Drop formation dynamics of constant low-viscosity,elastic fluids

The dynamics of drop formation under gravity has been investigated as a function of elasticity using a set of low-viscosity, ideal elastic fluids and an equivalent Newtonian glycerol–water solution. All solutions had the same shear viscosity, equilibrium surface tension, and density, but differed greatly in elasticity. The minimum drop radius in the early stages of drop formation (necking) was found to scale as expected from potential flow theory, independent of the elasticity of the solutions. Thus, during this stage of drop formation when viscous force is still weak, the dynamics are controlled by a balance between inertial and capillary forces, and there is no contribution of elastic stresses of the polymer. However, upon formation of the pinch regions, there is a large variation in the drop development to break-off observed between the various solutions. The elastic solutions formed secondary fluid threads either side of a secondary drop from the necked region of fluid between the upper and lower pinches, which were sustained for increasing amounts of time. The break-off lengths and times increase with increasing elasticity of the solutions. Evolution of the filament length is, however, identical in shape and form for all of the polymer solutions tested, regardless of differing elasticity. This de-coupling between filament growth rate and break-up time (or equivalently, final filament length at break-up) is rationalised. A modified force balance to that of Jones and Rees [48] is capable of correctly predicting the filament growth of these low-viscosity, elastic fluids in the absence of any elastic contributions due to polymer extension within the elongating filament. The elongation of the necked region of fluid (which becomes the filament) is dominated by the inertia of the drop, and is independent of the elasticity of the solution. However, elasticity does strongly influence the resistance of the pinch regions to break-off, with rapid necking resulting in extremely high rates of surface contraction on approach to the pinch point, initiating extension of the polymer chains within the pinch regions. This de-coupling phenomenon is peculiar to low-viscosity, elastic fluids as extension does not occur prior to the formation of the pinch points (i.e. just prior to break-up), as opposed to the high viscosity counterparts in which extension of polymers in solution may occur even during necking. Once steady-state extension of the polymers is achieved within the pinch at high extension rates, the thread undergoes elasto-capillary break-up as the capillarity again overcomes the viscoelastic forces. The final length at detachment and time-to-break-off (relative to the equivalent Newtonian fluid) is shown to be linearly proportional to the longest relaxation time of the fluid.     

Dynamics of horizontal viscoelastic fluid filaments

The tension force of a thinning high-molecular polymer solution filament is measured using the filament itself as a force sensor. The axial filament stresses and the effects of fluid flow from the filament into adjacent drops are estimated. It is shown that these effects are insignificant for polymer solutions in a low-viscosity solvent (water) but substantial for solutions in a high-viscosity fluid (glycerine). A modification of the standard rheological capillary filament method is proposed. This modification makes it possible to exclude any hypotheses concerning the stress distribution pattern within the filament. Periodic transverse oscillations of the filament axis are revealed and analyzed.     

Extensional flow of a viscoelastic filament governed by the FENE-CR model

A thin filament model is used to analyze the extensional flow of a viscoelastic thread governed by the FENE-CR model. The problem is solved numerically by finite differences using a third-order upwind scheme in space and a second order Runge-Kutta scheme in time. The behavior of the filament is controlled by the competing effects of surface tension and axial normal stresses which are characterized in terms of three-dimensional groups, the Deborah number De, the extensibility parameter L and the capillary number Ca. Surface tension has a destabilizing effect causing the filament to thin in the mid-section leading to a rupture. On the other hand normal stresses tend to stabilize the filament. If axial normal stresses are sufficiently large the filament deforms almost uniaxially due to strain hardening.     

Uniaxial extension of an isothermal fluid cylinder in the presence of inertia,surface tension and gravity

Effects of inertia, surface tension and gravity in the constant force stretching of isothermal cylindrical filaments of Newtonian, power-law and Maxwell-type fluids were analysed in Lagrangian coordinates. Solution for the purely gravitational extension of Newtonian fluid cylinder was found to be as simple as = 1 – C ₃ (1 – ) where designates the cross sectional area, the Lagrangian distance and the time. Analytical solutions were also available for the case of inertialess Newtonian and power-law fluids.A first-order backward differencing scheme and minimal computer time were sufficient to numerically analyse the constant force extension of Maxwell-type fluids in the presence of inertia, gravity and surface tension. Effects of inertia, surface tension and gravity on the severity of neck down occurring at either end of the filament are summarized in diagrams. The present approach is valid on any other constitutive model as far as there is a numerical scheme to analyse thehomogeneous extension of a cylinder of that particular fluid.     

Surface tension and buoyancy-driven flow in a non-isothermal liquid bridge

The Navier–Stokes–Boussinesq equations governing the transport of momentum, mass and heat in a non-isothermal liquid bridge with a temperature-dependent surface tension are solved using a vorticity-stream-function formulation together with a non-orthogonal co-ordinate transformation. The equations are discretized using a pseudo-unsteady semi-implicit finite difference scheme and are solved by the ADI method. A Picard-type iteration is adopted which consists of inner and outer iterative processes. The outer iteration is used to update the shape of the free surface. Two schemes have been used for the outer iteration; both use the force balance normal to the free surface as the distinguished boundary condition. The first scheme involves successive approximation by the direct solution of the distinguished boundary condition. The second scheme uses the artificial force imbalance between the fluid pressure, viscous and capillary forces at the free surface which arises when the boundary condition for force balance normal to the surface is not satisfied. This artificial imbalance is then used to change the surface shape until the distinguished boundary condition is satisfied. These schemes have been used to examine a variety of model liquid bridge situations including purely thermocapillary-driven flow situations and mixed thermocapillary- and bouyancy-driven flow.     

Spurious currents in finite element based level set methods for two‐phase flow

A study of spurious currents in continuous finite element based simulations of the incompressible Navier–Stokes equations for two‐phase flows is presented on the basis of computations on a circular drop in equilibrium. The conservative and the standard level set methods are used. It is shown that a sharp surface tension force, expressed as a line integral along the interface, can give rise to large spurious currents and oscillations in the pressure that do not decrease with mesh refinement. If instead a regularized surface tension representation is used, exact force balance at the interface is possible, both for a fully coupled discretization approach and for a fractional step projection method. However, the numerical curvature calculation introduces errors that cause spurious currents. Different ways to extend the curvature from the interface to the whole domain are discussed and investigated. The impact of using different finite element spaces and stabilization methods is also considered. Copyright © 2011 John Wiley & Sons, Ltd.     

Nonlinear analysis of the surface tension driven breakup of viscoelastic filaments

The surface tension driven breakup of viscoelastic filaments into droplets is qualitatively different from that of Newtonian liquid filaments. Disturbances on filaments of dilute polymer solutions often grow to a configuration consisting of nascent droplets connected by a thin ligament; the breakup time for this configuration is much longer than that predicted by extensions of Rayleigh's linear stability theory. We present here a nonlinear analysis of surface tension driven breakup of viscoelastic filaments using two complementary approaches that given equivalent results: a transient finite element solution and a one-dimensional thin filament approximation. We show that significant nonlinear effects lead to the experimentally-observed nascent droplet-ligament configuration, and we predict the entire evolution of the filament profile. Agreement with available experimental data for profile evolution and breakup of jets of Newtonian fluids and dilute polymer solutions is excellent.     

Experimental investigations into an isothermal spinning threadline: Extensional rheology of a Separan AP 30 solution in glycerol and water

Summary Experimental observations on a steady isothermally extending filament of a water/glycerol solution of Separan AP 30 are presented. Photographic records were analysed to give filament diameter (and hence filament speed) as a function of distance below the extrusion die (a glass capillary). Measurements of inline tension were also made. When effects of weight, surface tension and air drag were accounted for, the extensional stress at every point along the filament could be calculated. Results for stress versus extension rate are presented for various flow situations.Independent rheogoniometric measurements of simple shear viscosity, first and second normal stress differences, and of a crude relaxation time were also made at comparable rates of deformation.Comparison shows that apparent extensional viscosities are several orders of magnitude larger than corresponding simple shear viscosities. After discussion, no conclusion can be drawn about what constitutive equation is most suitable to describe the results.An analysis to predict air drag is given.With 18 figures     

基于移动粒子半隐式法的表面张力模拟

采用移动粒子半隐式法(MPS)模拟了受表面张力影响的自由面流动。表面张力的计算采取了一种较适合于MPS方法的表面自由能模型。方形液滴振荡和射流断裂的模拟结果分别与理论分析和试验结果一致,同时进行了三维射流注水模拟,从而验证了MPS方法结合该表面张力模型可以有效、方便地进行自由面流动中表面张力现象的模拟。     

A Resistance Model for Newtonian and Power-Law Non-Newtonian Fluid Transport in Porous Media

A theoretical model based on the force balance between pressure, viscous force, and inertia force is proposed to predict the flow resistance of Newtonian and power-law non-Newtonian fluids through porous packed beds. The present model takes inertia effect into consideration, and the flow regime can be extended from Darcy flow to non-Darcy flow. It is demonstrated that the present model can predict most available experimental data well. The present results are also compared to the Ergun equation and other drag correlations.     

Extensional dynamics of viscoplastic filaments: II. Drips and bridges

A model for the dynamics of slender filaments of Herschel–Bulkley fluid is used to explore viscoplastic dripping under gravity and thinning under controlled extension (liquid bridges). The conditions required for fluid to yield are delineated, and the subsequent thinning and progression to pinch-off are tracked numerically. Calculations varying the dimensionless parameters of the problem are presented to illustrate the effect of surface tension, rheology, inertia (for dripping) and gravity. The theoretical solutions are compared with laboratory experiments using aqueous solutions of Carbopol and Kaolin suspensions. For drips and bridges, experiments with Carbopol are well matched by the theory, using a surface tension equal to that of water, even in situations when the fluid is not slender. Experiments with Kaolin do not compare well with theory for physically plausible values of the surface tension. Implications for rheometry and surface-tension inference are discussed.     

Modelling and transient planar dynamics of suspended cables with moving mass

In this study, the 3D nonlinear equations of motion of the suspended cable with moving mass are obtained via the Hamilton principle, and its transient linear planar dynamics is investigated. Considering the quasi-static assumption, the condensed planar model accounting for the effect of the moving mass is derived, and it is then discretized by choosing the static deflection and sine series as shape functions. It is shown that this expansion shows good convergence features. The Newmark method is used to investigate the transient response. The effects of the inertia force, mass, sag and velocity of the moving mass on the transient dynamics of the suspended cable are systematically investigated. Finally, the horizontal tension of the suspended cable and the case of sequentially moving masses are examined.     

A new implicit surface tension implementation for interfacial flows

A new implementation of surface tension effects in interfacial flow codes is proposed which is both fully implicit in space, that is the interface never has to be reconstructed, and also semi‐implicit in time, with semi‐implicit referring to the time integration of the surface tension forces. The main idea is to combine two previously separate techniques to yield a new expression for the capillary forces. The first is the continuum surface force (CSF) method, which is used to regularize the discontinuous surface tension force term. The regularization can be elegantly implemented with the use of distance functions, which makes the level set method a suitable choice for the interface‐tracking algorithm. The second is to use a finite element discretization together with the Laplace–Beltrami operator, which enables simple reformulation of the surface tension term into its semi‐implicit equivalent. The performance of the new method is benchmarked against standard explicit methods, where it is shown that the new method is significantly more robust for the chosen test problems when the time steps exceed the numerical capillary time step restriction. Some improvements are also found in the average number of nonlinear iterations and linear multigrid steps taken while solving the momentum equations. Copyright © 2005 John Wiley & Sons, Ltd.     

液体环轴对称抛洒首次破碎的理论分析

对于水和无水乙醇环形轴对称抛洒实验的首次破碎结果的分析表明，在给定实验条件下，液体密度大约是气体的1000倍，液体首次破碎的过程分2个阶段:首先，界面不稳定性非线性发展，液体尖钉在惯性力和空气阻力的作用下，拉伸变细变长；然后在射流不稳定性的作用下，断裂成液体珠串。这个理论分析给出的首次破碎的平均液体直径与实验结果基本一致。理论分析表明，无水乙醇破碎液滴的平均直径比水液滴小，是无水乙醇表面张力值仅为水的1/3的一个必然结果。     

A second strain gradient elasticity theory with second velocity gradient inertia – Part II: Dynamic behavior

This paper is the sequel of a companion Part I paper devoted to the constitutive equations and to the quasi-static behavior of a second strain gradient material model with second velocity gradient inertia. In the present Part II paper, a multi-cell homogenization procedure (developed in the Part I paper) is applied to a nonhomogeneous body modelled as a simple material cell system, in conjunction with the principle of virtual work (PVW) for inertial actions (i.e. momenta and inertia forces), which at the macro-scale level takes on the typical format as for a second velocity gradient inertia material model. The latter (macro-scale) PVW is used to determine the equilibrium equations relating the (ordinary, double and triple) generalized momenta to the inertia forces. As a consequence of the surface effects, the latter inertia forces include (ordinary) inertia body forces within the bulk material, as well as (ordinary and double) inertia surface tractions on the boundary layer and (ordinary) inertia line tractions on the edge line rod; they all depend on the acceleration in a nonstandard way, but the classical laws are recovered in the case of no higher order inertia. The classical linear and angular momentum theorems are extended to the present context of second velocity gradient inertia, showing that the extended theorems—used in conjunction with the Cauchy traction theorem—lead to the local force and moment (stress symmetry) motion equations, just like for a classical continuum. A gradient elasticity theory is proposed, whereby the dynamic evolution problem for assigned initial and boundary conditions is shown to admit a Hamilton-type variational principle; the uniqueness of the solution is also discussed. A few simple applications to wave propagation and dispersion problems are presented. The paper indicates the correct way to describe the inertia forces in the presence of higher order inertia; it extends and improves previous findings by the author [Polizzotto, C., 2012. A gradient elasticity theory for second-grade materials and higher order inertia. Int. J. Solids Struct. 49, 2121–2137]. Overall conclusions are drawn at the end of the paper.     

Modeling surface tension of a two‐dimensional droplet using smoothed particle hydrodynamics

Surface tension plays a significant role at the dynamic interface of free‐surface flows especially at the microscale in capillary‐dominated flows. A model for accurately predicting the formation of two‐dimensional viscous droplets in vacuum or gas of negligible density and viscosity resulting from axisymmetric oscillation due to surface tension is solved using smoothed particle hydrodynamics composed of the Navier‐Stokes system and appropriate interfacial conditions for the free‐surface boundaries. The evolution of the droplet and its free‐surface interface is tracked over time to investigate the effects of surface tension forces implemented using a modified continuous surface force method and is compared with those performed using interparticle interaction force. The dynamic viscous fluid and surface tension interactions are investigated via a controlled curvature model and test cases of nonsteady oscillating droplets; attention is focused here on droplet oscillation that is released from an initial static deformation. Accuracy of the results is attested by demonstrating that (i) the curvature of the droplet that is controlled; (ii) uniform distribution of fluid particles; (iii) clean asymmetric forces acting on the free surface; and (iv) nonsteady oscillating droplets compare well with analytical and published experiment findings. The advantage of the proposed continuous surface force method only requires the use of physical properties of the fluid, whereas the interparticle interaction force method is restricted by the requirement of tuning parameters.     

Simulations of free surface flows with implementation of surface tension and interface sharpening in the two-fluid model

Two-fluid model used for free surface flows with large characteristic scales is improved; the smeared interface is sharpened with conservative level set method and the surface tension force with wetting angle is implemented. Surface tension force is split between two phases with several models. Detailed analysis showed the splitting of surface tension force with volume averaging as the most appropriate. The improved two-fluid model with interface sharpening and implemented surface tension is validated on several test cases. The pressure jump over a droplet interface test case showed that the pressure jump in simulation converges with grid refinement to the analytical one. The parasitic currents in simulation are one order of magnitude larger than in simulation with volume of fluid model. In the oscillating droplet test case the time period of oscillating droplet with initially ellipsoid or square shape is similar to the analytical time period. In the rising bubble test case, the rising bubble position, terminal velocity, and circularity are similar to the one observed in simulations with level set model. The wetting angle is implemented in the two-fluid model with interface sharpening and surface tension force. Model is tested in the simulation of droplet in contact with wall with different wetting angles.     

大射电天文望远镜悬索馈源系统驱动力的研究

针对大射电天文望远镜馈源系统中悬索呈现出的柔性特征，在已知馈源舱的位置，姿态，速度和加速度的情况下，将悬索离散为柔性索杆单元，建立了系统的逆运动学及逆动力学模型。分析计算了悬索上各离散点的位置、速度、加速度和惯性力，推导了在重力和惯性力共同作用下悬索的空间挠曲线方程，研究了悬索的刚体位移和弹性变形。在此基础上，通过刚体-柔性的迭代计算，求出悬索馈源系统的驱动力，为高精度的大射电天文望远镜提供了必要的设计参数。     

Reverse squeeze film flow in a continous flow system

The continuous-flow squeeze film apparatus has been adapted to permit flow to take place in either direction. This simulates normal and reversed squeezing flow between discs by having liquid moving through the lower plate, with neither plate moving. The liquid exudes from 1580 uniformly-distributed holes in the plate surface. All tests were performed at a temperature of 24.0°C.Water is used in early tests, and it is shown that the contribution to load bearing from the inertia of the fluid is comparable in reverse and normal flows; fluid inertia increases the force which would be required to move the plates in either direction. A novel “mirror image” graphical presentation is used.Tests using a dilute polymer solution show load enhancement effects for both normal and reversed squeeze film flow. The enhancement is roughly equal in both directions of flow, with no transient effects, and fluid elasticity increases the force which would be required to move the plates in either direction. It is suggested that the stress developed in the fluid is independent of the direction of flow.The significance of the tests regarding lubricating problems is mentioned; the important case of rapid load reversal requires further attention.