Boundary layer theory modified for analyzing finite-amplitude oscillations of a viscous liquid charged drop

The existing concepts of the boundary layer arising near the free surface of a viscous liquid, which is related to its periodic motion, are revised with the aim to calculate finite-amplitude linear oscillations of a viscous liquid charged drop. Equations complementing the boundary layer theory are derived for the vicinity of the oscillating free spherical surface of the drop. An analytical solution to these equations is found, comparison with an exact solution is made, and an estimate of the boundary layer thickness is obtained. The domain of applicability of the modified theory is defined.     

Modification of the boundary layer theory for calculating viscous drop oscillations in a uniform electrostatic field

Capillary oscillations on the free surface of a viscous conductive liquid drop placed in an electrostatic field are calculated. In an approximation linear in stationary deformation amplitude, the drop in this field has the shape of a spheroid extended along the field. The initial problem is modified and simplified in terms of the boundary layer theory by applying an approximation that is linear in the oscillation amplitude and quadratic in the eccentricity of the drop. The accuracy of the approximate solution relative to an exact one is estimated. It is shown that, with a rise in the electrostatic field strength (with an increase in the eccentricity of the drop) and in the viscosity of the liquid, the boundary layer at the free surface of the drop becomes thicker.     

Boundary layer theory modified for analysis of wave flow on the surface of a viscous liquid finite-thickness layer resting on a hard bottom

The theory of a boundary layer that is adjacent to the surface of an indefinitely deep viscous liquid and caused by its periodic motion is modified for analysis of finite-amplitude flow motion on the charged surface of a viscous conductive finite-thickness liquid layer resting on a hard bottom (the thickness of the layer is comparable to the wavelength). With the aim of adequately describing the viscous liquid flow, two boundary layers are considered: one at the free surface and the other at the hard bottom. The thicknesses of the boundary layers are estimated for which the difference between an exact solution and a solution to a model problem (stated in terms of the modified theory) may be set with a desired accuracy in the low-viscosity approximation. It is shown that the presence of the lower (bottom) boundary layer should be taken into account (with a relative computational error no more than 0.001) only if the thickness of the viscous layer does not exceed two wavelengths. For thicker layers, the bottom flow may be considered potential. In shallow liquids (with a thickness of two tenths of the wavelength or less), the upper (near-surface) and bottom layers overlap and the eddy flow entirely occupies the liquid volume. As the surface charge approaches a value that is critical for the onset of instability against the electric field negative pressure, the thicknesses of both layers sharply grow.     

Peculiarities of Acoustic Wave Reflection from a Boundary or Layer of a Two-Phase Medium

A mathematical model is presented for determining the oblique incidence of an acoustic wave at both a boundary and layer of a gas–drop mixture or a bubbly liquid of finite thickness. The basic wave reflection and transmission patterns are established for the incidence of a low-frequency acoustic wave at an interface between a pure gas and a gas–drop mixture, as well as between a pure and bubbly liquid. A range of varying volume fractions for a drop is determined, for which the zero value of the reflection coefficient is possible for low frequencies at oblique incidence. It is shown that the reflection coefficient will never be zero at angles of incidence above 24.5° from a gas–drop mixture at a pure gas boundary; however, when a wave is incident from a pure gas at a gas–drop mixture boundary, a zero reflection coefficient is possible for nonzero angles of incidence and the volume fraction of inclusions. The results of calculating reflection of an acoustic wave from a two-phase layer of a medium with a finite thickness are presented. It is established that the minimum reflection coefficient is possible depending on the perturbation frequency for a certain range of angles of incidence for the boundary or the layer of the gas–drop mixture, which is governed mainly by difference in densities between it and the pure gas.     

Modification of the boundary layer theory for analysis of finite-amplitude oscillations of a charged bubble in a viscous liquid

The prevailing concepts concerning the boundary layer near the free surface of a viscous liquid associated with oscillatory motion are modified for calculating finite-amplitude linear oscillations of a charged bubble in this liquid. Equations of the boundary layer theory for the neighbourhood of the oscillating free spherical surface of a charged bubble in a dielectric liquid are derived, their analytic solution is obtained and compared with the exact solution, and the thickness of the boundary layer is assessed. The range of applicability of the modified theory is determined.     

Thickness of a boundary layer attributed to the wave motion on the charged free surface of a viscous liquid

It is shown that the analytical estimator for the boundary layer thickness that contains the wave frequency in the denominator and is proposed for approximate calculation of the wave motion on the free surface of a viscous liquid cannot be formally applied to the wave motion on the uniformly charged liquid surface. The fact is that, when the surface charge density attains a value critical in terms for the Tonks-Frenkel instability, the wave frequency tends to zero. From the analysis of liquid motions near the electric charge critical density, a technique is proposed for calculating the thickness of a boundary layer attributed to flows of various kinds. It is found that the thickness of the boundary layer due to aperiodic flows with amplitudes exponentially growing with time (such flows take place at the stage of instability against the surface charge) does not exceed a few tenths of the wavelength, whereas the thickness of the boundary layer due to exponentially decaying liquid flows is roughly equal to the wavelength.     

On the theory of a boundary layer associated with wave motion on the free surface of a liquid

A modified theory of a boundary layer associated with a periodic capillary-gravitational motion on the free surface of an infinitely deep viscous liquid is proposed. The flow in the boundary layer is described in terms of a simplified (compared with the complete statement) model problem a solution to which correctly reflects the main features of an exact asymptotic solution: the rapid decay of the flow eddy part with depth of the liquid and insignificance of some terms appearing in the complete statement. The boundary layer thickness at which the discrepancy between the exact asymptotic solution and model solution is within a given margin is estimated.     

Modification of the boundary layer theory for analysis of wave motions in a cylindrical jet of a viscous liquid

The existent concepts of the boundary layer near the free surface of a viscous liquid, which is related to its periodic motion, are modified with the aim of analyzing the finite-amplitude wave motion on the surface of a thick charged jet of a viscous conducting liquid. To describe the flow in the boundary layer, a model problem is proposed that is simpler in statement compared with the complete problem and the solution of which uses the governing properties of the exact solution obtained in the low-viscosity asymptotics: the form of the dispersion relation, wave profile, and rate of velocity field viscous damping with time. An estimate is made of the boundary layer thickness at which the discrepancy between the exact solution and solution to the model problem (stated in terms of the theory proposed) falls into a given interval in the low-viscosity asymptotics. The domain of applicability of the modified theory is determined.     

Effect of longitudinal electric field on capillary instability of a thin axisymmetric layer of liquid dielectric coating a dielectric fiber

The flow of a viscous dielectric liquid surrounded with a gas is investigated in the process of capillary disintegration of a thin axisymmetric liquid layer on an undeformable cylindrical dielectric fiber in a uniform electric field is investigated. An asymptotic analysis of the system of equations and hydrodynamic boundary conditions written with allowance for surface ponderomotive forces is carried out for the case when the average thickness of the layer is much smaller than the radius of the fiber cross section. The problem of the transition of the liquid configuration from the state of a stationary cylindrical layer to the hydrodynamic state in the form of a regular sequence of drops is formulated. In this formulation, a nonlinear parabolic equation that describes the evolution of the local thickness of the layer on the time interval to the instant of drop formation is derived. The effect of the key parameters on the capillary instability is analyzed based on the linearized version of the resultant equation and the linearized electrostatic problem of calculating the field perturbations.     

On calculating the oscillations of a viscous liquid layer over a solid spherical core in terms of the boundary layer theory

The theory of a boundary layer near the periodically oscillating free surface of a spherical viscous liquid layer over a solid core (bottom) is modified. Two boundary layers are considered to adequately describe a liquid viscous flow in the system: one at the free surface of the liquid and the other at the solid bottom. The thicknesses of the boundary layers are estimated, which provide any given discrepancy between an exact solution to the model problem and a solution obtained in the small viscosity approximation. Taking into account the boundary layer near the solid bottom is shown to be significant only for lower oscillation modes. For higher modes, the flow near the core can be considered potential. In the case of lower modes and shallow liquid, the surface and bottom boundary layers overlap and an eddy flow occupies the entire volume of the liquid.     

Surface oscillations and the surface thickness of classical and quantum droplets

We calculate the contribution from surface oscillations to the surface thickness of a liquid drop. The results are compared to those for a surface made plane by gravity. We find that the zero-point motion of the surface modes gives a contribution to the surface thickness which is independent of the drop radius R, and is in good agreement for liquid He⁴ with other estimates of the surface thickness. For classical droplets, the mean square displacement of the surface varies as log R, in accord with the plane surface results.     

Properties of the adsorption layer in a binary liquid mixture near the critical point

A method for the determination of parameters of the surface layer (thickness and refraction index) at the liquid-vapor interface in binary liquid mixtures was developed. The parameters of the surface layer for the C₇H₁₄-C₇F₁₄ liquid mixture in the vicinity of the critical point, which was studied by means of ellipsometry in [1], were calculated using the proposed method. The temperature dependences of the thickness and refraction index that were determined in the homogeneous isotropic layer approximation at the interface liquid-vapor, show structural peculiarities that were not observed earlier. Their appearance is explained by the possible influence of hydrodynamic processes at the boundary.     

Incorporation of film theory in single droplet combustion model for prediction of precursor release in flame spray pyrolysis

The precursor release rate during flame synthesis has been shown to influence the uniformity of synthesized particles. However, its quantification through single droplet combustion modeling was based on its immediate release from the droplet without considering the effect of the mass boundary layer surrounding the droplet. Here, the film theory is applied with the single droplet combustion model to understand the precursor release in the droplet. The resulting mass boundary layer thickness is coupled with droplet temperature to qualitatively investigate precursor release in flame spray pyrolysis. It is shown that small droplets can enhance the precursor release rate due to their small mass boundary layer thickness and higher heating rate. Increasing the EHA content in the EHA/toluene solvent mixture reduces the mass boundary layer thickness and increases the droplet temperature due to EHA's low specific heat capacity. Using droplet sizes estimated by the phase Doppler interferometry, the model shows that the temporal droplet temperature profile remains relatively constant for six synthesis conditions. Concurrently, the mass boundary layer thickness is increased when the liquid and oxygen flow rates are reduced, and atomizing pressure drop is enhanced, resulting in the overall suppression of precursor release from the droplet and consequently increased formation of smaller-size primary particles. Insight into the relative tendencies of the pure gas-to-particle formation route during the synthesis was also derived as a function of the synthesis conditions. This new methodology for the characterization of precursor release is essential for a more accurate understanding and design of homogeneous nanomaterial using flame spray pyrolysis.     

Shielding of Prof-Fan cabin noise by the fuselage boundary layer

Recent flight tests of a Prop-Fan (advanced technology turboprop) model mounted on a business aircraft revealed noise levels on the fuselage surface considerably lower than was expected from theoretical calculations and other test experience. In this paper the role of the fuselage boundary layer in shielding the surface from noise via classical refraction effects is examined. In order to study the physical phenomenon with minimum mathematical complexity, the boundary layer is modeled by concentrating its shear in an infinitesimally thin layer displaced an effective boundary layer thickness from a rigid wall. Incident energy is assumed to arrive in plane waves. For the Prop-Fan model at its design cruise Mach number of 0·8, the theory indicates a strong shielding effect. The shielding diminishes at lower flight Mach number and for larger sound wavelengths. At full scale, the shielding effects will be less than in model scale because the wavelength of the dominant noise is larger relative to the fuselage boundary layer thickness.     

LES of spatially developing turbulent boundary layer over a concave surface

We revisit the problem of a spatially developing turbulent boundary layer over a concave surface. Unlike previous investigations, we simulate the combined effects of streamline curvature as well as curvature-induced pressure gradients on the turbulence. Our focus is on investigating the response of the turbulent boundary layer to the sudden onset of curvature and the destabilising influence of concave surface in the presence of pressure gradients. This is of interest for evaluating the turbulence closure models. At the beginning of the curve, the momentum thickness Reynolds number is 1520 and the ratio of the boundary layer thickness to the radius of curvature is δ₀/R = 0.055. The radial profiles of the mean velocity and turbulence statistics at different locations along the concave surface are presented. Our recently proposed curvature-corrected Reynolds Averaged Navier-Stokes (RANS) model is assessed in an a posteriori sense and the improvements obtained over the base model are reported. From the large Eddy simulation (LES) results, it was found that the maximum influence of concave curvature is on the wall-normal component of the Reynolds stress. The budgets of wall-normal Reynolds stress also confirmed this observation. At the onset of curvature, the effect of adverse pressure gradient is found to be predominant. This decreases the skin friction levels below that in the flat section.     

声悬浮条件下环己烷液滴的蒸发凝固

采用单轴式声悬浮方法研究了环己烷液滴的蒸发过程,发现环己烷液滴的蒸发可以使自身温度降至熔点以下并发生凝固.高速摄像实时观测表明,环己烷晶核开始形成于液滴赤道附近,并以枝晶方式长大,平均生长速度为12.5-160.4 mm/s.进一步研究发现,声悬浮条件下平均Sherwood数与平均Nusselt数的比值Sh/Nu是在自然对流条件下的1.3倍,这表明声流边界层有效提高了环己烷液滴的蒸发速率而对传热的促进作用相对较小,因而可以使液滴降至更低温度,进而发生凝固.据此,提出了挥发性液体在声悬浮条件下发生蒸发凝固的必要条件. 关键词： 声悬浮 声流 环己烷 蒸发凝固     

Partitioning of a heterotelechelic polystyrene to separate interfaces of thin films

Heterotelechelic deuteropolystyrenes have been synthesised with a tertiary amine functionality at one end and a fluorocarbon group at the other end of the polymer chain. A layer of this polymer, circa 120 ? thick, has been attached to the surface of a silicon substrate and subsequently covered with a much thicker layer of hydrogenous polystyrene. The combination has then been annealed at 413 K under vacuum for defined times and the subsequent distribution of the deutero heterotelechelic polymer determined using nuclear reaction analysis and neutron reflectometry. The influences of annealing time, molecular weight and thickness of the hydrogenous polymer have been examined. Nuclear reaction analysis showed that an excess of the heterotelechelic polymer formed at both interfaces with a larger excess remaining at the substrate-polymer interface. When the molecular weight of the hydrogenous polymer is lower than that of the deuteropolymer, the deutero layer is initially swollen by the hydrogenous polymer but the thickness then decreases as deutero polymer becomes detached from the silicon substrate and an additional excess layer is eventually formed at the vacuum-polymer surface. When the molecular weight of the hydrogenous polymer is higher, there is an initial shrinkage of the deuteropolymer layer, but the original thickness (∼ radius of gyration of the deuteropolymer) is regained on prolonged annealing. There is no evidence for bridging between the two interfaces by the heterotelechelic polymer. After five days annealing the volume fraction distribution of the deuteropolymer at the silicon substrate was well described by a self-consistent field model where the only adjustable parameter was the sticking energy of the tertiary amine group to the silicon substrate for which a value of 8k _B T was obtained. Comparison of the dependence of the equilibrium layer thickness of the deuteropolymer on the equilibrium grafting density at the silicon surface with the predictions of scaling theory for brush-like polymer layers suggested that the grafted molecules were in the ideal, unperturbed brush region. Received 12 October 2000 and Received in final form 27 March 2001     

Features of atomic and electronic structure of oxides on porous silicon surface according to XANES data

Based on synchrotron research of the fine structure main parameters of SiL _{2, 3} X-ray absorption edges (X-ray absorption near edge structure (XANES)) in porous silicon on boron-doped Si(100) wafers, the thickness of the surface oxide layer and the degree of distortions of the silicon-oxygen tetrahedron in this layer were estimated. The thickness of the oxide layer formed on the amorphous layer coating nanocrystals of porous silicon exceeds the thickness of the native oxide on the surface of Si(100) : P and Si(100) : B single-crystal (100) silicon wafers by several times. Distortion of the silicon-oxygen tetrahedron, i.e., the basic unit of silicon oxide, is accompanied by Si-O bond stretching and an increase in the angle between Si-O-Si bonds.     

Large deformation of liquid capsules enclosed by thin shells immersed in the fluid

The deformation of a liquid capsule enclosed by a thin shell in a simple shear flow is studied numerically using an implicit immersed boundary method. We present a thin-shell model for computing the forces acting on the shell middle surface during the deformation within the framework of the Kirchhoff–Love theory of thin shells. This thin-shell model takes full account of finite-deformation kinematics which allows thickness stretching as well as large deflections and bending strains. For hyperelastic materials, the plane-stress assumption is used to compute the hydrostatic pressure and the incompressibility condition yields the thickness strain component and the corresponding change in the thickness. The stresses developing over the cross-section of the shell are integrated over the thickness to yield the stress and moment resultants which are then used to compute the forces acting on the shell middle surface. The immersed boundary method is employed for calculating the hydrodynamics and fluid–structure interaction effects. The location of the thin shell is updated implicitly using the Newton–Krylov method. The present numerical technique has been validated by several examples including an inflation of a spherical shell and deformations of spherical and oblate spheroidal capsules in the shear flow.     

Effect of the marangoni convection on the injection mechanism of instability

Electroconvective instability of a nonisothermal layer of a weakly conductive liquid with a free boundary whose surface tension depends linearly on temperature is considered for the case where charge injection is performed through this surface. When calculating the unperturbed stationary distribution of the charge and field, we supposed that the injector is separated from the liquid by an air gap of finite thickness. It was, however, assumed when analyzing the stability of the system that the motion in the air gap has no effect on the motion in the liquid phase and the disturbances of the electric field and charge in the air gap decay rapidly because of its high conductivity.