The mechanism of drop formation from gas or vapour bubbles

This paper presents an interpretation of the mechanism of drop formation due to bubbles of gas or vapour travelling upwards through a liquid and collapsing at the liquid-gas interface. Size-distribution of entrained drops are obtained as a function of height and bubble sizes. Parabolic trajectories have been taken in consideration and an equation is given for the maximum height of drops.

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Zusammenfassung Eine Erklärung für die Bildung von Tropfen an der freien Flüssigkeitsoberfläche beim Zerplatzen von aufsteigenden Gas- oder Dampfblasen wird hier gegeben. Die Größenverteilung der abgeschleuderten Tropfen wird als Funktion der Schleuderhöhe und der Blasengröße erhalten. Parabolische Schleuderbahnen wurden berücksichtigt und eine Gleichung für die maximale Schleuderhöhe der Tropfen angegeben.

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Symbols surface tension - amplitude of the surface disturbance - _w density of liquid - a density of gas - kinematic viscosity - D _d diameter of droplet - D _j diameter of jet - E _d drop energy - E _j total energy absorbed by the jet - E _o energy release by collapsing bubble - E _p potential energy - E _R drag energy - E _s surface tension energy - F vertical force - G _j gravitational force acting on jet - H maximum trajectory height - L length of jet - L _opt length of separated upper portion of jet - l length unseparated lower portion of jet - M mass of water which must be set in motion to fill the crater - m mass of droplet - P _g gas pressure - P _l liquid pressure - P _i inside pressure - P _o outside pressure - R ₁,R ₂ curvatures of bubble - R _j radius of jet - R radius of spherical bubble - R* frictional drag force of jet - R _D frictional drag force of droplet - S ₁ surface tension force - t _F impulse time - t time of break-up of jet - U mean velocity acquired by the massM - v droplet velocity - V ₀ initial velocity of droplet - W ₁ gravitational force     

Fe-Si-Cr-Al四元系Fe_3Si基合金在水润滑条件下的摩擦学性能

利用Optimal SRV摩擦磨损试验机考察了Fe65Si25Cr5Al5、Fe70Si20Cr5Al5、Fe75Si15Cr5Al5和Fe80Si10Cr5Al5等几种Fe-Si-Cr-Al四元系Fe3Si与Si3N4配副时在水润滑条件下摩擦学性能.结果表明:Fe65Si25Cr5Al5和Fe70Si20Cr5Al5的磨损率低于纯Fe3Si和AISI 304不锈钢,同样,与这两种材料配副的Si3N4对偶材料磨损率也相对较低;Fe75Si15Cr5Al5和Fe80Si10Cr5Al5的磨损率则高于Fe3Si和AISI 304不锈钢.Fe70Si20Cr5Al5中Cr元素的化学活性高于其中Al元素的化学活性,材料表面富集的Cr2O3有效阻碍了Si3N4与水的摩擦化学反应,从而使Fe70Si20Cr5Al5在水环境中的摩擦学性能Fe3Si显著提高,在载荷分别为30、50、70和90 N条件下,Fe70Si20Cr5Al5比Fe3Si的磨损率分别降低了15.5%、20.2%、31.8%和38.2%;对偶材料Si3N4的磨损率则分别降低了67.9%、36.9%、46.6%和50.6%.     

2D numerical contributions for the study of non-cohesive sediment transport beneath tidal bores

2D numerical simulations of tidal bores were obtained using the OpenFOAM CFD software to solve the Navier–Stokes equations by means of the Finite Volume Method by applying a LES turbulence model. The trajectories of non-cohesive sediment particles beneath tidal bores were estimated using a tracker method. Using the fourth order Runge–Kutta scheme, the tracker method solves the Maxey and Riley equations, which requires the knowledge of the velocity field at time t. From 2D numerical simulations of tidal bores, we proposed a classification of tidal bores with respect to the Froude number Fr (or r the ratio of water depths). For a Froude number $1<Fr<1.43$ ($1<r<1.57$), the tidal bore is undular. For a Froude number $1.43<Fr<1.57$ ($1.57<r<1.75$), the tidal bore is partially breaking, which is similar to the transitional tidal bore defined by Furgerot (2014). And for a Froude number $Fr>1.57$ ($r>1.75$), the tidal bore is totally breaking. The numerical results of trajectories of non-cohesive sediment particles are similar to the type of trajectories given by the analytical model proposed by Chen et al. (2012) with some modifications to take into account the effects of gravity, elevation, and attenuation. The parameters of modified Chen's model, ${\beta }_1$, ${\beta }_2$ and ${\beta }_3$, are linearly proportional to the Froude number Fr. This is because the level of turbulence for undular tidal bores is low. The flow induced by an undular tidal bore is not complex. This physical phenomenon is quasi linear. The parameter ${\beta }_1$, related to the front celerity of the undular tidal bore, decreases when the Froude number Fr increases. The parameter ${\beta }_2$, related to the elevation, increases when the Froude number Fr increases. And the parameter ${\beta }_3$, related to the attenuation of the secondary waves, increases when the Froude number Fr increases.     

Recoverable strains and retardation times of monodisperse suspensions of silicon dioxide spheres in poly(dimethylsiloxane)

Steady-state viscosities η, steady-state recoverable strains γ _rs and characteristic retardation time τ _1/2 were measured for suspensions of monodisperse silicon dioxide (SiO₂) spheres in poly(dimethylsiloxane) (PDMS) with various volume fractions Φ of the suspended spheres at various creep stresses σ ₀. Two different regions are found in plots of η/η _m vs γ _rs, where η/η _m denotes the relative viscosity of the suspensions. In one region, η/η _m is proportional to γ _rs, while γ _rs is independent of η/η _m in the other region. In both regions, τ _1/2 is the functions of the shear strain rate in the steady-state of creep test independently of Φ. The origin of the elasticity is related to the ‘maximally distorted’ cages recovered owing to the repulsive interaction between the SiO₂ spheres and recovery of the cages in the shear-induced clusters of the suspended spheres.     

The cross sectional area difference method — a new technique for determination of particle concentration by laser doppler anemometry

A new technique for the determination of particle concentration from the signals of a laser Doppler anemometer (LDA) is described. It is based on a statistical relation between the number of Doppler periods, or the amplitude of the Doppler signals, and the particle concentration. The technique allows the mass flux of the dispersed phase of a two-phase flow to be obtained from the data set of a conventional one-dimensional (ID) LDA. The technique has been called the cross sectional area difference method. Simulations and first experimental results are presented and discussed.List of symbols a, b, c half-axes of measurement control volume (mcv) - a ₁, b ₁, c ₁ half-axes of detection volume - c _L velocity of light - d _m beam waist diameter - d _p particle diameter - d _pc diameter of the calibration particle - d _pmin minimum detectable particle diameter - e elementary charge - h Planck's constant - i number of particle size classes - k wavenumber - m visibility - m refractive index - n(d _p ) particle concentration - n(d _pi ) concentration of ith particle class - n vector of n(d _pi ) - q exponent of size dependence of G(d _p ) - v _x x-velocity component - x fringe spacing - y ₀, z ₀ coordinates of particle trajectory and cross sectional area - A cross sectional area of mcv - A matrix of A ₁ - a ₁ cross sectional area of detection volume - A ₁ difference of neighbouring cross sectional areas - C _A normalisation constant for linear graduation of amplitude - C _N normalisation constant for Doppler periods - C _scat non-size-dependent factor of G(d _p ) - C _x normalisation constant for nonlinear graduation of amplitude - F() power spectral density - G(d _p ) integral scattering function - H number of accumulated counts - H _max maximum number of accumulated counts - I amplitude of Doppler signal - I _max I for a particle passing through the origin of the mcv - I _s trigger level - K logarithmic amplitude ratio - K _max logarithmic amplitude ratio for I _s - K _x degree of linear class width of amplitude - K _A degree of nonlinear class width of amplitude - N number of Doppler periods - N _m number of Doppler periods required by signal validation - N _max N for a particle passing through the origin of the mcv - N ₀ fringe number inside mcv along x-axis - P _L laser power - S ₀ particle arrival rate - S ₁ trigger rate - S ₁ contribution to trigger rate coming from A ₁ - S ₁ vector of S _1i - S _1i contribution to trigger rate coming from ith class of distribution - _Q quantum efficiency - wavelength of laser light - off-axis angle - elevation angle - angular frequency - beam intersection angle - phase difference     

On a Crystalline Variational Problem, Part II:¶BV Regularity and Structure of Minimizers on Facets

For a nonsmooth positively one-homogeneous convex function φ:ℝ ⁿ → [0,+∞[, it is possible to introduce the class ?_φ (ℝ ⁿ ) of smooth boundaries with respect to φ, to define their φ-mean curvature κ_φ, and to prove that, for E∈?_φ (ℝ ⁿ ), κ_φ∈L ^∞(δE) [9]. Based on these results, we continue the analysis on the structure of δE and on the regularity properties of κ_φ. We prove that a facet F of δE is Lipschitz (up to negligible sets) and that κ_φ has bounded variation on F. Further properties of the jump set of κ_φ are inspected: in particular, in three space dimensions, we relate the sublevel sets of κ_φ on F to the geometry of the Wulff shape ?_φ≔{φ≤ 1 }. Accepted October 11, 2000?Published online 14 February, 2001     

Further Study on Pure Shear

It is known that the Cauchy stress tensor T is a pure shear when trT = 0. An elementary derivation is given for a coordinate system such that, when referred to this coordinate system, the diagonal elements of T vanish while the off-diagonal elements τ ₁, τ ₂, τ ₃, are the pure shears. The structure of τ _i (i = 1, 2, 3) depends on one non-dimensional parameter q = 54(detT)² / [tr(T ²)]³, 0 ≤ q ≤ 1. When q = 0, one of the three τ _i vanishes. A coordinate system can be chosen such that the remaining two have the same magnitude or one of the remaining two also vanishes. When q = 1, all three τ _i have the same magnitude. However, there is a one-parameter family of coordinate systems that gives the same three τ _i . For q ≠ 0 or 1, none of the three τ _i vanishes and the three τ _i in general have different magnitudes. Nevertheless, a coordinate system can be chosen such that two of the three τ _i have the same magnitude. ^★Professor Emeritus of University of Illinois at Chicago and Consulting Professor of Stanford University.     

Feasibility study of three-dimensional PIV by correlating images of particles within parallel light sheet planes

One approach to obtain information about the out-of-plane velocity component from PIV recordings is to analyze the height of the peak in the correlation plane. This value depends on the portion of paired particle images, which itself depends on the out-of-plane velocity component and on other parameters. To circumvent problems with other influences (e.g. background light, amount and size of images), images from another light sheet plane parallel to the first one were also captured for peak height normalization. Our experimental results show the feasibility of an out-of-plane velocity estimation by analyzing images of particles within parallel light sheets by spatial cross-correlation.List of Symbols C particle density in the flow - d particle image diameter - f ₀, f ₁ frames containing images of particles within the first light sheet at t=t ₀ (frame f ₀) and at t=t ₀ + t (frame f ₁) - f ₂ frame containing images of particles within the second light sheet parallel to the first one at t=t ₀ + 2t - F ₁ estimator of the loss of image pairs due to in-plane motion - F ₀ estimator of the loss of image pairs due to out-plane motion - F convolution of the particle image intensity distributions - K factor containing constant parameters in the correlation plane - M imaging magnification (image size/object size) - n ₀ number of particles in the measurement volume at t=t ₀ - n _0,1 number of particle image pairs in interrogation windows of f ₀ andf ₁ - n _1,2 number of particle image pairs in interrogation windows off ₁ and f ₂ - O _z overlap of the light sheets - R _C (s) convolution of the mean intensity distributions - R _D (s) correlation which gives the image displacement - R _F (s) fluctuating noise component of the cross correlation estimator - R _0,1(s _D ) cross-correlation peak height of interrogation windows off ₀ and f ₁ - R _1,2(s _iuD) cross-correlation peak height of interrogation windows of f ₁ and f ₂ - s two-dimensional separation vector in the correlation plane - s _D mean particle image displacement in the interrogation cell - t _e light pulse duration - t _f frame-transfer time of the video camera - u three-dimensional local flow velocity vector (u,v,w) - X _i position of the center of an interrogation window in the image plane (2d) - x _i position of the center of an interrogation volume in the flow (3d) - (z ₂ — Z ₁) displacement of the light sheets in z-direction - t separation time of the light pulses - x ₀ x-extension of an interrogation volume - y ₀ y-extension of an interrogation volume - z ₀ light sheet thickness The authors would like to thank DLR for supporting Markus Raffel's and Olaf Ronneberger's visit to Caltech (Center for Quantitative Visualisation), and the Office of Naval Research through the URI grant ONR-URI-N00014-92-J-1610. Dr. Alexander Weigand's generous offer of his experimental set-up and stimulating discussions with Dr. Jerry Westerweel and Dr. Thomas Roesgen are greatly appreciated. Special thanks also to Dr. Christian Willert for his advice regarding the modifications to the DPIV software.     

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

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

Three dimensional K-T z stress fields around the embedded center elliptical crack front in elastic plates

Through detailed three-dimensional (3D) finite element (FE) calculations, the out-of-plane constraints T_z along embedded center-elliptical cracks in mode I elastic plates are studied. The distributions of T_z are obtained near the crack front with aspect ratios (a/c) of 0.2, 0.4, 0.5, 0.6, 0.8 and 1.0. T_z decreases from an approximate value of Poisson ratio ν at the crack tip to zero with increasing normalized radial distances (r/a) in the normal plane of the crack front line, and increases gradually when the elliptical parameter angle ϕ changes from 0° to 90°at the same r/a. With a/c rising to 1.0, T_z is getting nearly independent of ϕ and is only related to r/a. Based on the present FE calculations for T_z, empirical formulas for T_z are obtained to describe the 3D distribution of T_z for embedded center-elliptical cracks using the least squares method in the range of 0.2≤a/c≤1.0. These T_z results together with the corresponding stress intensity factor K are well suitable for the analysis of the 3D embedded center-elliptical crack front field, and a two-parameter K-T_z principle is proposed. The project supported by the National Natural Science Foundation of China (50275073) The English text was polished by Keren Wang.     

An experimental study of the discharge coefficient of pipe perforations

Experiments were conducted to study the variation of the pressure loss coefficient of pipe perforations with geometrical parameters of the perforations and a Reynolds number based on the hydraulic diameter of an orifice representing the perforations. The experimental data are used to develop an empirical relationship between the head loss across the perforations and the geometrical and hydraulic parameters related to the perforations which was seen to give better predictions when the perforations are not very closely spaced. The experimental results reported herein correspond to the pipes of small perforated length, with downstream end of the pipe closed.List of symbols a area of the orifice - A total area of perforations - A ₁ inner area of pipe - A ₂ outlet area - C _r factor for static pressure regain - D diameter of the orifice - D _h hydraulic diameter - D _p internal diameter of the pipe - f_o friction factor for orifice surface - f_p friction factor for perforated pipe - g acceleration due to gravity - H water head inside the perforated pipe - H ₀ head outside the perforated pipe - H(A) experimental water head difference - H1 water head difference between inside and outside of perforations when A ₂ is outlet area and A ₁ is inside perforated surface area of pipe - H2 water head difference between inside and outside of perforations when A ₂ is outlet area and A ₁ is cross-sectional area of pipe - H3 water head difference between inside and outside of perforations when A ₂ is infinity and A ₁ is inside perforated surface area of pipe - H4 water head difference between inside and outside of perforations when A ₂ is infinity and A ₁ is cross-sectional area of pipe - H _t total head at the inlet of the perforated pipe - H₀ head loss across the orifice - H_f head loss due to surface friction - H_m head loss due to momentum reduction - K _f pressure loss coefficient for frictional losses - L _p perforated length of the pipe - n total number of orifices in a perforated pipe - N number of orifice rows - p pitch of perforations - P _a perimeter of the flow passage - P ₀ porosity of the perforated pipe - q flow rate through orifice - Q flow rate through perforations - Re ₀ Reynolds number based on orifice diameter - Re _p Reynolds number based on pipe diameter - T wall thickness of the pipe - v velocity of flow through the orifice Greek symbols V ₁ velocity of flow upstream of the orifice - V ₂ velocity of flow downstream of the orifice - (tou) coefficient depending on T/D ratio of the orifice - (zeeta) loss coefficient of fluid flow through perforated pipe - coefficient depending on the shape of the inlet edge of the orifice     

On a Differential Equation with State-Dependent Delay

We study a family of scalar differential equations with a single parameter a > 0 and delay r > 0. In the case of the constant delay r = 1 it is known that for parameters 0 < a < 1 the trivial solution of this family is asymptotically stable, whereas for a > 1 the trivial solution gets unstable, and a global center-unstable manifold connects the trivial solution to a slowly oscillating periodic orbit. Here, we consider a state-dependent delay r = r(x(t)) > 0 instead of the constant one, and generalize the result on the existence of slowly oscillating periodic solutions for parameters a > 1 under modest conditions on the delay function r.     

Augmentation of heat transfer in a bubble agitated vertical rectangular channel

This paper presents the results of an experimental study of convective heat transfer between three parallel vertical plates symmetrically spaced with and without bubble agitation to ascertain the degree of augmentation of the heat transfer coefficients due to agitation. The centre plate was electrically heated, while the other side plates were water-cooled forming two successive parallel vertical rectangular channels of dimensions 20 cm × 3.5 cm × 35 cm (length W, gap L, height H) each. At the bottom of the hot and cold plates air spargers were fitted. Water/ethylene glycol (100%) was used to fill the channels. The superficial gas velocity ranged from 0.0016 to 0.01 m/s. Top, bottom and sides of the channels were open to the water/ethylene glycol in the chamber which is the novel aspect of this study. Experimental data have been correlated as under: Natural convective heat transfer: Nu = 0.60 Gr ^0.29, r = 0.96, σ = 0.186, 1.17 E6 < Gr < 1.48 E7; Bubble agitated heat transfer: St = 0.11(ReFrPr ²)^−0.23, r = 0.82, σ = 0.002, 1.20 E−2 < (ReFrPr ²) < 1.36 E2.     

A numerical study of various rheological polydispersity measures

Model calculations were performed in order to investigate the sensitivity of various rheological polydispersity parameters for variations in the moments of the molar mass distribution (MMD) of linear polymers. Molar mass distributions were generated with the Gaussian and the Generalised exponential distribution functions, using a fixed weight average molar mass M _w and variable M _w /M _n and M _z /M _n . Assuming linear entangled polymeric chains, the linear viscoelastic properties were predicted by calculating the stress relaxation modulus of the consecutive monodisperse fractions with the BSW relaxation time spectrum and blending these curves with the double reptation blending rule. BSW relaxation parameters appropriate for polypropylene were used.  It was found that both the zero-shear viscosity and the so-called cross-over frequency, at which and are equal, depend mostly on M _w but also significantly on both M _w /M _n and M _z /M _w . By contrast, the steady-state compliance depends mainly on M _z /M _w , its functional dependence on moments of the MMD being best described by the Ferry equation.  None of the polydispersity parameters PI (from the modulus cross-over), MODSEP (the modulus separation) or PDR (from the shape of the flow curve), as introduced in literature depends solely on the polydispersity M _w /M _n . PI is the most sensitive indicator for this purpose. Finally, the parameters ER ( at a fixed low value of , MODSEP en DRI (from the shape of the flow curve) are shown to be good indicators for the weight (M _z /M _w ) of the high molar mass tail of the molar mass distribution. Received: 5 May 1998 Accepted: 30 July 1998     

Dynamical deformation of a flat liquid–liquid interface

The passage of solid spheres through a liquid–liquid interface was experimentally investigated using a high-speed video and PIV (particle image velocimetry) system. Experiments were conducted in a square Plexiglas column of 0.1 m. The Newtonian Emkarox (HV45 50 and 65% wt) aqueous solutions were employed for the dense phase, while different silicone oils of different viscosity ranging from 10 to 100 mPa s were used as light phase. Experimental results quantitatively reveal the effect of the sphere’s size, interfacial tension and viscosity of both phases on the retaining time and the height of the liquid entrained behind the sphere. These data were combined with our previous results concerning the passage of a rising bubble through a liquid–liquid interface in order to propose a general relationship for the interface breakthrough for the wide range of Mo ₁/Mo ₂ ∈ [2 × 10⁻⁵–5 × 10⁴] and Re ₁/Re ₂ ∈ [2 × 10⁻³–5 × 10²].