Natural frequencies and response of spinning liquid column with apparently sliding contact line

Summary The contact line of a liquid with a solid does in many cases—depending on the smoothness of the solid, the viscosity, the surface tension and the excitation force—apparently flow along the solid during oscillations. The influence of this effect upon the natural frequencies, the stability and the response of the system has been investigated at an oscillating and spinning cylindrical liquid column.List of symbols a radius of liquid bridge - h length of liquid bridge - I ₀,I ₁ modified Besselfunctions - J ₀,J ₁ Besselfunctions - p liquid pressure - r, ,z cylindrical polar coordinates - t time - u, v, w velocity distribution in rotating liquid - Weber number - axial excitation amplitude - elliptic case ( > 2 ₀) - hyperbolic case ( > 2 ₀) - liquid density - surface tension - liquid surface displacement - acceleration potential - ₀ rotational speed - axial forcing frequency - natural frequency of rotating system - _on natural frequency of harmonic axial response     

Natural damped frequencies and axial response of a rotating finite viscous liquid column

Summary For a finite solidly rotating cylindrical liquid column the damped natural axisymmetric frequencies have been determined. The liquid was considered incompressible and viscous. The cases of freely slipping edges and that of anchored edges have been treated. It was found that instability appears in a purely aperiodic root for the spinning liquid bridge. This is in contrast to the instability appearing in the damped oscillatory natural frequency of a nonspinning liquid column at . The spinning viscous liquid column exhibits the same instability as the frictionless liquid. It appears at for axisymmetric oscillations.List of symbols a radius of liquid column - I _m modified Bessel function of first kind and orderm - s complex frequency ( ) - r, ,z polar cylindrical coordinates - p pressure - t time - u, v, w radial-, azimuthal- and axial velocities of liquid, respectively - Weber number - h height of liquid column - dynamic viscosity of liquid - v kinematic viscosity of liquid (v=/) - density of liquid - surface tension of liquid - _r , _rz shear stress - (r, z, t) circulation - (r, z, t) streamfunction - ₀ angular velocity of liquid column about the axis of symmetry - (,t) free surface displacement     

Axisymmetric damped natural frequencies in a slowly spinning cylindrical container filled with viscous liquid

The axisymmetric fundamental natural damped frequency of the free surface in a slowly rotating cylindrical container is determined for incompressible viscous liquid. It was found that with increasing liquid height ratio h/a oscillation frequencies and decay magnitude both exhibit increased values. The influence of the surface tension parameter σ*=σa/?ν² shows for increasing magnitude increasing oscillation frequencies and decreasing decay magnitudes. For increasing Bond number Bo the oscillation frequency increases while the decay magnitude decreases. The effect of increasing the spin parameter ω₀=Ω₀ a ²/ν is manifested by an increase of the oscillation frequency and decay magnitude, i.e. for faster spin a liquid disturbance decays faster with larger oscillation frequency.     

Response of a spinning liquid column to pitch excitation

Summary The response of a solidly rotating liquid bridge consisting of inviscid liquid is determined for pitch excitation about its undisturbed center of mass. Free liquid surface displacement and velocity distribution has been determined in the elliptic (>2₀) and hyperbolic (<2₀) excitation frequency range.List of symbols a radius of liquid column - h length of column - I ₁ modified Besselfunction of first kind and first order - J ₁ Besselfunction of first kind and first order - r, ,z cylindrical coordinates - t time - u, v, w velocity distribution in radial-, circumferential-and axial direction resp. - mass density of liquid - free surface displacement - velocity potential - ₀ rotational excitation angle - ₀ velocity of spin - forcing frequency - _1n natural frequency - surface tension - acceleration potential - for elliptic range >2₀ - for hyperbolic range >2₀     

Viscous response of an anchored spinning liquid column to various axial excitations

Summary For a solidly rotating viscous cylindrical liquid column of finite length the response to axial synchronous, counter- and one-sided excitation is determined for anchored contact lines at the disc-rim. For a rotating column additional responses of inertial waves (hyperbolic range) appear for < 2 ₀, while in the elliptic range < 2 ₀ the sloshing response occurs. The various responses for the free surface displacement have been numerically evaluated. Only in the one-sided exitation case all resonance peaks appear, while for synchronous excitation only the odd resonances and for counter-excitation only the even resonance peaks occur.Notation a radius of column - h length of liquid bridge - I _n modified Bessel function - p liquid pressure - r, ,z cylindrical polar coordinates - t time - u, v, w velocity distribution - Weber number - z ₀ excitation amplitude - liquid density - surface tension - surface tension parameter - Ohnesorge number - liquid surface displacement - kinematic viscosity - ₀ rotational speed - dimensionless rotational speed - forcing frequency - dimensionless forcing frequency - dimensionless forcing frequency for non-viscous liquid - a= root of bi-cubic Eq.(33) - root of bi-cubic Eq.(33)     

Response of a spinning liquid column to axial excitation

Summary The response of a solidly rotating finite liquid column consisting of frictionless liquid is subjected to axial harmonic excitation. The response of the free liquid surface elevation and velocity distribution has been determined in the elliptic (>2 ₀) and hyperbolic frequency range (>2 ₀).Notation a radius of liquid bridge - h length of liquid bridge - I ₀,I ₁ modified Besselfunctions - J ₀,J ₁ Besselfunctions - P liquid pressure - r, ,z cylindrical polar coordinates - t time - u, v, w velocity distribution in rotating liquid - axial excitation amplitude - elliptic case (>2 ₀) - hyperbolic case (>2 ₀) - liquid density - surface tension - liquid surface displacement - acceleration potential - ₀ rotational speed - axial forcing frequency - natural frequency of rotating system - _02n –1 natural frequency of harmonic axial response With 8 Figures     

Break-up of a set of liquid threads under influence of surface tension

In this paper it is shown how the long-standing problem of the break-up of a cylindrical interface due to surface tension can be generalized to an arbitrary number of interacting interfaces in an arbitrary configuration. A system of immersed threads starting with two types of configurations is studied, i.e., a system of threads on a row and a system of threads at triangular vertices. From these cases, which are worked out in detail, it becomes clear how the stability of an arbitrary configuration can be determined. The (in)stability of the configuration is discussed in terms of the so-called disturbance growth rate. It turns out that the threads break up in specific phase patterns in which neighbouring threads are either in-phase or out-of-phase. For L threads, in principle 2^L phase patterns are possible. However, it is shown that the stability of the system directly follows from L so-called basic phase patterns. Special attention is paid to the special case of threads and fluid having equal viscosity. Then, the growth rate can be calculated analytically using Hankel transformations. An estimate for the growth rate in this case, which turns out to be quite sharp, is derived.     

Anchored edge pitching response of a spinning frictionless liquid column due to bottom and/or top excitation

The response of an anchored spinning frictionless liquid bridge to different pitching top and/or bottom excitation has been determined. The free surface displacement is obtained for synchronous, counter-excited and one-sided pitching excitation as a function of the dimensionless forcing frequency . The contactlines at the rim of the upper and lower discs are considered to be anchored. In synchronous pitching excitation only the even resonances show response peaks, while in a counter-excited system only odd resonances appear, which is in contrast to axial excitation modes. For one-sided excitation, all reasonances appear. Because of the different natural frequencies in direction of spin and opposite to it double peaks are present in the responses.     

Profile determination of a drop of liquid under surface tension,gravity and centrifugal forces

The steady-state shape of a drop of liquid under the action of surface tension, gravity and centrifugal forces is determined in this paper. The governing equations are derived from a variational principle on the total energy of the droplet. The steady-state shape is assumed to be axially symmetric, which allows describing it by means of its generator curve. This is approximated by a cubic parametric spline with suitable end conditions, and unknown supporting points. These are determined via the nonlinear least-square approximation of the arising overdetermined nonlinear algebraic system of equations. The procedure is illustrated with an example from the glass industry.     

液体表面张力测试仪的研究

针对传统液体表面张力测试仪的不足,研发了一种基于虚拟仪器技术的液体表面张力测试仪,并详细阐述了测量原理及其系统的实现方法.测试结果表明该测试仪可实现液体表面张力自动测试,读数时间短,有效避免了人为读数误差的影响,测量结果的一致性较好.     

Kinetic properties of surface electrons over liquid helium under strong electron-electron interaction

The mobility and lifetime of ground-level electrons are studied experimentally in a two-dimensional electron system at the surface of liquid helium at temperatures of0.4 to1.4 K and charge concentrations of (2.8–12.0) × 10⁸ cm^–2. It is shown that for fairly low temperature and high concentration where the frequency of interelectronic collisions is much higher than that of electron-ripplon ones, the so-called complete control condition is realized in the electron system, i.e., when the average drift velocity of electrons and the effective electron temperature can be introduced. This model is found to describe well the kinetic properties of surface electrons in the range of charge concentrations up to 4×10⁸ cm^–2, where the one-particle approach is no longer applicable. For the value of parameter =e²(n_s)^1/2/T=47 that corresponds to a strong electron-electron interaction, the lifetime is found to increase sharply and the short-range order typical of a liquid state appears in the two-dimensional electron layer.     

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.     

The surface tension of liquid aluminium-based alloys

In a systematic study, the surface tensions of the binary alloys Al–Fe and Al–Ni were investigated over a wide temperature and concentration range using electromagnetic levitation and the oscillating drop technique. Surface tensions were derived from the oscillation frequencies applying the formalism of Cummings and Blackburn. Temperature was measured by single-color pyrometry. Of particular interest in these alloys are melts corresponding to compositions of intermetallic phases, because potential ordering phenomena may influence all thermophysical properties. In both systems, an increase of the surface tension is observed at such concentrations. On the basis of partial excess Gibbs enthalpies, surface tensions can be calculated via the Butler equation and compared with experimental results. The agreement with our experimental data depends crucially on the quality of the thermodynamic potentials used. In addition, phenomenological models are also discussed, which describe the general trend correctly.     

Natural frequencies and axial response of a liquid layer under the influence of an axial steady micro-gravity field

Under the action of a steady axial gravity field a liquid layer experiences a change of natural frequencies in comparison to the annular layer. The natural frequencies exhibit a decrease with increasing axial Bond number and liquid height. This is particularly pronounced in the range of the lower modes. In addition the response of the liquid layer to axial excitation has been investigated and shows resonance peaks for all modes in contrast to those of the circular cylindrical layer. Increase of axial Bond number yields an increase of response magnitude.     

Concentration fluctuations and surface tension in liquid Au-Sn-Zn

Experimental thermodynamic data were applied to calculate the surface tension and the long-wavelength limit of the concentration-concentration fluctuations S_cc(0) in the ternary system Au-Sn-Zn. The fluctuations in the concentration of Zn and in particular those of Sn show marked maxima around Au₁₅Sn₃₀Zn₅₅ whereas for Au the fluctuations are smaller than those of a non-interacting ideal mixture. Over an extremely wide range of compositions the surface layer of the ternary is found to be primarily populated by Sn atoms. Excepting the alloys with larger amounts of Au (c_Au > 0.3) or high contents of Zn (c_Zn > 0.7) the surface tension is not very different from that of pure by Sn (0.52–0.60 Nm⁻¹). The progressive substitution of Sn by Zn at a constant value of c_Au≈ 0.3 leaves the surface tension practically constant up to c_Zn≈ 0.5. In sections of constant c_Au/c_Sn with c_Au/c_Sn > 0.3 the surface tension is observed to even pass through pronounced minima if larger amounts of Zn are added.     

Theoretical temperature-dependence surface tension of pure liquid gold

A calculation, based on statistical thermodynamics, of the surface tension of pure liquid gold, in the temperature range 1337-1653 K, is described. The calculated surface tension of liquid gold was found to be 1100 mJ/m² that agrees well with the reported experimental value (1149 mJ/m²). Results reveal that below 1480 K, the surface excess entropy changes by one order of magnitude (0.0151 mJ/m² K) as compared to the value above this temperature (0.1773 mJ/m² K), and thus predicting a structural changes at the surface of pure molten gold at high temperature.     

On the change of natural frequencies of a sloshing liquid by movable devices

Summary The present paper investigates the effectiveness of a movable device (mat), which is either immersed in an ideal fluid or covers the liquid free surface, to change the natural frequencies of a sloshing liquid in a circular cylindrical rigid tank. It is demonstrated that such a device reduces the frequencies of the free oscillations of the liquid substantially.

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Zur Änderung der Eigenfrequenzen einer schwingenden Flüssigkeit durch bewegliche Vorrichtungen

Zusammenfassung Vorliegende Arbeit untersucht die Wirksamkeit einer beweglichen Dämpfungs-vorrichtung (Matte) zur Herabsetzung der Eigenfrequenzen einer schwingenden reibungslosen Flüssigkeit mit freier Oberfläche in einem kreiszylindrischen starren Tank. Die Matte ist entweder in die Flüssigkeit eingebettet oder bedeckt deren Oberfläche. Es wird gezeigt, daß eine solche Vorrichtung die Frequenzen der freien Flüssigkeitsschwingungen wesentlich reduziert.

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With 3 Figures

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Presented at the Wissenschaftliche Jahrestagung GAMM, Delft, April 6–9, 1970.

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Presently at Institut für Mechanik, Techn. Hochschule Darmstadt, BRD.     

Thermocapillary flow in a liquid layer at minimum in surface tension

Summary In the present paper a class of similarity solutions for the two-dimensional Navier-Stokes and energy equations describing thermocapillary flows in a liquid layer of constant width and infinite extent is presented. The layer is bounded by a horizontal rigid plate from one side and opened to the ambient gas from the other one. The physical properties of the liquid are assumed to be constant except the surface tension which varies as a quadratic function with temperature. It is supposed that a constant temperature gradient exists along either the liquid free surface (case I) or the rigid boundary (case II).In both cases, by means of a similarity transformation, the equations of motion and energy are reduced to a system of three ordinary differential equations, one for the velocity and two for the temperature. The equation for the velocity can be solved separately from the other equations and its solution, found numerically, exists only for the Marangoni number less than a certain finite value. The solution of the whole system depends also on the Prandtl number. The solution of one of the temperature equations is presented in an analytical form and the other equation is solved numerically. Asymptotic formulas of the functions are also obtained for small and large Marangoni numbers. Flow pattern and temperature fields are presented. One convective roll exists in every semi-infinite layer. Fluid velocities at different points of the free surface are evaluated for an aqueous solution of n-heptanol and compared with those measured in the experiments.     

The surface tension of liquid pure aluminium and aluminium-magnesium alloy

This paper discusses the results of several experiments designed to further illustrate the recent findings of the present authors according to which, if a virtually leak-fee maximum bubble pressure system is used to measure the surface tension of liquid aluminium, a surface tension around 1100 mJ m^–2 is first obtained, decreasing to the oxidized value (around 865 mJ m^–2) as the experiment proceeds and oxygen enters the system mainly through the capillary walls. The peculiarities and difficulties inherent to the maximum bubble pressure method are illustrated. For instance, a study of the time needed for the surface tension to decrease to the oxidized value as a function of temperature reveals the important role played by the vapour pressure in the process. This is further illustrated by considering Al-Mg alloys, as magnesium has a different heat of vaporization and a much larger vapour pressure than aluminium at the measuring temperatures. Results for the changes in density and surface tension for the oxidized and unoxidized cases induced by magnesium (up to 8 wt%) are also presented and compared to previous data.     

Determination of the surface tension of liquid stainless steel

The surface tensions (σ) and temperature dependencies (dσ/dT) of several commercial 4-series ferritic stainless steels have been measured using the sessile drop technique on an Al₂O₃ plate over the temperature range 1789 to 1883 K in an atmosphere of high purity (P _{O 2} < 10^?19 MPa) argon gas. Precise densities of liquid stainless steels have also been obtained using the modified sessile drop method in order to calculate accurate values of the surface tension. The surface tensions of liquid stainless steels decreased markedly with increasing sulphur concentration in the steels. The variation of surface tensions of liquid stainless steels can be described by the following equation σ = 1790 ? 182 ln (1 + 260a _S) (mN/m) when only S is considered or σ = 1820 ? 304 ln (1 + 383a _O) ? 182ln (1 + 260a _S) (mN/m) when both S and O are considered. The equations apply to the following compositional ranges: mass%O = 0.0022–0.0064, mass%S = 0.0008–0.05. The temperature coefficient of the surface tension (dσ/dT) of liquid stainless steel was found to change from negative to positive at a sulphur concentration of about 30 mass ppm in the steel. Nitrogen was found to have little effect on the surface tension of liquid stainless steel.