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.     

Particles accelerate the detachment of viscous liquids

During detachment of a viscous fluid extruded from a nozzle, a filament linking the droplet to the latter is formed. Under the effect of surface tension, the filament thins until pinch-off and final detachment of the droplet. In this paper, we study the effect of the presence of individual particles trapped in the filament on the detachment dynamics using granular suspensions of small volume fractions (??<?6 %). We show that even a single particle strongly modifies the detachment dynamics. The particle perturbs the thinning of the thread, and a large droplet of fluid around the particle is formed. This perturbation leads to an acceleration of the detachment of the droplet compared to the detachment observed for a pure fluid. We quantify this acceleration for single particles of different sizes and link it to similar observations for suspensions of small volume fractions. Our study also gives more insight into particulate effects on detachment of denser suspensions and allows to explain the accelerated detachment close to final pinch-off observed previously (Bonnoit et al. Phys Fluids 24(4):043304, 2012).     

Coaxial-disk flow of an Oldroyd-B fluid: exact solution and stability

This paper reports an exact solution for the coaxial disk flow of an Oldroyd-B fluid. The flow is approximately generated by the parallel-plate viscometer. Asymptotic and numerical solutions are reported showing that there is a critical Weissenberg number based on the angular velocity and the Maxwellian relaxation time, above which the flow is unstable. A linearized stability analysis for the basic inertialess flow confirms this numerical instability and yields the critical Weissenberg number.     

Torsional flow stability of highly dilute polymer solutions

We examine the torsional flow stability, to axisymmetric disturbances, of a variety of multimode and non-linear constitutive models in a bounded parallel plate geometry. The analysis is facilitated by the construction of a regular perturbation scheme in the ratio of polymer to total viscosity. As a model for Boger fluids this corresponds to the assumption that the Boger fluid is highly dilute. The consequent simplification provided by the perturbation scheme allows us to examine the effects of a discrete spectrum of relaxation times, shear thinning, second normal stress difference, and finite extensional viscosity on the torsional instability.     

Justification of the drift-flux model for two-phase flow in a circular pipe

A flow of a gas-liquid dispersed mixture in a circular pipe with a variable inclination to the horizon, as applied to oil and gas flows in wells, is considered. Within the framework of a multi- fluid approach, the equations of an asymptotic drift-flux model, which contains an algebraic relation between the phase velocities and one momentum equation for the volume-averaged velocity of the mixture, are derived. It is shown that the drift-flux model in this formulation strictly follows from the balance laws under assumption of inertialess velocity slip of the phases in case of validity of one of the following conditions: (i) the dispersed-phase volume fraction is small; (ii) the phase velocity slip may be neglected; or (iii) the flow regime is inertialess and the acceleration of the mixture can be neglected. A numerical algorithm based on the SIMPLE method is implemented for solving the obtained equations of the drift-flux model. The possibility of modeling the gravitational segregation and the pressure buildup in a shut-in well and transient slug flows is demonstrated.     

Displacement of yield-stress fluids in a fracture

We consider a displacement of several yield-stress fluids in a Hele-Shaw cell. The topic is relevant to the development of a model for the flow of multiple phases inside a narrow fracture with application to hydraulically fracturing a hydrocarbon-bearing underground formation. Existing models for fracturing flows include only pure power-law models without yield stress, and the present work is aimed at filling this gap. The fluids are assumed to be immiscible and incompressible. We consider fluid advection in a plane channel in the presence of density gradients. Gravity is taken into account, so that there can be slumping and gravitational convection. We use the lubrication approximation so that governing equations are reduced to a 2D width-averaged system formed by the quasi-linear elliptic equation for pressure and transport equations for volume concentrations of fluids. The numerical solution is obtained using a finite-difference method. The pressure equation is solved using an iterative algorithm and the Multigrid method, while the transport equations are solved using a second-order TVD flux-limiting scheme with the superbee limiter. This numerical model is validated against three different sets of experiments: (i) gravitational slumping of fluids in a closed Hele-Shaw cell, (ii) viscous fingering of fluids with a high viscosity contrast due to the Saffman–Taylor (S–T) instability in a Hele-Shaw cell at microgravity conditions, (iii) displacement of Bingham fluids in a Hele-Shaw cell with the development of fingers due to the S–T instability. Good agreement is observed between simulations and laboratory data. The model is then used to investigate the joint effect of fingering and slumping. Numerical simulations show that the slumping rate of yield-stress fluid is significantly less pronounced than that of a Newtonian fluid with the same density and viscosity. If a low-viscosity Newtonian fluid is injected after a yield-stress one, the S–T instability at the interface leads to the development of fingers. As a result, fingers penetrating into a fluid with a finite yield stress locally decrease the pressure gradient and unyielded zones develop as a consequence.     

Asymptotic MHD flows in channels

A numerical and experimental study is made of the possibility of the onset of inertialess flow regimes in a conducting fluid in channels of complex geometry under the influence of a strong external field. The region of existence is determined for the inertialess regime from the Hartmann numbers and the MHD interaction parameter and asymptotic estimates are obtained for the drag coefficients.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 159–163, March–April, 1985.     

Computational study of buoyancy effects in a laminar starting jet

Vortical structures formed in evolving jets are important in applications such as fuel injection in diesel engines and fuel leaks. When the jet fluid is different from the ambient fluid, the buoyancy can play an important role in determining the jet flow structure, and hence, the entrainment and fluid mixing processes. In the present study, a jet of helium injected in air is investigated, with emphasis placed on delineating the buoyancy effects on vector–scalar fields during the starting phase. We utilize a computational model, previously validated to predict the flow field of low-density gas jets. The model incorporates finite volume approach to solve the transport equation of helium mass fraction coupled with conservation equations of mixture mass and momentum. Computations were performed for a laminar jet to characterize the advancing jet front, and to capture the formation and propagation of vortex rings and the related pinch-off process. Results show significant effects of buoyancy on jet advancement, as well as on vorticity and helium concentration in the core of the vortex rings.     

Interfacial instability of turbulent two-phase stratified flow: Pressure-driven flow and non-Newtonian layers

We derive a flat-interface model to describe the flow of two horizontal, stably stratified fluids, where the bottom layer exhibits non-Newtonian rheology. The model takes into account the yield stress and power-law nature of the bottom fluid. In the light of the large viscosity contrast assumed to exist across the fluid interface, and for large pressure drops in the streamwise direction, the possibility for the upper Newtonian layer to display fully developed turbulence must be considered, and is described in our model. We develop a linear-stability analysis to predict the conditions under which the flat-interface state becomes unstable, and pay particular attention to characterizing the influence of the non-Newtonian rheology on the instability. Increasing the yield stress (up to the point where unyielded regions form in the bottom layer) is destabilizing; increasing the flow index, while bringing a broader spectrum of modes into play, is stabilizing. In addition, a second mode of instability is found, which depends on conditions in the bottom layer. For shear-thinning fluids, this second mode becomes more unstable, and yet more bottom-layer modes can become unstable for a suitable reduction in the flow index. One further difference between the Newtonian and non-Newtonian cases is the development of unyielded regions in the bottom layer, as the linear wave on the interface grows in time. These unyielded regions form in the trough of the wave, and can be observed in the linear analysis for a suitable parameter choice.     

Draw resonance in film casting of viscoelastic fluids: A linear stability analysis

In order to understand the role of viscoelasticity on draw resonance in the isothermal film casting process, a steady state analysis and a linear stability analysis for three-dimensional flow disturbances have been conducted. The constitutive equation used is a modified convected Maxwell model, with shear-rate dependent viscosity and fluid characteristic time. The numerical results indicate that the flow is stable below a lower critical draw ratio and above an upper critical draw ratio. Shear thinning in viscosity reduces the lower critical draw ratio and somewhat increases the upper critical draw ratio—thereby enlarging the region of instability. Slower shear reduction in fluid characteristic time dramatically decreases the upper critical draw ratio but has no significant effect on the lower critical draw ratio; therefore, fluids with higher characteristic time are more stable.     

Flow of elasto-viscoplastic liquids through an axisymmetric expansion–contraction

In this work, we used a recently proposed constitutive model that addresses the elastic behavior of viscoplastic liquids. We used the equation in an expansion–contraction axisymmetric flow, and compared the results with experimental data found in the literature. Steady, inertialess numerical solutions were obtained by solving the balance equations of mass and momentum via the finite element method. We obtained the velocity and stress fields for different combinations of the governing parameters, and observed that elasticity leads to significant changes in the shape and position of the yield surface, affecting both the extra-stress and the rate of deformation fields in the cavity. The trends observed are in qualitative agreement with visualization results available in the literature.     

Instability of channel flow of a shear-thinning White–Metzner fluid

We consider the inertialess planar channel flow of a White–Metzner (WM) fluid having a power-law viscosity with exponent n. The case n = 1 corresponds to an upper-convected Maxwell (UCM) fluid. We explore the linear stability of such a flow to perturbations of wavelength k⁻¹. We find numerically that if n < n_c ≈ 0.3 there is an instability to disturbances having wavelength comparable with the channel width. For n close to n_c, this is the only unstable disturbance. For even smaller n, several unstable modes appear, and very short waves become unstable and have the largest growth rate. If n exceeds n_c, all disturbances are linearly stable. We consider asymptotically both the long-wave limit which is stable for all n, and the short-wave limit for which waves grow or decay at a finite rate independent of k for each n.The mechanism of this elastic shear-thinning instability is discussed.     

Viscoplastic flow between two approaching or parallel circular plates

The problem of the flow of a viscoplastic medium between two parallel circular plates in translatory coaxial relative motion is solved. The Bingham model [1] of a viscoplastic medium is assumed. The problem is solved in the inertialess thin layer approximation [2] for arbitrary values of the viscosity coefficient and yield stress.Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 1, pp. 9–17, January–February, 1996.     

旋转圆球入水空泡特性与流场结构的大涡模拟研究

圆球旋转入水过程对于基于先导物投放的新型入水降载方式具有重要研究价值. 采用大涡模拟方法结合均质多相流模型和VOF界面捕捉算法, 对低弗劳德数条件下疏水圆球高速旋转入水的自由运动过程进行了数值模拟, 研究了转速对入水空泡演化、流场结构和水动力特性的影响. 采用动网格与滑移网格技术实现圆球的自由运动, 并基于试验结果对比验证了数值模拟的可靠性与正确性. 旋转运动的升力效应导致圆球入水弹道发生偏转并从水面携带横向楔形射流侵入空泡内部. 采用入水砰击速度与转速进行归一化分析, 结果表明入水转速的增加显著改变了圆球的动力特性: 水平方向的速度和位移以及升力峰值都随入水转速的增加而变大, 但升力峰值受到入水速度的限制; 而垂直方向的速度和加速度以及空泡断裂深度几乎不受转速增加的影响, 并且空泡深闭合发生前圆球转速变化不大. 入水转速的增加也使液面飞溅环和空泡断裂的非对称性增强, 在较低转速时发生空泡表面闭合, 而在较高转速时则发生空泡深闭合. 对于空泡深闭合模式, 入水转速的增加带来更强的横向楔形射流, 并且抑制了空泡断裂产生的高压以及相应涡结构的生成, 致使圆球在入水砰击阶段承受更低的侧向压力.      

Nonlinear analysis of the deformation and breakup of viscous microjets using the method of lines

We present a numerical model for predicting the instability and breakup of viscous microjets of Newtonian fluid. We adopt a one‐dimensional slender‐jet approximation and obtain the equations of motion in the form of a pair of coupled nonlinear partial differential equations (PDEs). We solve these equations using the method of lines, wherein the PDEs are transformed to a system of ordinary differential equations for the nodal values of the jet variables on a uniform staggered grid. We use the model to predict the instability and satellite formation in infinite microthreads of fluid and continuous microjets that emanate from an orifice. For the microthread analysis, we take into account arbitrary initial perturbations of the free‐surface and jet velocity, as well as Marangoni instability that is due to an arbitrary variation in the surface tension. For the continuous nozzle‐driven jet analysis, we take into account arbitrary time‐dependent perturbations of the free‐surface, velocity and/or surface tension as boundary conditions at the nozzle orifice. We validate the model using established computational data, as well as axisymmetric, volume of fluid (VOF) computational fluid dynamic (CFD) simulations. The key advantages of the model are its ease of implementation and speed of computation, which is several orders of magnitude faster than the VOF CFD simulations. The model enables rapid parametric analysis of jet breakup and satellite formation as a function of jet dimensions, modulation parameters, and fluid rheology. Copyright © 2010 John Wiley & Sons, Ltd.     

Taylor–Dean instability of yield-stress fluids at large gaps

Centrifugal instability of Bingham fluids is investigated in Taylor–Dean flow when the gap size is large compared to the cylinders radii. To determine conditions for the onset of instability, an infinitesimal axisymmetric disturbance was introduced to the basic flow and its evolution in time was monitored using a normal-mode linear stability analysis. To avoid the problem with the stress discontinuity at the location of the yield surface(s), use was made of the Papanastasiou’s regularized variation of the Bingham model in order to obtain the basic flow velocity profiles. An eigenvalue problem was obtained for the exact Bingham fluid which was solved numerically using the collocation method. A plot of the neutral instability curve at different Bingham numbers suggests that the yield stress can have a stabilizing or destabilizing effect on Taylor–Dean flow depending on the sign and magnitude of the pressure gradient, and also on the sense of rotation of the two cylinders with respect to each other.     

Jet pinch-off and drop formation in immiscible liquid–liquid systems

The behavior of glycerin–water jets flowing into immiscible ambients of Dow Corning 200 fluid was investigated using laser induced fluorescence (LIF). Undistorted images were obtained by matching the index of refraction of the fluids. A sinusoidal perturbation was superposed on the flow to phase lock the drop formation. The forcing frequency dramatically affected the size, spacing, and number of drops that formed within a forcing cycle and the angle between drops and the jet interface just before pinch-off. Two fluid combinations were studied with similar density ratios, but viscosity ratios differing by a factor of 20. The viscosity ratio affected the jet stability as well as pinch-off angles and drop size. Received: 28 January 1999/Accepted: 20 January 2000     

Using filament stretching rheometry to predict strand formation and “processability” in adhesives and other non-Newtonian fluids

The spinning of polymeric fibers, the processing of numerous foodstuffs and the peel and tack characteristics of adhesives are all associated with the formation, stability and, ultimately, the longevity of thin fluid `strands'. This tendency to form strands is usually described in terms of the tackiness of the fluid or by heuristic concepts such as `stringiness' (Lakrout et al. J Adhesion 1999). The dynamics of such processes are complicated due to spatially and temporally non-homogeneous growth of extensional stresses, the action of capillary forces and the evaporation of volatile solvents. We describe the development and application of a simple instrument referred to as a microfilament rheometer (MFR) that can be used to readily differentiate between the dynamical response of different pressure-sensitive adhesive fluid formulations. The device relies on a quantitative observation of the rate of extensional thinning or `necking' of a thin viscous or viscoelastic fluid filament in which the solvent is free to evaporate across the free surface. This high-resolution measurement of the radial profile provides a direct indication of the ultimate time to break up of the fluid filament. This critical time is a sensitive function of the rheological properties of the fluid and the mass transfer characteristics of the solvent, and can be conveniently reported in terms of a new dimensionless quantity we refer to as a processability parameter P. We demonstrate the usefulness of this technique by presenting our results in the form of a case study in which we measure the visco-elasto-capillary thinning of slender liquid filaments for a number of different commercial polymer/solvent formulations and relate this to the reported processing performance of the materials. We also compare the MFR observations with the prediction of a simple 1D theory derived from the governing equations that model the capillary thinning of an adhesive filament. Received: 22 December 1999/Accepted: 4 January 2000     

Analysis of power-law self-similar solutions to the problem of hydraulic fracture crack formation

The problem of hydraulic fracture crack propagation in a porous medium is studied in the approximation of small crack opening and the inertialess flow of an incompressible Newtonian hydraulic fracturing fluid inside the crack. A one-parameter family of power-law self-similar solutions is considered in order to determine the crack width evolution, the fluid velocity in the crack, and the seepage depth in the case of high and low seepage rates through the soil when a fluid flow rate is given at the crack inlet.     

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.