Liquid Wedge Research Articles

Fizeau interferometry-based measurement of the photo-elastic coefficient and the cut-off wavelength in bent standard single-mode optical fiber

New technique based on multiple-beam Fizeau fringes to study the influence of bending on a standard single-mode optical fiber which has a total minimum dispersion at 1.31 μm, is described. The technique is used to determine some of the single-mode optical fiber properties such as the photo-elastic coefficient of the cladding material. Estimation of the cut-off wavelength is needed to support the propagation of the fundamental HE ∥ mode in the bent fiber is investigated. The mathematical analysis to describe the fringe shifts in case of a matching and a non-matching liquid wedge interferometer is given. Mathematical expressions to determine the photo-elastic coefficient and the cut-off wavelength from the interference fringe shift due to bending are derived. The determined value of the photo-elastic coefficient is 0.229, which is in good agreement with the literature. In a regime of curvature between 7.5 and 19 mm, three additional modes are allowed to be guided due to the decrease of the cladding index in the tensile region. Micro-interferograms are given for illustration.

Liquid Entrainment from the Meniscus of a Liquid Wedge by a Moving Horizontal Plate

A hydrodynamic model is proposed for smear formation during the drawing of a film from the meniscus of a limited-volume liquid wedge. It was assumed that the regime of the flow is capillary and the characteristic time of the film formation is small compared to the characteristic time of the change in the meniscus curvature. At small time intervals, the film thickness was determined according to the Landau–Levich–Derjaguin method. At large time intervals, the curvature of the meniscus is described by a linear differential equation for the dynamics of the liquid wedge volume. Under these conditions, the film thickness changes along the plate according to a linear law with the tangent coefficient depending on the capillary number according to a power law, and the profile of the longitudinal section of the smear is close to a right triangle.

Interferometric examination of the effect of bending on some single-mode fibre parameters

The numerical aperture, the acceptance angle, the dispersion per unit length and the modal dispersion are important parameters in optical fibre characterization. Optical anisotropy produced in fibres undergoing bending gives valuable information for the characterization of these fibres. The present work provides a sensitive and analytical new technique to examine the influence of bending on single-mode optical fibres measuring, with a high spatial resolution, the radial changes in the previous fibre parameters due to bending. Multiple-beam Fizeau fringes crossing perpendicularly a bent fibre immersed in a matching silvered liquid wedge have been used successfully to determine the variation in the fibre index profile as a function of the fringe shift with accuracy 1 × 10−4. Mathematical expressions are given utilizing the measured fringe shift to characterize the bent fibre and to determine its radial parameter profiles.

Interferometric determination of gamma radiation effects on optical parameters of a GRIN optical fibre

Multiple-beam white-light interference fringes across an irradiated GRIN fibre, of length 2 cm and immersed in a liquid wedge interferometer that is followed by a grating spectrograph, are produced. These fringes are used to investigate the effect of gamma radiation, with different doses from 0.01 to 1 MGy, on the optical parameters of both the cladding (pure silica) and the core (germanium-doped silica) of the GRIN fibre. Optical parameters such as refractive index, refractive index difference, numerical aperture, V-number, index profile, chromatic dispersion and zero-dispersion wavelength are determined and studied. A two-term Sellmeier dispersion relation is adopted and fitted to the experimental data of the refractive index. Then, a dose-dependent Sellmeier formula is deduced to represent the effect of gamma radiation on different optical parameters.

Application of fringes of equal chromatic order for investigating the effect of temperature on optical parameters of a GRIN optical fibre

Multiple-beam white light interference fringes across a GRIN optical fibre immersed in a liquid wedge interferometer placed in front of a grating spectrograph are produced. They are used to investigate the effect of temperature on the optical fibre parameters of both the cladding (pure SiO2) and the core (GeO2-doped SiO2). A two-term Sellmeier dispersion relation is adopted and fitted to the experimental refractive index data. Then a temperature-dependent Sellmeier formula is deduced to study the effect of temperature on the refractive index of the fibre and its dispersion in addition to all of its optical parameters. These parameters are the average energy gap, the variation of the thermo-optic coefficient with photon energy, the refractive index difference, the numerical aperture, the normalized frequency (V-number), the number of modes, the material dispersion and the zero-dispersion wavelength.

On the profile of the liquid wedge underneath a growing vapour bubble and the reversal of the wall heat flux

Combining mass and energy balances, a differential equation for the profile of a liquid wedge underneath a vapour bubble that is growing on a solid surface is derived. It connects the spatial and temporal changes of the film (wedge) thickness with the spatial temperature changes and velocity of liquid at the interface. Specifying particular conditions, the equation reduces to those from the literature. The paper brings further an illustrative explanation of why the wall heat flux in the wedge region may reverse its direction.

Dissipation in dynamics of a moving contact line.

The dynamics of the deformations of a moving contact line is studied assuming two different dissipation mechanisms. It is shown that the characteristic relaxation time for a deformation of wavelength 2pi/|k| of a contact line moving with velocity v is given as tau(-1)(k)=c(v)|k|. The velocity dependence of c(v) is shown to depend drastically on the dissipation mechanism: we find c(v)=c(v=0)-2v for the case in which the dynamics is governed by microscopic jumps of single molecules at the tip (Blake mechanism), and c(v) approximately c(v=0)-4v when viscous hydrodynamic losses inside the moving liquid wedge dominate (de Gennes mechanism). We thus suggest that the debated dominant dissipation mechanism can be experimentally determined using relaxation measurements similar to the Ondarcuhu-Veyssie experiment [T. Ondarcuhu and M. Veyssie, Nature 352, 418 (1991)].

Three-dimensional refractive index profile of a GRIN optical waveguide using multiple beam interference fringes

Multiple beam Fizeau fringes across a GRIN fiber immersed in a silvered liquid wedge have been used to obtain a three-dimensional refractive index profile of optical fiber core. An image processing system has been used to analyze the fringe shape across the fiber cross-section. A simple mathematical approach has been applied utilizing the resulting experimental data to estimate the fiber three-dimensional refractive index profile. The tomographic back projection technique has been applied.

Contact angles for evaporating liquids predicted and compared with existing experiments

The stationary meniscus of an evaporating, perfectly wetting system exhibits an apparent contact angle Θ which vanishes with the applied temperature difference ΔT, and is maintained for ΔT > 0 by a small-scale flow driven by evaporation. Existing theory predicts Θ and the heat flow q∗ from the contact region as the solution of a free-boundary problem. Though that theory admits the possibility that Θ and q∗ are determined at the same scale, we show that, in practice, a separation of scales gives the theory an inner and outer structure; Θ is determined within an inner region contributing a negligible fraction of the total evaporation, but q∗ is determined at larger scales by conduction across an outer liquid wedge subtending an angle Θ. The existence of a contact angle can thus be assumed for computing the heat flow; the problems for Θ and q∗ decouple. We analyse the inner problem to derive a formula for Θ as a function of ΔT and material properties; the formula agrees closely with numerical solutions of the existing theory. Though microphysics must be included in the model of the inner region to resolve a singularity in the hydrodynamic equations, Θ is insensitive to microphysical detail because the singularity is weak. Our analysis shows that Θ is determined chiefly by the capillary number Ca = μlVl/σ based on surface tension σ, liquid viscosity μl and a velocity scale Vl set by evaporation kinetics. To illustrate this result of our asymptotic analysis, we show that computed angles lie close to the curve Θ = 2.2Ca1/4; a small scatter of ±15% about that curve is the only hint that Θ depends on microphysics. To test our scaling relation, we use film profiles measured by Kim (1994) to determine experimental values of Θ and Ca; these are the first such values to be published for the evaporating meniscus. Agreement between theory and experiment is adequate; the difference is less than ±40% for 9 of 15 points, while the scatter within experimental values is ±25%.

Application of the Method of Generalized Rays to Acoustic Waves in a Liquid Wedge Over Elastic Bottom

A range-dependent problem of shallow water acoustics is modeled by a liquid wedge over fast-speed elastic substratum, and the solution is derived based on the method of generalized rays, where the total acoustic field is decomposed into a sum of partial waves, comprising the wave emitted from the point source and the multi-reflected waves. The contribution of each partial wave at a given time is represented by a single integral. The diffracted waves arising from scattering at the apex of the wedge are neglected in this otherwise exact solution. The pressure records are evaluated at the downslope receiver when the pressure pulse applied at a point situated within the wedge has impulsive or harmonic time history. It was found that the arrival of the source pulse is preceded by the group of head wave arrivals, and this early portion of the record (the ground wave) includes considerable pressure build-ups. For a Heaviside source-pulse, we have found that the pressure records of the partial waves exhibit a singular behavior such as the logarithmic singularity at the arrival time of the totally reflected spherical wave. This singular behavior was not found for less sharp input, that is, a triangular source-pulse. We have also found that for a causal harmonic input of lower frequency the ground wave stands out very clearly, whereas for that of higher frequency it is almost suppressed, and for both frequencies the response becomes more or less steady-state soon after the arrival of a few partial waves.

APPLICATION OF THE METHOD OF GENERALIZED RAYS TO ACOUSTIC WAVES IN A LIQUID WEDGE OVER ELASTIC BOTTOM

A range-dependent problem of shallow water acoustics is modeled by a liquid wedge over fast-speed elastic substratum, and the solution is derived based on the method of generalized rays, where the total acoustic field is decomposed into a sum of partial waves, comprising the wave emitted from the point source and the multi-reflected waves. The contribution of each partial wave at a given time is represented by a single integral. The diffracted waves arising from scattering at the apex of the wedge are neglected in this otherwise exact solution. The pressure records are evaluated at the downslope receiver when the pressure pulse applied at a point situated within the wedge has impulsive or harmonic time history. It was found that the arrival of the source pulse is preceded by the group of head wave arrivals, and this early portion of the record (the ground wave) includes considerable pressure build-ups. For a Heaviside source-pulse, we have found that the pressure records of the partial waves exhibit a singular behavior such as the logarithmic singularity at the arrival time of the totally reflected spherical wave. This singular behavior was not found for less sharp input, that is, a triangular source-pulse. We have also found that for a causal harmonic input of lower frequency the ground wave stands out very clearly, whereas for that of higher frequency it is almost suppressed, and for both frequencies the response becomes more or less steady-state soon after the arrival of a few partial waves.

The flow and heat transfer in the wedge-shaped liquid film formed during the growth of a vapour bubble

After providing simple equations that should govern the behaviour of a wedge-shaped liquid film sandwiched between the vapour in a growing bubble and the heated wall, we discuss some questions that may be important for modelling heat transfer in nucleate boiling. Considered are the curvature of the film surface and the liquid flow in the film, for example. We further examine the consequences of some simplifications adopted in models from the literature. Our analysis indicates that the bubble interface is concave-convex in the region of the three-phase-line (TPL). The convex curvature is caused by strong evaporation at the tip of the liquid wedge under no slip condition of the liquid on the wall surface. The ultra-thin layer of this wedge adhering to the wall is practically motionless and hence incapable of supplying the quantity of liquid to the film tip necessary for evaporation to occur rapidly at a fixed TPL. Based on this fact, we are led to the conclusion that, under common boiling conditions, the TPL and the film interface are not motionless, as was assumed in very recent models of bubble growth. These models are shown in the present paper to be inconsistent. Namely, the liquid flow orthogonal to the heated surface is ignored within the hydrodynamics, but this flow is indeed included into the energy balance of the models.

Dynamique d'étalement d'un liquide rhéofluidifiant

The rate of spreading of a drop on a rigid substrate is governed by viscous dissipation in the liquid, the capillary driving force being compensated by the braking force resulting from viscous shearing in the liquid. In the case where the liquid is not newtonian but shear thinning or pseudoplastic, a deviation from the law of P.-G. de Gennes is observed, in particular a slower spreading kinetics corresponding to an increase of the liquid viscosity as the spreading speed decreases. In this study, this result is interpreted from the rheological behaviour of the non-newtonian liquid and a modified equation describing the spreading dynamics is proposed. This equation is obtained from two different calculations, either in considering an average viscosity or in expressing the distribution of speed in a shear thinning liquid wedge. The shape, slightly aspherical, of non-newtonian liquid drops having a size smaller than the capillary length, is also very simply interpreted, observing that the liquid viscosity increases from the edge to the center of drops during spreading, near the solid surface.

A liquid wedge as generating technique for Lamb and Rayleigh waves

A model is developed to describe the excitation of Lamb and Rayleigh waves by a Gaussian beam incident on a liquid/air/solid corner. Analytical expressions for the displacement fields of the generated modes are derived by applying a mode theory that accounts for the contribution of the reflected waves. Numerical computations illustrate the optimum conditions under which the strongest stimulation of Lamb and Rayleigh waves may be expected.

Determination of the material dispersion and the index profile parameters of a GRIN optical waveguide applying white light interference fringes

Multiple-beam white light fringes across a graded-index optical waveguide enclosed in a liquid wedge interferometer and followed by a grating spectrograph are obtained. An accurate approach for interferogram analysis and data reduction is applied. The material dispersions of the immersion liquid, the cladding and the core of the fibre across the visible spectrum are deducted. Also the index profile parameters of the fibre are evaluated. A modified single-term Sellmeier dispersion formula is adopted and fitted to the experimental data.

Modal analysis of Lamb wave generation in elastic plates by liquid wedge transducers

A modal analysis is presented to describe the excitation of Lamb waves in an elastic plate using a liquid wedge transducer. Analytical expression for the displacement of a given mode is derived for the excitation by a uniform bounded beam. In contrast to previous studies, the contribution of the reflected wave is included in the input exciting forces using a perturbation theory. The conversion efficiency, defined as the ratio of the guided mode power to the incident power, is related to a single parameter which depends on the rate of attenuation due to leakage from the guided wave into the liquid wedge. Numerical results relevant to the fundamental Lamb modes are obtained as a function of frequency for various incident beam widths and plate thickness. Using optical interferometric detection, direct measurements of the Lamb modes displacements have been carried out in aluminium plates to verify the theoretical analysis.

Simulation of plunging wave impact on a vertical wall

We present a numerical study of the impact of a two-dimensional plunging wave on a rigid vertical wall in the context of potential flow. The plunging wave impinging the wall is generated using a mixed-Eulerian-Lagrangian (MEL) boundary-integral scheme. The initial stage of the impact is characterized by an oblique impact of a liquid wedge on the wall and is solved using a similarity solution. Following the initial impact, the MEL simulation is continued to capture the transient impact process. The effect of an air cushion trapped between the plunger and the wall is considered. In addition to details such as temporal evolutions and surface profiles, the main interests are the maximum impact pressure on the wall and its rise time. To arrive at appropriate scaling laws for these, simulations are performed and correlations are explored for a broad range of local plunging wave kinematic and geometric parameters. To assess the present results, direct comparisons are made with the experiment of Chan & Melville (1988). Reasonable quantitative agreement is obtained and likely sources for discrepancies are identified and discussed.

Estimation of the index profile parameters of a GRIN optical waveguide and fringe synthesis applying multiple beam interference

Multiple beam Fizeau fringes across a GRIN fibre immersed in a silvered liquid wedge have been obtained using matching and non-matching liquids. An image processing system has been used to analyse the fringe shape across the fibre cross section. A simple mathematical approach has been applied, utilizing the resulting experimental data to estimate the fibre index profile parameters. A detailed synthesis of the experimentally secured fringes across a GRIN fibre has been obtained by analysing the fringe shape into its components. In the case of a matching liquid it is found to be composed of half an ellipse and a saddle, another half ellipse is added for non-matching liquids. This is achieved utilizing the index profile parameters estimated.

Multiple-beam interferometric studies on optical fibres

The fringe shape across a graded index (GRIN) fibre immersed in a silvered liquid wedge, when analysed to its elements, is found to be the result of two half ellipses and a saddle. For a step index fibre, the elements contributing to its fringe shape are merely two half ellipses. Using a matching liquid cancels the outer half ellipse in both cases. The central dip contributes an extra half ellipse and a saddle over the dip but in the reverse direction away from the wedge apex. Such analyses furnishes the basis of a method to deduce the characteristics of GRIN fibres, namely alpha and Delta n, which is tested on liquids of different refractive indices.

Dynamics of partial wetting

Two broad classes of models have been used to describe the motion of a contact line when the contact angle θ deviates from the equilibrium value θ e : a) an Eyring approach, emphasizing the microscopic jump of a single molecule at the tip. b) a hydrodynamic approach, concentrating on the viscous losses inside the liquid wedge of angle θ. In the present review, we compare the predictions from both models, for two critical experiments: 1) The pull out of a vertical plate from a fluid at rest -showing (for finite θ e) a critical velocity V c above which the plate is completely wet. 2) The velocity of growth of a dry patch for a non wettable surface covered by a flat liquid film -which turns out to vary like θ e 3 at small θ e. The net conclusion is that, at small θ e and for low velocities V, the hydrodynamic losses dominate, while at large θ e and large V, the molecular features are probably important.