History force on a sphere in a weak linear shear flow

A numerical study of history forces acting on a spherical particle in a linear shear flow, over a range of finite Re, is presented. In each of the cases considered, the particle undergoes rapid acceleration from Re₁ to Re₂ over a short-time period. After acceleration, the particle is maintained at Re₂ in order to allow for clean extraction of drag and lift kernels. Good agreement is observed between current drag kernel results and previous investigations. Furthermore, ambient shear is found to have little influence on the drag kernel. The lift kernel is observed to be oscillatory, which translates to a non-monotonic change in lift force to the final steady state. In addition, strong dependence on the start and end conditions of acceleration is observed. Unlike drag, the lift history kernel scales linearly with Reynolds number and shear rate. This behavior is consistent with a short-time inviscid evolution. A simple expression for the lift history kernel is presented.     

Viscoelasticity-induced migration of a rigid sphere in confined shear flow

Suspensions of solid particles in liquids are often made to flow in devices with characteristic dimensions comparable to that of the suspended particles, the so-called confined situation, as in the case of several microfluidic applications. Combination of confinement with viscoelasticity of the suspending liquid can lead to peculiar effects. In this paper we present the first 3D simulation of the dynamics of a particle suspended in a viscoelastic liquid under imposed confined shear flow. The full system of equations is solved through the finite element method. A DEVSS/SUPG formulation with a log-representation of the conformation tensor is implemented, assuring stable and convergent results up to high flow rates. Particle motion is handled through an ALE formulation. To optimize the computational effort and to reduce the remeshing and projection steps required when the mesh becomes too distorted, a rigid motion of the grid in the flow direction is performed, so that, in fact, the particle moves along the cross-streamline direction only.Confinement and viscoelasticity are found to induce particle migration, i.e., transverse motion across the main flow direction, towards the closest wall. Under continuous shearing, three different dynamical regimes are recognized, related to the particle-wall distance. A simple heuristic argument is given to link the cross-flow migration to normal stresses in the suspending liquid.The analysis is then extended to a time-dependent shear flow imposed by periodically inverting the direction of wall motion. A slower migration is found for higher forcing frequency. A peculiar effect arises if the inversion period is chosen close to the fluid relaxation time: the migration velocity oscillates around zero, and the overall migration is suppressed. Such novel prediction of a dynamic instability scenario, with the particle escaping the center plane of the channel, and many features of the computed results, are in nice agreement with recent experiments reported in the literature [14].     

Simultaneous effect of rotation and natural convection on the flow about a liquid sphere

The problem of mixed convection around a liquid sphere that experiences a rotation about its axis parallel to the free stream is studied numerically using a finite- difference technique. The coupled boundary-layer energy and momentum equations are numerically solved over a wide range of Grashof number that represents the cases of aiding and opposing free convection and for wide range of the spin parameter Ta/Re². The surface of the sphere also rotates as a result of the shear stress exerted from the external flow of air. The effect of both parameters on the velocity components as well as the temperature within the thermal boundary-layer is presented. Results show that increasing the aiding free convection and the spin parameter cause increases in the shear stress and the local heat transfer coefficient.     

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.     

Stress-strain state of a half-space with shear displacement and rotation of a rigid sphere partly embedded in it

Institute of Mechanics, Academy of Sciences of the Ukrainian SSR, Kiev. Translated from Prikladnaya Mekhanika, Vol. 27, No. 1, pp. 24–31, January, 1991.     

Effect of rotation on the electrogasdynamic characteristics of flow in a cylindrical channel

Scientific Research Institute of Applied Mathematics and Mechanics, 634050 Tomsk. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, No. 2, pp. 3–7, March–April, 1995     

Effect of blowing on the resistance of a sphere in laminar flow of viscous fluid

Using the method of matching asymptotic expansions [1–3], a stationary field of velocities is obtained in the vicinity of a sphere for Reynolds numbers R and R computed from the blowing velocity and the fluid flow, respectively; they satisfy the relations R² 1 and R 1. It is also shown that for intensive blowing (R 1), the resistive force is considerably smaller than that found by using the Stokes formula. For weak blowing the results are in good agreement with the solution of Oseen.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskai Fiziki, No. 3, pp. 110–114, May–June, 1972.     

Steady slow rotation of a sphere in a visco-elastic liquid

Summary In this paper, we obtain the flow due to slow steady rotation of a sphere in a visco-elastic liquid characterized by the constitutive relation given by Rivlin. The non-Newtonian effects are strongly dependent on a non-dimensional parameter K independent of the angular velocity of the sphere. If 1<K3, we notice four vortices symmetrically placed around the sphere. When K lies outside this range, the direction of the flow pattern is the same as that in the Newtonian case but displaced towards the sphere as K decreases. Also the expression for the couple on the sphere has been obtained which depends on K.     

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.     

Influence of viscoelasticity on drop deformation and orientation in shear flow: Part 1. Stationary states

The influence of matrix and droplet viscoelasticity on the steady deformation and orientation of a single droplet subjected to simple shear is investigated microscopically. Experimental data are obtained in the velocity–vorticity and velocity–velocity gradient plane. A constant viscosity Boger fluid is used, as well as a shear-thinning viscoelastic fluid. These materials are described by means of an Oldroyd-B, Giesekus, Ellis, or multi-mode Giesekus constitutive equation. The drop-to-matrix viscosity ratio is 1.5. The numerical simulations in 3D are performed with a volume-of-fluid algorithm and focus on capillary numbers 0.15 and 0.35. In the case of a viscoelastic matrix, viscoelastic stress fields, computed at varying Deborah numbers, show maxima slightly above the drop tip at the back and below the tip at the front. At both capillary numbers, the simulations with the Oldroyd-B constitutive equation predict the experimentally observed phenomena that matrix viscoelasticity significantly suppresses droplet deformation and promotes droplet orientation. These two effects saturate experimentally at high Deborah numbers. Experimentally, the high Deborah numbers are achieved by decreasing the droplet radius with other parameters unchanged. At the higher capillary and Deborah numbers, the use of the Giesekus model with a small amount of shear-thinning dampens the stationary state deformation slightly and increases the angle of orientation. Droplet viscoelasticity on the other hand hardly affects the steady droplet deformation and orientation, both experimentally and numerically, even at moderate to high capillary and Deborah numbers.     

Elastico-viscous boundary-layer flow on the surface of a sphere

Summary The Prandtl boundary-layer theory is extended for an idealized elastico-viscous liquid. The boundary layer equations are solved approximately by Kármán-Pohlhausen technique for the case of a sphere. It is shown that the increase in the elasticity of the liquid causes a shift in the point of separation towards the forward stagnation point.

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Zusammenfassung Die Prandtlsche Grenzschicht-Theorie wird für eine idealisierte viskoelastische Flüssigkeit erweitert. Die Grenzschichtgleichungen werden für den Fall einer angeströmten Kugel näherungsweise mit Hilfe der Kármán-Pohlhausen-Methode gelöst. Es wird gezeigt, daß das Anwachsen der Flüssigkeitselastizität eine Verschiebung des Ablösepunktes auf den vorderen Staupunkt hin zur Folge hat.

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Nomenclature b ^ik arbitrary contravariant tensor - D non-dimensional boundary layer thickness - g _ik metric tensor of a fixed coordinate system - K curvature at any point on the generating curve - K ₀ elastico-viscous parameter - p arbitrary hydrostatic pressure - p _ik stress tensor - p ^ik part of stress tensor associated with the change of shape of material - R radius of the sphere - r radius of any transverse cross-section of the sphere - t time - U potential velocity around the body - U stream-velocity at a large distance from the body - u, w velocity components along (x, z) directions respectively - x distance measured along a generating line from the forward stagnation point - z distance measured along a normal to the surface - non-dimensional elastico-viscous parameter - density of the liquid - boundary layer thickness - convected time derivative - ₀ limiting viscosity for very small changes in deformation velocity - angle measured along the transverse direction - x/R - v kinematic coefficient of viscosity - T _s shearing stress on the surface of the sphere With 2 figures and 1 table     

Effect of rotation on Goertler vortices in the boundary layer flow on a curved surface

This paper presents a numerical prediction of the formation of Goertler vortices on a curved surface with effect of rotation. The criterion of flow visualization marking the onset position of Goertler vortices is employed in the present paper. For facilitating the numerical study, the computation is carried out in the transformed x and ηplane. The results show that the onset position characterized by the Goertler number, depends on the rotation number Ro, the Prandtl number and the wave number. The value of critical Goertler number increases with the increase in negative rotation, while the value of Goertler number decreases with the increase in positive rotation on a concave surface. On the contrary, the value of critical Goertler number decreases with the increase in negative rotation on a convex surface. The obtained critical Goertler number and wave number are compared with the previous theoretical and experimental data. Copyright © 2002 John Wiley & Sons, Ltd.     

Super-rotation flow in a precessing sphere

The super rotation here means that the majority of fluid inside a precessing sphere rotates around the precession axis with angular velocity larger than that of the precession rotation itself. This phenomenon observed experimentally and numerically is explained to be driven by a cooperative interplay between the Coriolis force, the pressure gradient and the spherical geometry in the boundary layer.     

Steady rotation of a composite sphere in a concentric spherical cavity

The problem of steady rotation of a compositesphere located at the centre of a spherical container has beeninvestigated.A composite particle referred to in this paperis a spherical solid core covered with a permeable sphericalshell.The Brinkman’s model for the flow inside the composite sphere and the Stokes equation for the flow in the spherical container were used to study the motion.The torque experienced by the porous spherical particle in the presence ofcavity is obtained.The wall correction factor is calculated.In the limiting cases,the analytical solution describing thetorque for a porous sphere and for a solid sphere in an unbounded medium are obtained from the present analysis.     

Effect of confinement on droplet deformation in shear flow

The dynamics of a single droplet under shear flow between two parallel plates is investigated by using the immersed boundary method. The immersed boundary method is appropriate for simulating the drop-ambient fluid interface. We apply a volume-conserving method using the normal vector of the surface to prevent mass loss of the droplet. In addition, we present a surface remeshing algorithm to cope with the distortion of droplet interface points caused by the shear flow. This mesh quality improvement in conjunction with the volume-conserving algorithm is particularly essential and critical for long time evolutions. We study the effect of wall confinement on the droplet dynamics. Numerical simulations show good agreement with previous experimental results and theoretical models.     

Forces on a porous sphere spinning in a fluid flow

Exact expressions are found for the drag (modified Stokes force) and the lift (modified Magnus force) on a porous sphere spinning slowly in a viscous fluid flowing slowly and uniformly past it.     

Slow rotation of a sphere in a confined volume of a dilute gas

The axially symmetric motion of a gas in a volume confined between an external immobile surface of rotation and a coaxial surface of a rotating sphere is considered. A solution is obtained by the moment method based on the Boltzmann equation with a collision integral of Maxwellian molecules. The gas-velocity distribution and an expression for the friction torque exerted on the sphere are obtained for arbitrary Knudsen numbers and for an arbitrary shape of the outer surface. The proportionality of the gas slip velocity over the surface of the sphere to the friction strain is shown. The friction torque is investigated for specific shapes of the outer surface. The motion of a gas filling the space between concentric spheres, each of which rotates about an arbitrary axis, is treated. In the limiting case of small Knudsen numbers the expressions obtained are compared with the corresponding results for a continuous medium.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 117–124, July–August, 1978.The authors are grateful to the participants of the seminars guided by G. I. Petrov and A. M. Golovnyi for discussions concerning this work.