Non-newtonian secondary flows in ducts of rectangular cross-section

Summary A numerical study of the secondary flows which occur in the laminar pressure-driven motion of dilute polymer solutions in ducts of rectangular cross-section is presented. The full nonlinear equations of motion for a Maxwell fluid with shear thinning are solved by an explicit finite-difference technique. Results are presented in an inertial framing which indicate the existence of the previously observed eight-vortex secondary flow in ducts of various aspect ratios. In addition, the effect of spanwise rotations on the strength and the structure of the secondary flow is examined. The results indicate that even the mere presence of the diurnal rotation of the earth can give rise to a double-vortex secondary flow which is of a comparable order of magnitude to the viscoelastic secondary flows in dilute polymer solutions. Specific computations are presented which illustrate the distortional effect that the imposition of a spanwise rotation has on the viscoelastic secondary flow structure in square ducts. The implications that these results can have on laboratory experiments are discussed briefly along with other possible applications to rotating and curved flows.With 15 Figures     

Convective heat transfer in the entrance region of a rectangular duct with two indented sides

This study presents a numerical solution to convective heat transfer in laminar flow in the thermal entrance region of a rectangular duct with two indented sides. The flow is considered to be hydrodynamically fully developed and thermally developing laminar flow of incompressible, Newtonian fluids with constant thermal properties. The ducts are subjected to a constant wall temperature. An algebraic technique is used to discretize the solution domain and a boundary-fitted coordinate system is numerically developed. The governing equations in the boundary-fitted coordinates are solved by the control volume-based finite difference method. Distribution of the bulk temperature and the Nusselt number along the direction of flow is calculated and presented graphically. Also calculated is the thermal entrance length of the rectangular ducts with two indented sides. The parameters, such as the friction factor times the Reynolds number, and the Nusselt number for the fully developed flow and thermally developing flow are obtained.List of symbols a half-width of duct [m] - A cross-sectional area [m²] - b half-height of duct [m] - C _p specific heat [kJ kg^-1 K^-1] - D _h hydraulic diameter [m] - f skin friction factor = - h _z local heat transfer coefficient [Wm^–2 K^–1] - h _T asymptotic heat transfer coefficient [Wm^–2 K^–1] - J Jacobian matrix of transformation, Eq. (20) - k thermal conductivity [Wm^–1 K^–1] - L thermal entrance length [m] - L ^* dimensionless thermal entrance length =L/D _h RePr - L1 maximum number of grids in direction - M1 maximum number of grids in direction - Nu _T asymptotic Nusselt number =h _T D _h /k - p pressure [n m^–2] - P circumferential length [m] - Pr Prandtl number = /_T - Re Reynolds number =w _m D _h / - T temperature [K] - T _b bulk temperature [K] - T _i inlet temperature - T _w circumferential duct wall temperature [K] - w velocity [ms^–1] - w _m mean velocity [ms^–1] - W dimensionless velocity = – - W _m dimensionless mean velocity - x, y transveral coordinates [m] - X,Y dimensionless transversal coordinates =      

Thermal conductivity of amorphous Teflon (AF 1600) at high pressure

The thermal conductivity, λ of amorphous Teflon AF 1600 [poly(1,3-dioxole-4,5-difluoro-2,2-bis(trifluoromethyl)-co-tetrafluoroethylene)] has been measured at pressures up to 2 GPa in the temperature range 93–392 K. At 295 K and atmospheric pressure, we obtained λ=0.116, W·m⁻¹·K⁻¹. The bulk modulus was measured up to 1.0 GPa in the temperature range 150–296 K and the combined data yielded the following values ofg=(∂ln λ ∂lnp) _r :2.8±0.2 at 296 K, 3.0±0.2 at 258 K, 3.0±0.2 at 236 K. 3.4±0.2 at 200 K. and 3.4±0.2 at 150 K.     

The effects of temperature gradient and growth rate on the microstructure of directionally solidified Sn–3.5Ag eutectic solder

The mechanical properties (microhardness, tensile strength) of alloys are controlled by their microstructure, which depends strongly on temperature gradient (G) and growth rate (V). Thus, it is important to understand the relationships among G, V and microstructure (rod eutectic) of Sn–Ag solders. The Sn–3.5 wt% Ag eutectic alloy was directionally solidified upward with a constant growth rate, V (16.5 μm/s) at different temperature gradients, G (1.43–4.28 K/mm) and with a constant temperature gradient, G (3.93 K/mm) at different growth rates, V (8.3–500 μm/s) in a Bridgman–type directional solidification furnace. The rod spacings (λ) have been measured from both longitudinal section (parallel to the growth direction, λ _L ) and transverse section (perpendicular to the growth direction, λ _T ) of the samples. The undercooling values (ΔT) were calculated by using V, λ and system parameters (K ₁ and K ₂). It was found that the values of λ (λ _T , λ _L ) decrease while V and G are increasing. The relationships between rod spacing and solidification parameters (G and V) were obtained by linear regression analysis. The dependences of eutectic spacings λ on undercooling (ΔT) are also analyzed. λ² V, λΔT, ΔTV ^−0.5 and ΔTG ^−0.5 values were determined by using λ, ΔT, V and G values. The results obtained in this work are compared with the Jackson–Hunt eutectic theory and the similar experimental works. The experimental l_\textT ² \textV \lambda_{\text{T}}^{ 2} {\text{V}} value (159.3 μm³/s) is slightly lower than the result 174.6 μm³/s calculated from Jackson–Hunt eutectic theory.     

The Si-O bonding in Na2O-TiO-SiO2 and K2O-TiO2-SiO2 glasses as studied by SiKβ X-ray fluorescence and infra-red absorption spectroscopy

The SiKβ fluorescence X-ray and infra-red absorption spectra of Na₂O-TiO₂-SiO₂ and K₂O-TiO₂-SiO₂ glasses have been measured in order to study the Si-O bonding as a function of the composition of glass. The chemical shifts of the SiKβ peak relative to that of silica glass [Δλ=λ(SiO₂)-λ(specimen)>0] and the wavenumbers of the infra-red absorption peaks around 950cm⁻¹ and 1100 cm⁻¹ have been determined from the spectra. The SiKβ chemical shift in both series of glasses increased with increasing total content of R₂O and TiO₂ and with decreasing ratio of TiO₂/(K₂O+TiO₂), where R₂O represents Na₂O or K₂O. The infra-red peaks around 950 cm⁻¹ and 1100 cm⁻¹ shifted towards lower wavenumbers with increasing total content of R₂O and TiO₂ and with decreasing ratio of TiO₂/(R₂O+TiO₂). These experimental facts were interpreted as showing that the Si-O bond in glass was weakened as the total content of R₂O and TiO₂ increased and that the Si-O bond was weakened by R₂O more than by TiO₂.     

Two-phase flow regimes in round, square and rectangular tubes during condensation of refrigerant R134a

An experimental investigation of two-phase flow mechanisms during condensation of refrigerant R134a in six small diameter round (4.91 mm), square (D_h=4 mm, α=1), and rectangular (4×6 and 6×4 mm: D_h=4.8 mm, α=0.67 and 1.5; 2×4 and 4×2 mm: D_h=2.67 mm, α =0.5 and 2) was conducted. Unique experimental techniques and test sections were developed to enable the documentation of the flow mechanisms during phase change. For each tube under consideration, flow mechanisms were recorded over the entire range of qualities for five different refrigerant mass fluxes between 150 and 750 kg m⁻² s⁻¹. The flow mechanisms were categorized into four different flow regimes: intermittent flow, wavy flow, annular flow, and dispersed flow. In addition, the large amount of data enabled the delineation of several different flow patterns within each flow regime, which provides a clearer understanding of the different modes of two-phase flow. Transition lines between the respective flow patterns and regimes on these maps were established based on the experimental data. It was found that for similar hydraulic diameters, flow regime transitions are not very strongly dependent on tube shape or aspect ratio. These maps and the transition lines can be used to predict the particular flow pattern or regime that will be established for a given mass flux, quality and tube geometry.     

λCalculi with Explicit Substitutions Preserving Strong Normalization

This paper studies preservation of β-strong normalization by two different confluent λ-calculi with explicit substitutions defined in [96]; the particularity of these calculi, called λ _d and λ _dn respectively, is that both have a (weak) composition operator for substitutions. We apply an abstract simulation technique to reduce preservation of β-strong normalization of λ _d and λ _dn to that of another calculus, called λ _f having no composition operator. Then, preservation of β-strong normalization of λ _f is shown using the same technique as in [2]. As a consequence, λ _d and λ _dn become the first λ-calculi with explicit substitutions having (weak) composition and preserving β-strong normalization. As an aside, we also show how to apply our technique to reduce preservation of β-strong normalization of the calculus λ _v in [20] to that of λ _f . Received: August 19, 1997; revised version: November 13, 1998     

Mixed convection flow near an axisymmetric stagnation point on a vertical cylinder

The mixed convection flow near an axisymmetric stagnation point on a vertical cylinder is considered. The equations for the fluid flow and temperature fields reduce to similarity form that involves a Reynolds number R and a mixed convection parameter λ, as well as the Prandtl number σ. Numerical solutions are obtained for representative values of these parameters, which show the existence of a critical value λ _c  = λ _c (R, σ) for the existence of solutions in the opposing (λ < 0) case. The variation of λ _c with R is considered. In the aiding (λ > 0) case solutions are possible for all λ and the asymptotic limit λ → ∞ is obtained. The limits of large and small R are also treated and the nature of the solution in the asymptotic limit of large Prandtl number is briefly discussed.     

Heat transport in dilute mixtures of3He in superfluid4He

We report new measurements of the effective thermal conductivity K_eff and relaxation time τ in dilute mixtures of³He in superfluid⁴He, with molar concentrationsX≤10⁻³. The temperature range extended fromT≈1.4 K toT _λ. Both K_cff and τ are found to agree with theoretical predictions, in contrast to previous experiments where significant differences were observed. A new thermal conductivity cell design was used which almost completely eliminates extraneous volumes and surfaces, and the earlier results are explained in relation to these design changes.     

Kinetics of isothermal reactive diffusion between solid Fe and liquid Al

The kinetics of the reactive diffusion between solid Fe and liquid Al was experimentally observed using Fe/Al diffusion couples. The diffusion couples were prepared by an isothermal bonding technique and then immediately annealed in the temperature range of T = 1053–1093 K for various times up to t = 600 s. Owing to annealing, an intermetallic layer with a rather uniform thickness is produced at the Fe/Al interface in the diffusion couple and grows into the solid Fe specimen. The intermetallic layer consists of Fe₂Al₅ and FeAl₃, and the thickness is much smaller for FeAl₃ than for Fe₂Al₅. Hence, the growth of the intermetallic layer is predominantly governed by Fe₂Al₅. The total thickness, l, of the intermetallic layer increases with increasing annealing time, t, according to the parabolic relationship l ² = Kt. This may mean that the growth of the intermetallic layer is controlled by volume diffusion. If the temperature dependence of the parabolic coefficient K is expressed by the equation K = K ₀exp(−Q _K /RT), K ₀ = 126 m²/s and Q _K  = 248 kJ/mol are obtained from the experimental values of K at T = 1053–1093 K by the least-squares method. A mathematical model was used to evaluate the interdiffusion coefficient, D, of Fe₂Al₅ from K. The evaluation provides D ₀ = 2.55 × 10³ m²/s and Q = 259 kJ/mol for the dependence of D on T described as D = D ₀exp(−Q/RT). Thus, K is one fifth of D at T = 1053–1093 K, and Q _K does not necessarily coincide with Q. D is about two orders of magnitude greater for the values evaluated in the present study than for those extrapolated from the previously reported result at T = 823–913 K. The microstructure observation in a previous study suggests that such large values of D are attributed to the boundary diffusion in the intermetallic layer as well as the crystallographic anisotropy of Fe₂Al₅.     

Emissive characteristics of mesa-stripe lasers (λ=3.0–3.6 μm) made from InGaAsSb/InAsSbP double heterostructures

The directional patterns, current-voltage characteristics, and spectral characteristics of mesastripe lasers with InGaAsSb active layers, emitting at λ=3.0–3.6 μm (77 K) and having threshold currents ≥15 mA (j _th≥200 A/cm²), are investigated. The maximum output power is 1.4 mW (λ∼3.3 μm), the differential quantum efficiency ∼3%(τ=5–30 μs, f=500 Hz) for lasing in a longitudinal mode with beam divergences ΔΘ∥∼15° and ΔΘ ⊥ ∼30°. The relationship of the differential quantum efficiency to the order of the spatial mode of the lasing is demonstrated. A single-mode, current-tunable (−30 cm⁻¹/A) laser is used to measure the transmission of methane in the region of the ν ₃ absorption band. Pis’ma Zh. Tekh. Fiz. 24, 40–45 (June 26, 1998)     

Eliminating nuisance parameters: two characterizations

Given a statistical modelP = {P_θ : θ ∈ x} and a surjective functiong: ϑ→Λ the problem of transformingP into a new modelQ= {_λ : λ ∈ Λ} indexed by Λ is investigated. Two characterizations are given for those modelsQ of the form Q_λ = ∫ P_θ π_λ(dθ), where π_λ is some probability such that π_λ(g=λ)=1. The first is related to a geometric property ofQ, while the second rests on the inferential implications of adoptingQ. Also, in the first π_λ is allowed to be finitely additive, while in the second π_λ is σ-additive. Finally, integrated likelihoods are revisited in light of the second characterization.     

Improvements in the Dynamic Plane Source Method

Three sources of errors in the extended dynamic plane source (EDPS) method caused by the discrepancy between experiment and model are analyzed. The source effect is eliminated by introducing the nuisance parameter R ₀ and the surface effect by a surrounding vacuum. The original model assumes a constant heating power but a constant current is used in the experiment. Suppression of this effect leads to a new solution of the heat equation designated as the constant-current model. The measurements on polymethylmetacrylate (PMMA) in vacuum, evaluated by the constant-current model, provided results of λ = 0.191 W.m⁻¹.K⁻¹ and a = 0.118 ×10⁻⁶ m ².s ^{− 1}, which are in good agreement with published values. The total standard uncertainty was estimated as 1.5 % for both thermophysical properties.     

Cyclic fatigue-crack propagation in a silicon carbide whisker-reinforced alumina composite: role of load ratio

The characteristics of subcritical crack growth by cyclic fatigue have been examined in a silicon carbide whisker-reinforced alumina composite, with specific reference to the role of load ratio (ratio of minimum to maximum applied stress intensity, R=K _min/K _max); results are compared with similar subcritical crack-growth data obtained under constant load conditions (static fatigue). Using compact-tension samples cycled at ambient temperatures, cyclic fatigue-crack growth has been measured over six orders of magnitude from ∼10⁻¹¹–10⁻⁵ m cycle⁻¹ at load ratios ranging from 0.05–0.5. Growth rates (da/dN) display an approximate Paris power-law dependence on the applied stress-intensity range (ΔK), with an exponent varying between 33 and 50. Growth-rate behaviour is found to be strongly dependent upon load ratio; the fatigue threshold, ΔK _TH, for example, is found to be increased by over 80% at R=0.05 compared to R=0.5. These results are rationalized in terms of a far greater dependency of growth rates on K _max(da/dN ∞ K _max ³⁰ ) compared to ΔK(da/dN ∞ ΔK ⁵), in contrast to fatigue behaviour in metallic materials where generally the reverse is true. Micromechanisms of crack advance underlying such behaviour are discussed in terms of timedependent crack bridging involving either matrix grains or unbroken whiskers.     

Electrical properties of (K0.5Na0.5)1? x Ag x NbO3 lead-free piezoelectric ceramics

(K_0.5Na_0.5)_1−x Ag _x NbO₃ lead-free piezoelectric ceramics have been fabricated by an ordinary ceramic technique. The results of XRD reveal that Ag⁺ diffuses into the K_0.5Na_0.5NbO₃ lattices to form a new solid solution with an orthorhombic perovskite structure and the solubility of Ag⁺ into A-sites of K_0.5Na_0.5NbO₃ is about 0.20. The ceramics can be well-sintered at 1,100–1,110 °C. The partial substitution of Ag⁺ for A-site ion (K_0.5Na_0.5)⁺ decreases slightly both paraelectric cubic-ferroelectric tetragonal (T _C) and ferroelectric tetragonal-ferroelectric orthorhombic phase transition temperatures (T _O−T). The ferroelectricity of the ceramics becomes weak at high Ag⁺ concentration. The ceramic with x = 0.10 possesses optimum electrical properties: d ₃₃ = 135 pC/N, k _P = 0.43, k _t = 0.46, ε _r = 470, tanδ = 3.39%, and T _C = 394 °C.     

Thermal equation of state of natural chromium spinel up to 26.8 GPa and 628 K

A pressure–volume–temperature data set has been obtained for natural chromium spinel, using synchrotron X-ray diffraction with a resistance heated diamond-anvil cell (RHDAC). The unit cell parameter of the chromium spinel was measured by energy dispersive X-ray diffraction up to pressures of 26.8 GPa and temperatures of 628 K. No phase change has been observed. The observed P–V–T data were fit to the high-temperature Birch-Murnaghan equation of state, with V ₀ fixed at its experimental value, yields K ₀ = 209 ± 9 GPa, (∂K/∂T)_P = −0.056 ± 0.035 GPa K⁻¹, and α₀ = 7±1 × 10⁻⁵ K⁻¹. The temperature derivative of the bulk modulus (∂K/∂T)_P of chromium spinel is determined here for the first time. The obtained K ₀ is slightly higher than the previous results of synthetic spinel. We suggest that Fe²⁺–Mg²⁺ substitution is responsible for the high bulk modulus of chromium spinel.     

Notch effect and brittle-ductile transition

We present the fresults of the investigation of the effect of notches on the brittle-ductile transition in the process of fracture of structural materials. It is shown that the local elastoplastic state of a material in the vicinity of the notch can be described by using the corresponding stress intensity factorsK _I ^σ , strain intensity factorsK _i ^ε , and strain-energy intensity factorsK _I ^ω . We prove that the factorsK _I ^σ andK _I ^ε are linear functions of the notch radius ρ on the coordinates (wherei=σ or ε) and propose to characterize the brittle-ductile transition in the process of fracture of the material by the slopes of these functions. Bay Zoltan Institute of Logistics and Production Systems, Miskolc-Tapolca, Hungary. Published in Fizyko-Khimichna Mekhanika Materialiv, Vol. 34, No. 5, pp. 27–33, September–October, 1998.     

Specific Heat of 4He Confined in Cylindrical Micro-Channels and Near the Superfluid Transition

We present new measurements of the specific heat C _P of ⁴He confined in cylindrical micro-channels of diameter L=1.89 μm at saturated vapor pressure and near the bulk superfluid-transition temperature T _λ . The results, when combined with the specific-heat exponent α=−0.01264, and with the correlation-length exponent ν=0.6709 obtained from α by hyper-scaling, support the departure from a universal, size independent, scaling function reported before by Lipa et al. [J. Low Temp. Phys. 124(3–4): 443, 2001] for L=8.17 μm and 0.26 μm. In agreement with Lipa et al., we found that a size independent effective scaling function can be recovered within experimental resolution when an empirical effective correlation-length exponent is used. As expected from theory, well away from T _λ the temperature dependence of the suppression of C _P by the confining medium is consistent with a power law with an exponent equal to α+ν=0.658. The universal ratio of the suppression at equal distances below and above the transition was measured to be R _f2=2.3±0.2. This result is consistent with the theoretical value R _f2=2.06 obtained by Bhattacharyya and Bhattacharjee and is close also to the ratio R _ξ ≃2.7 of the correlation-length amplitudes.     

Thermal Conductivity of Standard Sands II. Saturated Conditions

A non-stationary thermal probe technique was used to measure the thermal conductivity of three saturated standard sands (Ottawa sand C-109, Ottawa sand C-190, and Toyoura sand) in a range of soil porosities (n) from 0.32 to 0.42, and temperatures (T) from 25 °C to 70 °C. The sand thermal conductivities at full saturation (λ _sat) increased with decreasing n (increasing compaction, 1 − n). In addition, a declining λ _sat(T) _n=const trend was observed. The peak λ _sat values and highest decreasing rate of λ _sat with T were observed at the heaviest compaction and lowest tested T. This trend gradually diminished with increasing T and expanding volume of water (larger n) due to the markedly lower ability of water to conduct heat than quartz. A series-parallel model, containing three parallel paths of heat flow (through continuous solids, continuous fluid, and solids plus fluid in series), was successfully applied to predicted λ _dry and λ _sat data. The model by de Vries, with new fitted grain shape values, also closely followed measured λ _sat data. The corresponding square root of the relative mean squared errors varied from 2.9 % to 3.4 % for C-109, from 1.9 % to 3.0 % for C-190, and from 2.3 % to 2.4 % for Toyoura sand. The use of a weighted geometric mean model also provided good λ _sat estimates with errors ranging from 3.1 % to 3.5 % for C-109 and C-190 and 8.3 % for Toyoura sand. This paper also discusses a successful attempt to model λ _sat as a product of thermal conductivity of the solid fraction (quartz plus other minerals) and a thermal conductance factor of water.     

Effect of antimony on the growth kinetics of aluminium-silicon eutectic alloys

On treating aluminium-silicon alloy with 0.2 wt% Sb, it was revealed that antimony refines the eutectic structure by reducing the interflake spacing rather than acting as a modifier. The growth mechanism is similar to the unmodified Al=Si flake structure, giving the relationships of the type T=K ₁ V ^0.51 and =K ₂ V ^–0.4, where K ₁ and K ₂ are constants at high solidification rate, the transition from flake to fibre is observed. However, this transition occurs at lower velocity compared to quench modification of the pure alloy. The high magnitude of undercooling measured with the antimony-treated alloy is attributed to constitutional undercooling, which leads to extra refinement of the Al-Si eutectic structure.