Selective withdrawal of polymer solutions: Experiments

Selective withdrawal refers to the process of drawing one or both components of stratified fluids through a tube placed near their interface. This paper reports an experimental study of selective withdrawal of viscous and viscoelastic liquids under air. The key mechanism of interest is how the viscoelasticity in the bulk liquid affects the evolution of the free surface. This is investigated by comparing the interfacial behavior between a Newtonian silicone oil and two dilute polymer solutions. While the surface undergoes smooth and gradual deformation for Newtonian liquids, for the polymer solutions there is a critical transition where the surface forms a cusp from which an air jet emanates toward the suction tube. This transition shows a hysteresis when the flow rate or location of the tube is varied. In the subcritical state, the surface of polymer solutions deform much more than its Newtonian counterpart but the deformation is more localized. The interfacial behavior of the polymer solutions can be attributed to the large polymer stress that develops under the surface because of predominantly extensional deformation.     

Interface stabilization in two-layer channel flow by surface heating or cooling

The linear stability of plane Poiseuille flow of two immiscible Newtonian liquids in a differentially heated channel is considered. The equations of motion and energy are fully coupled via temperature-dependent fluid-viscosity coefficients. A long-wave asymptotic formulation of the stability problem is presented together with numerical results for disturbances of arbitrary wavelength. Two combinations of immiscible liquids are analyzed: silicone/water and oil/water (water at the bottom layer in both cases). It is shown that an imposed wall temperature difference can be stabilizing or destabilizing depending on the disturbance wavenumber and layer thickness ratio. Interfacial tension has a stabilizing effect on the interface. Stabilizing influence of interfacial tension is observed at intermediate and large wavenumbers. Most importantly, for certain ranges of the controlling dimensionless parameters, stable interfaces at all disturbance wavelengths can be attained.     

流场中聚合物共混体系液滴形变的理论模型

讨论了两相聚合物共混体系中，悬浮于另一种牛顿（或粘弹）液体中的牛顿（或粘弹）液滴的形变理论模型．影响液滴形变的主要因素有两相的组成、粘度比和弹性比、动态界面张力、临界界面张力系数，外流场形式及其强度．对于两相均为牛顿流体的体系，理论预测能够与实验相符；对于两相（或其中一相）为粘弹流体的体系，由于弹性的影响而使液滴形变的研究变得复杂，理论模型尚需完善．建立完整的液滴形变理论模型还需深入研究界面层、微观分子形变、液滴之间及液滴和连续相介质之间的相互作用对液滴形变的影响     

A model for film-forming with Newtonian and shear-thinning fluids

The formation of a thin film by (i) the slow penetration of a gas bubble into a liquid filled tube, (ii) the withdrawal of a planar substrate from a liquid filled gap, is investigated theoretically for the cases of both Newtonian and shear-thinning liquids; the latter conforming to either a power–law or Ellis model. Formulated as a boundary value problem underpinned by lubrication theory, the analysis gives rise to a system of ordinary differential equations which are solved numerically subject to appropriate boundary conditions. For Newtonian liquids comparison of the predicted residual film thickness for a wide range of capillary number, Ca  (10⁻⁴, 10), is made with others obtained using existing expressions, including the classical one of Bretherton, in the region of parameter space over which they apply. In the case of (i), prediction of the behaviour of the residual fluid fraction and gap-to-film thickness ratio, for a Newtonian liquid and one that is shear-thinning and modelled via a power–law, is found to be in particularly good agreement with experimental data for Ca < 0.2. For (ii), both shear-thinning models are utilized and contour plots of residual film thickness generated as a function of Ca and the defining parameters characteristic of each model.     

Numerical simulations of non-isothermal three-dimensional flows in an extruder by a finite-volume method

This study considers numerical applications of a finite-volume method to steady non-isothermal flows in geometries close to a single-screw extruder. Two geometrical configurations of the channel, with gap and zero gap, are investigated. The simulations concern incompressible fluids obeying different constitutive equations: Newtonian, generalized Newtonian with shear-thinning properties (Carreau–Yasuda law), and two viscoelastic differential models, the upper convected maxwell (UCM) and the Phan–Thien/Tanner (PTT). The temperature dependence is described by a Williams–Landel–Ferry (WLF) equation. For discretizing the equations and unknowns, we use a staggered grid with a QUICK scheme for the convective-type terms and solve the set of governing equations by a decoupled algorithm, stabilized by a pseudo-transient stress term and an elastic viscous stress splitting (EVSS) technique, in the viscoelastic case for the UCM model. The numerical results enable us to state the influence of temperature and rheological properties on the flow characteristics in the geometries investigated and underline the complex behaviour of the materials in such configurations.     

Anomalous rolling of spheres down an inclined plane

A sphere in air will roll down a plane that is tilted away from the vertical. The only couple acting about the point of contact between the sphere and the plane is due to the component of the weight of the sphere along the plane, provided that air friction is negligible. If on the other hand the sphere is immersed in a liquid, hydrodynamic forces will enter into the couples that turn the sphere, and the rotation of the sphere can be anomalous, i.e., as if rolling up the plane while it falls. In this paper we shall show that anomalous rolling is a characteristic phenomenon that can be observed in every viscoelastic liquid tested so far. Anomalous rolling is normal for hydrodynamically levitated spheres, both in Newtonian and viscoelastic liquids. Normal and anomalous rolling are different names for dry and hydrodynamic rolling. Spheres dropped at a vertical wall in Newtonian liquids are forced into anomalous rotation and are pushed away from the wall while in viscoelastic liquids, they are forced into anomalous rotation, but are pushed toward the wall. If the wall is inclined and the fluid is Newtonian, the spheres will rotate normally for dry rolling, but the same spheres rotate anomalously in viscoelastic liquids when the angle of inclination from the vertical is less than some critical value. The hydrodynamic mechanisms underway in the settling of circular particles in a Newtonian fluid at a vertical wall are revealed by an exact numerical simulation based on a finite-element solution of the Navier-Stokes equations and Newton's equations of motion for a rigid body.     

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.     

Laminar flow in three-dimensional square–square expansions

In this work we investigate the three-dimensional laminar flow of Newtonian and viscoelastic fluids through square–square expansions. The experimental results obtained in this simple geometry provide useful data for benchmarking purposes in complex three-dimensional flows. Visualizations of the flow patterns were performed using streak photography, the velocity field of the flow was measured in detail using particle image velocimetry and additionally, pressure drop measurements were carried out. The Newtonian fluid flow was investigated for the expansion ratios of 1:2.4, 1:4 and 1:8 and the experimental results were compared with numerical predictions. For all expansion ratios studied, a corner vortex is observed downstream of the expansion and an increase of the flow inertia leads to an enhancement of the vortex size. Good agreement is found between experimental and numerical results. The flow of the two non-Newtonian fluids was investigated experimentally for expansion ratios of 1:2.4, 1:4, 1:8 and 1:12, and compared with numerical simulations using the Oldroyd-B, FENE-MCR and sPTT constitutive equations. For both the Boger and shear-thinning viscoelastic fluids, a corner vortex appears downstream of the expansion, which decreases in size and strength when the elasticity of the flow is increased. For all fluids and expansion ratios studied, the recirculations that are formed downstream of the square–square expansion exhibit a three-dimensional structure evidenced by a helical flow, which is also predicted in the numerical simulations.     

Laminar jet contraction and velocity distribution in immiscible liquid-liquid systems

The flow behavior of a Newtonian liquid jet injected vertically into an immiscible Newtonian liquid phase is analyzed. Boundary-layer type approximations are used to simplify the general equations, and an approximate momentum-integral type numerical solution is obtained. This solution predicts the velocity distribution in each phase and the jet radius. The effects on jet behavior of the five dimensionless groups needed to characterize the gravitational, interfacial tension and viscous forces are shown. In particular the importance of the continuous phase viscosity is demonstrated. Experimental measurements of jet radius confirm the essential features of the analysis and illustrate the shortcomings of the approximate solution.     

Effect of trigonometric sine,square and triangular wave-type time-periodic gravity-aligned oscillations on Rayleigh–Bénard convection in Newtonian liquids and Newtonian nanoliquids

The influence of trigonometric sine, square and triangular wave-types of time-periodic gravity-aligned oscillations on Rayleigh–Bénard convection in Newtonian liquids and in Newtonian nanoliquids is studied in the paper using the generalized Buongiorno two-phase model. The five-mode Lorenz model is derived under the assumptions of Boussinesq approximation, small-scale convective motion and some slip mechanisms. Using the method of multiscales, the Lorenz model is transformed to a Ginzburg–Landau equation the solution of which helps in quantifying the heat transport through the Nusselt number. Enhancement of heat transport in Newtonian liquids due to the presence of nanoparticles/nanotubes is clearly explained. The study reveals that all the three wave types of gravity modulation delay the onset of convection and thereby to a diminishment of heat transport. It is also found that in the case of trigonometric sine type of gravity modulation heat transport is intermediate to that of the cases of triangular and square types. The paper is the first such work that attempts to theoretically explain the effect of three different wave-types of gravity modulation on onset of convection and heat transport in the presence/absence of nanoparticles/nanotubes. Comparing the heat transport by the single-phase and by the generalized two-phase models, the conclusion is that the single-phase model under-predicts heat transport in nanoliquids irrespective of the type of gravity modulation being effected on the system. The results of the present study reiterate the findings of related experimental and numerical studies.

     

Computing non‐Newtonian fluid flow with radial basis function networks

This paper is concerned with the application of radial basis function networks (RBFNs) for solving non‐Newtonian fluid flow problems. Indirect RBFNs, which are based on an integration process, are employed to represent the solution variables; the governing differential equations are discretized by means of point collocation. To enhance numerical stability, stress‐splitting techniques are utilized. The proposed method is verified through the computation of the rectilinear and non‐rectilinear flows in a straight duct and the axisymmetric flow in an undulating tube using Newtonian, power‐law, Criminale–Ericksen–Filbey (CEF) and Oldroyd‐B models. The obtained results are in good agreement with the analytic and benchmark solutions. Copyright © 2005 John Wiley & Sons, Ltd.     

Separating forces in blade coating of viscous and viscoelastic liquids

Coating of viscous and viscoelastic liquids is examined both theoretically and experimentally. A single simple geometry, a blade over a rotating roll, is considered. A perturbation solution to the Navier-Stokes equations yields a lubrication theory with first order corrections for curvature and inertia. A numerical solutions by the Finite Element Method (FEM) is compared to the analytical solutions. For Newtonian fluids, agreement between these mathematical models, and data on blade loading, is quite good.The effect of a non-Newtonian viscosity is explored by adopting a purely viscous power law model. The zeroth-order (lubrication) equations are solved by the method of Steidler and Horowitz, and predictions for coating thickness and blade loading agree quite well with those obtained from a FEM solution of the full equations of motion for a power law fluid. Data on blade loading, obtained using a strongly elastic polymer solution, are compared to these mathematical models, and discrepancies are noted.     

Purely elastic interfacial instabilities in superposed flow of polymeric fluids

Purely elastic interfacial stability of superposed plane Poiseuille flow of polymeric liquids has been investigated utilizing both asymptotic and numerical techniques. It is shown that these instabilities are caused by an unfavorable jump in the first normal stress difference across the fluid interface. To determine the significance of these instabilities in finite experimental geometries, a comparison between the maximum growth rates of purely elastic instabilities with instabilities driven primarily by a viscosity or a combined viscosity and elasticity difference is made. Based on this comparison, it is shown that purely elastic interfacial instabilities can play a major role in superposed flow of polymeric liquids in finite experimental geometries.     

Three-dimensional arbitrary Lagrangian–Eulerian numerical prediction method for non-linear free surface oscillation

A numerical prediction method has been proposed to predict non-linear free surface oscillation in an arbitrarily-shaped three-dimensional container. The liquid motions are described with Navier–Stokes equations rather than Laplace equations which are derived by assuming the velocity potential. The profile of a liquid surface is precisely represented with the three-dimensional curvilinear co-ordinates which are regenerated in each computational step on the basis of the arbitrary Lagrangian–Eulerian (ALE) formulation. In the transformed space, the governing equations are discretized on a Lagrangian scheme with sufficient numerical accuracy and the boundary conditions near the liquid surface are implemented in a complete manner. In order to confirm the applicability of the present computational technique, numerical simulations are conducted for the free oscillations of viscid and inviscid liquids and for highly non-linear oscillation. In addition, non-linear sloshing motions caused by horizontal and vertical excitations and a transition from non-linear sloshing to swirling are numerically predicted in three-dimensional cylindrical containers. Conclusively, it is shown that these sloshing motions associated with high non-linearity are reasonably predicted with the present numerical technique. © 1998 John Wiley & Sons, Ltd.     

Numerical simulation of landslide impulsive waves by incompressible smoothed particle hydrodynamics

An incompressible‐smoothed particle hydrodynamics (I‐SPH) formulation is presented to simulate impulsive waves generated by landslides. The governing equations, Navier–Stokes equations, are solved in a Lagrangian form using a two‐step fractional method. Landslides in this paper are simulated by a submerged mass sliding along an inclined plane. During sliding, both rigid and deformable landslides mass are considered. The present numerical method is examined for a rigid wedge sliding into water along an inclined plane. In addition solitary wave generated by a heavy box falling inside water, known as Scott Russell wave generator, which is an example for simulating falling rock avalanche into artificial and natural reservoirs, is simulated and compared with experimental results. The numerical model is also validated for gravel mass sliding along an inclined plane. The sliding mass approximately behaves like a non‐Newtonian fluid. A rheological model, implemented as a combination of the Bingham and the general Cross models, is utilized for simulation of the landslide behaviour. In order to match the experimental data with the computed wave profiles generated by deformable landslides, parameters of the rheological model are adjusted and the numerical model results effectively match the experimental results. The results prove the efficiency and applicability of the I‐SPH method for simulation of these kinds of complex free surface problems. Copyright © 2007 John Wiley & Sons, Ltd.     

Computation of 2D Navier–Stokes equations with moving interfaces by using GMRES

In this paper, we present a novel numerical algorithm to compute two‐dimensional (2D) viscous interfacial flows governed by the incompressible Navier–Stokes equations together with interfacial conditions. The essential idea is to use the generalized minimum residual (GMRES) method to efficiently solve the large algebraic system resulting from the temporal and spatial discretizations. With this algorithm, moving interfaces can be captured with high accuracy and viscous effects on wave motion can be studied in detail. Copyright © 2006 John Wiley & Sons, Ltd.     

A finite element analysis of the steady laminar entrance flow in a 90° curved tube

A standard Galerkin finite element penalty function method is used to approximate the solution of the three-dimensional Navier–Stokes equations for steady incompressible Newtonian entrance flow in a 90° curved tube (curvature ratio δ = 1/6) for a triple of Dean numbers (κ = 41, 122 and 204). The computational results for the intermediate Dean number (κ = 122) are compared with the results of laser–Doppler velocity measurements in an equivalent experimental model. For both the axial and secondary velocity components, fair agreement between the computational and experimental results is found.     

Numerical simulation and design optimization of an electrohydrodynamic pump for dielectric liquids

With an ever increasing demand for more effective heat sinks, liquid based electronic cooling has become a new prospect in the field. The present study introduces an electrohydrodynamic (EHD) pump with a simple design for dielectric liquids which have potential applications for electronic cooling. The pump consists of an eccentrically sandwiched wire electrode placed at the horizontal centerline between two parallel flat-plate electrodes. The EHD flow of dielectric liquid induced by the space charge generated due to the Onsager effect was obtained by the numerical solution of the Poisson–Nernst–Planck equations for ion transport and the Navier–Stokes equations for fluid flow. Good agreement obtained in the comparison of the numerical and the experimental results of velocity for the centrally sandwiched wire electrode case confirmed the validity of the numerical results. For a fixed voltage, the pump flow rate depends on the eccentricity of the wire electrode with respect to the plate electrodes and also with the electrode dimensions. By using the Taguchi method an optimum design for the EHD pump is obtained considering the wire electrode diameter, the flat plate electrode length and the eccentricity (the horizontal distance between the centers of wire and flat-plate electrodes) as the design parameters for fixed channel dimensions.     

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.     

Effect of time-periodic vertical oscillations of the Rayleigh–Bénard system on nonlinear convection in viscoelastic liquids

A study of heat transport in Rayleigh–Bénard convection in viscoelastic liquids with/without gravity modulation is made using a most minimal representation of Fourier series and a representation with higher modes. The Oldroyd-B constitutive relation is considered. The resulting non-autonomous Lorenz model (generalized Khayat–Lorenz model of four modes and seven modes) is solved numerically using the adaptive-grid Runge–Kutta–Fehlberg45 method to quantify the heat transport. The effect of gravity modulation is shown to be stabilizing there by leading to a situation of reduced heat transfer. The Deborah number is shown to have an antagonistic influence on convection compared to the stabilizing effect of modulation amplitude and elastic ratio. The results in respect of Maxwell, Rivlin–Ericksen and Newtonian liquids are obtained as particular cases of the present study. A transformation of the momentum equations illustrates the equivalence of present approach and the one due to Khayat that uses normal stresses explicitly.