Time dependent flow in equimolar micellar solutions: transient behaviour of the shear stress and first normal stress difference in shear induced structures coupled with flow instabilities

Recently we studied time dependent structural changes that are coupled with flow instabilities (Fischer 1998; Wheeler 1998; Fischer 2000). Within a stability analysis, a classification scheme for the feedback circuit of coupled shear-induced structure and flow instabilities was derived by Schmitt et al. (1995) and applied to our samples. Here, inhomogeneous flow layers of different concentration and viscosity are generated by shear-induced diffusion (spinodal demixing) and, as consequence, one no longer observes a homogeneous solution but a type of shear banding that is seen here for the first time. In this paper we present the behaviour of the first normal stress difference observed in the critical shear-rate regime where transient shear-induced structure is coupled with flow instability. Similar to the oscillations of the shear stresses (strain-controlled rheometer) one observes oscillations in the first normal stress difference. This behaviour indicates that elastic structures are built up and destroyed while the shear-induced structures occur and that the induced phase is more elastic than the initial one. Oscillations of shear stress and first normal stress difference are in phase and indicate that both phenomena are caused by the same mechanism. Received: 30 June 1999/Accepted: 14 December 1999     

Flow structures,nonlinear inertial waves and energy transfer in rotating spheres

We investigate flow structures, nonlinear inertial waves and energy transfer in a rotating fluid sphere, using a Galerkin spectral method based on helical-wave decomposition (HWD). Numerical simulations of flows in a sphere are performed with different system rotation rates, where a large-scale forcing is employed. For the case without system rotation, the intense vortex structures are tube-like. When a weak rotation is introduced, small-scale structures are reduced and vortex tubes tend to align with the rotation axis. As the rotation rate increases, a large-scale anticyclonic vortex structure is formed near the rotation axis. The structure is shown to be led by certain geostrophic modes. When the rotation rate further increases, a cyclone and an anticyclone emerge from the top and bottom of the boundary, respectively, where two quasi-geostrophic equatorially symmetric inertial waves dominate the flow. Based on HWD, effects of spherical confinement on rotating turbulence are systematically studied. It is found that the forward cascade becomes weaker as the rotation increases. When the rotation rate becomes larger than some critical value, dual energy cascades emerge, with an inverse cascade at large scales and a forward cascade at small scales. Finally, the flow behavior near the boundary is studied, where the average boundary layer thickness gets smaller when system rotation increases. The flow behavior in the boundary layer is closely related to the interior flow structures, which create significant mass flux between the boundary layer and the interior fluid through Ekman pumping.     

Strain stiffening of non-colloidal hard sphere suspensions dispersed in Newtonian fluid near liquid-and-crystal coexistence region

Concentrated hard sphere suspensions often show an interesting nonlinear behavior, called strain stiffening, in which the viscosity or modulus starts to increase at critical strain amplitude. Sudden increase of rheological properties is similar to shear thickening; however, the particle dynamics in the strain stiffening under oscillatory shear flow does not necessarily coincide with the mechanism of shear thickening under step shear flow. In this study, we have systematically investigated the nonlinear rheology of non-colloidal (>1???m) hard sphere suspensions dispersed in Newtonian fluid near liquid-and-crystal coexistence region in order to better understand the strain stiffening behavior. The suspensions near liquid-and-crystal coexistence region are known to locally form the closed packing structure. The critical strain amplitude which is the onset of strain stiffening was different for the storage and loss modulus. But they converged to each other as the suspension forms a more crystalline structure. The critical strain amplitude was independent of medium viscosity, imposed angular frequency, and particle size, but was strongly dependent upon particle volume fraction. The onset of strain stiffening was explained in terms of shear-induced collision due to particle motion in the closed packing structure. Nonlinear stress wave-forms, which reflect the micro-structural change, were observed with the onset of strain stiffening. During the strain stiffening, enhanced elastic stress before and after flow reversal was observed which originates from changes in the suspension microstructure. Nonlinearity of the shear stress in terms of Fourier intensity was extremely increased up to 0.55. Beyond the strain stiffening, the suspension responded liquid-like and the nonlinearity decreased but the elastic shear stress was still indicating the microstructure rearrangement within a cycle.     

Self-similar solution of the unsteady flow in the stagnation point region of a rotating sphere with a magnetic field

The unsteady flow and heat transfer of a viscous incompressible electrically conducting fluid in the forward stagnation point region of a rotating sphere in the presence of a magnetic field are investigated in this study. The unsteadiness in the flow field is caused by the velocity at the edge of the boundary layer and the angular velocity of the rotating sphere, both varying continuously with time. The system of ordinary differential equations governing the flow is solved numerically. For some particular cases, an analytical solution is also obtained. It is found that the surface shear stresses in x- and y-directions and the surface heat transfer increase with the acceleration, the magnetic and the rotation parameters whether the magnetic field is fixed relative to the fluid or body, except that the surface shear stress in x-direction and the surface heat transfer decrease with increasing the magnetic parameter when the magnetic field is fixed relative to the body. For a certain value of the acceleration parameter, the surface shear stress in the x-direction vanishes while the surface shear stress in the y-direction and the surface heat transfer remain finite. Also, below a certain value of the acceleration parameter, reverse flow occurs in the x-component of the velocity profile. Received on 18 May 1998     

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.     

Study on aerodynamic force acting on a sphere with and without boundary layer trips around the critical Reynolds number with a magnetic suspension and balance system

Aerodynamic force acting on a sphere for five kinds of boundary layer trips around the critical Reynolds number, together with the force on a smooth sphere, was successfully measured. This was achieved using JAXA’s 60-cm Magnetic Suspension and Balance System after performing detailed simulations and adjusting the sphere mass and its control parameters. The minimum drag coefficient of a smooth sphere was evaluated around 0.19 in the support-interference-free condition. No hysteresis was observed for the drag coefficient in the critical range for tested sphere with boundary layer trips. Using three serially connected 2nd-order Butterworth low-pass filters, an inertia force oscillating at less than 15 Hz was evaluated from the measured model position, and the unsteady aerodynamic force acting on the sphere was also evaluated with reasonable accuracy. Two kinds of oscillatory aerodynamic forces appeared in the critical range depending on the sphere surface condition: a force rotating around an axis parallel to the uniform flow for both a smooth sphere and a sphere with axially symmetric 0.17-mm-high backward step, and an oscillating force in the plane including the axis parallel to the flow for a sphere with axially symmetric step implemented with 0.35–mm-thick tape with wrinkles acting as small vortex generators. There was also observed a force irregularly rotating through less than 180° in the range about a sphere axis parallel to the flow for a smooth sphere in the supercritical range.     

The motion of a closely-fitting sphere in a fluid-filled tube

Singular perturbation techniques are used to investigate the slow, asymmetric flow around a sphere positioned eccentrically within a long, circular, cylindrical tube filled with viscous fluid. The results apply to situations in which the sphere occupies virtually the entire cross section of the cylinder, so that the clearance between the particle and tube wall is everywhere small compared with both the sphere and tube radii. The technique is an improvement over conventional “lubrication-theory” analyses.Asymptotic expansions, valid for small dimensionless clearances, are obtained for the hydrodynamic force, torque and pressure drop for flow past a stationary sphere, as well as for the case of a sphere translating or rotating in an otherwise quiescent fluid. These expansions are employed to predict the macroscopic behavior of both a neutrally-buoyant sphere suspended in a Poiseuille flow, and a sedimenting sphere in a vertical tube.The results find application in capillary blood flow, pipeline transport of encapsulated materials, and falling-ball viscometers.     

Stress diffusion and high order viscoelastic effects in the 3D flow past a sedimenting sphere subject to orthogonal shear

The effect of polymer stress diffusion in the unbounded flow past a sedimenting, freely rotating, rigid sphere subject to shear in a plane perpendicular to the direction of sedimentation is investigated analytically. Steady state, creeping, incompressible, and isothermal flow is assumed. For viscoelastic fluids following the Oldroyd-B constitutive model, three-dimensional results for the velocity vector, pressure, and viscoelastic extra-stress tensor are derived by including an artificial diffusion term in the constitutive equation and using regular perturbation theory with the small parameter being the Deborah number. The analytical solution reveals that the influence of the stress diffusion term on the results may be significant (and sometimes unexpected) and strongly depends on the magnitude of the dimensionless diffusion coefficient. For instance, it is shown that the critical Deborah number, below which a physical solution arises, decreases with the increase in the diffusion coefficient. Also, comparison against simulation results from the literature shows excellent agreement up to shear Weissenberg number (defined as the product of the imposed shear rate with the single relaxation time of the fluid) approximately equal to unity.     

Numerical analysis of the rotating viscous flow approaching a solid sphere

A numerical simulation is performed to investigate the flow induced by a sphere moving along the axis of a rotating cylindrical container filled with the viscous fluid. Three‐dimensional incompressible Navier–Stokes equations are solved using a finite element method. The objective of this study is to examine the feature of waves generated by the Coriolis force at moderate Rossby numbers and that to what extent the Taylor–Proudman theorem is valid for the viscous rotating flow at small Rossby number and large Reynolds number. Calculations have been undertaken at the Rossby numbers (R_o) of 1 and 0.02 and the Reynolds numbers (R_e) of 200 and 500. When R_o=O(1), inertia waves are exhibited in the rotating flow past a sphere. The effects of the Reynolds number and the ratio of the radius of the sphere and that of the rotating cylinder on the flow structure are examined. When R_o ? 1, as predicted by the Taylor–Proudman theorem for inviscid flow, the so‐called ‘Taylor column’ is also generated in the viscous fluid flow after an evolutionary course of vortical flow structures. The initial evolution and final formation of the ‘Taylor column’ are exhibited. According to the present calculation, it has been verified that major theoretical statement about the rotating flow of the inviscid fluid may still approximately predict the rotating flow structure of the viscous fluid in a certain regime of the Reynolds number. Copyright © 2004 John Wiley & Sons, Ltd.     

Molecular orientation, “Region I” shear thinning and the cholesteric phase in aqueous hydroxypropylcellulose under shear

Birefringence and flow visualization are used to study molecular orientation, texture, and the cholesteric nature of a 60 wt% aqueous hydroxypropylcellulose solution at low to moderate shear rates. There is a sharp transition in behavior at a shear rate near 0.5 s^–1. Below this rate, the sample shows “Region I” shear thinning, takes on a frosted appearance, has low flow-induced orientation, and exhibits faint optical diffraction characteristic of cholesteric liquid crystals. Upon flow cessation from low rates, a highly organized cholesteric phase showing bright optical diffraction is re-formed in around 5min. Above the critical shear rate, the sample exhibits much higher orientation and a striped texture which is readily apparent with or without polarized light illumination. Upon flow cessation, molecular orientation decays significantly for up to around 400 s; however, the striped textures established during shear persist. It is suggested that persistence of the cholesteric phase under shear is responsible for the occurrence of Region I shear thinning at low shear rates in this solution.     

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.     

Effect of molecular weight and shear on phase diagram of PS/PVME blend

The molecular weight dependence of the lower critical solution temperature of polystyrene (PS) and poly(vinyl methyl ether) blends was studied by laser light transmission. The temperature of phase separation was found to decrease with increasing PS molecular weight. In the steady shear flow conditions and near the critical temperature, shear was found to enhance fluctuations of concentration at relatively small shear rates, whereas it suppresses such fluctuations at high shear rates. The shift in Flory-interaction parameter Δ_χ was calculated from experimental data and its sign was used to predict shear-induced mixing or shear-induced demixing under flow field. The obtained experimental results were compared to Criado-Sancho et al. and Clarke-Mcleish models.     

Viscosity enhancement in the flow of hydrolysed poly(acrylamide) saline solutions around spheres: implications for enhanced oil recovery

We have studied dilute aqueous solutions of hydrolysed poly(acrylamide), in various ionic environments, in flow around single spheres and around two spheres aligned on the axis of flow. The spheres are held on flexible cantilevers, while the polymer solutions, or solvent, are drawn past at controlled flow rates. We estimate the specific viscosities of the various solutions as a function of the strain rate over strain rates encompassing both the shear thinning and extension thickening regimes. For flow of solutions without added salts around a single sphere, we observe shear thinning followed by a significant increase in the non-Newtonian viscosity with increasing strain rate. The shear thinning reduces the maximal extensional viscosities of the solutions, which has important implications regarding the effectiveness of hydrolysed poly(acrylamide) in oil field applications. For flow of polymer solutions around two axially aligned spheres, we observe a significant reduction in the non-Newtonian forces experienced by the downstream sphere in comparison to the upstream sphere. We consider that this is salient to the understanding of non-Newtonian viscosification in porous media flow.     

Spherical Couette flow of a viscoelastic fluid Part I: Experimental study of the inner sphere rotation

Flow of viscoelastic fluid contained between a stationary outer sphere and a rotating inner sphere is studied experimentally. In the present investigation, relatively low-concentration polyacrylamide-water solutions are used as viscoelastic fluid, and for the sake of comparison glycerin-water solutions are used as the Newtonian fluid. In experiments, measurements of the rotational torque and the velocity profile in the meridional plane of the spherical gap are made. Various transition phenomena of flow modes that are unique to viscoelastic fluids are investigated by flow visualization for a wide range of rotational Reynolds numbers. Experimental results revealed that a roll-cell-like structure (banded radial structure) caused by elastic instability is generated in the polar region, and that it propagates toward the equatorial region when rotation of the inner sphere is increased, resulting in the formation of two distinct regions: the elastic dominant region and the inertial dominant region. Flow modes are classified and the critical Reynolds numbers are obtained for different gap widths. A correlation is obtained for the torque data in the regime before the onset of instability.     

Unsteady laminar MHD flow and heat transfer in the stagnation region of an impulsively spinning and translating sphere in the presence of buoyancy forces

 An analysis has been carried out to determine the development of momentum and heat transfer occurring in the laminar boundary layer of an incompressible viscous electrically conducting fluid in the stagnation region of a rotating sphere caused by the impulsive motion of the free stream velocity and the angular velocity of the sphere. At the same time the wall temperature is also suddenly increased. This analysis includes both short and long-time solutions. The partial differential equations governing the flow are solved numerically using an implicit finite-difference scheme. There is a smooth transition from the short-time solution to the long-time solution. The surface shear stresses in the longitudinal and rotating directions and the heat transfer are found to increase with time, magnetic field, buoyancy parameter and the rotation parameter. Received on 27 January 2000     

Viscosity measurements with a magnetoviscometer in the zero-shear and transition region of polypropylenes

Streaming of a non-Newtonian fluid around a sphere is of special importance not only for measuring viscosities with falling spheres, but also for many problems connected with polymer processing. Using the mentioned measuring principle, attention has to be paid to the following points: The sphere is moving in a fluid (melt) of finite extension which requires the application of wall and perhaps end corrections. These are possibly not the same for Newtonian and non-Newtonian fluids. To calculate the viscosity with the help of Stokes law the steady-state velocity is necessary, and it is essential, how long it takes the sphere to reach it. To compare our results with other data, an average shear rate has to be calculated, since there is no uniform shear rate around the sphere. Velocities being very low in our experiments result in very small Reynolds numbers (Re < 10^–3), which allows the application of Stokes law practically without corrections.The experiments were performed at zero shear and in the transition region above. It turned out, that it is usually not possible to extrapolate from shear-dependent viscosity data to zero-shear viscosity.Dedicated to Prof. A. Neckel on the occasion of his 60th birthday     

Numerical Investigation of the Flow Past a Rotating Golf Ball and Its Comparison with a Rotating Smooth Sphere

Large-eddy simulations are conducted for a rotating golf ball and a rotating smooth sphere at a constant rotational speed at the subcritical, critical and supercritical Reynolds numbers. A negative lift force is generated in the critical regime for both models, whereas positive lift forces are generated in the subcritical and supercritical regimes. Detailed analysis on the flow separations on different sides of the models reveals the mechanism of the negative Magnus effect. Further investigation of the unsteady aerodynamics reveals the effect of rotating motion on the development of lateral forces and wake flow structures. It is found that the rotating motion helps to stabilize the resultant lateral forces for both models especially in the supercritical regime.     

Stokes flow of a rotating sphere around the axis of a circular orifice or a circular disk

This paper presents an infinite series solution to the creeping flow equations for the axisymmetric motion of a sphere of arbitrary size rotating in a quiescent fluid around the axis of a circular orifice or a circular disk whose diameters are either larger or smaller than that of the sphere. Numerical tests of the convergence are passed and the comparison with the exact solution and other computational results shows an agreement to five significant figures for the torque coefficients in both cases. The torque coefficients are obtained for the sphere located up to a position tangent to the wall plane containing either the orifice or the disk. It is concluded that the torque coefficients of the sphere and the disk are monotonically increasing with the decrease of the distance from the disk or the orifice plane in both cases.     

A comparative assessment of fluid flow and heat transfer parameters in laminar flow around a circular cylinder and a sphere

Theoretical studies have been carried out for a comparative assessment of hydrodynamic boundary layer thickness, displacement thickness and shear stress at the wall for laminar flow around a circular cylinder and a sphere with the help of the approximate method due to Karman and Pohlhausen for two dimensional flow and the method as applied to bodies of revolution based on the work of F. W. Scholkemeier, respectively. Thermal boundary layer thickness and Nusselt number have been evaluated around the surface of the solids. Comparison is made with available solutions. The graphical presentation of the results depicts a concise and relative assessment of fluid flow and heat-transfer parameters for flow around cylinder and sphere.     

Motion of a permeable shell in a spherical container filled with non-Newtonian fluid

This paper presents an analytical study of creeping motion of a permeable sphere in a spherical container filled with a micro-polar fluid. The drag experienced by the permeable sphere when it passes through the center of the spherical container is studied.Stream function solutions for the flow fields are obtained in terms of modified Bessel functions and Gegenbauer functions. The pressure fields, the micro-rotation components,the drag experienced by a permeable sphere, the wall correction factor, and the flow rate through the permeable surface are obtained for the frictionless impermeable spherical container and the zero shear stress at the impermeable spherical container. Variations of the drag force and the wall correction factor with respect to different fluid parameters are studied. It is observed that the drag force, the wall correction factor, and the flow rate are greater for the frictionless impermeable spherical container than the zero shear stress at the impermeable spherical container. Several cases of interest are deduced from the present analysis.