Viscosity of a Newtonian fluid calculated from the deformation of droplets covered with a surfactant under a linear shear flow

The viscosity of small fluid droplets covered with a surfactant is determined using drop deformation techniques. This method, proposed by Hu and Lips, is here extended to the case of the presence of a surface-active adsorpted at the liquid–liquid interface, to consider more general scenarios. In these experiments, a droplet is sheared by another immiscible fluid of known viscosity, both Newtonian liquids. From the steady-state deformation and retraction mechanisms, the droplet viscosity is calculated using an equation derived from the theories of Taylor and Rallison. Although these theories were expressed for surfactant-free interfaces, they can be applied when a surfactant is present in the system if the sheared droplet reaches reliable steady-state deformations and the surfactant attains its equilibrium adsorption concentration. These determinations are compared to bulk viscosities measured in a rheometer for systems with different viscosity ratios and surfactant concentrations. Very good agreement between both determinations is found for drops more viscous than the continuous phase.     

Dynamics of a Taylor bubble in steady and pulsatile co-current flow of Newtonian and shear-thinning liquids in a vertical tube

A computational analysis is carried out to ascertain the effects of steady and pulsatile co-current flow, on the dynamics of an air bubble rising in a vertical tube containing water or a solution of Carboxymethylcellulose (CMC) in water. The mass fraction (m_f) of CMC in the solution is varied in the range 0.1% ⩽ m_f ⩽ 1% to accommodate zero-shear dynamic viscosities in the range 0.009–2.99 Pa-s. It was found that the transient and time-averaged velocities of Taylor bubbles are independent of the bubble size under both steady as well as pulsatile co-current flows. The lengths of the Taylor bubbles under the Newtonian conditions are found to be consistently greater than the corresponding shear-thinning non-Newtonian conditions for any given zero-shear dynamic viscosity of the liquid. In contrast to observations in stagnant liquid columns, an increase in the dynamic viscosity of the liquid (under Newtonian conditions) results in a concomitant increase in the bubble velocity, for any given co-current liquid velocity. In shear-thinning liquids, the change in the bubble velocity with an increase in m_f is found to be relatively greater at higher co-current liquid velocities. During pulsatile shear-thinning flows, distinct ripples are observed to occur on the bubble surface at higher values of m_f, the locations of which remain stationary with reference to the tube for any given pulsatile flow frequency, while the bubble propagated upwards. In such a pulsatile shear-thinning flow, a localised increase in dynamic viscosity is accompanied near each ripple, which results in a localised re-circulation region inside the bubble, unlike a single re-circulation region that occurs in Newtonian liquids, or shear-thinning liquids with low values of m_f. It is also seen that as compared to frequency, the amplitude of pulsatile flow has a greater influence on the oscillating characteristics of the rising Taylor bubble. The amplitude of oscillation in the bubble velocity increases with an increase in the CMC mass fraction, for any given value of pulsatile flow amplitude.     

基于水平集方法的毛管上升特征研究

毛管上升现象与许多行业密切相关,系统地对此现象进行研究具有重大意义。与传统理论研究方法不同,本文使用N-S方程耦合水平集方法模拟毛管气液上升行为。通过与简化条件的解析解对比,验证了模拟方法的可靠性。此外,详细地研究了毛管振荡现象,并分析了影响毛管振荡行为的主要因素。结果表明,水平集方法能够精确地表征毛管振荡现象,与数值解相比具有更高的精度。毛管长度的增加能够减弱液柱振荡,主要归结于非湿相气体的粘滞力作用;湿相密度和湿相粘度同样对毛管振荡现象影响显著。湿相密度越大,惯性力越大,促进了毛管振荡;而湿相粘度变大,会增大粘滞力作用,因此减弱了毛管振荡现象。毛管振荡是由多种影响因素共同控制的,流体的惯性力是造成毛管振荡的主要原因,而粘滞力是减弱毛管振荡行为的主要因素,使液柱振荡逐渐衰减,并稳定至平衡高度。     

A microtube viscometer with a thermostat

The viscometer presented in this paper is suitable for measuring the viscosity of liquids in micro-litre quantities. It consists of a micro-flow experimental system with a thermostat. Using the measurements of the flow rates and pressure drops of a liquid passing through a microtube, the liquids viscosity can be calculated from on Hagen-Poiseuille theory. After calibration, the viscometer was used to measure viscosities of deionized water and ethyl alcohol at temperatures ranging from 0 to 40 °C. For both test liquids, the relative deviation of the measured values from those quoted in the literature (obtained using other viscometers) was less than 2.6%. The relative uncertainty of the experimental system was reduced to ±1.8% using the relative measuring method. Due to the micro-scale of the test section, only a micro-litre quantity of liquid is needed for a test; this is a potential advantage for measurement of bio-liquid viscosities.     

On the viscosity of magnetic fluid with low and moderate solid fraction

The design of a pressurized capillary rheometer operating at prescribed temperature is described to measure the viscosity of magnetic fluids (MFs) containing Fe3O4 magnetic nanoparticles (MNPs). The equipment constant of the rheometer was obtained using liquids with predetermined viscosities. Experimentally measured viscosities were used to evaluate different equations for suspension viscosities. Deviation of measured suspension viscosities from the Einstein equation was found to be basically due to the influence of spatial distribution and aggregation of Fe3O4 MNPs. By taking account of the coating layer on MNPs and the aggregation of MNPs in MFs, a modified Einstein equation was proposed to fit the experimental data. Moreover, the influence of external magnetic field on viscosity was also taken into account. Viscosities thus predicted are in good agreement with experimental data. Temperature effect on suspension viscosity was shown experimentally to be due to the shear-thinning behavior of the MFs.     

A rheological theory for liquid crystal thin films

A macroscopic rheological theory for compressible isothermal nematic liquid crystal films is developed and used to characterize the interfacial elastic, viscous, and viscoelastic material properties. The derived expression for the film stress tensor includes elastic and viscous components. The asymmetric film viscous stress tensor takes into account the nematic ordering and is given in terms of the film rate of deformation and the surface Jaumann derivative. The material function that describes the anisotropic viscoelasticity is the dynamic film tension, which includes the film tension and dilational viscosities. Viscous dissipation due to film compressibility is described by the anisotropic dilational viscosity. Three characteristic film shear viscosities are defined according to whether the nematic orientation is along the velocity direction, the velocity gradient, or the unit normal. In addition the dependence of the rheological functions on curvature and film thickness has been identified. The rheological theory provides a theoretical framework to future studies of thin liquid crystal film stability and hydrodynamics, and liquid crystal foam rheology. Received: 9 October 2000 Accepted: 6 April 2001     

On the viscosity of magnetic fluid with low and moderate solid fraction

The design of a pressurized capillary rheometer operating at prescribed temperature is described to measure the viscosity of magnetic fluids （MFs） containing Fe3O4 magnetic nanoparticles （MNPs）. The equipment constant of the rheometer was obtained using liquids with predetermined viscosities. Experimentally measured viscosities were used to evaluate different equations for suspension viscosities. Deviation of measured suspension viscosities from the Einstein equation was found to be basically due to the influence of spatial distribution and aggregation of Fe3O4 MNPs. By taking account of the coating layer on MNPs and the aggregation of MNPs in MFs, a modified Einstein equation was proposed to fit the experimental data. Moreover, the influence of external magnetic field on viscosity was also taken into account. Viscosities thus predicted are in good agreement with experimental data. Temperature effect on suspension viscosity was shown experimentally to be due to the shear-thinning behavior of the MFs.     

Viscous flow through the swirl chamber of a pressure-swirl atomizer

We study theoretically the viscous flow through the swirl chamber of a pressure-swirl atomizer. The aim is to predict quantitatively the flow rate of a given liquid driven by a given pressure difference across the atomizer in order to explain the counter-intuitive experimental finding that, for moderate liquid dynamic viscosities, at a given driving pressure difference and atomizer geometry, a higher liquid viscosity results in a higher flow rate. The concept for the flow analysis is to subdivide the flow field in the swirl chamber into zones allowing for neglect of velocity components or boundary-layer simplification. The result is a quantitative prediction of the liquid flow rate for a given driving pressure difference and atomizer geometry, and for given liquid properties relevant for the discharge from the atomizer. Flow rates are compared to experimental data from various sources and show good agreement. Another part of the results is the diameter of the air core formed around the symmetry axis of the swirl chamber, which is of sub-millimetric order here. This result is compared to different experimental correlations and also shows very good agreement. For small values of the swirl velocity and/or very high liquid viscosities, the air core breaks down. The phenomenon of air core break down has been analysed, and it is shown that for high viscosities the air core breaks down due to weak swirl velocity.     

Weakly nonlinear instability of the flow of two immiscible liquids with different viscosities in a pipe

The weakly nonlinear stability due to an axisymmetric disturbance of two immiscible liquids with different viscosities in a pipe is worked out. The most important conclusions drawn from this work are the independence of the Reynolds number in the stability criterion, the existence of stable waves with sawtooth profile when the more viscous fluid is located at the core and it occupies most of the pipe, and the destabilizing effect of the surface tension, effect that can be overtaken by the stabilizing effect due to the difference in viscosity of the two fluids, when the viscosity of the ring fluid is much smaller than the viscosity of the core fluid and the thickness of the ring flow is smaller than the thickness of the core flow.     

Coupling dynamic analysis of a liquid-filled spherical container subject to arbitrary excitation

Using spherical coordinates, the coupling nonlin- ear dynamic system of a liquid-filled spherical tank, which can be excited discretionarily, is deduced by the H-O variational principle, and the viscous damping is introduced via the liquid dissipation function. The kinetic equations of the coupling system are deduced by the relationship between the velocity of liquid particles and the disturbed liquid surface equation. Normal differential equations are obtained through the Galerkin method. An equivalent mechanical model is developed for liquid sloshing in a spherical tank subject to arbitrary excitation. The fixed and slosh masses, as well as the spring and damping constants, are determined in such a way as to satisfy the principle of equivalence. Numerical simulations illustrate the theoretical results in this paper as well.     

Dynamic scaling of unsteady shear-thinning non-Newtonian fluid flows in a large-scale model of a distal anastomosis

Dimensional analysis has been applied to an unsteady pulsatile flow of a shear-thinning power-law non-Newtonian liquid. An experiment was then designed in which both Newtonian and non-Newtonian liquids were used to model blood flow through a large-scale (38.5 mm dia.), simplified, rigid arterial junction (a distal anastomosis of a femorodistal bypass). The flow field within the junction was obtained by Particle Imaging Velocimetry and near-wall velocities were used to calculate the wall shear stresses. Dimensionless wall shear stresses were obtained at different points in the cardiac cycle for two different but dynamically similar non-Newtonian fluids; the good agreement between the measured dimensionless wall shear stresses confirm the validity of the dimensional analysis. However, blood exhibits a constant viscosity at high-shear rates and to obtain complete dynamic similarity between large-scale experiments and life-scale flows, the high-shear viscosity also needs to be included in the analysis. How this might be done is discussed in the paper.     

Two-dimensional study of drop deformation under simple shear for Oldroyd-B liquids

The effect of viscoelasticity on the deformation of a circular drop suspended in a second liquid in shear is investigated with direct numerical simulations. A numerical algorithm based on the volume-of-fluid method for interface tracking is implemented in two dimensions with the Oldroyd-B constitutive model for viscoelastic liquids. The code is verified against a normal mode analysis for the stability of two-layer flow in a channel; theoretical growth rates are reproduced for the interface height, velocity and stress components. Drop simulations are performed for drop and matrix liquids of different viscosities and elasticities. A new feature is found for the case of equal viscosity, when the matrix liquid is highly elastic and surface tension is low; hook-like structures form at the drop tips. This is due to the growth of first normal stress differences that occur slightly above the front tip and below the back tip as the matrix elasticity increases above a threshold value.     

The stability of two stratified non-newtonian liquids in couette flow

Consideration is given to the stability of the interface between two Oldroyd liquids with shear-dependent viscosities, flowing in distinct layers while undergoing plane Couette flow. Results are presented as regions of stability in the plane determined by the logarithms of the viscosity and depth ratios. The work of previous authors for two Newtonian, power-law and constant-viscosity Oldroyd liquids is revealingly presented in a similar fashion. It is found that the dependence of the viscosities on shear-rate can drastically affect the regions of interfacial stability in a way over and above that due to just a change in the effective viscosity ratio. It is also found that for the Oldroyd liquids this viscosity variation affects the stability when it is present in the less-viscous layer.     

Effect of a viscous liquid loading on Love wave propagation

This paper describes a theory of surface Love waves propagating in a layered elastic waveguide loaded on its surface by a viscous (Newtonian) liquid. An analytical expression for the complex dispersion equation of Love waves has been established. The real and imaginary parts of the complex dispersion equation were separated and resulting system of nonlinear algebraic equations was solved numerically. The influence of the viscosity of liquid on the dispersion curves of phase velocity, the wave attenuation and the distribution of the Love wave amplitude is analyzed numerically. The propagation loss is produced only by the viscosity of liquids. Elastic layered waveguide is assumed to be loss-less. The numerical solutions show the dependence of the phase velocity change, the wave attenuation and the wave amplitude distribution in terms of the liquid viscosity and the wave frequency. The results of the investigations are fundamental and can be applied in the design and development of liquid viscosity sensors and biosensors, in Non-Destructive Testing (NDT) of materials, in geophysics and seismology.     

Evaporation of pure liquids with increased viscosity in a falling film evaporator

The present study investigated fluid dynamics and heat transfer of viscous pure liquids in a falling film evaporator. This is of special benefit as it avoids mass transfer effects on the evaporation behaviour. Experiments at a single-tube glass falling film evaporator were conducted. It allowed a full-length optical film observation with a high-speed camera. Additionally the evaporator was equipped with a slotted weir distribution device. Test fluids provided viscosities ranging from μ = 0.3 to 41 mPa s. The Reynolds number was between 0.7 and 1,930. Surface evaporation and the transition to nucleate boiling were studied to gain information about the film stability at maximum wall superheat. A reliable database for laminar and laminar-wavy viscous single component films was created. The experimental results show a significant enhancement in the wave development due to the film distribution. A wavy flow with different wave velocities was superposed to the film in each liquid load configuration without causing a film breakdown or dry spots on the evaporator tube. It was found that nucleate boiling can be allowed without causing film instabilities over a significant range of wall superheat.     

Effect of Liquid Viscosity on Dispersion of Quasi-Lamb Waves in an Elastic-Layer–Viscous-Liquid-Layer System

The dispersion curves are constructed and propagation of quasi-Lamb waves are studied for wide range of frequencies based on the Navier–Stokes three-dimensional linearized equations for a viscous liquid and linear equations of the classical theory of elasticity for an elastic layer. For a thick liquid layer, the effect of the viscosity of the liquid and the thickness of elastic and liquid layers on the phase velocities and attenuation coefficients of quasi-Lamb modes is analyzed. It is shown that in the case of a thick liquid layer for all modes, there are elastic layers of certain thickness with minimal effect of liquid viscosity on the phase velocities and attenuation coefficients of modes. It is also discovered that for some modes, there are both certain thicknesses and certain ranges of thickness where the effect of liquid viscosity on the phase velocities and attenuation coefficients of these modes is considerable. We ascertain that liquid viscosity promotes decrease of the penetration depth of the lowest quasi-Lamb mode into the liquid. The developed approach and the obtained results make it possible to ascertain for wave processes the limits of applicability of the model of ideal compressible fluid. Numerical results in the form of graphs are adduced and analyzed.     

Instability analysis of nonlinear surface waves in a circular cylindrical container subjected to a vertical excitation

Singular perturbation theory of two-time scale expansions was developed both in inviscid and weak viscous fluids to investigate the motion of single surface standing wave in a liquid-filled circular cylindrical vessel, which is subject to a vertical periodical oscillation. Firstly, it is assumed that the fluid in the circular cylindrical vessel is inviscid, incompressible and the motion is irrotational, a nonlinear evolution equation of slowly varying complex amplitude, which incorporates cubic nonlinear term, external excitation and the influence of surface tension, was derived from solvability condition of high-order approximation. It shows that when forced frequency is low, the effect of surface tension on mode selection of surface wave is not important. However, when forced frequency is high, the influence of surface tension is significant, and can not be neglected. This proved that the surface tension has the function, which causes free surface returning to equilibrium location. Theoretical results much close to experimental results when the surface tension is considered. In fact, the damping will appear in actual physical system due to dissipation of viscosity of fluid. Based upon weakly viscous fluids assumption, the fluid field was divided into an outer potential flow region and an inner boundary layer region. A linear amplitude equation of slowly varying complex amplitude, which incorporates damping term and external excitation, was derived from linearized Navier–Stokes equation. The analytical expression of damping coefficient was determined and the relation between damping and other related parameters (such as viscosity, forced amplitude and depth of fluid) was presented. The nonlinear amplitude equation and a dispersion, which had been derived from the inviscid fluid approximation, were modified by adding linear damping. It was found that the modified results much reasonably close to experimental results. Moreover, the influence both of the surface tension and the weak viscosity on the mode formation was described by comparing theoretical and experimental results. The results show that when the forcing frequency is low, the viscosity of the fluid is prominent for the mode selection. However, when the forcing frequency is high, the surface tension of the fluid is prominent. Finally, instability of the surface wave is analyzed and properties of the solutions of the modified amplitude equation are determined together with phase-plane trajectories. A necessary condition of forming stable surface wave is obtained and unstable regions are illustrated.     

Dispersion of linear gravity waves on a viscoelastic fluid in an horizontal canal

A characteristic equation is derived that describes the spatial decay of linear surface gravity waves on Maxwell fluids. Except at small frequencies, the derived dispersion relation is different from the temporal decay dispersion relation which is normally studied within fluid mechanics. The implications for waves on viscous Newtonian fluids using the two different dispersion relations is briefly discussed. The wave number is measured experimentally as function of the frequency in a horizontal canal. Seven Newtonian fluids and four viscoelastic liquids with constant viscosity have been used in the experiments. The spatial decay theory for Newtonian fluids fits well to the experimental data. The model and experiments are used to determine limits for the Maxwell fluid time numbers for the four viscoelastic liquids. As a result of low viscosity it was not possible within this study to obtain these time numbers from oscillatory experiments. Therefore, a comparison of surface gravity wave experiments with theory is applicable as a method to evaluate memory times of low viscosity viscoelastic fluids.     

端承粘弹性桩纵向振动的轴对称解析解

将桩和土体视为粘弹性介质,在频率域求得了桩轴对称纵向振动时动力响应的解析解,进而得到桩头复刚度的解析表达式.通过数值计算,给出了桩头动刚度因子和等效阻尼随激励频率的响应,考察了土层粘性、桩长径比、桩土模量比、桩泊松比和粘性等参数对桩头刚度因子和等效阻尼的影响.研究表明:由于考虑了桩的径向变形效应以及土层对桩的径向力作用,轴对称解析解的桩头动刚度因子和等效阻尼与经典Euler杆模型桩的桩头动刚度因子和阻尼有较大区别.因此,经典Euler杆模型桩的适用范围具有一定的局限性,在若干情形下应采用基于三维粘弹性的桩纵向振动轴对称模型进行分析.     

Film thickness in blade coating of viscous and viscoelastic liquids

Coating of viscous and viscoelastic liquids is examined both theoretically and experimentally. A rigid blade, accurately positioned over a rotating roll, provides an experimental system in which coating thickness is measured as a function of geometric parameters. A perturbation solution to the Navier—Stokes equations yields a lubrication theory which shows agreement with the data to an extent depending on the specific geometry.The effect of a non-Newtonian viscosity is explored by adopting a purely viscous power-law model. The lubrication equations are solved by the method of Horowitz and Steidler [1], and predict an increase in coating thickness relative to the Newtonian case. Data for viscoelastic fluids show both an increase and a decrease in coating thickness compared with Newtonian liquids depending on the relative magnitude of shear thinning and elastic effects.