Role of shear-thinning and viscoelasticity on the onset of instability of glass fiber suspensions in the concentric spherical gap between an inner rotating hemisphere and an outer stationary sphere

An experimental study investigated the effects of glass fiber suspensions on the onset of instability in the non-Newtonian fluid flow in the concentric spherical gaps between an inner rotating hemisphere and an outer stationary whole sphere. Glass fibers with different aspect ratios were mixed with a macromolecule polymeric solution to obtain different suspension fluids. For comparison, the pure macromolecule polymeric fluid was also investigated. The torques on the inner sphere were measured for various spherical gaps and various rotational Reynolds numbers. The onset of instability of the polymeric fluid flows was delayed by adding glass fibers to the polymeric solution for all tested gap ratios.     

An experimental investigation of upstream flow characteristics of viscoelastic fluids in an annular die entry

Flow behaviour of viscoelastic polymer solutions on the upstream side of an annular die entry has been experimentally investigated and compared with published results on entry flow in circular die. Stable and unstable flow patterns were observed depending on the magnitudes of Reynolds and elasticity numbers. The latter number represents the relative effects of elastic and inertial forces. The stable flow region consists of an elasticity-controlled vortex growth regime and an inertia-controlled divergent flow regime. These two flow regimes have also been observed in circular entry flow. The criteria for the onset of flow instability in an annulus, based on the maximum vortex size, agree qualitatively with various criteria proposed for polymer metls and solutions in circular entry flow. The unstable flow patterns revealed a two-stage instability with a metastable region in between. The first-stage instability is characterised by low frequency disruption of the stationary vortex; while the second-stage instability, which occurs at high Reynolds numbers, is characterised by high frequency random distortion of the flow field.     

Creation of very-low-Reynolds-number chaotic fluid motions in microchannels using viscoelastic surfactant solution

Solutions of flexible high-molecular-weight polymers or some kinds of surfactant are viscoelastic fluids. The elastic stress is induced in such viscoelastic fluid flows and grows nonlinearly with the flow-rate resulting in many particular flow phenomena, including purely elastic instability. The purely elastic instability can even result in a kind of chaotic fluid motion, the so-called elastic turbulence, which is a recently discovered flow phenomenon and arises at arbitrarily small Reynolds number. By using viscoelastic surfactant solution, we attempted to create the peculiar chaotic fluid motions in several specially designed microchannels in which flows with curvilinear streamlines can be generated. The viscoelastic working fluids were aqueous solutions of surfactant, CTAC/NaSal (cetyltrimethyl ammonium chloride/sodium salicylate). CTAC solutions with weight concentration of 200 ppm (part per million) and 1000 ppm, respectively, at room temperature were tested. For comparison, water flows in the same microchannels were also visualized. The Reynolds numbers for all the microchannel flows were quite small (for solution flows, the Reynolds numbers were the order of or smaller than one) and the flow should be definitely laminar for Newtonian fluid. It was found that the regular laminar flow patterns for low-Reynolds-number Newtonian fluid flow in different microchannels were strongly deformed in solution flows: either asymmetrical flow structures or time-dependent vortical fluid motions appeared. These chaotic flow phenomena were considered to be induced by the viscoelasticity of the CTAC solutions. Discussions about the potential applications using such kind of chaotic fluid motions were also made.     

Drag of a rotating sphere

We consider the problem of steady incompressible viscous fluid flow about a rotating sphere, with the flow specified on a sphere of finite radius, which reduces to the solution of the complete Navier-Stokes equations.The dimensionless stream functions and circulai velocity are sought in the form of series in powers of the Reynolds numbers, which converge for small values of this number. Recurrence formulas are derived for determining the coefficients of these series. The pressure, rotational resistance torque, and drag are determined. It is established that the rotating sphere has higher drag than a stationary sphere. The leading term of the series in powers of the Reynolds number for the drag and resistive torque is calculated.     

An experimental investigation of the flow about a sphere rotating in a rivlin-ericksen fluid

Tangential and radial velocity profiles were measured for the flow about a sphere rotating slowly in a Newtonian fluid, contained in a rectangular tank. Velocities were determined from enlarged streak photographs of aluminium particles moving in a collimated “sheet” of light, at several planes through the flow field. Similar velocity profiles were measured for the flow of a 1.50% Natrosol 250 H solution about two spheres of different diameters rotating in two different sized rectangular tanks. A set of velocity distributions were also measured for a sphere rotating in a 0.9% Natrosol 250 H solution. A dye tracer study of the flow about a sphere rotating in this liquid is presented as well. Both Natrosol solutions exhibited viscoelastic behaviour. The Newtonian fluid study was carried out at a Reynolds number of 1.2 and the viscoelastic fluid studies were within the Reynolds number range of 0.05–1.24.The zero shear viscosities of the Natrosol solutions were measured using the falling-sphere method. The non-Newtonian material parameters were obtained by fitting the theoretical curves to the measured velocity data. The values of the elastic and shear thinning parameters for the two liquids obtained in the different geometrical and dynamical situations are compared.     

Finite element analysis of the axially symmetric motion of an incompressible viscous fluid in a spherical annulus

This paper presents results obtained by employing a modified Galerkin finite element method to analyse the steady state flow of a fluid contained between two concentric, rotating spheres. The spheres are assumed to be rigid and the cavity region between the spheres is filled with an incompressible, viscous, Newtonian fluid. The inner sphere is constrained to rotate about a vertical axis with a prescribed angular velocity, while the outer sphere is fixed. Results for the circumferential function Ω, streamfunction ψ, vorticity function ζ and inner boundary torque T₁ are presented for Reynolds numbers Re ? 2000 and radius ratios 0.1 ? α ? 0.9. The method proved effective for obtaining results for a wide range of radius ratios (0.1 ? α ? 0.9) and Reynolds numbers (0 ? Re ? 2000). Previous investigators who employed the finite difference method experienced difficulties in obtaining results for cases with radius ratios α ? 0.2, except for small Reynolds numbers (Re ? 100). Results for Ω, Ψ, ζ and T₁ obtained in this study for radius ratios 0.8 ≤ α ≤ 0.9 verified the development of Taylor vortices reported by other investigators. The research indicates that the method may be useful for analysing other non-linear fluid flow problems.     

Flow of a shear-thinning liquid in a cylindrical container with a rotating end wall

The flow patterns produced by rotating one end wall of a circular cylinder completely filled with a strongly shear-thinning viscoelastic liquid have been investigated using the laser-induced fluorescence flow visualization technique. An intense toroidal vortex is produced in the vicinity of the rotating end wall with outward spiraling flow over the end wall itself. This vortex drives a second countercirculating vortex of low intensity in the region of the stationary end wall. Under some circumstances an axial jet of fluid is observed moving away from the rotating end wall. This jet showed evidence of instability, whereas all flows were otherwise completely steady. The double-vortex structure is different from those recently observed in either a Newtonian or slightly shear-thinning liquid or in the low Reynolds number flow of an elastic liquid. There are, however, similarities with older work for a viscoelastic liquid at relatively high Reynolds numbers. The observations highlight the suitability of the cylinder/rotating end wall configuration as a sensitive test case for computational work.     

Stability of planar stagnation flow of a highly viscoelastic fluid

Linear stability analysis is used to predict the onset of instabilities in inertialess viscoelastic planar stagnation flow. Beyond a critical value of the dimensionless flow rate, or Deborah number, the creeping base flow of similarity type, which is valid in the limit of vanishingly small Reynolds numbers, becomes unstable to localized three-dimensional disturbances. Stability calculations of the local similarity type viscoelastic flow in a small region near the stagnation plane are reported for the quasi-linear Oldroyd-B constitutive equation. The stability results for a range of Deborah numbers and viscosity ratio are presented to explore systematically the effects of elasticity and other rheological properties. The onset of instability and the temporal and spatial characteristics of the secondary flow predicted here resemble other purely elastic instabilities measured and predicted for viscoelastic flows in other simple and complex geometries with curved streamlines.     

Purely elastic instabilities in three-dimensional cross-slot geometries

Creeping and low Reynolds number flows of an upper-convected Maxwell (UCM) fluid are investigated numerically in a three-dimensional orthogonal cross-slot geometry. We analyze two different flow configurations corresponding to uniaxial extension and biaxial extension, and assess the effects of extensional flow type, Deborah and Reynolds numbers on flow dynamics near the interior stagnation point. Using these two flow arrangements the amount of stretch and compression near the stagnation point can be varied, providing further insights on the viscoelastic flow instability mechanisms in extensionally dominated flows with an interior stagnation point. The uniaxial extensional flow arrangement leads to the onset of a steady flow asymmetry, followed by a second purely elastic flow instability that generates an unsteady flow at higher flow rates. On the other hand, for the biaxial extension flow configuration a symmetric flow is observed up to the critical Deborah number when the time-dependent purely elastic instability sets in, without going through the steady symmetric to steady asymmetric transition.     

Sphere settling in an aging yield stress fluid: link between the induced flows and the rheological behavior

The flow pattern induced by the settling of a non-Brownian sphere in a fluid depends on the rheological properties of that fluid. For instance, at small Reynolds numbers, the pattern presents a fore–aft symmetry in a Newtonian fluid, whereas, in some viscoelastic polymer solutions, it can exhibit a negative wake, i.e., an upward flow in the sphere’s wake. This study is an experimental work on the settling of a sphere in a suspension of a synthetic colloidal clay, laponite. The fluid is a yield stress fluid that ages, i.e., whose rheological properties evolve over time. We show that the settling velocity of a given sphere, as well as the induced flow pattern, are strongly modified as the fluid ages. In particular, the flow pattern asymmetry increases with the age of the fluid, and a negative wake eventually forms. We relate those modifications to rheological measurements and suggest, in line with works dealing with polymer solutions, that it is the increase in the fluid viscoelasticity that is responsible for the formation of a negative wake. The flow field measurements are also compared with flow-induced birefringence measurements, and we show that very slow relaxation processes are involved in the sphere settling.     

Instability of the flow around an impacting sphere

Results from an experimental and numerical study of the flow generated by a sphere immersed in fluid, impacting normally without rebound on a solid wall, are presented. The parameters are the running distance before impact and the sphere Reynolds number. For running lengths less than 7.5 diameters, the sphere wake before impact is axisymmetric in the form of an attached vortex ring. After impact, this ring overtakes the sphere and spreads out along the wall. For Reynolds numbers below 1000, the flow remains axisymmetric at all times. For higher values, perturbations of azimuthal wavenumbers 20–25 are observed on the vortex ring, leading to a breakdown of the flow. We analyse different hypotheses concerning the origin of this instability, with the conclusion that a centrifugal instability mechanism is likely to be acting in this flow. Comparisons are made with the flow involving an isolated vortex ring approaching a wall. Numerical simulations of this case have revealed that two distinct instability mechanisms are operating, one of which appears to be similar to the centrifugal instability observed for the sphere impact.     

Numerical modeling of viscoelastic counter flows using the pressure correction method

Counter flows of a viscoelastic fluid described by the rheological Oldroyd model in crossshaped channels are investigated. The modeling is based on the pressure correction method in a convenient-in-use form and with a simple topology of the computation grid and formally proved convergence. It is shown that, starting from certain threshold values of the Weissenberg numbers, the flow pattern in the stabilization stage exhibits considerable changes following two different mechanisms, depending on the Reynolds number. In particular, at low Reynolds numbers (less than 0.1) the flows involve vortex-like structures near the central point, where at the same time anomalies in normal stress distributions are observable. The similarity of these structures with the elastic instability phenomena, which were previously observed in the experimental realizations of the counter flows of this type and in other processes, is shown. To demonstrate the numerical procedure convergence, the results of calculations with different computation grid steps varied on a wide range are presented. In the context of the problem considered the general features of elastic instability are discussed.     

Effect of fluid temperature and uniformly accelerated rotation of the inner sphere on the selection of the flow pattern in a spherical layer

The process of the selection of one of the two flow patterns possible in the hysteresis region, when the Reynolds number is varied in different directions, and differing with respect to the azimuthal wavenumber, 3 or 4, is experimentally investigated. The flow pattern selection proceeds under the influence of an increase in the rotation velocity of the inner sphere at a constant acceleration, the post-acceleration velocity remaining constant. The spherical layer thickness is equal to the inner sphere radius and the outer boundary is fixed. It is established that there is a time lag between the beginning (end) of the sphere acceleration and the beginning (end) of the variation in the measured azimuthal velocity component. It is found that the acceleration necessary for one flow pattern to be replaced by the other significantly depends not only on the Reynolds numbers at which the acceleration begins and ends but also on the fluid temperature in the layer. It is shown that the temperature dependence can be attributed to the variation in the Reynolds number corresponding to the position of the hysteresis boundary when the working fluid viscosity is varied in the layer.     

Flow past a rotating sphere in a Bingham plastic fluid,up to a Reynolds number of 10,000

The flow induced by a sphere rotating inside a non-Newtonian, Bingham fluid has been investigated numerically. The rotating sphere is enclosed in a concentric cubic box with solid boundaries. The Bingham number varied between 0.01 and 100,000 and the Reynolds number varied between 0.01 and 10,000. The torque increases with the Bingham number and reaches an asymptotic state at large Bn. The torque is independent of the Reynolds number at high Bn. The yielded region around the sphere has been determined and an empirical equation is found for its extent.     

Particle-liquid mass transfer in viscoelastic media

The problem of mass transfer from a solid sphere to a viscoelastic fluid has been examined theoretically. It is shown that fluid elasticity increases marginally the mass transfer rate in the creeping flow regime. This will have serious implications on the mass transfer from bubbles if impurities are present. Some conclusions on mass transfer at high Reynolds numbers are also offered.     

Response of viscoelastic fluids in extensional flow generated by a pulsating sphere

A simple analysis of the periodic extensional flow generated by a pulsating sphere in an infinite sea of viscoelastic fluid has been carried out. The general procedure is illustrated by two specific constitutive equations: the corotational Jeffreys fluid and the Oldroyd fluid model B. The response of these fluids is reflected in the temporal variation of the pressure on the surface of the sphere, with Reynolds and Deborah numbers and parameters of the constitutive equations as independent variables. For the case of pulsation with infinitesimal amplitude the fluid response is summarised in the form of pressure amplitude and phase lag versus Deborah number plots. The role of the pulsating flow in the characterisation of viscoelastic fluids and the extension of the procedure to other constitutive equations are briefly discussed.     

Drag and Separation Flow Past Solid Sphere with Porous Shell at Moderate Reynolds Number

Axisymmetric viscous, two-dimensional steady and incompressible fluid flow past a solid sphere with porous shell at moderate Reynolds numbers is investigated numerically. There are two dimensionless parameters that govern the flow in this study: the Reynolds number based on the free stream fluid velocity and the diameter of the solid core, and the ratio of the porous shell thickness to the square root of its permeability. The flow in the free fluid region outside the shell is governed by the Navier–Stokes equation. The flow within the porous annulus region of the shell is governed by a Darcy model. Using a commercially available computational fluid dynamics (CFD) package, drag coefficient and separation angle have been computed for flow past a solid sphere with a porous shell for Reynolds numbers of 50, 100, and 200, and for porous parameter in the range of 0.025–2.5. In all simulation cases, the ratio of b/a was fixed at 1.5; i.e., the ratio of outer shell radius to the inner core radius. A parametric equation relating the drag coefficient and separation point with the Reynolds number and porosity parameter were obtained by multiple linear regression. In the limit of very high permeability, the computed drag coefficient as well as the separation angle approaches that for a solid sphere of radius a, as expected. In the limit of very low permeability, the computed total drag coefficient approaches that for a solid sphere of radius b, as expected. The simulation results are presented in terms of viscous drag coefficient, separation angles and total drag coefficient. It was found that the total drag coefficient around the solid sphere as well as the separation angle are strongly governed by the porous shell permeability as well as the Reynolds number. The separation point shifts toward the rear stagnation point as the shell permeability is increased. Separation angle and drag coefficient for the special case of a solid sphere of radius r = a was found to be in good agreement with previous experimental results and with the standard drag curve.     

Large Deborah number flows around confined microfluidic cylinders

Viscoelastic flow around a confined cylinder at high Deborah numbers is studied using microfluidic channels. By varying fluid properties and flow rates, a systematic study of the roles of elasticity and inertia is accomplished. Two new elastic flow instabilities that occur at high Deborah numbers are identified. A downstream instability of disordered and temporally varying streamlines is observed at a Deborah number above 10. This instability is a precursor to an unsteady vortex that develops upstream of the cylinder at higher Deborah numbers. Both instabilities occur at moderate Reynolds numbers but are fundamentally elastic. The size and steadiness of the upstream vortex are primarily controlled by the Deborah and the elasticity number.     

Hydrodynamic force and torque models for a particle moving near a wall at finite particle Reynolds numbers

This research work is aimed at proposing models for the hydrodynamic force and torque experienced by a spherical particle moving near a solid wall in a viscous fluid at finite particle Reynolds numbers. Conventional lubrication theory was developed based on the theory of Stokes flow around the particle at vanishing particle Reynolds number. In order to account for the effects of finite particle Reynolds number on the models for hydrodynamic force and torque near a wall, we use four types of simple motions at different particle Reynolds numbers. Using the lattice Boltzmann method and considering the moving boundary conditions, we fully resolve the flow field near the particle and obtain the models for hydrodynamic force and torque as functions of particle Reynolds number and the dimensionless gap between the particle and the wall. The resolution is up to 50 grids per particle diameter. After comparing numerical results of the coefficients with conventional results based on Stokes flow, we propose new models for hydrodynamic force and torque at different particle Reynolds numbers. It is shown that the particle Reynolds number has a significant impact on the models for hydrodynamic force and torque. Furthermore, the models are validated against general motions of a particle and available modeling results from literature. The proposed models could be used as sub-grid scale models where the flows between particle and wall can not be fully resolved, or be used in Lagrangian simulations of particle-laden flows when particles are close to a wall instead of the currently used models for an isolated particle.     

Flow of wormlike micelle solutions past a confined circular cylinder

In this paper, we present the results of an investigation into the flow of a series of viscoelastic wormlike micelle solutions past a confined circular cylinder. Although this benchmark flow has been studied in great detail for polymer solutions, this paper reports the first experiments to use a viscoelastic wormlike micelle solution as the test fluid. The flow kinematics, stability and pressure drop were examined for two different wormlike micelle solutions over a wide range of Deborah numbers and cylinder to channel aspect ratios. A combination of particle image velocimetry and pressure drop measurements were used to characterize the flow kinematics, while flow-induced birefringence measurements were used to measure the micelle deformation and alignment in the flow. The pressure drop was found to decrease initially due to the shear thinning of the test fluid before increasing at higher flow rates as elastic effects begin to dominate the flow. Above a critical Deborah number, an elastic instability was observed for just one of the test fluids studied, the other remained stable for all Deborah number tested. Flow-induced birefringence and velocimetry measurements showed that observed instability originates in the extensional flow in the wake of the cylinder and appears not as periodic counter-rotating vortices as has been observed in the flow of polymer solutions past circular cylinders, but as a chaotic rupture event in the wake of the cylinder that propagates axially along the cylinder. Reducing the cylinder to channel aspect ratio and the degree of shearing introduced by the channel walls had a weak impact on the stability of the flow. These measurements, when taken in conjunction with previous work on flow of wormlike micelle solutions through a periodic array of cylinders, definitively show that the instability can be attributed to a breakdown of the wormlike micelle solutions in the extensional flow in the wake of the cylinder.