Liquid Wedge Research Articles

Splash jet generated by collision of two liquid wedges

AbstractA complete nonlinear self-similar solution that characterizes the impact of two liquid wedges symmetric about the velocity direction is obtained assuming the liquid to be ideal and incompressible, with negligible surface tension and gravity effects. Employing the integral hodograph method, analytical expressions for the complex potential and for its derivatives are derived. The boundary value problem is reduced to two integro-differential equations in terms of the velocity modulus and angle to the free surface. Numerical results are presented in a wide range of wedge angles for the free surface shapes, streamline patterns, and pressure distributions. It is found that the splash jet may cause secondary impacts. The regions with and without secondary impacts in the plane of the wedge angles are determined.

Opto-Mechanical Structure Design of the Image Stabilization Collimator

The collimator has an extensive application as an experiment equipment that can provide an infinity goal in the instrument alignment and testing.This paper provides a new type of collimator with image stabilization function in consideration of human disturbance factors, including environment micro-vibration and so on. Liquid surface which appears natural level under gravity imaging optical path of collimator can emerge through liquid refraction and reflex later. And we should make use of this phenomenon. Mathematical modeling and analysis on the principle of dynamic imaging of the liquid wedge have been made to design several optical principle charts of the collimator with image stabilization function.One of the image stabilization optical principle structure styles has been designed based on its mechanical structure.An experiment prototype is produced and is made more practical.

Exact solutions of the problem of free-boundary unsteady flows

Some approach to the solution of boundary value problems for finding functions that are analytical in a wedge is proposed. If the ratio of the angle at the wedge vertex to the number π is rational, then the boundary value problem is reduced to the finite system of ordinary differential equations. Such approach, applied to the problem of inertial motion of a liquid wedge, made it possible to sum the series with small denominators arising in the problem and find four exact examples of self-similar flows with a free boundary.

Asymmetric impact between liquid and solid wedges

The hydrodynamic problem of impact between a solid wedge and a liquid wedge is analysed. The liquid is assumed to be ideal and incompressible; gravity and surface tension effects are ignored. The flow generated by the impact is assumed to be irrotational and therefore can be described by the velocity potential theory. The solution procedure is based on the analytical derivation of the complex-velocity potential in a parameter plane and the function mapping conformally the parameter plane onto the similarity plane. The mapping function is found as a combination of the derivatives of the complex potential in the similarity and parameter planes, through the integral equations for mixed and homogeneous boundary-value problems in terms of the velocity modulus and the velocity angle with the fluid boundary, together with the dynamic and kinematic boundary conditions. These equations are solved through a numerical method. The procedure is first verified through comparisons with some known results. Simulations are then made for a variety of cases, and detailed results are presented in terms of the free surface shape, streamlines, pressure distribution on the wetted solid surface, and contact angles between the free surface and the body surface.

Multiple-beam interference fringes applied to investigate the index profile of the optical fiber dip

A numerical-based phase shift method is presented to study optical fibers having a dip in the refractive index profile at the center of the core. Mathematical expressions for the shape of multiple-beam Fizeau fringes crossing a graded-index optical fiber (GRIN) comprising a central graded-index dip or GRIN fiber with a constant refractive index dip, immersed in a silvered liquid wedge, are derived. The index profile parameters of the central dip are determined by comparing the theoretical fringe shape across the fiber cross-section and the experimental fringe shape obtained from micro-interferograms of the wedge interferometer.

Under-ice seiches

A closed, ice-covered water body containing water with homogeneous density distribution is considered. No-slip conditions are specified for flow velocity at the lower ice boundary and on the bed. Two variants of boundary conditions are considered on the side boundary: the boundary is either a solid vertical wall with a finite liquid depth or the liquid wedges out to zero depth values. Ice either is attached fast to the shore or is separated from it by an open-water zone. A basic fourth order equation is derived for the amplitude of ice oscillations. The introduction of friction results in the appearance of reduced depth. Eigenvalue problem is considered for evaluating seiche periods. For the case when the liquid wedges out at the shore, the basic equation becomes singular at water body boundaries. A lake with a longitudinal depth profile, which can be approximated by a parabola, is considered. The solution is found by the method of matched asymptotic expansions. In the inner domain, beyond the boundary layers, the equation is reduced to Legendre equation, which yields a new relationship for the spectrum of seiche oscillations both in the presence of ice and in an open lake with varying depth. Boundary layers appear at the margins of the lake, where the liquid wedges out; a solution is found for these layers.

Development of ultrasonic waveguide sensor for under-sodium inspection in a sodium-cooled fast reactor

The reactor core and in-vessel structures of a sodium-cooled fast reactor (SFR) cannot be examined visually due to the opaque sodium. The examination of the in-vessel structures is possible using ultrasonics to penetrate the sodium. A plate-type ultrasonic waveguide sensor using a leaky Lamb wave (A 0 mode) has been developed for under-sodium visual inspection of the reactor core and in-vessel structures. In the plate waveguide sensor, the A 0 leaky Lamb wave is utilized for the single mode generation and the effective radiation capability in a fluid. The liquid wedge is applied for the generation of the A 0 mode in the low frequency range. The long pulse tone-burst excitation should be applied to minimize the dispersion effect in 10 m long distance propagation of the A 0 Lamb wave. And a novel technique which is capable of steering a radiation beam of a waveguide sensor without a mechanical movement of the waveguide sensor has been suggested. The control of the beam angle can be achieved by a frequency tuning method of the excitation pulse in the dispersive low frequency range of the A 0 Lamb wave. A 10 m long ultrasonic waveguide sensor module which consists of a plate waveguide, a liquid wedge, an ultrasonic sensor, and an acoustical shielding protection tube has been designed and manufactured. The possibility of applying the ultrasonic waveguide sensor module to an under-sodium visual inspection has been investigated. The experimental tests such as the long distance propagation test of A 0 Lamb wave, the beam profile measurements, and C-scanning experiments in water have been carried out for the performance of the ultrasonic waveguide sensor. The feasibility of the ultrasonic waveguide sensor technique has been successfully demonstrated.

Assessment of fatigue damage in solid plates through the ultrasonic Lamb wave approach

Changes (degradations) in the mechanical properties of solid plates induced by cyclic fatigue loading will influence the features of ultrasonic Lamb wave propagation, such as dispersion and attenuation. This paper has qualitatively analyzed the feasibility of using the amplitude-frequency characteristics and the stress wave factors (SWFs) of ultrasonic Lamb wave propagation to assess fatigue damage in solid plates. Liquid wedge transducers located on the surface of solid plates tested are used to generate and detect the Lamb wave signals. Based on the Ritec-SNAP ultrasonic measurement system, the experimental setup for assessing the degree of fatigue damage in solid plates using ultrasonic Lamb wave approach has been established. For several rolled aluminum sheets subjected to tension-tension cyclic loading, the experimental examinations have been performed for the relationships between the amplitude-frequency characteristics of ultrasonic Lamb wave propagation and the numbers of loading cycles (denoted by N), as well as the correlations between the Lamb wave SWFs and N. The experimental results show that the Lamb wave SWFs decrease monotonously and sensitively with the increment of cycles of fatigue loading. Based on the correlations between the Lamb wave SWFs and N, it is further verified that ultrasonic Lamb wave propagation combined with the Lamb wave SWFs can be used to effectively assess early fatigue damage in solid plates.

Entrainment of a film on a surface from the meniscus of a liquid wedge during coating

The shape evolution of an entrained film from the meniscus of a liquid wedge is studied, both experimentally and theoretically. The liquid wedge is formed by a droplet of liquid injected between a substrate and a tilted plate. When the substrate moves relative to the tilted plate with a constant velocity, a film of a constant slope is entrained on it, while another film remains on the tilted plate. The numerical and analytical investigation of the process provides the dependence of the length and slope of the entrained film after the end of drawing process, as well as the maximum thickness of the film on the tilted plate, on the capillary number. The length of the entrained film was found to be minimal for infinitely large capillary numbers when the surface tension effects are negligibly small. Experimental data confirm the predicted characteristic geometry of the film for capillary numbers up to 0.75.

Wave propagation in a fluid wedge over a solid half-space – Mesh-free analysis with experimental verification

Interaction between a bounded ultrasonic beam and a liquid wedge over a solid half-space is studied. A semi-analytical technique called distributed point source method (DPSM) is adopted for modeling the ultrasonic field in a wedge shaped fluid structure over a solid half space. This study is important for analyzing the ultrasonic waves used for the non-destructive inspections of partially immersed structures. It is also useful for studying the effect of underwater ultrasonic or acoustic wave experiments on marine lives near the shore. The problem geometry considers a bounded acoustic beam striking a fluid–solid interface between a fluid wedge and a solid half-space at steady-state. Solution of this problem is beyond the scope of the currently available analytical methods when the beam is bounded. Only numerical method (boundary element method (BEM) or finite element method (FEM)) based packages (e.g. PZFlex) are in principle capable of modeling ultrasonic fields in such structures. At high frequencies FEM and BEM based packages require huge amount of computation memory and time for their executions that DPSM technique can avoid. Effect of the angle of strike and the fluid wedge angle variation on the wave propagation characteristics is studied. Theoretical predictions are compared with some experimental results.

Range and cross-range propagation effects in a liquid wedge overlaying an elastic bottom

The penetrable-wedge model, a constant density isospeed layer of fluid with a pressure-release horizontal surface and a sloping elastic bottom, provides an extensive insight into the role of the ocean bottom in acoustic propagation from an underwater source. In particular, it is a realistic model of a rock-bottom ocean near a shoreline that accounts for horizontal refraction and allows for a ground wave. This paper discusses some new results for small and large range propagation for two penetrable-wedge models: one where the shear wave speed in the bottom is lower than the sound speed in the fluid and the other where the shear wave speed is higher. The operational representation of the ray-integral solution for the acoustic field from a point source in a penetrable wedge of fluid is further developed to the stage at which one can compute the exact, other than the omission of diffraction at the wedge apex, pressure response curve, as recorded at a receiver, due to an arbitrary time variation of the pressure at the source in a 3° wedge. The three-dimensional propagation effects are examined for range transmission when the receivers are located up-slope and down-slope of the source, and for cross-range transmission when the receivers are located cross-slope of the source.

Modeling of the fringe shift in multiple beam interference for glass fibers

A quadratic model is suggested to describe the fringe shift occurred due to the phase variations of uncladded glass fiber introduced between the two plates of the liquid wedge interferometer. The fringe shift of the phase object is represented in the harmonic term which appears in the denominator of the Airy distribution formula of Fabry-Perot’s interferometer. A computer program is written to plot the computed fringe shifts of the described model.

Angle-Tunable Liquid Wedge Prism Driven by Electrowetting

We developed a tunable wedge prism by electrowetting actuation. Our prism can shift the field of view to the left or to the right and is useful for small cameras and endoscopes because its field of view can be changed without the need for large instruments. It consists of two plates that face each other, and a liquid is sandwiched between them. The liquid and plates form a wedge when the angle between the plates is changed. We used electrowetting to change the angle between the plates. The wedge prism is small because the liquid driven by electrowetting works as a prism. In addition, the prism produces a clear field of view because its optical flatness is determined by the flatness of the plates.

Two-dimensional liquid column and liquid droplet impact on a solid wedge

The hydrodynamic impact due to a two-dimensional (2D) liquid column or a 2D liquid droplet hitting on a solid wedge is analysed. The problem is solved using the complex velocity potential together with the boundary element method. A stretched coordinate system is used, which is defined through the ratio of the normal Cartesian coordinate system to an appropriately chosen time-varying length scale. Numerical simulations are first made for impact by a liquid wedge. The results from the time-domain method are found to be in a good agreement with the similarity solution. Simulations are also made for impact by an elliptic droplet. A condition for bisection of the droplet is introduced, which is found to provide stable and converged results.

Forced Detachment of Immersed Elastic Rubber Beads

We investigate the strength of adhesion and the dynamics of detachment of elastic beads (Young's modulus E approximately 1 MPa) adhering to a horizontal solid surface in a viscous liquid. The beads are initially compressed on the surface. Their unbinding is imposed by fast vertical stretching (above a certain threshold value). The decrease in the contact radius is monitored by interferential microscopy. We find that the dynamics of detachment involves three steps: (i) fast elastic decompression, (ii) slow adhesive detachment, and (iii) catastrophic rupture. They can be interpreted by a transfer of the Johnson Kendall Roberts (JKR) energy toward viscous losses in the liquid wedge, near the rubber/solid/liquid (R/S/L) contact line.

The radiation mode theory in ultrasonics

This paper describes the history and the state of the art in radiation mode theory (RMT) in ultrasonics. The RMT originates from electromagnetism in which it has proved to be very efficient in the field of wave guides and discontinuities. In ultrasonics, the RMT made its entrance only a decade ago and has already proved to be very efficient in describing the interaction of sound with discontinuities such as a step on a plate, a liquid wedge, the extremity of a plate and much more. It is likely that the development of the RMT for two-dimensional (2-D) isotropic media has come almost to an end. This paper lists the results obtained so far. Further extensions to more complicated media are to be expected in the coming decade.

Wetting morphologies at microstructured surfaces

The wetting of microstructured surfaces is studied both experimentally and theoretically. Even relatively simple surface topographies such as grooves with rectangular cross section exhibit a large variety of different wetting morphologies as observed by atomic force microscopy. This polymorphism arises from liquid wedge formation along the groove corners and from contact line pinning along the groove edges. A global morphology diagram is derived that depends only on two system parameters: (i) the aspect ratio of the groove geometry and (ii) The contact angle of the underlying substrate material. For microfluidics, the most interesting shape regimes involve extended liquid filaments, which can grow and shrink in length while their cross section stays essentially constant. Thus, any method by which one can vary the contact angle can be used to switch the length of the filament, as is demonstrated in the context of electrowetting.

Spreading of nanofluids driven by the structural disjoining pressure gradient.

This paper discusses the role of the structural disjoining pressure exerted by nanoparticles on the spreading of a liquid film containing these particles. The origin of the structural disjoining pressure in a confined geometry is due to the layering of the particles normal to the confining plane and has already been traced to the net increase in the entropy of the system in previous studies. In a recent paper, Wasan and Nikolov (Nature, 423 (2003) 156) pointed out that the structural component of the disjoining pressure is strong enough to move a liquid wedge; this casts a new light on many applications-most notably, detergency. While the concept of spreading driven by the disjoining pressure is not new, the importance of the structural disjoining pressure arises from its long-range nature (as compared to the van der Waals' force), making it an important component of the overall force balance near the contact line. In this paper, we report on a parametric study of the spreading phenomena by examining the effects of nanoparticle size, concentration and polydispersity on the displacement of an oil-aqueous interface with the aqueous bulk containing nanoparticles. The solution of the extended Laplace-Young equations for the profile of the meniscus yields the position of the nominal contact line under the action of the structural disjoining pressure. Simulations show that the displacement of the contact line is greater with a high nanoparticle volume fraction, small particles for the same volume fraction, monodispersed (in size) particles rather than polydispersed particles and when the resisting capillary pressure is small, i.e., when the interfacial tension is low and/or the radius of the dispersed phase drop/bubble is large.

An exact representation of the image field in a perfect wedge

A solution is presented for the image component of the three-dimensional (3-D) acoustic wave-field from an impulsive point source in a liquid wedge with impenetrable boundaries (mixed Dirichlet-Neumann) modeling shallow water over sloping bottom (shallow-water wedge). This exact and complete solution is in the form of a sum of partial waves including the pulse emitted from the source plus a finite number of pulses reflected off the wedge boundaries, and it is generally valid for all ranges and depths. The 3-D acoustic propagation effects in an impenetrable wedge are examined for range propagation when the receivers are located directly up-slope and down-slope of the source and at the source-point itself, and for cross-range propagation when the receivers are located directly cross-slope of the source. When the receiver recedes up-slope from the source, the duration of the pressure response is shortened, and the number of pulses arriving along the indirect ray paths (the backscattered pulses) decreases. When the receiver recedes down-slope from the source, the response duration is constant, and the number of backscattered pulses also decreases. As the source and the receiver separate in the cross-range-direction, the response duration is so shortened that the arrival of the last-arriving pulse approaches that of the first-arriving pulse in the limit (i.e., when the separation between the source and the receiver is “large” in the cross-range-direction), and the number of backscattered pulses remains constant.

A new solution for the acoustic pressure‐field in a perfect wedge

AbstractA time domain analytical solution is presented for the acoustic pressure‐field due to an impulsive point source in a liquid wedge with a rigid sloping bottom. The horizontal boundary of the wedge is a pressure‐release surface. The wedge is isovelocity, and it is three‐dimensional (3‐D) in that the source and the receiver are not restricted to the plane perpendicular to the wedge apex. This 3‐D wedge with perfectly reflecting (i.e., impenetrable) boundaries, the 3‐D perfect wedge, is the simplest theoretical model of a shallow water environment in which the ocean bottom has a constant slope, but it is inadequate in that it cannot account for acoustic penetration of the bottom typical for a real ocean floor.In a perfect wedge, the acoustic field produced by a point source consists of two components: the diffraction field due to scattering at the apex, which under certain conditions vanishes [1], and the image field including the wave emitted by the source and a number of waves reflected off the wedge boundaries. These waves with spherical wave fronts propagate along the stationary time paths (the ray paths). The repeated reflections from the sloping bottom introduce the curvature into the projection of the ray path onto the horizontal boundary, provided that the first segment of this path originating at the source has a cross‐slope component. This path curvature in the horizontal is known as “horizontal, or bathymetric refraction” [2]. The ray path that leaves the source propagating up‐slope may be turned around to reach the receiver propagating back down‐slope. This change in propagation direction, known as “backscattering” [2], is due to multiple bottom reflections. The receiver can thus be reached along one of the two ray paths: the direct path or the indirect path which is turned around on approaching the apex. The image field in the wedge is thus three‐dimensional, i.e., it depends on three spatial coordinates: the range (normal to the apex), the cross‐range (parallel to the apex), and the depth coordinate.