Mathematical expression of pressure gradient in the flow of spherical capsules less dense than water

This study yielded a mathematical expression to calculate the pressure gradient (ΔP/L)_m of the flow of a spherical capsule train. An experimental investigation was carried out to determine pressure drops of two-phase mixture flow of spherical ice capsules and water inside the pipelines of cooling systems. Instead of ice capsules, spherical capsules made of polypropylene material whose density (870 kg/m³) is similar to that of ice were used in the experiments. Flow behavior of the spherical capsules, 0.08 m outer diameter, was observed in the measuring section inside plexiglass pipes, 0.1 m inner diameter (ID) and 6 m in length; pressure drops were measured on the 4 m section. The investigation was carried out in the 1.2 × 10⁴ < Re < 1.5 × 10⁵ range and under transport concentration (C_tr) by 5–30%. Dimensionless numbers of the physical event were found out by conducting a dimensional analysis, so that mixture density was expressed in terms of specific gravity and in situ concentration. After arriving at certain conclusions based on the relevant experimental findings and observations, empirical and mathematical models which can be used for calculation of the pressure gradient were developed. Comparison of the mathematical model with the experimental findings revealed that pressure drop values deviated by 2.7% on average for 2.5 × 10⁴ < Re < 1.5 × 10⁵.     

On Pressurized Curvilinearly Orthotropic Circular Disk, Cylinder and Sphere Made of Radially Nonuniform Material

A unified analysis is presented for the elastic response of a pressurized cylindrically anisotropic hollow disk under assumed conditions of plane stress, or a hollow cylinder under plane strain conditions, and a spherically anisotropic hollow sphere, made of material which is nonuniform in the radial direction according to the power law relationship. The solution for a cylinder under generalized plane strain is also presented. Two parameters play a prominent role in the analysis: the material nonuniformity parameter m, and the parameter ?? which accounts for the combined effects of material anisotropy, represented by the specified parameters (??, ??, ??), and material nonuniformity, represented by the parameter m. The radial and circumferential stresses are the linear combinations of two power functions of the radial coordinate, whose exponents (n ₁ and n ₂) depend on the parameters m and ??. New light is added to the stress amplification and shielding under combined effects of curvilinear anisotropy and radial nonuniformity. Different loading combinations are considered, including the equal pressure at both boundaries, and the uniform pressure at the inner or the outer boundary. While the stress state for the equal pressure loading is uniform in the case of isotropic uniform material (m=0, ??=1), and for one particular radially nonuniform and anisotropic material, it is strongly nonuniform for a general anisotropic or nonuniform material. If the aspect ratio of the inner and outer radii decreases (small hole in a large disk/cylinder or sphere), the magnitude of the circumferential stress at the inner radius increases for n ₁>0 (stress amplification), and decreases for n ₁<0 (stress shielding). Both can be achieved by various combinations of the material parameters m, ??, ??, and ??. While the stress amplification in the case of a pressurized external boundary occurs readily, it occurs only exceptionally in the case of a pressurized internal boundary. The effects of material parameters on the displacement response are also analyzed. The approximate character of the plane stress solution of a pressurized thin disk is discussed and the results are compared with those obtained by numerical solution of the exact three-dimensional disk model.     

Terminal velocity of a Taylor drop in a vertical pipe

A scaling analysis based on the field equations for two phases and the jump conditions at the interface is carried out to deduce a balance of forces acting on a Taylor drop rising through stagnant liquid in a vertical pipe. The force balance is utilized to deduce a functional form of an empirical correlation of terminal velocity of the Taylor drop. Undetermined coefficients in the correlation are evaluated by making use of available correlations for two limiting cases, i.e. extremely high and low Reynolds number Taylor bubbles in large pipes. Terminal velocity data obtained by interface tracking simulations are also used to determine the coefficients. The proposed correlation expresses the Froude number Fr as a function of the drop Reynolds number Re_D, the Eötvös number Eo_D and the viscosity ratio μ^*. Comparisons between the correlation, simulations and experimental data confirm that the proposed correlation is applicable to Taylor drops under various conditions, i.e., 0.002 < Re_D < 4960, 4.8 < Eo_D < 228, 0 ? μ^* ? 70, 1 < N < 14700, −12 < log M < 4, and d/D < 1.6, where N is the inverse viscosity number, M the Morton number, d the sphere-volume equivalent drop diameter and D the pipe diameter.     

Moiré method for in-plane and out-of-plane displacement measurements

A high-frequency phase grating on a specimen surface is illuminated by oblique beams. Recordings of two diffracted beams by holographic interferometry reveal the wavefront changesN _a andN _b. In-plane displacements are proportional toN _a?N_b, while out-of-plane displacements are proportional toN _a+N_b. Experimental corroboration is presented.     

Axisymmetric thermal stress in a thin circular disk by the method of matched asymptotic expansions

Using the method of matched asymptotic expansions, a solution is constructed for thin disks with stress-free edges which consists of an analytical solution in the interior and a boundary layer correction. The analytical solution is shown to be accurate to order (h/a)² compared to unity, and is that given by the classical theory of thin plates. There is no boundary layer in the case of a clamped disk, so that the thin plate solution is valid throughout. The solution of the boundary layer equations was obtained numerically by solving the finite difference form of the governing equations using an iterative scheme called “Dynamic Relaxation”.     

On the quenching of steel and zircaloy spheres in water-based nanofluids with alumina,silica and diamond nanoparticles

The quenching curves (temperature vs time) for small (∼1 cm) metallic spheres exposed to pure water and water-based nanofluids with alumina, silica and diamond nanoparticles at low concentrations (?0.1 vol%) were acquired experimentally. Both saturated (ΔT_sub = 0 °C) and highly subcooled (ΔT_sub = 70 °C) conditions were explored. The spheres were made of stainless steel and zircaloy, and were quenched from an initial temperature of ∼1000 °C. The results show that the quenching behavior in nanofluids is nearly identical to that in pure water. However, it was found that some nanoparticles accumulate on the sphere surface, which results in destabilization of the vapor film in subsequent tests with the same sphere, thus greatly accelerating the quenching process. The entire boiling curves were obtained from the quenching curves using the inverse heat transfer method, and revealed that alumina and silica nanoparticle deposition on the surface increases the critical heat flux and minimum heat flux temperature, while diamond nanoparticle deposition has a minimal effect on the boiling curve. The possible mechanisms by which the nanoparticles affect the quenching process were analyzed. It appears that surface roughness increase and wettability enhancement due to nanoparticle deposition may be responsible for the premature disruption of film boiling and the acceleration of quenching. The basic results were also confirmed by quench tests with rodlets.     

Array realization of complex-source beam

An array is constructed to radiate the far-field pattern of a single complex point source. For any nonzero error tolerance, the physical dimension of the array is smaller (sometimes much smaller) than the diameter of the branch-cut disk of the complex point source. The inverse source problem is formulated for non-resonant arrays with reactive zones that do not extend significantly beyond the physical dimensions of the array. Both time-harmonic and pulsed beams are considered. In numerical examples, each array element consists of real point sources in an end-fire configuration.     

Fluid forces on a very low Reynolds number airfoil and their prediction

This paper presents the measurements of mean and fluctuating forces on an NACA0012 airfoil over a large range of angle (α) of attack (0-90°) and low to small chord Reynolds numbers (Re_c), 5.3 × 10³-5.1 × 10⁴, which is of both fundamental and practical importance. The forces, measured using a load cell, display good agreement with the estimate from the LDA-measured cross-flow distributions of velocities in the wake based on the momentum conservation. The dependence of the forces on both α and Re_c is determined and discussed in detail. It has been found that the stall of an airfoil, characterized by a drop in the lift force and a jump in the drag force, occurs at Re_c ? 1.05 × 10⁴ but is absent at Re_c = 5.3 × 10³. A theoretical analysis is developed to predict and explain the observed dependence of the mean lift and drag on α.     

Subcooled pool boiling of water on a downward-facing stainless steel disk in a gap

Pool boiling is experimentally studied on a relatively large downward-facing surface with heated stainless steel disk diameters of D = 100 and 300 mm in confined space at atmospheric pressure using water as the working fluid. The bulk working fluid is subcooled. The gap size s can be adjusted to 10, 15 and 20 mm for D = 100 mm and 0.9–77 mm for D = 300 mm. We found that pool boiling under the present condition is far weaker than that occurring on an upward-facing surface. Furthermore, we found that the larger the diameter of the stainless steel plate, the weaker the pool boiling heat transfer. The heat transfer rate may be predicted by the Kutateladze correlation for s > 20 mm and by its modified form for s < 20 mm for both D = 100 and D = 300 mm. Two different typical bubble circulation motions are found. Type I motion occurs with a probability of 90.9% and type II occurs with a probability of 9.1% according to the statistical calculations. Most coalesced bubble diameters are from 90 to 100 mm for D = 100 mm and from 100 to 200 mm for D = 300 mm.     

On the N-wave equations and soliton interactions in two and three dimensions

Several important examples of the N-wave equations are studied. These integrable equations can be linearized by formulation of the inverse scattering as a local Riemann-Hilbert problem (RHP). Several nontrivial reductions are presented. Such reductions can be applied to the generic N-wave equations but mainly the 3- and 4-wave interactions are presented as examples. Their one and two-soliton solutions are derived and their soliton interactions are analyzed. It is shown that additional reductions may lead to new types of soliton solutions. In particular the 4-wave equations with ?₂ × ?₂ reduction group allow breather-like solitons. Finally it is demonstrated that RHP with sewing function depending on three variables t, x and y provides some special solutions of the N-wave equations in three dimensions.     

Simulation of springback: Through-thickness integration

The number of through-thickness integration points (N_IP) required for accurate springback analysis following sheet forming simulation using shell elements is a subject of confusion and controversy. Li and Wagoner recommended, in 1999, based on a finite element analysis (FEA) of draw-bending springback, the use of 25 integration points (IP), with up to 51 IP required to ensure accuracies of 1%. Several researchers have since reported that N_IP between 5 and 11 are adequate, or even that 7 or 9 IP are optimal, with reduced accuracy for more IP. These apparent contradictions are addressed with an analytical model of elasto-plastic bending under tension, followed by elastic springback. The fractional error in the evaluated bending moment, which is equal to the fractional error in springback, was determined by comparing three numerical integration schemes, with various N_IP, to the closed-form result. The results illustrate the oscillatory nature of numerical integration error with small parametric changes, such that fortuitous agreement can be obtained in isolated simulations where the number of integration points is inadequate. The concept of an assured error limit is introduced as well as a maximum error limit (for a range of generally unknown sheet tensions). The assured error limit varies with the integration scheme, N_IP, bending ratio (R/t), and sheet tension. Guidelines for the number of integration points required for given error tolerances are reported to allow practitioners to choose numerical parameters appropriately.     

Reissner-Sagoci problem for a homogeneous coating on a functionally graded half-space

This paper is concerned with the axisymmetric elastostatic problem related to the rotation of a rigid punch which is bonded to the surface of a nonhomogeneous half-space. The half-space is composed of an isotropic homogeneous coating in the form of layer, which is attached to the functionally graded half-space. The shear modulus of the FGM is assumed to vary in the direction of axis Oz normal to the boundary as μ₁(z) = μ₀(1 + αz)^β, where μ₀, α, β are positive constants. The punch undergoes rotation due to the action of the internal loads. By using Hankel's integral transforms, the mixed boundary value problem is reduced to dual integral equations, and next, to a Fredholm's integral equation of the second kind, which is solved numerically for the case of β = 2. The final results show the effect of non-homogeneity on the shear stresses and an unknown moment of punch rotation.     

Experimental study on the heat transfer coefficient of water flow boiling in mini/microchannels

The heat transfer coefficients of the evaporative water flow in mini/microchannels are studied experimentally to explore the novel heat dissipation for high power electronics. Two sets of parallel channels which are 61 channels with hydraulic diameter of 0.293 mm and 20 channels with hydraulic diameter of 1.2 mm are investigated respectively. The inlet and outlet temperatures of fluids, and the temperatures beneath the channels are measured to calculate the heat dissipation of the evaporative water in channels. The experiments are carried out with the mass flow rates range from 11.09 kg/(m² s) to 44.36 kg/(m² s) for minichannels and 49.59 kg/(m² s) to 198.37 kg/(m² s) for microchannels. The effective heat flux range from 5 W/cm² to 50 W/cm², and the resulted outlet vapor qualities range from 0 to 0.8. The relations of the heat transfer coefficient with heat flux and vapor quality are analyzed according to the results. The experimental heat transfer coefficients are compared with the prediction of latest developed correlations. A new correlation takes the effect of Bond number is proposed, and be verified that it is effective to predict the heat transfer coefficient of both minichannels and microchannels in a large range of vapor qualities.     

Stochastic finite element nonlinear free vibration analysis of piezoelectric functionally graded materials beam subjected to thermo-piezoelectric loadings with material uncertainties

In this paper, second order statistics of large amplitude free flexural vibration of shear deformable functionally graded materials (FGMs) beams with surface-bonded piezoelectric layers subjected to thermopiezoelectric loadings with random material properties are studied. The material properties such as Young’s modulus, shear modulus, Poisson’s ratio and thermal expansion coefficients of FGMs and piezoelectric materials with volume fraction exponent are modeled as independent random variables. The temperature field considered is assumed to be uniform and non-uniform distribution over the plate thickness and electric field is assumed to be the transverse components E _z only. The mechanical properties are assumed to be temperature dependent (TD) and temperature independent (TID). The basic formulation is based on higher order shear deformation theory (HSDT) with von-Karman nonlinear strain kinematics. A C ⁰ nonlinear finite element method (FEM) based on direct iterative approach combined with mean centered first order perturbation technique (FOPT) is developed for the solution of random eigenvalue problem. Comparison studies have been carried out with those results available in the literature and Monte Carlo simulation (MCS) through normal Gaussian probability density function.     

Effect of third- and fourth-order moments on the modeling of unresolved transition arrays

The impact of the third (skewness) and fourth (kurtosis) reduced centered moments on the statistical modeling of E1 lines in complex atomic spectra is investigated through the use of Gram–Charlier, Normal Inverse Gaussian and Generalized Gaussian distributions. It is shown that the modeling of unresolved transition arrays with non-Gaussian distributions may reveal more detailed structures, due essentially to the large value of the kurtosis. In the present work, focus is put essentially on the Generalized Gaussian, the power of the argument in the exponential being constrained by the kurtosis value. The relevance of the new statistical line distribution is checked by comparisons with smoothed detailed line-by-line calculations and through the analysis of 2p → 3d transitions of recent laser or Z-pinch absorption measurements. The issue of calculating high-order moments is also discussed (Racah algebra, Jucys graphical method, semi-empirical approach…).     

On the generalized Cauchy function and new Conjecture on its exterior singularities

This article studies on Cauchy’s function f (z) and its integral, (2πi)J[ f (z)] ≡■C f (t)dt/(t z) taken along a closed simple contour C, in regard to their comprehensive properties over the entire z = x + iy plane consisted of the simply connected open domain D + bounded by C and the open domain D outside C. (1) With f (z) assumed to be C n (n < ∞-times continuously differentiable) z ∈ D + and in a neighborhood of C, f (z) and its derivatives f (n) (z) are proved uniformly continuous in the closed domain D + = [D + + C]. (2) Cauchy’s integral formulas and their derivatives z ∈ D + (or z ∈ D ) are proved to converge uniformly in D + (or in D = [D +C]), respectively, thereby rendering the integral formulas valid over the entire z-plane. (3) The same claims (as for f (z) and J[ f (z)]) are shown extended to hold for the complement function F(z), defined to be C n z ∈ D and about C. (4) The uniform convergence theorems for f (z) and F(z) shown for arbitrary contour C are adapted to find special domains in the upper or lower half z-planes and those inside and outside the unit circle |z| = 1 such that the four general- ized Hilbert-type integral transforms are proved. (5) Further, the singularity distribution of f (z) in D is elucidated by considering the direct problem exemplified with several typ- ical singularities prescribed in D . (6) A comparative study is made between generalized integral formulas and Plemelj’s formulas on their differing basic properties. (7) Physical sig- nificances of these formulas are illustrated with applicationsto nonlinear airfoil theory. (8) Finally, an unsolved inverse problem to determine all the singularities of Cauchy function f (z) in domain D , based on the continuous numerical value of f (z) z ∈ D + = [D + + C], is presented for resolution as a conjecture.     

Bifurcations in the wake of a thick circular disk

Using DNS, we investigate the dynamics in the wake of a circular disk of aspect ratio χ = d/w = 3 (where d is the diameter and w the thickness) embedded in a uniform flow of magnitude U ₀ perpendicular to its symmetry axis. As the Reynolds number Re = U ₀ d/ν is increased, the flow is shown to experience an original series of bifurcations leading to chaos. The range Re ${\in}$ [150, 218] is analysed in detail. In this range, five different non-axisymmetric regimes are successively encountered, including states similar to those previously identified in the flow past a sphere or an infinitely thin disk, as well as a new regime characterised by the presence of two distinct frequencies. A theoretical model based on the theory of mode interaction with symmetries, previously introduced to explain the bifurcations in the flow past a sphere or an infinitely thin disk (Fabre et al. in Phys Fluids 20:051702, 2008), is shown to explain correctly all these results. Higher values of the Reynolds number, up to 270, are also considered. Results indicate that the flow encounters at least four additional bifurcations before reaching a chaotic state.     

Motion of singles bubbles moving under a slightly inclined surface through stationary liquids

The influence of the liquid properties on the dynamical bubble shape and on the bubble motion has been investigated for bubbles moving under a downward facing inclined surface. The Morton number Mo varied from 2.59 × 10⁻¹¹ to 2.52 × 10⁺⁰¹. The Bond number Bo covered the range from 10 to 150 and the surface inclination angle θ was varied from 2° to 6°. To cover the wide range of Mo, several liquids such as glycerine, propanediol, water and isopropanol were used. The results have shown that the relation Fr = Fr(Bo, Mo, θ) is not adequate to describe the bubble motion, where Fr is the terminal Froude number. The choice of the terminal Reynolds number Re as the dependent parameter, allowed the clarification of the role of the Morton number on the bubble motion. At a given Bond number, the bubble Reynolds number decreases monotonously with the Morton number. Furthermore, an empirical correlation Re = Re(Bo, Mo, θ) is given that can be readily used in the mathematical modelling of bubble laden flows under solids.     

Conjugate heat transfer in the unbounded flow of a viscoelastic fluid past a sphere

This work addresses the conjugate heat transfer of a simplified PTT fluid flowing past an unbounded sphere in the Stokes regime (Re = 0.01). The problem is numerically solved with the finite-volume method assuming axisymmetry, absence of natural convection and constant physical properties. The sphere generates heat at a constant and uniform rate, and the analysis is conducted for a range of Deborah (0 ≤ De ≤ 100), Prandtl (10⁰ ≤ Pr ≤ 10⁵) and Brinkman (0 ≤ Br ≤ 100) numbers, in the presence or absence of thermal contact resistance at the solid–fluid interface and for different conductivity ratios (0.1 ≤ κ ≤ 10). The drag coefficient shows a monotonic decrease with De, whereas the normalized stresses on the sphere surface and in the wake first increase and then decrease with De. A negative wake was observed for the two solvent viscosity ratios tested (β = 0.1 and 0.5), being more intense for the more elastic fluid. In the absence of viscous dissipation, the average Nusselt number starts to decrease with De after an initial increase. Heat transfer enhancement relative to an equivalent Newtonian fluid was observed for the whole range of conditions tested. The dimensionless temperature of the sphere decreases and becomes more homogeneous when its thermal conductivity increases in relation to the conductivity of the fluid, although small changes are observed in the Nusselt number. The thermal contact resistance at the interface increases the average temperature of the sphere, without affecting significantly the shape of the temperature profiles inside the sphere. When viscous dissipation is considered, significant changes are observed in the heat transfer process as Br increases. Overall, a simplified PTT fluid can moderately enhance heat transfer compared to a Newtonian fluid, but increasing De does not necessarily improve heat exchange.     

Experimental investigation of two-phase flow splitting in an equal-sided impacting tee junction with inclined outlets

Phase-distribution data have been generated for two-phase (air-water) flow splitting at an impacting tee junction with a horizontal inlet and inclined outlets. This investigation also considered the possibility of full separation at the junction and the effect of the outlet angle of inclination on partial separation at various inlet conditions. A flow loop with the ability to incline the outlets from horizontal to vertical was constructed. The operating conditions were as follows: test section inside diameter (D) of 13.5 mm, nominal junction pressure (P_s) of 200 kPa (abs), near ambient temperature (T_s), inlet superficial gas velocities (J_G1) ranging from 2.0 to 40 m/s, inlet superficial liquid velocities (J_L1) ranging from 0.01 to 0.18 m/s, inlet qualities (x₁) ranging from 0.1 to 0.9, mass split ratios (W₃/W₁) from 0 to 1.0, and inlet flow regimes of stratified, wavy, and annular. The data reveal that the degree of maldistribution of the phases depended on the inlet conditions, the mass split ratio at the junction, and the inclination angle of the outlets.