A boundary integral method for two-dimensional (non)-Newtonian drops in slow viscous flow

A boundary integral method for the simulation of the time-dependent deformation of Newtonian or non-Newtonian drops suspended in a Newtonian fluid is developed. The boundary integral formulation for Stokes flow is used and the non-Newtonian stress is treated as a source term which yields an extra integral over the domain of the drop. The implementation of the boundary conditions is facilitated by rewriting the domain integral by means of the Gauss divergence theorem. To apply the divergence theorem smoothness assumptions are made concerning the non-Newtonian stress tensor. The correctness of these assumptions in actual simulations is checked with a numerical validation procedure. The method appears mathematically correct and the numerical algorithm is second order accurate. Besides this validation we present simulation results for a Newtonian drop and a drop consisting of an Oldroyd-B fluid. The results for Newtonian and non-Newtonian drops in two dimensions indicate that the steady state deformation is quite independent of the drop-fluid. The deformation process, however, appears to be strongly dependent on the drop-fluid. For the non-Newtonian drop a mechanical model is developed to describe the time-dependent deformation of the cylinder for small capillary numbers.     

The turbulent flow of non-Newtonian fluids in the absence of anomalous wall effects

For Newtonian fluids, the engineering predictions for pressure drop in turbulent pipe flow are well established. However, in the case of non-Newtonian liquids, only a few design techniques have been proposed and these do not share a common basis with the approach for Newtonian systems. This present work attempts to provide a common basis for both Newtonian and non-Newtonian systems in situations where anomalous wall effects are absent. Previously published experimental data suggest that if the Reynolds number is calculated on the basis of the apparent viscosity at the wall then the standard Newtonian correlations can be used for the prediction of pressure drop. The use of the wall viscosity in defining the Reynolds number also serves as a test for anomalous behaviour. Any departure of the experimental data from the Newtonian turbulent friction factor correlation indicates anomalous behaviour.     

An experimental study of drop migration in shear flow between concentric cylinders

The phenomenon of migration of liquid drops in Couette flow between concentric cylinders due to non-Newtonian fluid properties and shape deformation has been studied experimentally. The results agree very well with the theory of Chan and Leal, which included the effect of hydrodynamic interaction with the bounding walls, and that of velocity profile curvature in a Couette device. Significant observations that were not reported in previous studies include the migration of a deformable Newtonian drop to an equilibrium position between the centerline and the inner rotor, and the competition between normal stresses and shape deformation effects for the case of a Newtonian drop in a non-Newtonian fluid.     

Computational analysis of binary collisions of shear-thinning droplets

Scale-reduced models of transport processes and reactive mixing in sprays require improved closure laws, taking into account the characteristic features of elementary spray processes. The present paper investigates binary droplet collisions as such an elementary process. In the case of shear-thinning liquids considered here, this requires a profound understanding of the influence of the non-Newtonian fluid rheology on the flow inside the colliding drops and the collision complex dynamics. We employ direct numerical simulations based on the Volume-of-Fluid method to study these collisions. The results give a quantitative prediction of the resulting droplet collision diameter as well as a qualitative prediction of the complete time evolution. During collisions, extremely thin fluid lamellae appear inside the expanding complex. These have to be accounted for in a physically sound simulation and we apply a stabilization of the lamella to keep it from rupturing. The simulations show that in all considered cases an effective constant viscosity can be found a posteriori which leads to the same collision dynamics. But this effective viscosity is neither the mean nor the minimum viscosity.     

Turbulent flow of Newtonian and shear-thinning liquids through a sudden axisymmetric expansion

 Measurements are reported for the turbulent flow through a sudden expansion of a moderately elastic shear-thinning liquid and also for two Newtonian liquids. The differences in the mean velocity fields for the two fluid types are relatively small, including the length of the recirculation region which is essentially unaffected by the fluid rheology. Although turbulent kinetic energy levels for the non-Newtonian fluids are always lower than for the Newtonian fluids, no significant difference is found in the relative contributions to the turbulent kinetic energy of the axial, radial and tangential normal stresses. Since the vorticity thicknesses are much the same for all flows, viscoelasticity appears to be responsible for the reduced levels of turbulent kinetic energy for the non-Newtonian fluids. Received: 6 November 1998/Accepted: 27 January 1999     

Diffuse-interface simulations of drop coalescence and retraction in viscoelastic fluids

Drop dynamics plays a central role in defining the interfacial morphology in two-phase complex fluids such as emulsions and polymer blends. In such materials, the components are often microstructured complex fluids themselves. To model and simulate drop behavior in such systems, one has to deal with the dual complexity of non-Newtonian rheology and evolving interfaces. Recently, we developed a diffuse-interface formulation which incorporates complex rheology and interfacial dynamics in a unified framework. This paper uses a two-dimensional implementation of the method to simulate drop coalescence after head-on collision and drop retraction from an elongated initial shape in a quiescent matrix. One of the two phases is a viscoelastic fluid modeled by an Oldroyd-B equation and the other is Newtonian. For the parameter values examined here, numerical results show that after drop collision, film drainage is enhanced when either phase is viscoelastic and drop coalescence happens more readily than in a comparable Newtonian system. The last stage of coalescence is dominated by a short-range molecular force in the model that is comparable to van der Waals force. The retraction of drops from an initial state of zero-velocity and zero-stress is hastened at first, but later resisted by viscoelasticity in either component. When retracting from an initial state with pre-existing stress, produced by cessation of steady shearing, viscoelasticity in the matrix hinders retraction from the beginning while that in the drop initially enhances retraction but later resists it. These results and the physical mechanisms that they reveal are consistent with prior experimental observations.     

The stratified flow of gas and non-newtonian liquid in horizontal pipes

Predictions of pressure drop and holdup are presented for the stratified flow of gas and non-Newtonian liquid obeying the Ostwald-de Waele power law model. The model of Taitel & Dukler (1976) for gas/Newtonian liquid flow is extended to liquids possessing either shear-thinning or shear-thickening laminar flow behaviour and computed results are given for flow behaviour indices in the range $\text{0.1 ≤ n ≤ 2}$. In particular, conditions are defined for drag reduction of the liquid flow by the presence of the gas. It is concluded that drag reduction occurs over the largest ranges of liquid and gas flow rates at the lowest $\text{n}$ values, provided that liquid flow remains laminar, but that maximum drag reduction may be expected for shear-thickening liquids with $\text{n}$ values of 2 or greater. Ratios of the liquid flow rate in the presence of gas to that for liquid flow alone under a constant pressure gradient are also presented. These ratios frequently exceed unity and are greatest for highly shear-thinning liquids.Although the Taitel & Dukler approach is consistent with experiments on gas/Newtonian liquid flow, and, in addition, appears to be valid for immiscible Newtonian liquid-liquid systems, provided that the viscosity ratio of the two phases is at least five, experiments are required to confirm its applicability for gas/non-Newtonian systems.     

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.     

Models for the deformation of a single ellipsoidal drop: a review

Dilute polymer blends and immiscible liquid emulsions are characterized by a globular morphology. The dynamics of a single drop subjected to an imposed flow field has been considered to be a valuable model system to get information on dilute blends. This problem has been studied either theoretically by developing exact theories for small drop deformations or by developing simplified models often based on phenomenological assumptions. In this paper, a critical overview of the available models for the dynamics of a single drop is presented, discussing four different systems, namely the Newtonian system, where a single Newtonian drop is immersed in an infinite Newtonian matrix; the non-Newtonian system, where at least one of the components, the drop fluid or the matrix one, is non-Newtonian; the confined Newtonian system, where the matrix is confined and wall effects alter the drop dynamics; and the confined non-Newtonian system.     

Fractional Flow Theory Applicable to Non-Newtonian Behavior in EOR Processes

The method of characteristics, or fractional-flow theory, is extremely useful in understanding complex Enhanced Oil Recovery (EOR) processes and in calibrating simulators. One limitation has been its restriction to Newtonian rheology except in rectilinear flow. Its inability to deal with non-Newtonian rheology in polymer and foam EOR has been a serious limitation. We extend fractional flow methods for two-phase flow to non-Newtonian fluids in one-dimensional cylindrical flow, where rheology changes with distance from injection well. The fractional flow curve is then a function of position and we analyze the characteristic equations for two applications—polymer and foam floods. For polymer flooding, we present a semi-analytical solution for the changing fractional flow curve where characteristics and shocks collide. The semi-analytical solution is shown to give good agreement with the finite-difference simulation thus helping us understand the development and resolution of shocks. We discuss two separate cases of foam injection with or without preflush. We observe that the fractional flow solutions are more accurate than finite-difference simulations on a comparable grid and hence the method can be used to calibrate simulators. For SAG (alternating-slug) foam injection, characteristics and shocks collide, making the fractional-flow solution complex. Nonetheless, one can solve exactly for changing mobility near the well, to greater accuracy than with conventional simulation. The fractional-flow method extended to non-Newtonian flow can be useful both for its insights for scale-up of laboratory experiments and to calibrate computer simulators involving non-Newtonian EOR. It can also be an input to streamline simulations.     

A note on the translation of a thin rod inside a cylinder

The problem of a thin rod moving longitudinally along the axis of symmetry of a cylindrical vessel is examined for Newtonian and non-Newtonian liquids. For non-Newtonian fluids, the inelastic power-law type solution predicts the experimental results particularly well. On account of wall effects, the induced pressure gradients are much greater for a Newtonian fluid than for a viscoelastic fluid. In fact, in the latter case, they may be considered negligible when the radius of the inner cylinder is small compared to the one of the outer cylinder.     

Creeping flow of viscoplastic materials through a planar expansion followed by a contraction

In this work, the creeping flow of a viscoplastic fluid through a planar channel with an expansion followed by a contraction is analyzed numerically. The solution of the conservation equations of mass and momentum is obtained via the finite volume method. In order to model the non-Newtonian behavior of the fluid, it was used the generalized Newtonian fluid constitutive equation. The viscosity function was the one proposed by Souza Mendes and Dutra [Souza Mendes, P.R., Dutra, E.S.S., 2004. Viscosity function for yield-stress liquids. Appl. Rheol. 14, 296–302]. The yielded and unyielded regions are obtained for several combinations of rheological parameters. The influence of these parameters on pressure drop through the cavity is also obtained and analyzed.     

On the flow of Newtonian and non-Newtonian liquids through corrugated pipes

Rheologica Acta - Consideration is given to the flow of Newtonian and non-Newtonian liquids through a “corrugated” pipe of circular cross section whose radius sinusoidally along its...     

Pulsatile flows of Leslie–Ericksen liquid crystals

Capillary pulsatile flows of calamitic (rod-like) and discotic nematic liquid crystals are analyzed using the Leslie–Ericksen equations for low-molar mass liquid crystals, using computational, analytical, and scaling methods. The dependence of flow-enhancement and power requirement on frequency, amplitude, pressure drop wave-form, molecular geometry is characterized. The unique roles of orientation-dependent local viscosity and backflow (orientation-driven flow) on flow-enhancement and power requirement are elucidated. The local viscosity effect is shown to be a significant factor in flow-enhancement at all pressure drops, but only affects power requirement at higher pressure drops. Backflow has weak effects on flow-enhancement and large effects on power requirements at low average pressure drops. Amplitude, frequency, and molecular geometry effects are clearly manifested through viscosity and backflow. A detailed comparison with predictions for power law fluids shows a clear correspondence between these non-Newtonian fluids and nematic liquid crystals. The unique distinguishing feature of pulsatile flows of liquid crystals is found to be backflow, such that power increases with increasing frequency, a featured that does not exist in other non-Newtonian fluids due to lack of a strong flow driven by restructuring/re-orientation processes. Future use of these new results may include measurements of viscoelastic parameters that control backflow.     

A unified compatibility method for exact solutions of non-linear flow models of Newtonian and non-Newtonian fluids

Criteria are established for higher order ordinary differential equations to be compatible with lower order ordinary differential equations. Necessary and sufficient compatibility conditions are derived which can be used to construct exact solutions of higher order ordinary differential equations subject to lower order equations. We provide the connection to generalized groups through conditional symmetries. Using this approach of compatibility and generalized groups, new exact solutions of non-linear flow problems arising in the study of Newtonian and non-Newtonian fluids are derived. The ansatz approach for obtaining exact solutions for non-linear flow models of Newtonian and non-Newtonian fluids is unified with the application of the compatibility and generalized group criteria.     

Computations of breakup modes in laminar compound liquid jets in a coflowing fluid

We present a numerical investigation of breakup modes of an axisymmetric, laminar compound jet of immiscible fluids, which flows in a coflowing immiscible outer fluid. We use a front-tracking/finite difference method to track the unsteady evolution and breakup of the compound jet, which is governed by the Navier–Stokes equations for incompressible Newtonian fluids. Numerical results show that depending on parameters such as the Reynolds number Re (in the range of 5–30) and Weber Number We (in the range of 0.1–0.7), based on the inner jet radius and inner fluid properties, the compound jet can break up into drops in various modes: inner dripping–outer dripping (dripping), inner jetting–outer jetting (jetting), and mixed dripping–jetting. Decreasing Re or increasing We promotes the jetting mode. The transition from dripping to jetting is also strongly affected by the velocity ratios, U₂₁ (intermediate to inner velocities) and U₃₁ (outer to inner velocities). Increasing U₂₁ makes the inner jet thinner and stretches the outer jet and thus promotes jetting. In contrast, increasing U₃₁ thins the outer jet, and thus, when the inner jet is dripping, the outer jet can break up into drops in the mixed dripping–jetting mode. Continuously increasing U₃₁ results in thinning both inner and outer jets and thus produces small drops in the jetting mode. In addition, starting from dripping, a decrease in the interfacial tension ratio of the outer to inner interfaces results in the mixed dripping–jetting and jetting modes. These modes produce various types of drops: simple drops, and compound drops with a single inner drop (single-core compound drops) or a few inner drops (multi-core compound drops).     

Studies on two-phase co-current air/non-Newtonian shear-thinning fluid flows in inclined smooth pipes

In this work, co-current flow characteristics of air/non-Newtonian liquid systems in inclined smooth pipes are studied experimentally and theoretically using transparent tubes of 20, 40 and 60 mm in diameter. Each tube includes two 10 m long pipe branches connected by a U-bend that is capable of being inclined to any angle, from a completely horizontal to a fully vertical position. The flow rate of each phase is varied over a wide range. The studied flow phenomena are bubbly flow, stratified flow, plug flow, slug flow, churn flow and annular flow. These are observed and recorded by a high-speed camera over a wide range of operating conditions. The effects of the liquid phase properties, the inclination angle and the pipe diameter on two-phase flow characteristics are systematically studied. The Heywood–Charles model for horizontal flow was modified to accommodate stratified flow in inclined pipes, taking into account the average void fraction and pressure drop of the mixture flow of a gas/non-Newtonian liquid. The pressure drop gradient model of Taitel and Barnea for a gas/Newtonian liquid slug flow was extended to include liquids possessing shear-thinning flow behaviour in inclined pipes. The comparison of the predicted values with the experimental data shows that the models presented here provide a reasonable estimate of the average void fraction and the corresponding pressure drop for the mixture flow of a gas/non-Newtonian liquid.     

A 2D boundary element method for simulating the deformation of axisymmetric compound non‐Newtonian drops

The boundary integral formulation of the solution to the Stokes equations is used to describe the deformation of small compound non‐Newtonian axisymmetric drops suspended in a Newtonian fluid that is subjected to an axisymmetric flow field. The non‐Newtonian stress is treated as a source term in the Stokes equations, which yields an extra integral over the domains containing non‐Newtonian material. By transforming the integral representation for the velocity to cylindrical co‐ordinates and performing the integration over the azimuthal direction analytically, the dimension of the problem can be reduced from three to two. A boundary element method for the remaining two‐dimensional problem aimed at the simulation of the deformation of such axisymmetric compound non‐Newtonian drops is developed. Apart from a numerical validation of the method, simulation results for a drop consisting of an Oldroyd‐B fluid and a viscoelastic material are presented. Moreover, the method is extended to compound drops that are composed of a viscous inner core encapsulated by a viscoelastic material. The simulation results for these drops are verified against theoretical results from literature. Moreover, it is shown that the method can be used to identify the dominant break‐up mechanism of compound drops in relation to the specific non‐Newtonian character of the membrane. Copyright © 1999 John Wiley & Sons, Ltd.     

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.     

Viscous flow of a suspension of liquid drops in a cylindrical tube

Viscous flow in a circular cylindrical tube containing an infinite line of viscous liquid drops equally spaced along the tube axis is considered under the assumption that a surface tension, sufficiently large, holds the drops in a nearly spherical shape. Three cases are considered: (1) axial translation of the drops, (2) flow of the external fluid past a line of stationary drops, and (3) flow of external fluid and liquid drops under an imposed pressure gradient. Both fluids are taken to be Newtonian and incompressible, and the linearized equations of creeping flow are used.The results show that both drag and pressure drop per sphere increase as the spacing increases at fixed radius and also increase as the radius of the drop increases. The presence of the internal motion reduces the drag and pressure gradients in all cases compared to rigid spheres, particularly for drops approaching the size of the tube.