _v (anisotropy ratio f?=?I _v/I _h?<?1). The head gradient and the velocity are uniform in the mean, parallel to bedding, and of constant and given as J and U, respectively. Transport is ruled by advection, which typically overwhelms pore scale dispersion in the breakthrough curve (BTC) determination. In the present study we analyze the impact of anisotropy f on the BTC of a passive solute, which is related to the mass flux??? (t, x) at a control plane at x. While a considerable body of literature dealt with weakly heterogeneous formations ( ${\sigma _{Y}^{2} <1 }$ ), the present study addresses the case of ${\sigma _{Y}^{2} >1 }$ , which is of interest for many aquifers and is more difficult to solve either numerically or by approximations. We approach the three dimensional problem by modeling the structure as an ensemble of densely packed oblate spheroids of semi-major and semi-minor axis R and f R, respectively, and independent lognormal K, submerged in a matrix of uniform conductivity K _ef, the effective conductivity of the ensemble. The detailed numerical simulations of transport show that the BTC is insensitive to the value of the anisotropy ratio f, i.e.,??? (t, x) I _h/U depends only on ${\sigma _{Y}^{2}}$ (except for small differences in the tail). This important result implies that transport, as quantified by BTCs or spatial longitudinal mass distributions, can be modeled accurately by the much simpler solutions developed in the past for isotropic media, like e.g., the semi-analytical self-consistent approximation.     

An experimental study on turbulence modification in the near-wall boundary layer of a dilute gas-particle channel flow

Turbulence modifications of a dilute gas-particle flow are experimentally investigated in the lower boundary layer of a horizontal channel by means of a simultaneous two-phase PIV measurement technique. The measurements are conducted in the near-wall region with y ⁺?<?250 at Re _τ (based on the wall friction velocity u _τ and half channel height h)?=?430. High spatial resolution and small interrogation window are used to minimize the PIV measurement uncertainty due to the velocity gradient near the wall. Polythene beads with the diameter of 60?μm (d _p ⁺ ?=?1.71, normalized by the fluid kinematic viscosity ν and u _τ) are used as dispersed phase, and three low mass loading ratios (Φ _m ) ranging from 10^?4 to 10^?3 are tested. It is found that the addition of the particles noticeably modifies the mean velocity and turbulent intensities of the gas-phase, as well as the turbulence coherent structures, even at Φ _m ?=?0.025?%. Particle inertia changes the viscous sublayer of the gas turbulence with a smaller thickness and a larger streamwise velocity gradient, which increases the peak value of the streamwise fluctuation velocity ( $ u_{\text{rms}}^{ + } $ ) of the gas-phase with its location shifting to the wall. Particle sedimentation increases the roughness of the bottom wall, which significantly increases the wall-normal fluctuation velocity ( $ v_{\text{rms}}^{ + } $ ) and Reynolds shear stress ( $ - \langle u^{ \prime } v^{\prime } \rangle^{ + } $ ) of the gas-phase in the inner region of the boundary layer (y ⁺?<?10). Under effect of particle–wall collision, the Q2 events (ejections) of the gas-phase are slightly increased by particles, while the Q4 events (sweeps) are obviously decreased. The spatial scale of the coherent structures near the wall shrinks remarkably with the presence of the particles, which may be attributed to the intensified crossing-trajectory effects due to particle saltation near the bottom wall. Meanwhile, the $ v_{\text{rms}}^{ + } $ and $ - \langle u^{ \prime } v^{\prime } \rangle^{ + } $ of the gas-phase are significantly reduced in the outer region of the boundary layer (y ⁺?>?20).     

Laminar free convection heat transfer from a horizontal ring

The analytical solution of laminar free convective heat transfer in an unlimited space from an isothermal horizontal ring with an adiabatic plug is presented. The results of theoretical considerations are presented as relation of the Nusselt and Rayleigh numbers: $$Nu_D = 1.151 \cdot (Ra_D )^{1/5} \cdot \Phi (\phi _0 )$$ \] where Φ(φ₀) is a function of shape coefficient of the ring (φ₀=d/D). The solution presented has been verified experimentally with rings of constant external diameter (D=0.06 [m]) and various internal diameters (d=0, 0.01, 0.02, 0.04 and 0.05 [m]). The fluid tested was glycerin. The theoretical predictions agree well with the experimental results.     

Dean-coupled inertial migration and transient focusing of particles in a curved microscale pipe flow

The Dean-coupled inertial migration of neutrally buoyant spherical particles that are suspended in a curved microscale pipe flow was experimentally investigated in the range of $ 6.4 \le {\text{Re}} \le 129 $ and $ 1.69 \le De \le 34.1 $ . The three-dimensional positions of the particles were measured by using digital holographic microscopy. The diameter of the microtube was 350???m, and the ratios of the tube diameter (D) to the particle diameter (d) were D/d?=?12, 23, 35, and 50. Over a critical value of the Focusing number (F _C), the particles were initially tubular-pinched at the entrance of the curved region. The detailed structures of the Segré?CSilberberg annulus as well as its deformation attributed to secondary flow were analyzed. Diverse agglomeration patterns of particles corresponding to the various flow conditions were observed. The optimal conditions that induced the particles to focus at a certain lateral position were determined.     

Γ-Convergence Analysis of Systems of Edge Dislocations: the Self Energy Regime

This paper deals with the elastic energy induced by systems of straight edge dislocations in the framework of linearized plane elasticity. The dislocations are introduced as point topological defects of the displacement-gradient fields. Following the core radius approach, we introduce a parameter ${\varepsilon > 0}$ representing the lattice spacing of the crystal, we remove a disc of radius ${\varepsilon}$ around each dislocation and compute the elastic energy stored outside the union of such discs, namely outside the core region. Then, we analyze the asymptotic behaviour of the elastic energy as ${\varepsilon \rightarrow 0}$ , in terms of Γ-convergence. We focus on the self energy regime of order ${\log\frac{1}{\varepsilon}}$ ; we show that configurations with logarithmic diverging energy converge, up to a subsequence, to a finite number of multiple dislocations and we compute the corresponding Γ-limit.