Interaction of free-surface waves with a floating dock

A new method to describe the interaction of waves with a rigid or flexible dock, with zero draft, is derived. By means of Green's theorem an integral equation along the platform for either the velocity potential or the deflection is obtained. In the two-dimensional case this equation is solved by means of a superposition of exponential functions. With a specific choice of the Green function the integration with respect to the space coordinate can be carried out analytically. The integration left is the integration in the k-plane that occurs in the chosen Green function. Subsequently the contour of this integral is modified in the complex plane. This results at first in a dispersion relation for the phase functions in the expansion. Then the set of algebraic equations for the amplitude coefficients follows from the same singularity analysis in the complex plane. These equations are very simple and easy to solve. In contrast to the classical approach of eigen-mode expansions, there is no need to split the problem in a symmetric and antisymmetric one. An other advantage is that the transmission and reflection coefficients are determined seperately by means of Green's theorem, applied at the free surface in the far field. The method is first explained for the semi-infinite rigid dock, followed by the rigid strip, the moving strip and the flexible moving platform. In the appendix it is explained how to derive a set of algebraic equations in the case when the incident wave is not perpendicular to the strip.     

Generation of water waves and bores by impulsive bottom flux

The inviscid free-surface flow due to an impulsive bottom flux on constant depth is investigated analytically and numerically. The following classes of two-dimensional flow are considered: an upwelling flow which is uniform over a half-plane, a line source/sink, and a dipole aligned along the bottom. The bottom flux is turned on impulsively and may decay with time. The fully nonlinear problem is solved numerically. A small-time asymptotic expansion to third order is found for the nonlinear problem. An asymptotic large-time solution is found for the linearized problem. A steady source will generate a pair of symmetric bores, and their breaking is investigated. A steady sink generates a depression wave if it is weak, and dip instability if it is strong. Wave breaking will occur for intermediate sink strengths. A decaying source emits solitary waves.     

Nonlinear capillary free-surface flows

Capillary free-surface flows are considered. The fluid is taken to be inviscid and incompressible and the flow to be irrotational. Particular attention is devoted to two-dimensional flows for which the free surfaces intersect rigid walls. These include cavitating flows and local flows at the front of a small object (probe or insect) moving at the surface of a fluid. A general study of the effect of surface tension on the possible singularities which can occur at the separation points is presented. The results confirm and generalise previous findings on the subject.     

Interaction of free-surface waves with floating flexible strips

The method developed by the author to derive a set of algebaic equations to solve the interaction of free-surface waves with a single floating rigid or flexible two-dimensional platform with small draft is extended to the case that the platform consists of strips with different constant flexural rigidity and mass. The method is based on the application of Green's theorem, with a specific choice of the Green function to arrive at a differential-integral equation along the platform. This equation can be solved exactly by means of superposition of exponential functions, a standard method to solve a set of linear differential equations. After integration with respect to the space coordinate the residue theorem leads to both the dispersion relation along each individual strip and an algebraic equation for the coefficients. Due to very fast convergence with respect to the number of coefficients taken into account the series are truncated. Depending on the water-depth, in each series three to ten terms are taken into account. Results are shown for a structure consisting of several strips that are tightly connected and for disjoint strips. In the latter case the computation of the water level between the strips is also computed. The water level and the reflection and transmission coefficients are not unknowns in the algebraic equation, but are computed afterwards by means of Green's theorem.     

On the use of boundary-integral equation methods for unsteady free-surface flows

The Mixed Eulerian-Lagrangian Methods (MEL) forfree-surface potential flows solved by boundary-integral equations (BIEs) is considered, and the diffusion and dispersion errors are studied in the discrete linearized problem. The diffusion error is the base for the stability analysis of the scheme; both the errors give indications on the accuracy of the numerical solution. The study is divided into two steps: comparison of the discrete dispersion relation with the analytical solution and coupling with different time-integration schemes. In particular, a stability analysis of the Runge-Kutta and Taylor-expansion schemes, previously not given in the literature, is addressed. It is shown that MEL methods based on first- and second-order explicit Runge-Kutta and Taylor-expansion schemes are unstable, regardless of the technique adopted to discretize the BIEs. Higher-order Runge-Kutta and Taylor-expansion schemes lead to conditionally stable methods. Known results for explicit, implicit and explicit-implicit Euler schemes are recovered by the present analysis. The theoretical predictions of the errors are confirmed for two different boundary-element techniques: a high-order panel method based on B-Splines to solve for the velocity potential and a spectrally-accurate method based on the Euler-McLaurin summation formula to solve directly for the velocity field.     

Self-similar behaviour in the coagulation equations

This paper describes solutions to the Smoluchowski coagulation equations with power-law kernels in both constant-mass and constant-monomer cases. Exact solutions are obtained in special cases by a generating function approach. For more general kernels, the large-time behaviour is obtained by use of matched asymptotics. Numerical results are also given, which confirm the asymptotic analysis.     

Withdrawal from a fluid of finite depth through a line sink, including surface-tension effects

The steady withdrawal of an inviscid fluid of finite depth into a line sink is considered for the case in which surface tension is acting on the free surface. The problem is solved numerically by use of a boundary-integral-equation method. It is shown that the flow depends on the Froude number, F _B=m(gH ³ _B)^–1/2, and the nondimensional sink depth =H _S/H _B, where m is the sink strength, g the acceleration of gravity, H _B is the total depth upstream, H _S is the height of the sink, and on the surface tension, T. Solutions are obtained in which the free surface has a stagnation point above the sink, and it is found that these exist for almost all Froude numbers less than unity. A train of steady waves is found on the free surface for very small values of the surface tension, while for larger values of surface tension the waves disappear, leaving a waveless free surface. It the sink is a long way off the bottom, the solutions break down at a Froude number which appears to be bounded by a region containing solutions with a cusp in the surface. For certain values of the parameters, two solutions can be obtained.     

A new variant for measuring the surface tension of liquid metals and alloys by the oscillating drop method

The theoretical background of the oscillating drop technique for measuring surface tension is briefly presented and the different analysis procedures are cited. A new method is described for obtaining oscillation frequencies by fast fourier transformation (FFT) of the pyrometer voltage signals from temperature measurements at the top of the levitated sample. The results on the first experiments on liquid nickel are in a good agreement with the hterature data.Paper presented at the Fourth International Workshop on Subsecond Thermophysics. June 27–29, 1995, Köln, Germany.     

Waves and singularities in nonlinear capillary free-surface flows

The problem is a free-surface flow of a fluid, emerging from a semi-infinite container. The fluid is assumed to be inviscid, incompressible and the flow to be two dimensional and irrotational. When surface tension is neglected the free surface leaves the wall of the container tangentially. We show that when surface tension is taken into account, there is, in general, a train of waves on the free surface and a discontinuity in slope where the free surface separates from the wall of the container. These new solutions include, as particular cases, previously obtained solutions for which the free surface is waveless in the far field. Although the calculations are presented for a special flow configuration, the results are general and apply to other potential free surface flows where a free-surface intersects a rigid wall.     

The surface tension meniscus of liquid helium

A technique is described for calculating the shape of the surface tension meniscus at a vertical wall, in the presence of Van der Waals forces. The method of analytic continuation is used to compute a solution of the relevant differential equation. Results (believed accurate to better than 0.5%) are presented for He³ and for He⁴ at a variety of film flow rates. It is found that the shape of the classical meniscus remains unaltered over most of its length, but it is displaced horizontally relative to the wall by a flow dependent amount. The results are contrasted with those from a previous calculation.     

Surface tension of aqueous binary solutions

Measurements of surface tension have been performed to determine the effects of both temperature and concentration on the surface tension of aqueous solutions of D-Sorbitol, potassium chloride, and ammonium chloride. A differential capillary-rise method was employed for the measurements. The results showed that the surface tension of test solutions increased as the temperature decreased and that the surface tension of chloride solutions increased with an increase in its concentration, while for D-Sorbitol solution the surface tension decreased with increasing concentration. Correlation equations for the surface tension of three aqueous binary solutions as a function of temperature and concentration were determined.     

Surface tension of low-temperature aqueous solutions

Measurements of the surface tension have been carried out to determine the effects of both temperature and concentration on the surface tension of aqueous solutions of sodium chloride, propylene glycol, and ethylene glycol. A differential capillary-rise method was employed for the measurements. The results show that the surface tension of the ethylene glycol solution and the propylene glycol solution increases as the concentration of the solution decreases, while for the sodium chloride solution the surface tension increases monotonically as the concentration increases. The surface tension of the liquids was found to be an almost-linear function of temperature from 20°C to just above the freezing temperature. Equations for the surface tension of the three binary aqueous solutions as a function of temperature and concentration are presented.     

Interfacial tension of critical liquid mixtures in the presence of a surfactant

The interfacial tension of a liquid-liquid interface in the critical region has been obtained as a function of the surfactant concentration on the basis of the Landau-Ginzburg model for critical phenomena. It is shown that surfactants may reduce significantly the interfacial tension in addition to the well-known near-critical reduction.     

Free-surface flow of a fluid body with an inner circular cylinder in impulsive motion

An analytical theory and numerical computations are developed for the two-dimensional free-surface flow of an initially circular layer of inviscid fluid surrounding a rigid circular cylinder. The two cylinders are initially concentric. The fluid packet is released from rest and the flow suddenly starts forced by gravity and by the simultaneous impulsive motion of the inner body. A small-time expansion of the fully nonlinear free-surface problem is developed and a closed-form solution is found up to third order for an arbitrary radius of the rigid cylinder. For the gravitational flow around the body at rest, the solution is extended up to fourth order. Free-surface profiles and hydrodynamic forces on the cylinder are calculated and discussed against numerical solutions of the exact unsteady nonlinear problem. Some basic features, such as the formation of an almost uniform layer surrounding the upstream side of the body, are captured by the theory quite well and only later on in time significant quantitative differences appear. Similarly, the behaviour of hydrodynamic loads is rather well predicted during initial stages preceding larger fluctuations observed on a longer time-scale.     

On the fracture pressure of liquid metals

Fracture pressure of six liquid metals, Pb, Al, Cu, Ni, Ti and Fe have been calculated at their melting points in three different ways; (i) by extrapolating molecular dynamic (MD) data reported in the literature for various temperatures to their melting points, (ii) by using the fracture pressure equation developed by Fisher and (iii) by using the work of nucleation suggested by Fletcher. Results have shown that the Fisher equation and MDs estimates agree closely, whereas the ones based on the work of nucleation are systematically lower than the other two. In all cases, calculated fracture pressures are several orders of magnitude different from those assumed in the literature, emphasizing the extent of weakening by extrinsic factors in liquid metals.     

Surface tension of aqueous binary solutions at low temperatures

The paper is concerned with measurements of the surface tension of aqueous binary solutions at low temperatures. The effects of both temperature and concentration on the surface tension of CaCl₂, NaClO₃, and propylene glycol have been investigated. A differential capillary-rise method was employed for the measurements. The results showed that the surface tension of CaCl₂ and NaClO₃ increases monotonically as the concentration of the solution increases, while for the propylene glycol solution the surface tension decreases with increasing concentration. The surface tension of the testing liquids was found to be an almost-linear function of temperature from 20°C to just above the freezing temperature. Equations for the surface tension of the three aqueous binary solutions as a function of temperature and concentration are presented. Paper presented at the Fourth Asian Thermophysical Properties Conference, September 5–8, 1995, Tokyo, Japan.     

液体表面张力系数与浓度的关系实验研究

为探究液体表面张力与液体浓度的关系,利用压阻式力敏传感器,采用拉脱法,测试室温下水及不同浓度的蔗糖水、盐水、酒精、肥皂水的表面张力系数。实验结果表明:蔗糖水和盐水的表面张力系数随浓度的增大而增加,而酒精、肥皂水的表面张力系数随浓度的增大而减小。     

高温液相添加剂对纳米LiBr溶液表面张力和沸腾温度的影响

通过在纳米溴化锂(LiBr)溶液中添加表面活性剂降低溶液沸腾温度,从而有助于大幅度应用工业余热、废热等低品味热源。尽管影响沸腾温度的因素较多,但是溶液的表面张力是一个重要的因素,首先对异辛醇(C8H18O)、壬醇(C9H20O)和癸醇(C10H22O)3种高沸点液相添加剂对纳米LiBr溶液的表面张力的影响做了深入研究,并进一步研究了相对应的沸腾温度。并且通过实验发现随着烷醇中烷基数增加,表面张力更低,使得沸腾温度也进一步降低。     

液态纯铁1 550℃的粘度及表面张力与结构的相关性

用θ-θ液态金属X射线衍射仪研究了液态纯铁在1 550℃的结构,根据实验所得双体分布函数,利用Iida和Reynolds的公式对液态纯铁的动力粘度和表面张力进行了计算.结果表明,液态纯铁在1 550℃的动力粘度η=6.44mPa@s,表面张力γ=2.04N/m;根据公式r/η= ,液态纯铁的动力粘度和表面张力之间吻合较好;液体结构、动力粘度和表面张力三者间存在密切联系.