Viscoelastic fluid behavior in annulus using Giesekus model

An analytical solution is derived for the steady state, laminar, axial, fully developed flow of a viscoelastic fluid obeying the Giesekus model without any retardation time in a concentric annulus.An approximation is used for the estimation of radial normal stress. The influence of Deborah number (De) and the mobility factor (α) on the velocity profile, axial pressure gradient are investigated and results show strong effects of mobility factor and Deborah number on above parameters.     

Finite element calculation of viscoelastic flow in a journal bearing: II. Moderate eccentricity

Finite element calculations of two-dimensional flows of viscoelastic fluids in a journal bearing geometry reported in an earlier paper (J. Non-Newt. Fluid Mech. 16 (1984) 141-172) are extended to higher eccentricity (ρ = 0.4); at this higher eccentricity flow separation occurs in the wide part of the gap for a Newtonian fluid. Calculations for the second-order fluid (SOF), upper-convected Maxwell (UCM), and the Giesekus models are continued in increasing Deborah number for each model until either a limit point is reached or oscillations in the solution make the numerical accuracy too poor to warrant proceeding. No steady solutions to the UCM model were found beyond a limit point De_c, as was the case for results at low eccentricities. The value of De_c was moderately stabel to mesh refinement. A limit point also terminated the calculations with a SOF model, in contradiction to the theorems for uniqueness and existence for this model. The critical value of De increased drastically with increasing refinement of the mesh, as expected for solution pathology caused by approximation error. Calculations for the Giesekus fluid with the mobility parameter α ≠ O showed no limit points, but failed when irregular oscillations destroyed the quality of the solution. The behavior of the recirculation region of the flow and the load on the inner cylinder were very sensitive to the value of α used in the Giesekus model. The recirculation disappeared at low values of De except when the mobility parameter α was so small that the viscosity was almost constant over the range of shear rates in the calculations. The recirculation persisted over the entire range of accessible De for the UCM fluid, the limit of α = O of the Giesekus model. The behavior of the recirculation is coupled directly to the viscosity by calculations with an inelastic fluid with the same viscosity predicted by the Giesekus model.     

Influence of viscoelasticity on drop deformation and orientation in shear flow: Part 1. Stationary states

The influence of matrix and droplet viscoelasticity on the steady deformation and orientation of a single droplet subjected to simple shear is investigated microscopically. Experimental data are obtained in the velocity–vorticity and velocity–velocity gradient plane. A constant viscosity Boger fluid is used, as well as a shear-thinning viscoelastic fluid. These materials are described by means of an Oldroyd-B, Giesekus, Ellis, or multi-mode Giesekus constitutive equation. The drop-to-matrix viscosity ratio is 1.5. The numerical simulations in 3D are performed with a volume-of-fluid algorithm and focus on capillary numbers 0.15 and 0.35. In the case of a viscoelastic matrix, viscoelastic stress fields, computed at varying Deborah numbers, show maxima slightly above the drop tip at the back and below the tip at the front. At both capillary numbers, the simulations with the Oldroyd-B constitutive equation predict the experimentally observed phenomena that matrix viscoelasticity significantly suppresses droplet deformation and promotes droplet orientation. These two effects saturate experimentally at high Deborah numbers. Experimentally, the high Deborah numbers are achieved by decreasing the droplet radius with other parameters unchanged. At the higher capillary and Deborah numbers, the use of the Giesekus model with a small amount of shear-thinning dampens the stationary state deformation slightly and increases the angle of orientation. Droplet viscoelasticity on the other hand hardly affects the steady droplet deformation and orientation, both experimentally and numerically, even at moderate to high capillary and Deborah numbers.     

Uniform flow of viscoelastic fluids past a confined falling cylinder

Uniform steady flow of viscoelastic fluids past a cylinder placed between two moving parallel plates is investigated numerically with a finite-volume method. This configuration is equivalent to the steady settling of a cylinder in a viscoelastic fluid, and here, a 50% blockage ratio is considered. Five constitutive models are employed (UCM, Oldroyd-B, FENE-CR, PTT and Giesekus) to assess the effect of rheological properties on the flow kinematics and wake patterns. Simulations were carried out under creeping flow conditions, using very fine meshes, especially in the wake of the cylinder where large normal stresses are observed at high Deborah numbers. Some of the results are compared with numerical data from the literature, mainly in terms of a drag coefficient, and significant discrepancies are found, especially for the constant-viscosity constitutive models. Accurate solutions could be obtained up to maximum Deborah numbers clearly in excess of those reported in the literature, especially with the PTT and FENE-CR models. The existence or not of a negative wake is identified for each set of model parameters.     

BUBBLE CHARACTERISTICS IN A TWO-DIMENSIONAL VERTICALLY VIBRO-FLUIDIZED BED

Measurement of bubble size and local average bubble rise velocity was carried out in a vertically sinusoidal vibro-fluidized bed. Glass beads of Geldart group B particles were fluidized at different gas velocities, while the bed was vibrated at different frequencies and amplitudes to study their effects on the bubble behavior. This is compared with the case of no vibration in a two-dimensional bed and it is concluded that with vibration the local average bubble size, dbav, decreases significantly, especially at minimum bubbling velocity. The average bubble size increases slightly with increasing vibration frequency and amplitude. The local average bubble rise velocity is higher than that with no vibration, though with increasing vibration frequency and amplitude, it does not change significantly.     

An experimental and numerical investigation of the dynamics of microconfined droplets in systems with one viscoelastic phase

The dynamics of single droplets in a bounded shear flow is experimentally and numerically investigated for blends that contain one viscoelastic component. Results are presented for systems with a viscosity ratio of 1.5 and a Deborah number for the viscoelastic phase of 1. The numerical algorithm is a volume-of-fluid method for tracking the placement of the two liquids. First, we demonstrate the validation of the code with an existing boundary integral method and with experimental data for confined systems containing Newtonian components. This is followed by numerical simulations and experimental data for the combined effect of geometrical confinement and component viscoelasticity on the droplet dynamics after startup of shear flow at a moderate capillary number. The viscoelastic liquids are Boger fluids, which are modeled with the Oldroyd-B constitutive model and the Giesekus model. Confinement substantially increases the viscoelastic stresses and the elongation rates in and around the droplet. We show that the latter can be dramatic for the use of the Oldroyd-B model in confined systems with viscoelastic components. A sensitivity analysis for the choice of the model parameters in the Giesekus constitutive equation is presented.     

Practical comparison of differential viscoelastic constitutive equations in finite element analysis of planar 4:1 contraction flow

Finite element modeling of planar 4:1 contraction flow (isothermal incompressible and creeping) around a sharp entrance corner is performed for favored differential constitutive equations such as the Maxwell, Leonov, Giesekus, FENE-P, Larson, White-Metzner models and the Phan Thien-Tanner model of exponential and linear types. We have implemented the discrete elastic viscous stress splitting and streamline upwinding algorithms in the basic computational scheme in order to augment stability at high flow rate. For each constitutive model, we have obtained the upper limit of the Deborah number under which numerical convergence is guaranteed. All the computational results are analyzed according to consequences of mathematical analyses for constitutive equations from the viewpoint of stability. It is verified that in general the constitutive equations proven globally stable yield convergent numerical solutions for higher Deborah number flows. Therefore one can get solutions for relatively high Deborah number flows when the Leonov, the Phan Thien-Tanner, or the Giesekus constitutive equation is employed as the viscoelastic field equation. The close relationship of numerical convergence with mathematical stability of the model equations is also clearly demonstrated.     

Numerical study of Saffman–Taylor instability in immiscible nonlinear viscoelastic flows

In this paper, a numerical solution for Saffman–Taylor instability of immiscible nonlinear viscoelastic-Newtonian displacement in a Hele–Shaw cell is presented. Here, a nonlinear viscoelastic fluid pushes a Newtonian fluid and the volume of fluid method is applied to predict the formation of two phases. The Giesekus model is considered as the constitutive equation to describe the nonlinear viscoelastic behavior. The simulation is performed by a parallelized finite volume method (FVM) using second order in both the spatial and the temporal discretization. The effect of rheological properties and surface tension on the immiscible Saffman–Taylor instability are studied in detail. The destabilizing effect of shear-thinning behavior of nonlinear viscoelastic fluid on the instability is studied by changing the mobility factor of Giesekus model. Results indicate that the fluid elasticity and capillary number decrease the intensity of Saffman–Taylor instability.     

Parametric Analysis for a Single Collapsing Bubble

This work presents a sensitivity analysis for cavitation processes, studying in detail the effect of various model parameters on the bubble collapse. A complete model (Hauke et al. Phys Rev E 75:1–14, 2007) is used to obtain how different parameters influence the collapse in SBSL experiments, providing some clues on how to enhance the bubble implosion in real systems. The initial bubble radius, the frequency and the amplitude of the pressure wave are the most important parameters determining under which conditions cavitation occurs. The range of bubble sizes inducing strong implosions for different frequencies is computed; the initial radius is the most important parameter characterized the intensity of the cavitation processes. However, other parameters like the gas and liquid conductivity or the liquid viscosity can have an important effect under certain conditions. It is shown that mass transfer processes play an important role in order to correctly predict the trends related with the effect of the liquid temperature, which translates into the bubble dynamics. Moreover, under some particular circumstances, evaporation can be encountered during the bubble collapse; this can be profitably exploited in order to feed reactants when the most extreme conditions inside the bubbles are reached. Thus, this paper aims at providing a global assessment of the effect of the different parameters on the entire cycle of a single cavitating spherical bubble immersed in an ultrasonic field. This work has been partially supported by Ministerio de Ciencia y Tecnologia, under grant number CTM2004-06184-C02-02.     

Finite element computations of viscoelastic two-phase flows using local projection stabilization

A three-field local projection stabilized (LPS) finite element method is developed for computations of a three-dimensional axisymmetric buoyancy driven liquid drop rising in a liquid column where one of the liquid is viscoelastic. The two-phase flow is described by the time-dependent incompressible Navier-Stokes equations, whereas the viscoelasticity is modeled by the Giesekus constitutive equation in a time-dependent domain. The arbitrary Lagrangian-Eulerian (ALE) formulation with finite elements is used to solve the governing equations in the time-dependent domain. Interface-resolved moving meshes in ALE allows to incorporate the interfacial tension force and jumps in the material parameters accurately. A one-level LPS based on an enriched approximation space and a discontinuous projection space is used to stabilize the numerical scheme. A comprehensive numerical investigation is performed for a Newtonian drop rising in a viscoelastic fluid column and a viscoelastic drop rising in a Newtonian fluid column. The influence of the viscosity ratio, Newtonian solvent ratio, Giesekus mobility factor, and the Eötvös number on the drop dynamics are analyzed. The numerical study shows that beyond a critical Capillary number, a Newtonian drop rising in a viscoelastic fluid column experiences an extended trailing edge with a cusp-like shape and also exhibits a negative wake phenomena. However, a viscoelastic drop rising in a Newtonian fluid column develops an indentation around the rear stagnation point with a dimpled shape.     

An approximate solution for the Couette–Poiseuille flow of the Giesekus model between parallel plates

An approximate analytical solution is derived for the Couette–Poiseuille flow of a nonlinear viscoelastic fluid obeying the Giesekus constitutive equation between parallel plates for the case where the upper plate moves at constant velocity, and the lower one is at rest. Validity of this approximation is examined by comparison to the exact solution during a parametric study. The influence of Deborah number (De) and Giesekus model parameter (α) on the velocity profile, normal stress, and friction factor are investigated. Results show strong effects of viscoelastic parameters on velocity profile and normal stress. In addition, five velocity profile types were obtained for different values of α, De, and the dimensionless pressure gradient (G).     

An initially spherical bubble rising near a vertical wall

The dynamics of a single rising bubble in the vicinity of a vertical wall is explored via three-dimensional numerical simulations. A finite volume method is used to solve the incompressible Navier–Stokes equations. The gas–liquid interface is reconstructed by volume-of-fraction (VOF) method. The trajectory, velocity, shape and vorticity of the bubble are analyzed in detail. The numerical results show that the presence of the wall imposes a repulsion on the bubble and that the bubble migrates away from the wall upon release. The onset of bubble path oscillation is found to occur earlier than for a freely rise counterpart and also at a lower Galilei number. Interestingly, we find that the vertical wall serves as a destabilizing factor in the wall-normal direction but a stabilizing factor in the spanwise direction. The increase of bubble inertia is discovered to enhance the influence of the wall. Furthermore, the bubble oscillations seem insensitive to the variation of the initial bubble-wall distance.     

The axisymmetric vibrations of a cylindrical incompressible liquid column with a gas bubble

The axisymmetric vibrations of an ideal incompressible liquid column in a rigid circular cylindrical vessel with a spherical gas bubble pulsating near the position of dynamic equilibrium are considered. The boundary-value problem for the liquid velocity potential and the equations for the vibrations of the gas bubble are solved under the conditions on the free surface, sidewall, and the boundary of the gas body. For the case of small amplitudes, the resonance frequencies of the system are determined, and the pressure field in the liquid column is constructed. The results are compared with data known for the gas-accumulation model, data obtained without allowance for the boundedness of the liquid, and experimental data. National Technical University (KPI), Kiev, Ukraine. Translated from Prikladnaya Mekhanika, Vol. 36, No. 7, pp. 74–80, July, 2000.     

Level set method for numerical simulation of a cavitation bubble,its growth,collapse and rebound near a rigid wall

A level set method of non-uniform grids is used to simulate the whole evolution of a cavitation bubble, including its growth, collapse and rebound near a rigid wall. Single-phase Navier–Stokes equation in the liquid region is solved by MAC projection algorithm combined with second-order ENO scheme for the advection terms. The moving interface is captured by the level set function, and the interface velocity is resolved by “one-side” velocity extension from the liquid region to the bubble region, complementing the second-order weighted least squares method across the interface and projection inside bubble. The use of non-uniform grid overcomes the difficulty caused by the large computational domain and very small bubble size. The computation is very stable without suffering from large flow-field gradients, and the results are in good agreements with other studies. The bubble interface kinematics, dynamics and its effect on the wall are highlighted, which shows that the code can effectively capture the “shock wave”-like pressure and velocity at jet impact, toroidal bubble, and complicated pressure structure with peak, plateau and valley in the later stage of bubble oscillating. The project supported by the National Natural Science Foundation of China (10272032 and 10672043). The English text was polished by Keren Wang.     

Attempts to find a molecular theory for which the high-frequency dynamic viscosity is less than the solvent viscosity

It has been found that for some dilute polymer solutions the dynamic viscosity at very high frequencies is less than the zero-shear-rate solvent viscosity. Such an effect cannot be explained by the usual kinetic theories using bead-spring-rod models. Here we examine several modifications of the kinetic theories that might be expected to explain the experimental facts.This paper is dedicated to Prof. Hanswalter Giesekus on the occasion of his retirement as Editor of Rheologica Acta.     

水平均流中细管排放气泡的三维数值模拟

在液体为无粘不可压，流动有势和气体遵循完全气体绝热关系的假定下，本文应用边界积分方程方法数值模拟了水平均流中垂直细管排放气泡的三维动力学问题，边界采用高阶有限元表达。文中介绍了有关泡面法向矢量、切向速度、曲率和接触线等的计算技术。与已知解的比较，表明了这一数值方法的高精度和优越性。算例显示了水平均流对于气泡形状和体积的影响     

A numerical study on nonlinear dynamics of three-dimensional time-depended viscoelastic Taylor-Couette flow

In this paper, three-dimensional viscoelastic Taylor-Couette instability between concentric rotating cylinders is studied numerically. The aim is to investigate and provide additional insight about the formation of time-dependent secondary flows in viscoelastic fluids between rotating cylinders. Here, the Giesekus model is used as the constitutive equation. The governing equations are solved using the finite volume method (FVM) and the PISO algorithm is employed for pressure correction. The effects of elasticity number, viscosity ratio, and mobility factor on various instability modes (especially high order ones) are investigated numerically and the origin of Taylor-Couette instability in Giesekus fluids is studied using the order of magnitude technique. The created instability is simulated for large values of fluid elasticity and high orders of nonlinearity. Also, the effect of elastic properties of fluid on the time-dependent secondary flows such as wave family and traveling wave and also on the critical conditions are studied in detail.     

A numerical and experimental study of a collapsing bubble-induced droplet ejector

This paper aims to study a novel drop-on-demand droplet generation mechanism in which the oscillation and deformation of a non-equilibrium bubble in close proximity to a free surface induce an axisymmetric liquid spike on the free surface. The evolution of the liquid spike and its deformation due to the effect of surface tension force lead to the formation of a droplet. The free surface can be accorded by either a circular hole on a horizontal flat plate or by the top opening/nozzle of a vertical cylinder. A high-speed camera capable of obtaining images at a frame rate of 15,000 fps is utilized to observe the droplet formation process. Numerical simulations corresponding to the experiments are performed using the boundary integral spatial solution coupled with the time integration, i.e., a mixed Eulerian–Lagrangian method. In the experiments the bubble is generated using a very low voltage (only 55 V) in contrast to the relatively much higher voltages usually employed in reported works. This is very attractive from a safety viewpoint and accords great simplification of the setup. A comparison is made between the numerical and experimental results. A reasonable agreement has been found. The influences of the main design parameters, namely, the bubble-free surface distance and the dimension of the hole/nozzle on the bubble dynamics and on the droplet formation process are discussed and the conditions of the bubble dynamics under which a satellite-free droplet can be generated are sought. Furthermore, the effects of different geometries, namely, the horizontal flat plate and the vertical cylinder on the bubble dynamics and on the droplet features are examined. One important feature of the proposed actuation mechanism is the capability of producing droplets much smaller than the nozzle size. The possible applications of this mechanism are those where the accurate direction of the ejected droplet is of great importance such as inkjet printing.      

Bubbles in liquids with phase transition

In the forthcoming second part of this paper a system of balance laws for a multi-phase mixture with many dispersed bubbles in liquid is derived where phase transition is taken into account. The exchange terms for mass, momentum and energy explicitly depend on evolution laws for total mass, radius and temperature of single bubbles. Therefore in the current paper we consider a single bubble of vapor and inert gas surrounded by the corresponding liquid phase. The creation of bubbles, e.g. by nucleation is not taken into account. We study the behavior of this bubble due to condensation and evaporation at the interface. The aim is to find evolution laws for total mass, radius and temperature of the bubble, which should be as simple as possible but consider all relevant physical effects. Special attention is given to the effects of surface tension and heat production on the bubble dynamics as well as the propagation of acoustic elastic waves by including slight compressibility of the liquid phase. Separately we study the influence of the three phenomena heat conduction, elastic waves and phase transition on the evolution of the bubble. We find ordinary differential equations that describe the bubble dynamics. It turns out that the elastic waves in the liquid are of greatest importance to the dynamics of the bubble radius. The phase transition has a strong influence on the evolution of the temperature, in particular at the interface. Furthermore the phase transition leads to a drastic change of the water content in the bubble. It is shown that a rebounding bubble is only possible, if it contains in addition an inert gas. In Part 2 of the current paper the equations derived are sought in order to close the system of equations for multi-phase mixture balance laws for dispersed bubbles in liquids involving phase change.     

Growth of a gas bubble in a supersaturated and slightly compressible liquid at low Mach number

In this paper, the growth of a gas bubble in a supersaturated and slightly compressible liquid is discussed. The mathematical model is solved analytically by using the modified Plesset and Zwick method. The growth process is affected by: sonic speed in the liquid, polytropic exponent, diffusion coefficient, initial concentration difference, surface tension, viscosity, adjustment factor and void fraction. The famous formula of Plesset and Zwick is produced as a special case of the result at some values of the adjustment factor. Moreover, the resultant formula is implemented to the case of the growth of underwater gas bubble.