Shock wave kinetics of Fe + NO based on Fe, O, and N atom measurements

The reaction of Fe atoms with NO was studied behind incident shock waves in the temperature range of 780–1,020 K at pressures between 0.3 and 1.2 bar. Atomic-resonance-absorption spectroscopy (ARAS) was applied for the time-resolved measurement of Fe , N, and O atoms in gas mixtures containing Fe(CO)₅ and NO, highly diluted in argon. The experiments showed a Fe-atom consumption without an associated O- or N-atom formation which can be explained by a recombination of Fe and NO:

Sensitivity Study of Simulation Parameters Controlling CO<Subscript>2</Subscript> Trapping Mechanisms in Saline Formations

The primary purpose of this study is to understand quantitative characteristics of mobile, residual, and dissolved CO₂ trapping mechanisms within ranges of systematic variations in different geologic and hydrologic parameters. For this purpose, we conducted an extensive suite of numerical simulations to evaluate the sensitivities included in these parameters. We generated two-dimensional numerical models representing subsurface porous media with various permutations of vertical and horizontal permeability (k _v and k _h), porosity (f{\phi}), maximum residual CO₂ saturation (S_gr^max{S_{\rm gr}^{\max}}), and brine density (ρ _br). Simulation results indicate that residual CO₂ trapping increases proportionally to k_v, k_h, S_gr^max{k_{\rm v}, k_{\rm h}, S_{\rm gr}^{\max}} and ρ _br but is inversely proportional to f.{\phi.} In addition, the amount of dissolution-trapped CO₂ increases with k _v and k _h, but does not vary with f{\phi } , and decreases with S_gr^max{S_{\rm gr}^{\max}} and ρ _br. Additionally, the distance of buoyancy-driven CO₂ migration increases proportionally to k _v and ρ _br only and is inversely proportional to k_h, f{k_{\rm h}, \phi } , and S_gr^max{S_{\rm gr}^{\max}} . These complex behaviors occur because the chosen sensitivity parameters perturb the distances of vertical and horizontal CO₂ plume migration, pore volume size, and fraction of trapped CO₂ in both pores and formation fluids. Finally, in an effort to characterize complex relationships among residual CO₂ trapping and buoyancy-driven CO₂ migration, we quantified three characteristic zones. Zone I, expressing the variations of S_gr^max{S_{\rm gr}^{\max}} and k _h, represents the optimized conditions for geologic CO₂ sequestration. Zone II, showing the variation of f{\phi} , would be preferred for secure CO₂ sequestration since CO₂ has less potential to escape from the target formation. In zone III, both residual CO₂ trapping and buoyancy-driven migration distance increase with k _v and ρ _br.     

Boundary Regularity in Variational Problems

We prove that, if ${u : \Omega \subset \mathbb{R}^n \to \mathbb{R}^N}We prove that, if u : W ì \mathbbRⁿ ? \mathbbR^N{u : \Omega \subset \mathbb{R}^n \to \mathbb{R}^N} is a solution to the Dirichlet variational problem

minwòW F(x, w, Dw) dx     subject  to     w o u0  on  ?W,\mathop {\rm min}\limits_{w}\int_{\Omega} F(x, w, Dw)\,{\rm d}x \quad {\rm subject \, to} \quad w \equiv u_0\; {\rm on}\;\partial \Omega,      4. Der Einfluß der Prandtlzahl auf den Wärmeübergang und Druckverlust künstlich aufgerauhter Strömungskanäle      E. Burck 《Heat and Mass Transfer》1969,2(2):87-98 The influence of the Prandtl number on heat transfer and pressure drop characteristics of artificially roughened test sections has been investigated experimentally in the Prandtl number range from 3 to 180. For integral roughenesses and fully roughened test sections the efficiency η=ε Nu /ε ζ can be described by the Prandtl number and the roughness parameter \(k_{\text{S}}^ + = Re{\text{(}}k_{\text{S}} /d_{\text{h}} )\sqrt \zeta /8\) . The relation between the efficiency η, the Prandtl numberPr and the roughness parameterk s + can be expressed by the following empirical relation: $$\eta = \log \frac{{Pr^{{\text{0,33}}} }}{{k_{\text{S}}^{ + {\text{ 0,243}}} }} - 0,32 \cdot 10^{ - 3} k_{\text{S}}^ + {\text{ log }}Pr + {\text{1,25}}{\text{.}}$$ With this relation for the heat transfer and friction characteristics of smooth and rough channels it is possible to calculate the increase of heat transfer for rough channels by means of pressure drop measurements which are necessary to determine the friction factor ζ and the equivalent sand roughness depth; provided that heat transfer and friction characteristics of the respective smooth channel are known.      5. A shock tube study on the thermal decomposition of CS2 based on S(3P) and S(1D) concentration measurements      D. Woiki  P. Roth 《Shock Waves》1994,4(2):95-99 The thermal decomposition of CS2 highly diluted in Ar was studied behind reflected shock waves by monitoring time-dependent absorption profiles of S(3P) and S(1D) using atomic resonance absorption spectroscopy (ARAS). The rate coefficient of the reaction:       6. Regularity of Relaxed Minimizers of Quasiconvex Variational Integrals with (<Emphasis Type="Italic">p</Emphasis>, <Emphasis Type="Italic">q</Emphasis>)-growth      Thomas Schmidt 《Archive for Rational Mechanics and Analysis》2009,193(2):311-337       7. Multiple Blow-Up Phenomena for the Sinh-Poisson Equation      Massimo Grossi  Angela Pistoia 《Archive for Rational Mechanics and Analysis》2013,209(1):287-320 We consider the sinh-Poisson equation $$(P) _ \lambda - \Delta{u} = \lambda \, {\rm sinh} \, u \quad {\rm in} \, \Omega, \quad u = 0 \quad {\rm on} \, \partial\Omega$$ , where Ω is a smooth bounded domain in ${\mathbb{R}^2}$ and λ is a small positive parameter. If ${0 \in \Omega}$ and Ω is symmetric with respect to the origin, for any integer k if λ is small enough, we construct a family of solutions to (P) λ , which blows up at the origin, whose positive mass is 4πk(k?1) and negative mass is 4πk(k + 1). This gives a complete answer to an open problem formulated by Jost et al. (Calc Var PDE 31(2):263–276, 2008).      8. Forced convection heat transfer and friction characteristics in multi-exit -rectangular package with inclined tube bundles      M. -Y. Wen 《Heat and Mass Transfer》2005,41(10):921-930 An experiment was carried out to investigate the characteristics of the heat transfer and pressure drop for forced convection airflow over tube bundles that are inclined relative to the on-coming flow in a rectangular package with one outlet and two inlets. The experiments included a wide range of angles of attack and were extended over a Reynolds number range from about 250 to 12,500. Correlations for the Nusselt number and pressure drop factor are reported and discussed. As a result, it was found that at a fixed Re, for the tube bundles with attack angle of 45 ° has the best heat transfer coefficient, followed by 60, 75 and 90 °, respectively. This investigation also introduces the factors which can be used for finding the heat transfer and the pressure drop factor on the tube bundles positioned at different angles to the flow direction. Moreover, no perceptible dependence of Cand C on Re was detected. In addition, flow visualizations were explored to broaden our fundamental understanding of the heat transfer for the present study.      9. Fluid flow and heat transfer in microchannels with rectangular cross section      Jung-Yeul Jung  Ho-Young Kwak 《Heat and Mass Transfer》2008,44(9):1041-1049 Forced convective heat transfer coefficients and friction factors for flow of water in microchannels with a rectangular cross section were measured. An integrated microsystem consisting of five microchannels on one side and a localized heater and seven polysilicon temperature sensors along the selected channels on the other side was fabricated using a double-polished-prime silicon wafer. For the microchannels tested, the friction factor constant obtained are values between 53.7 and 60.4, which are close to the theoretical value from a correlation for macroscopic dimension, 56.9 for D h  = 100 μm. The heat transfer coefficients obtained by measuring the wall temperature along the micro channels were linearly dependent on the wall temperature, in turn, the heat transfer mechanism is strongly dependent on the fluid properties such as viscosity. The measured Nusselt number in the laminar flow regime tested could be correlated by which is quite different from the constant value obtained in macrochannels.      10. An experimental investigation of pore-scale distributions of nonaqueous phase liquids at residual saturation      Alex S. Mayer  Cass T. Miller 《Transport in Porous Media》1993,10(1):57-80 A set of experiments was conducted to analyze the size distribution of nonaqueous phase liquid (NAPL) blobs as a function of aqueous phase velocities, NAPL densities, and interfacial tension between the aqueous phase and the NAPL. Blob size distributions were analyzed in a model cell after a state of residual saturation was achieved, using image analysis methods. Blob lengths parallel to the direction of flow and parallel to the direction of gravity ranged over about two orders of magnitude, while blob areas varied over about three orders of magnitude. Form factors, which are a measure of blob shapes, varied over a single order of magnitude, with a median form factor of about 0.6. Blobs with smaller areas occurred much more frequently than larger blobs. However, the few larger blobs contributed to the majority of the total NAPL saturation in the model cell. The geometric standard deviations of blob length parallel to the direction of gravity and square root of area exhibited inverse relationships to Bond number (Bo), for 1.3×10–2–2. Blob length parallel to the direction of flow and area exhibited a weak inverse correlation with capillary number (Ca), for 3×10–8–5.      11. On the Reynolds number scaling of vorticity production at no-slip walls during vortex-wall collisions      G. H. Keetels  W. Kramer  H. J. H. Clercx  G. J. F. van Heijst 《Theoretical and Computational Fluid Dynamics》2011,25(5):293-300 Recently, numerical studies revealed two different scaling regimes of the peak enstrophy Z and palinstrophy P during the collision of a dipole with a no-slip wall [Clercx and van Heijst, Phys. Rev. E 65, 066305, 2002]: Z μ Re0.8{Z\propto{\rm Re}^{0.8}} and P μ Re2.25{P\propto {\rm Re}^{2.25}} for 5 × 102 ≤ Re ≤ 2 × 104 and Z μ Re0.5{Z\propto{\rm Re}^{0.5}} and P μ Re1.5{P\propto{\rm Re}^{1.5}} for Re ≥ 2 × 104 (with Re based on the velocity and size of the dipole). A critical Reynolds number Re c (here, Rec ? 2×104{{\rm Re}_c\approx 2\times 10^4}) is identified below which the interaction time of the dipole with the boundary layer depends on the kinematic viscosity ν. The oscillating plate as a boundary-layer problem can then be used to mimick the vortex-wall interaction and the following scaling relations are obtained: Z μ Re3/4, P μ Re9/4{Z\propto{\rm Re}^{3/4}, P\propto {\rm Re}^{9/4}} , and dP/dt μ Re11/4{\propto {\rm Re}^{11/4}} in agreement with the numerically obtained scaling laws. For Re ≥ Re c the interaction time of the dipole with the boundary layer becomes independent of the kinematic viscosity and, applying flat-plate boundary-layer theory, this yields: Z μ Re1/2{Z\propto{\rm Re}^{1/2}} and P μ Re3/2{P\propto {\rm Re}^{3/2}}.      12. On the Regularity Conditions of Suitable Weak Solutions of the 3D Navier–Stokes Equations      Dongho Chae 《Journal of Mathematical Fluid Mechanics》2010,12(2):171-180 Let v and ω be the velocity and the vorticity of the a suitable weak solution of the 3D Navier–Stokes equations in a space-time domain containing z0=(x0, t0)z_{0}=(x_{0}, t_{0}), and let Qz0,r = Bx0,r ×(t0 -r2, t0)Q_{z_{0},r}= B_{x_{0},r} \times (t_{0} -r^{2}, t_{0}) be a parabolic cylinder in the domain. We show that if either $\nu \times \frac{\omega}{|\omega|} \in L^{\gamma,\alpha}_{x,t}(Q_{z_{0},r})$\nu \times \frac{\omega}{|\omega|} \in L^{\gamma,\alpha}_{x,t}(Q_{z_{0},r}) with $\frac{3}{\gamma} + \frac{2}{\alpha} \leq 1, {\rm or} \omega \times \frac{\nu} {|\nu|} \in L^{\gamma,\alpha}_{x,t} (Q_{z_{0},r})$\frac{3}{\gamma} + \frac{2}{\alpha} \leq 1, {\rm or} \omega \times \frac{\nu} {|\nu|} \in L^{\gamma,\alpha}_{x,t} (Q_{z_{0},r}) with \frac3g + \frac2a ￡ 2\frac{3}{\gamma} + \frac{2}{\alpha} \leq 2, where Lγ, αx,t denotes the Serrin type of class, then z0 is a regular point for ν. This refines previous local regularity criteria for the suitable weak solutions.      13. Fully Developed Forced Convection in a Parallel Plate Channel with a Centered Porous Layer      Ozgur Cekmer  Moghtada Mobedi  Baris Ozerdem  Ioan Pop 《Transport in Porous Media》2012,93(1):179-201 In this study, fully developed heat and fluid flow in a parallel plate channel partially filled with porous layer is analyzed both analytically and numerically. The porous layer is located at the center of the channel and uniform heat flux is applied at the walls. The heat and fluid flow equations for clear fluid and porous regions are separately solved. Continues shear stress and heat flux conditions at the interface are used to determine the interface velocity and temperature. The velocity and temperature profiles in the channel for different values of Darcy number, thermal conductivity ratio, and porous layer thickness are plotted and discussed. The values of Nusselt number and friction factor of a fully clear fluid channel (Nu cl = 4.12 and fRe cl = 24) are used to define heat transfer increment ratio (eth = Nup/Nucl)({\varepsilon _{\rm th} =Nu_{\rm p}/Nu_{\rm cl})} and pressure drop increment ratio (ep = fRep/fRecl )({\varepsilon_{\rm p} =fRe_{\rm p}/fRe_{\rm cl} )} and observe the effects of an inserted porous layer on the increase of heat transfer and pressure drop. The heat transfer and pressure drop increment ratios are used to define an overall performance (e = eth/ep)({\varepsilon = \varepsilon_{\rm th}/\varepsilon_{\rm p})} to evaluate overall benefits of an inserted porous layer in a parallel plate channel. The obtained results showed that for a partially porous filled channel, the value of e{\varepsilon} is highly influenced from Darcy number, but it is not affected from thermal conductivity ratio (k r) when k r > 2. For a fully porous material filled channel, the value of e{\varepsilon} is considerably affected from thermal conductivity ratio as the porous medium is in contact with the channel walls.      14. Study of pulsating flow in close-coupled catalyst manifolds using phase-locked hot-wire anemometry      T.?PersoonsEmail author  E.?Van den Bulck  S.?Fausto 《Experiments in fluids》2004,36(2):217-232       15. The Connection Between Isometries and Symmetries of Geodesic Equations of the Underlying Spaces      Tooba Feroze  F. M. Mahomed  Asghar Qadir 《Nonlinear dynamics》2006,45(1-2):65-74 A connection between the symmetries of manifolds and differential equations is sought through the geodesic equations of maximally symmetric spaces, which have zero, constant positive or constant negative curvature. It is proved that for a space admitting so(n+1) or so(n,1) as the maximal isometry algebra, the symmetry of the geodesic equations of the space is given by so( or (where d 2 is the two-dimensional dilation algebra), while for those admitting (where represents semidirect sum) the algebra is sl(n+2). A corresponding result holds on replacing so(n) by so(p,q) with p+q = n. It is conjectured that if the isometry algebra of any underlying space of non-zero curvature is h, then the Lie symmetry algebra of the geodesic equations is given by , provided that there is no cross-section of zero curvature at the point under consideration. If there is a flat subspace of dimension m, then the symmetry group becomes ).      16. The Asymptotic Finite-dimensional Character of a Spectrally-hyperviscous Model of 3D Turbulent Flow      Joel Avrin 《Journal of Dynamics and Differential Equations》2008,20(2):479-518 We obtain attractor and inertial-manifold results for a class of 3D turbulent flow models on a periodic spatial domain in which hyperviscous terms are added spectrally to the standard incompressible Navier–Stokes equations (NSE). Let P m be the projection onto the first m eigenspaces of A =−Δ, let μ and α be positive constants with α ≥3/2, and let Q m =I − P m , then we add to the NSE operators μ A φ in a general family such that A φ≥Q m A α in the sense of quadratic forms. The models are motivated by characteristics of spectral eddy-viscosity (SEV) and spectral vanishing viscosity (SVV) models. A distinguished class of our models adds extra hyperviscosity terms only to high wavenumbers past a cutoff λ m0 where m 0 ≤ m, so that for large enough m 0 the inertial-range wavenumbers see only standard NSE viscosity. We first obtain estimates on the Hausdorff and fractal dimensions of the attractor (respectively and ). For a constant K α on the order of unity we show if μ ≥ ν that and if μ ≤ ν that where ν is the standard viscosity coefficient, l 0 = λ1−1/2 represents characteristic macroscopic length, and is the Kolmogorov length scale, i.e. where is Kolmogorov’s mean rate of dissipation of energy in turbulent flow. All bracketed constants and K α are dimensionless and scale-invariant. The estimate grows in m due to the term λ m /λ1 but at a rate lower than m 3/5, and the estimate grows in μ as the relative size of ν to μ. The exponent on is significantly less than the Landau–Lifschitz predicted value of 3. If we impose the condition , the estimates become for μ ≥ ν and for μ ≤ ν. This result holds independently of α, with K α and c α independent of m. In an SVV example μ ≥ ν, and for μ ≤ ν aspects of SEV theory and observation suggest setting for 1/c within α orders of magnitude of unity, giving the estimate where c α is within an order of magnitude of unity. These choices give straight-up or nearly straight-up agreement with the Landau–Lifschitz predictions for the number of degrees of freedom in 3D turbulent flow with m so large that (e.g. in the distinguished-class case for m 0 large enough) we would expect our solutions to be very good if not virtually indistinguishable approximants to standard NSE solutions. We would expect lower choices of λ m (e.g. with a > 1) to still give good NSE approximation with lower powers on l 0/l ε, showing the potential of the model to reduce the number of degrees of freedom needed in practical simulations. For the choice , motivated by the Chapman–Enskog expansion in the case m = 0, the condition becomes , giving agreement with Landau–Lifschitz for smaller values of λ m then as above but still large enough to suggest good NSE approximation. Our final results establish the existence of a inertial manifold for reasonably wide classes of the above models using the Foias/Sell/Temam theory. The first of these results obtains such an of dimension N > m for the general class of operators A φ if α > 5/2. The special class of A φ such that P m A φ = 0 and Q m A φ ≥ Q m A α has a unique spectral-gap property which we can use whenever α ≥ 3/2 to show that we have an inertial manifold of dimension m if m is large enough. As a corollary, for most of the cases of the operators A φ in the distinguished-class case that we expect will be typically used in practice we also obtain an , now of dimension m 0 for m 0 large enough, though under conditions requiring generally larger m 0 than the m in the special class. In both cases, for large enough m (respectively m 0), we have an inertial manifold for a system in which the inertial range essentially behaves according to standard NSE physics, and in particular trajectories on are controlled by essentially NSE dynamics.       17. Lower Semicontinuity for Integral Functionals in the Space of Functions of Bounded Deformation Via Rigidity and Young Measures      Filip Rindler 《Archive for Rational Mechanics and Analysis》2011,202(1):63-113 We establish a general weak* lower semicontinuity result in the space BD(Ω) of functions of bounded deformation for functionals of the form $ {ll} \,\mathcal{F}(u) := &\int_\Omega f (x, \mathcal{E} u) \;{\rm d} x + \int_\Omega f^\infty \left( x, \frac{{\rm d} E^s u}{{\rm d} |{E^s u}|} \right) \;{\rm d} |{E^s u}| \\ &+ \int_{\partial \Omega} f^\infty \left( x, u|_{\partial \Omega} \odot n_\Omega \right) \;{\rm d} \mathcal{H}^{d-1}, \qquad u \in {\rm BD}(\Omega). $ \begin{array}{ll} \,\mathcal{F}(u) := &\int_\Omega f (x, \mathcal{E} u) \;{\rm d} x + \int_\Omega f^\infty \left( x, \frac{{\rm d} E^s u}{{\rm d} |{E^s u}|} \right) \;{\rm d} |{E^s u}| \\ &+ \int_{\partial \Omega} f^\infty \left( x, u|_{\partial \Omega} \odot n_\Omega \right) \;{\rm d} \mathcal{H}^{d-1}, \qquad u \in {\rm BD}(\Omega). \end{array}      18. Verlauf der Fluidtemperaturen im Querstromrohrbündel      Th. Bes  W. Roetzel 《Heat and Mass Transfer》1993,28(8):457-463 A new analytical method is presented for the determination of temperature distribution and effectiveness of heat transfer in different cross-flow arrangements. As an improvement over the known analytical solutions simplified energy balance equations are used. This simplification consists only in their notation but not in representation of the exact energy balance. The demanded accuracy of the thermal analysis is achieved by proper selection of a mean value of fluid temperature outside the tube. For each tubei in the bundle a mean value ? i of the dimensionless fluid temperature outside the tube is introduced according to $$\vartheta _i = \omega \vartheta _{i + {1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}} + (1 - \omega )\vartheta _{i - {1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}} $$ where ? i?1/2 and ? i+1/2 are the local temperatures in front and behind thei-th tube, respectively. With symbol ω a weight coefficient is denoted $$\omega = {1 \mathord{\left/ {\vphantom {1 {(1 - e^{ - NTU_{{ \bot \mathord{\left/ {\vphantom { \bot n}} \right. \kern-\nulldelimiterspace} n}} } )}}} \right. \kern-\nulldelimiterspace} {(1 - e^{ - NTU_{{ \bot \mathord{\left/ {\vphantom { \bot n}} \right. \kern-\nulldelimiterspace} n}} } )}} - {n \mathord{\left/ {\vphantom {n {NTU_ \bot }}} \right. \kern-\nulldelimiterspace} {NTU_ \bot }}$$ The number of tube rows in the bundle isn and the number of transfer units of the outside stream is NTU⊥. Through the introduction of the weight coefficient ω, the mathematical operations related to calculation of the temperature field are radically simplified. This enabled the development of the procedure, valid for three codirected cross-flow arrangements (Fig. 1) and any numbern of the rows.      19. On the Aleksandrov–Bakel’man–Pucci Estimate for Some Elliptic and Parabolic Nonlinear Operators      Roberto Argiolas  Fernando Charro  Ireneo Peral 《Archive for Rational Mechanics and Analysis》2011,202(3):875-917       20. A New Approach to the Fundamental Theorem of Surface Theory      Philippe G. Ciarlet  Liliana Gratie  Cristinel Mardare 《Archive for Rational Mechanics and Analysis》2008,188(3):457-473 The fundamental theorem of surface theory classically asserts that, if a field of positive-definite symmetric matrices (a αβ ) of order two and a field of symmetric matrices (b αβ ) of order two together satisfy the Gauss and Codazzi-Mainardi equations in a simply connected open subset ω of , then there exists an immersion such that these fields are the first and second fundamental forms of the surface , and this surface is unique up to proper isometries in . The main purpose of this paper is to identify new compatibility conditions, expressed again in terms of the functions a αβ and b αβ , that likewise lead to a similar existence and uniqueness theorem. These conditions take the form of the matrix equationwhere A 1 and A 2 are antisymmetric matrix fields of order three that are functions of the fields (a αβ ) and (b αβ ), the field (a αβ ) appearing in particular through the square root U of the matrix field The main novelty in the proof of existence then lies in an explicit use of the rotation field R that appears in the polar factorization of the restriction to the unknown surface of the gradient of the canonical three-dimensional extension of the unknown immersion . In this sense, the present approach is more “geometrical” than the classical one. As in the recent extension of the fundamental theorem of surface theory set out by S. Mardare [20–22], the unknown immersion is found in the present approach to exist in function spaces “with little regularity”, such as , p > 2. This work also constitutes a first step towards the mathematical justification of models for nonlinearly elastic shells where rotation fields are introduced as bona fide unknowns.      设为首页 | 免责声明 | 关于勤云 | 加入收藏 Copyright©北京勤云科技发展有限公司  京ICP备09084417号

Der Einfluß der Prandtlzahl auf den Wärmeübergang und Druckverlust künstlich aufgerauhter Strömungskanäle

The influence of the Prandtl number on heat transfer and pressure drop characteristics of artificially roughened test sections has been investigated experimentally in the Prandtl number range from 3 to 180. For integral roughenesses and fully roughened test sections the efficiency η=ε _Nu /ε _ζ can be described by the Prandtl number and the roughness parameter \(k_{\text{S}}^ + = Re{\text{(}}k_{\text{S}} /d_{\text{h}} )\sqrt \zeta /8\) . The relation between the efficiency η, the Prandtl numberPr and the roughness parameterk _s ⁺ can be expressed by the following empirical relation: $$\eta = \log \frac{{Pr^{{\text{0,33}}} }}{{k_{\text{S}}^{ + {\text{ 0,243}}} }} - 0,32 \cdot 10^{ - 3} k_{\text{S}}^ + {\text{ log }}Pr + {\text{1,25}}{\text{.}}$$ With this relation for the heat transfer and friction characteristics of smooth and rough channels it is possible to calculate the increase of heat transfer for rough channels by means of pressure drop measurements which are necessary to determine the friction factor ζ and the equivalent sand roughness depth; provided that heat transfer and friction characteristics of the respective smooth channel are known.     

A shock tube study on the thermal decomposition of CS2 based on S(3P) and S(1D) concentration measurements

The thermal decomposition of CS₂ highly diluted in Ar was studied behind reflected shock waves by monitoring time-dependent absorption profiles of S(³P) and S(¹D) using atomic resonance absorption spectroscopy (ARAS). The rate coefficient of the reaction:

Multiple Blow-Up Phenomena for the Sinh-Poisson Equation

We consider the sinh-Poisson equation $$(P) _ \lambda - \Delta{u} = \lambda \, {\rm sinh} \, u \quad {\rm in} \, \Omega, \quad u = 0 \quad {\rm on} \, \partial\Omega$$ , where Ω is a smooth bounded domain in ${\mathbb{R}^2}$ and λ is a small positive parameter. If ${0 \in \Omega}$ and Ω is symmetric with respect to the origin, for any integer k if λ is small enough, we construct a family of solutions to (P) _λ , which blows up at the origin, whose positive mass is 4πk(k?1) and negative mass is 4πk(k + 1). This gives a complete answer to an open problem formulated by Jost et al. (Calc Var PDE 31(2):263–276, 2008).     

Forced convection heat transfer and friction characteristics in multi-exit -rectangular package with inclined tube bundles

An experiment was carried out to investigate the characteristics of the heat transfer and pressure drop for forced convection airflow over tube bundles that are inclined relative to the on-coming flow in a rectangular package with one outlet and two inlets. The experiments included a wide range of angles of attack and were extended over a Reynolds number range from about 250 to 12,500. Correlations for the Nusselt number and pressure drop factor are reported and discussed. As a result, it was found that at a fixed Re, for the tube bundles with attack angle of 45 ° has the best heat transfer coefficient, followed by 60, 75 and 90 °, respectively. This investigation also introduces the factors which can be used for finding the heat transfer and the pressure drop factor on the tube bundles positioned at different angles to the flow direction. Moreover, no perceptible dependence of Cand C on Re was detected. In addition, flow visualizations were explored to broaden our fundamental understanding of the heat transfer for the present study.     

Fluid flow and heat transfer in microchannels with rectangular cross section

Forced convective heat transfer coefficients and friction factors for flow of water in microchannels with a rectangular cross section were measured. An integrated microsystem consisting of five microchannels on one side and a localized heater and seven polysilicon temperature sensors along the selected channels on the other side was fabricated using a double-polished-prime silicon wafer. For the microchannels tested, the friction factor constant obtained are values between 53.7 and 60.4, which are close to the theoretical value from a correlation for macroscopic dimension, 56.9 for D _h  = 100 μm. The heat transfer coefficients obtained by measuring the wall temperature along the micro channels were linearly dependent on the wall temperature, in turn, the heat transfer mechanism is strongly dependent on the fluid properties such as viscosity. The measured Nusselt number in the laminar flow regime tested could be correlated by which is quite different from the constant value obtained in macrochannels.     

An experimental investigation of pore-scale distributions of nonaqueous phase liquids at residual saturation

A set of experiments was conducted to analyze the size distribution of nonaqueous phase liquid (NAPL) blobs as a function of aqueous phase velocities, NAPL densities, and interfacial tension between the aqueous phase and the NAPL. Blob size distributions were analyzed in a model cell after a state of residual saturation was achieved, using image analysis methods. Blob lengths parallel to the direction of flow and parallel to the direction of gravity ranged over about two orders of magnitude, while blob areas varied over about three orders of magnitude. Form factors, which are a measure of blob shapes, varied over a single order of magnitude, with a median form factor of about 0.6. Blobs with smaller areas occurred much more frequently than larger blobs. However, the few larger blobs contributed to the majority of the total NAPL saturation in the model cell. The geometric standard deviations of blob length parallel to the direction of gravity and square root of area exhibited inverse relationships to Bond number (Bo), for 1.3×10^–2–2. Blob length parallel to the direction of flow and area exhibited a weak inverse correlation with capillary number (Ca), for 3×10^–8–5.     

On the Reynolds number scaling of vorticity production at no-slip walls during vortex-wall collisions

Recently, numerical studies revealed two different scaling regimes of the peak enstrophy Z and palinstrophy P during the collision of a dipole with a no-slip wall [Clercx and van Heijst, Phys. Rev. E 65, 066305, 2002]: Z μ Re^0.8{Z\propto{\rm Re}^{0.8}} and P μ Re^2.25{P\propto {\rm Re}^{2.25}} for 5 × 10² ≤ Re ≤ 2 × 10⁴ and Z μ Re^0.5{Z\propto{\rm Re}^{0.5}} and P μ Re^1.5{P\propto{\rm Re}^{1.5}} for Re ≥ 2 × 10⁴ (with Re based on the velocity and size of the dipole). A critical Reynolds number Re _c (here, Re_c ? 2×10⁴{{\rm Re}_c\approx 2\times 10^4}) is identified below which the interaction time of the dipole with the boundary layer depends on the kinematic viscosity ν. The oscillating plate as a boundary-layer problem can then be used to mimick the vortex-wall interaction and the following scaling relations are obtained: Z μ Re^3/4, P μ Re^9/4{Z\propto{\rm Re}^{3/4}, P\propto {\rm Re}^{9/4}} , and dP/dt μ Re^11/4{\propto {\rm Re}^{11/4}} in agreement with the numerically obtained scaling laws. For Re ≥ Re _c the interaction time of the dipole with the boundary layer becomes independent of the kinematic viscosity and, applying flat-plate boundary-layer theory, this yields: Z μ Re^1/2{Z\propto{\rm Re}^{1/2}} and P μ Re^3/2{P\propto {\rm Re}^{3/2}}.     

On the Regularity Conditions of Suitable Weak Solutions of the 3D Navier–Stokes Equations

Let v and ω be the velocity and the vorticity of the a suitable weak solution of the 3D Navier–Stokes equations in a space-time domain containing z₀=(x₀, t₀)z_{0}=(x_{0}, t_{0}), and let Q_z0,r = B_x0,r ×(t₀ -r², t₀)Q_{z_{0},r}= B_{x_{0},r} \times (t_{0} -r^{2}, t_{0}) be a parabolic cylinder in the domain. We show that if either $\nu \times \frac{\omega}{|\omega|} \in L^{\gamma,\alpha}_{x,t}(Q_{z_{0},r})$\nu \times \frac{\omega}{|\omega|} \in L^{\gamma,\alpha}_{x,t}(Q_{z_{0},r}) with $\frac{3}{\gamma} + \frac{2}{\alpha} \leq 1, {\rm or} \omega \times \frac{\nu} {|\nu|} \in L^{\gamma,\alpha}_{x,t} (Q_{z_{0},r})$\frac{3}{\gamma} + \frac{2}{\alpha} \leq 1, {\rm or} \omega \times \frac{\nu} {|\nu|} \in L^{\gamma,\alpha}_{x,t} (Q_{z_{0},r}) with \frac3g + \frac2a ￡ 2\frac{3}{\gamma} + \frac{2}{\alpha} \leq 2, where L^{γ, α}_x,t denotes the Serrin type of class, then z₀ is a regular point for ν. This refines previous local regularity criteria for the suitable weak solutions.     

Fully Developed Forced Convection in a Parallel Plate Channel with a Centered Porous Layer

In this study, fully developed heat and fluid flow in a parallel plate channel partially filled with porous layer is analyzed both analytically and numerically. The porous layer is located at the center of the channel and uniform heat flux is applied at the walls. The heat and fluid flow equations for clear fluid and porous regions are separately solved. Continues shear stress and heat flux conditions at the interface are used to determine the interface velocity and temperature. The velocity and temperature profiles in the channel for different values of Darcy number, thermal conductivity ratio, and porous layer thickness are plotted and discussed. The values of Nusselt number and friction factor of a fully clear fluid channel (Nu _cl = 4.12 and fRe _cl = 24) are used to define heat transfer increment ratio (e_th = Nu_p/Nu_cl)({\varepsilon _{\rm th} =Nu_{\rm p}/Nu_{\rm cl})} and pressure drop increment ratio (e_p = fRe_p/fRe_cl )({\varepsilon_{\rm p} =fRe_{\rm p}/fRe_{\rm cl} )} and observe the effects of an inserted porous layer on the increase of heat transfer and pressure drop. The heat transfer and pressure drop increment ratios are used to define an overall performance (e = e_th/e_p)({\varepsilon = \varepsilon_{\rm th}/\varepsilon_{\rm p})} to evaluate overall benefits of an inserted porous layer in a parallel plate channel. The obtained results showed that for a partially porous filled channel, the value of e{\varepsilon} is highly influenced from Darcy number, but it is not affected from thermal conductivity ratio (k _r) when k _r > 2. For a fully porous material filled channel, the value of e{\varepsilon} is considerably affected from thermal conductivity ratio as the porous medium is in contact with the channel walls.     

The Connection Between Isometries and Symmetries of Geodesic Equations of the Underlying Spaces

A connection between the symmetries of manifolds and differential equations is sought through the geodesic equations of maximally symmetric spaces, which have zero, constant positive or constant negative curvature. It is proved that for a space admitting so(n+1) or so(n,1) as the maximal isometry algebra, the symmetry of the geodesic equations of the space is given by so( or (where d ₂ is the two-dimensional dilation algebra), while for those admitting (where represents semidirect sum) the algebra is sl(n+2). A corresponding result holds on replacing so(n) by so(p,q) with p+q = n. It is conjectured that if the isometry algebra of any underlying space of non-zero curvature is h, then the Lie symmetry algebra of the geodesic equations is given by , provided that there is no cross-section of zero curvature at the point under consideration. If there is a flat subspace of dimension m, then the symmetry group becomes ).     

The Asymptotic Finite-dimensional Character of a Spectrally-hyperviscous Model of 3D Turbulent Flow

We obtain attractor and inertial-manifold results for a class of 3D turbulent flow models on a periodic spatial domain in which hyperviscous terms are added spectrally to the standard incompressible Navier–Stokes equations (NSE). Let P _m be the projection onto the first m eigenspaces of A =−Δ, let μ and α be positive constants with α ≥3/2, and let Q _m =I − P _m , then we add to the NSE operators μ A _φ in a general family such that A _φ≥Q _m A ^α in the sense of quadratic forms. The models are motivated by characteristics of spectral eddy-viscosity (SEV) and spectral vanishing viscosity (SVV) models. A distinguished class of our models adds extra hyperviscosity terms only to high wavenumbers past a cutoff λ _m0 where m ₀ ≤ m, so that for large enough m ₀ the inertial-range wavenumbers see only standard NSE viscosity. We first obtain estimates on the Hausdorff and fractal dimensions of the attractor (respectively and ). For a constant K _α on the order of unity we show if μ ≥ ν that and if μ ≤ ν that where ν is the standard viscosity coefficient, l ₀ = λ₁^−1/2 represents characteristic macroscopic length, and is the Kolmogorov length scale, i.e. where is Kolmogorov’s mean rate of dissipation of energy in turbulent flow. All bracketed constants and K _α are dimensionless and scale-invariant. The estimate grows in m due to the term λ _m /λ₁ but at a rate lower than m ^3/5, and the estimate grows in μ as the relative size of ν to μ. The exponent on is significantly less than the Landau–Lifschitz predicted value of 3. If we impose the condition , the estimates become for μ ≥ ν and for μ ≤ ν. This result holds independently of α, with K _α and c _α independent of m. In an SVV example μ ≥ ν, and for μ ≤ ν aspects of SEV theory and observation suggest setting for 1/c within α orders of magnitude of unity, giving the estimate where c _α is within an order of magnitude of unity. These choices give straight-up or nearly straight-up agreement with the Landau–Lifschitz predictions for the number of degrees of freedom in 3D turbulent flow with m so large that (e.g. in the distinguished-class case for m ₀ large enough) we would expect our solutions to be very good if not virtually indistinguishable approximants to standard NSE solutions. We would expect lower choices of λ _m (e.g. with a > 1) to still give good NSE approximation with lower powers on l ₀/l _ε, showing the potential of the model to reduce the number of degrees of freedom needed in practical simulations. For the choice , motivated by the Chapman–Enskog expansion in the case m = 0, the condition becomes , giving agreement with Landau–Lifschitz for smaller values of λ _m then as above but still large enough to suggest good NSE approximation. Our final results establish the existence of a inertial manifold for reasonably wide classes of the above models using the Foias/Sell/Temam theory. The first of these results obtains such an of dimension N > m for the general class of operators A _φ if α > 5/2. The special class of A _φ such that P _m A _φ = 0 and Q _m A _φ ≥ Q _m A ^α has a unique spectral-gap property which we can use whenever α ≥ 3/2 to show that we have an inertial manifold of dimension m if m is large enough. As a corollary, for most of the cases of the operators A _φ in the distinguished-class case that we expect will be typically used in practice we also obtain an , now of dimension m ₀ for m ₀ large enough, though under conditions requiring generally larger m ₀ than the m in the special class. In both cases, for large enough m (respectively m ₀), we have an inertial manifold for a system in which the inertial range essentially behaves according to standard NSE physics, and in particular trajectories on are controlled by essentially NSE dynamics.      

Lower Semicontinuity for Integral Functionals in the Space of Functions of Bounded Deformation Via Rigidity and Young Measures

We establish a general weak* lower semicontinuity result in the space BD(Ω) of functions of bounded deformation for functionals of the form

Verlauf der Fluidtemperaturen im Querstromrohrbündel

A new analytical method is presented for the determination of temperature distribution and effectiveness of heat transfer in different cross-flow arrangements. As an improvement over the known analytical solutions simplified energy balance equations are used. This simplification consists only in their notation but not in representation of the exact energy balance. The demanded accuracy of the thermal analysis is achieved by proper selection of a mean value of fluid temperature outside the tube. For each tubei in the bundle a mean value ? _i of the dimensionless fluid temperature outside the tube is introduced according to $$\vartheta _i = \omega \vartheta _{i + {1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}} + (1 - \omega )\vartheta _{i - {1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}} $$ where ? _i?1/2 and ? _i+1/2 are the local temperatures in front and behind thei-th tube, respectively. With symbol ω a weight coefficient is denoted $$\omega = {1 \mathord{\left/ {\vphantom {1 {(1 - e^{ - NTU_{{ \bot \mathord{\left/ {\vphantom { \bot n}} \right. \kern-\nulldelimiterspace} n}} } )}}} \right. \kern-\nulldelimiterspace} {(1 - e^{ - NTU_{{ \bot \mathord{\left/ {\vphantom { \bot n}} \right. \kern-\nulldelimiterspace} n}} } )}} - {n \mathord{\left/ {\vphantom {n {NTU_ \bot }}} \right. \kern-\nulldelimiterspace} {NTU_ \bot }}$$ The number of tube rows in the bundle isn and the number of transfer units of the outside stream is NTU_⊥. Through the introduction of the weight coefficient ω, the mathematical operations related to calculation of the temperature field are radically simplified. This enabled the development of the procedure, valid for three codirected cross-flow arrangements (Fig. 1) and any numbern of the rows.     

A New Approach to the Fundamental Theorem of Surface Theory

The fundamental theorem of surface theory classically asserts that, if a field of positive-definite symmetric matrices (a _αβ ) of order two and a field of symmetric matrices (b _αβ ) of order two together satisfy the Gauss and Codazzi-Mainardi equations in a simply connected open subset ω of , then there exists an immersion such that these fields are the first and second fundamental forms of the surface , and this surface is unique up to proper isometries in . The main purpose of this paper is to identify new compatibility conditions, expressed again in terms of the functions a _αβ and b _αβ , that likewise lead to a similar existence and uniqueness theorem. These conditions take the form of the matrix equation