Studies on the axisymmetric sphere–sphere interaction problem in Newtonian and non-Newtonian fluids

In this research, experimental studies have been performed on the hydrodynamic interaction between two spheres by using particle image velocimetry and measuring the force between the spheres. To approach the system as a resistance problem, a servo-driving system was set-up by assembling a microstepping motor, a ball screw and a linear motion guide for the particle motion. Glycerin and a dilute solution of polyacrylamide in glycerin were used as Newtonian and non-Newtonian fluids, respectively. The polymer solution behaves like a Boger fluid when the concentration is 1000 ppm or less. The experimental results were compared with the asymptotic solution of Stokes equation. The result shows that fluid inertia and unsteadiness play important roles in the particle–particle interaction in the Newtonian fluid. This implies that the motion of two particles in suspension is not reversible even in the Newtonian fluid. In the non-Newtonian fluid, in addition to inertial effect, normal stress differences and viscoelasticity play important roles as expected. In dilute solutions weak shear thinning and the migration of polymer molecules in the inhomogeneous flow field also appear to affect the physics of the problem.     

The instantaneous slow flow of a visco-elastic fluid between two concentric spheres

Summary A theoretical analysis is made of the flow of an incompressible viscoelastic fluid contained between two concentric spheres when the outer sphere is moved instantaneously in a given direction, whilst the inner sphere remains at rest. The solution is developed by successive approximations, the first corresponding to the instantaneous slow flow of a Newtonian viscous fluid. By allowing the radius of the outer sphere to approach infinity, the result obtained can be used to give an approximate solution to the equations of motion of a visco-elastic fluid flowing slowly past a fixed sphere.     

Nonuniform convective Couette flow

An exact solution describing the convective flow of a vortical viscous incompressible fluid is derived. The solution of the Oberbeck–Boussinesq equation possesses a characteristic feature in describing a fluid in motion, namely, it holds true when not only viscous but also inertia forces are taken into account. Taking the inertia forces into account leads to the appearance of stagnation points in a fluid layer and counterflows, as well as the existence of layer thicknesses at which the tangent stresses vanish on the lower boundary. It is shown that the vortices in the fluid are generated due to the nonlinear effects leading to the occurrence of counterflows and flow velocity amplification, compared with those given by the boundary conditions. The solution of the spectral problem for the polynomials describing the tangent stress distribution makes it possible to explain the absence of the skin friction on the solid surface and in an arbitrary section of an infinite layer.     

The slow motion of two touching fluid spheres along their line of centers

The creeping motion along their line of centers of two fluid spheres in contact is analyzed. An exact solution is presented. Corrections to the Hadamard—Rybezynski equation are tabulated for various particle radii ratios and particle fluid to external fluid viscosity ratios. In the limit of infinite particle viscosity, these corrections are shown to agree with previous calculations for rigid spheres.     

Motion of two pulsating spheres in an ideal incompressible fluid

The problem of the interaction of two pulsating spheres in an ideal incompressible fluid was first investigated in detail by Bjerknes [1]. However, his and subsequent studies on this subject [2–5] were restricted to the interaction forces between the spheres, whereas the law of their motion was not considered because of the much greater complexity of the corresponding problem. The aim of the present paper is to find an approximate analytic solution to the problem of the motion of two pulsating spheres in an ideal incompressible fluid filling the entire space exterior to the spheres under the assumption that the flow of the fluid is irrotational.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 159–162, May–June, 1983.     

Flow of a viscous fluid past an array of spheres

The Stokes flow of a viscous incompressible fluid through a periodic array of impenetrable spheres with linear friction on the boundary is considered. A solution and an expression for the drag are obtained to terms of order c^5/3 compared with unity (c is the volume concentration of the spheres). The proposed algorithm permits solution with any required degree of accuracy. The solution contains as limits the cases of perfect slip and no-slip on the surfaces of the spheres. In the problem with the no-slip condition, an asymptotically exact lower bound for the drag, which is valid for all values of the concentration c, is constructed.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 37–44, July–August, 1981.     

Combined effects of non-Newtonian couple stresses and fluid inertia on the squeeze film characteristics between a long cylinder and an infinite plate

On the basis of the Stokes micro-continuum theory together with the averaged inertia principle, the combined effects of non-Newtonian couple stresses and convective fluid inertia forces on the squeeze film motion between a long cylinder and an infinite plate are presented. A closed-form solution has been derived for squeeze film characteristics including the film pressure, the load capacity and the response time. Comparing with the Newtonian-lubricant non-inertia case, the combined effects of couple stresses and convective inertia forces provide an increase in the film pressure, the load capacity and the response time. In addition, the quantitative effects of couple stresses and convective inertia forces are more pronounced for cylinder–plate system operating at a larger couple stress parameter and film Reynolds number, as well as a smaller squeeze film height. To guide the use of the present study, a numerical example is also illustrated for engineers when considering both the effects of non-Newtonian couple stresses and fluid convective inertia forces.     

Axisymmetric deformation of poroelastic shells of revolution

A linear theory is developed for axisymmetric deformation of thin poroelastic shells of revolution. With fluid solid coupling included through Biot's consolidation theory, results are presented for cylindrical shells with an oscillating internal pressure and various surface boundary conditions on the fluid. First, the effects of fluid flow and shell inertia on the stretching behavior are studied through a separation of variables solution. Then, the bending behavior near a clamped edge is examined through an asymptotic solution of a matrix form of the governing equations. The results show that the asymptotic solution is accurate in the low frequency range, when the loading time is large compared to the consolidation time. In addition, for the examples studied, the fluid flow influences the membrane more than the bending behavior, but damping due to flow resistance is limited near resonance.     

Momentum and heat transfer over a continuous moving surface in a non-Darcian fluid

The momentum and heat transfer characteristics associated with the boundary layer on a continuous moving flat surface in a non-Darcian fluid have been investigated exploiting a local similarity solution procedure. The full boundary layer equations, which describe the effects of convective inertia, solid boundary, and porous inertia in addition to the Darcy flow resistance, were solved using novel transformed variables, deduced from a scale analysis on the momentum and energy conservation equations. Details are provided for the effects of convective inertia and porous inertia on the velocity and temperature profiles. The resulting friction and heat transfer characteristics are found to be substantially different from those of forces convection over a stationary flat plate. Furthermore, useful asymptotic expressions for the local Nusselt number are presented in consideration of possible physical limiting conditions.     

Viscosity of a Suspension with a Cubic Array of Spheres in a Shear Flow

Viscous fluid flow past an infinite periodic array of rigid spheres of the same radius is considered. A solution of the Stokes equations periodic in three variables is obtained for viscous incompressible flow with a linear velocity profile. The solution takes into account the hydrodynamic interaction of an infinite number of particles in the array. An expression for the effective viscosity of a suspension with a cubic array of particles is obtained.     

Slow Motion of an Assemblage of Porous Spherical Shells Relative to a Fluid

The body-force-driven motion of a homogeneous distribution of spherically symmetric porous shells in an incompressible Newtonian fluid and the fluid flow through a bed of these shell particles are investigated analytically. The effect of the hydrodynamic interaction among the porous shell particles is taken into account by employing a cell-model representation. In the limit of small Reynolds number, the Stokes and Brinkman equations are solved for the flow field around a single particle in a unit cell, and the drag force acting on the particle by the fluid is obtained in closed forms. For a suspension of porous spherical shells, the mobility of the particles decreases or the hydrodynamic interaction among the particles increases monotonically with a decrease in the permeability of the porous shells. The effect of particle interactions on the creeping motion of porous spherical shells relative to a fluid can be quite significant in some situations. In the limiting cases, the analytical solution describing the drag force or mobility for a suspension of porous spherical shells reduces to those for suspensions of impermeable solid spheres and of porous spheres. The particle-interaction behavior for a suspension of porous spherical shells with a relatively low permeability may be approximated by that of permeable spheres when the porous shells are sufficiently thick.     

Influence of inertia,topography and gravity on transient axisymmetric thin‐film flow

This study examines theoretically the development of early transients for axisymmetric flow of a thin film over a stationary cylindrical substrate of arbitrary shape. The fluid is assumed to emerge from an annular tube as it is driven by a pressure gradient maintained inside the annulus, and/or by gravity in the axial direction. The interplay between inertia, annulus aspect ratio, substrate topography and gravity is particularly emphasized. Initial conditions are found to have a drastic effect on the ensuing flow. The flow is governed by the thin‐film equations of the ‘boundary‐layer’ type, which are solved by expanding the flow field in terms of orthonormal modes in the radial direction. The formulation is validated upon comparison with the similarity solution of Watson (J. Fluid Mech 1964; 20 :481) leading to an excellent agreement when only 2–3 modes are included. The wave and flow structure are examined for high and low inertia. It is found that low‐inertia fluids tend to accumulate near the annulus exit, exhibiting a standing wave that grows with time. This behaviour clearly illustrates the difficulty faced with coating high‐viscosity fluids. The annulus aspect is found to be influential only when inertia is significant; there is less flow resistance for a film over a cylinder of smaller diameter. For high inertia, the free surface evolves similarly to two‐dimensional flow. The substrate topography is found to have a significant effect on transient behaviour, but this effect depends strongly on inertia. It is observed that the flow of a high‐inertia fluid over a step‐down exhibits the formation of a secondary wave that moves upstream of the primary wave. Gravity is found to help the film (coating) flow by halting or prohibiting the wave growth. The initial film profile and velocity distribution dictate whether the fluid will flow downstream or accumulate near the annulus exit. Copyright © 2004 John Wiley & Sons, Ltd.     

Numerical study of the axially symmetric motion of an incompressible viscous fluid in an annulus between two concentric rotating spheres

The research reported herein involved the study of the transient motion of a system consisting of an incompressible Newtonian fluid in an annulus between two concentric, rotating, rigid spheres. The primary purpose of the research was to study the use of a numerical method for analysing the transient motion that results from the interaction between the fluid in the annulus and the spheres which are started suddenly by the action of prescribed torques. The problems considered in this research included cases where: (a) one or both spheres rotate with prescribed constant angular velocities and (b) one sphere rotates due to the action of an applied constant or impulsive t?orque. In this research the coupled solid and fluid equations were solved numerically by employing the finite difference technique. With the approach adopted in this research, only the derivatives with respect to spatial variables were approximated with the use of the finite difference formulae. The steady state problem was also solved as a separate problem (for verification purposes), and the results were compared with those obtained from the solution of the transient problem. Newton's algorithm was employed to solve the algebraic equations which resulted from the steady state problem, and the Adams fourth-order predictor–corrector method was employed to solve the ordinary differential equations for the transient problem. Results were obtained for the streamfunction, circumferential function, angular velocity of the spheres and viscous torques acting on the spheres as a function of time for various values of the system dimensionless parameters.     

Viscous Flow Past a Periodic Array of Spheres

Viscous flow past an infinite periodic array of rigid spheres is considered. The hydrodynamic interaction of all the particles in the array is taken into account. An analytical solution of the problem is proposed. The forces exerted by the fluid on the array particles are calculated and an expression for the velocity of fluid filtration through the array is obtained. The results are compared with the previous theoretical and experimental results.     

Slow viscoelastic radial flow between parallel disks

A perturbation analysis is presented for the steady-state radial flow of a third-order fluid between two parallel disks. The results include previous perturbation analyses in which various other rheological models were used. The pressure drop needed to maintain the radial flow is less than that for the Newtonian creeping-flow solution because of fluid inertia and shear-thinning viscosity, whereas the normal stresses have the opposite effect. Possible use of the “radial flow viscometer” for experimental evaluation of second-order constants is also discussed. Finally, molecular stretching in the flow system is examined using the elastic dumbbell model for a polymer molecule.     

Influence of an acoustic wave on a system of two spheres in a viscous fluid

In this article we formulate and solve the problem of the influence of radiation forces (forces created by the radiation pressure) on two spheres in a viscous fluid during the transmission of an acoustic wave. On the basis of these forces we investigate the nature of the interaction between the spheres as determined by the mutual disturbance of the flow fields around them as a result of interference between the primary and secondary waves reflected from the spheres. A previously proposed [2] approach is used in the investigations. The radiation force acting on one of the spheres is filtered by averaging the convolution of the stress tensor in the fluid with the unit normal to the surface of the sphere over a time interval and over the surface of the sphere. The stresses in the fluid are represented, to within second-order quantities in the parameters of the wave field, in terms of the velocity potentials obtained from the solution of the linear problem of the diffraction of the primary wave by the free spheres. The diffraction problem is formulated and solved within the framework of the theory of linear viscoelastic solids [6]. The case of an ideal fluid has been studied previously [3–5, 7]. Radiation forces are one of the causes of the relative drift of solid particles situated in a fluid in an acoustic field.S. P. Timoshenko Institute of Mechanics, Academy of Sciences of Ukraine, Kiev. Translated from Prikladnaya Mekhanika, Vol. 30, No. 2, pp. 33–40, February, 1994.     

The indentation of a Newtonian fluid in a finite cylindrical container by a right circular cylinder

The indentation of the free surface of a Newtonian fluid in a finite cylindrical container by a right circular cylinder is considered. It is assumed that weight and inertia effects are negligible compared to viscous effects. A finite difference technique is used to obtain approximate values for initial velocities, pressures, and stresses at any point in the fluid as well as an estimate of the force required to indent the fluid with a given velocity. The solution obtained forms the basis for a primary indenter viscometer for very viscous fluids which have viscosities in the range of 10⁴–10¹⁰ poises.     

Drift of spheres in a rotating fluid

The drift of spheres in a rotating fluid is investigated. The problem is studied experimentally and numerically using the Galerkin method. It is shown that for small angular velocities of the fluid Ω the drift velocity of the spheres is almost independent of Ω, but once a certain threshold value Ω_* is attained the drift velocity rapidly decreases. The experimental dependence of the translational velocity of the sphere on the fluid angular velocity is explained on the basis of a theoretical analysis.     

Squeeze flow of interstitial Herschel–Bulkley fluid between two rigid spheres

Interaction between two spheres with an interstitial fluid is crucial in discrete element modeling for simulating the behaviors of ‘wet’ particulate materials. The normal viscous force of squeeze flow between two arbitrary rigid spheres with an interstitial Herschel–Bulkley fluid was studied on the basis of Reynolds’ lubrication theory, resulting in analytical integral expressions of pressure distribution and the viscous force between the two spheres. According to the variation of shear stress, the fluid was divided into yielding and unyielding regions, followed by a discussion on the thickness of the two regions. The result of this paper could be reduced to either the power-law fluid or the Bingham fluid case.     

Freely rising light solid spheres

This paper examines the behavior of spheres rising freely in a Newtonian fluid when the ratio between the density of the spheres and that of the surrounding fluid is about 0.02. High-speed imaging is used to reconstruct three-dimensional trajectories of the rising spheres. From the analysis of the trajectories the magnitudes of the drag and lift forces exerted by the surrounding fluid are deduced. It is argued that the two main contributions to the drag force are (i) a viscous drag that may be estimated from the standard drag curve by evaluating the Reynolds number using the actual value of the velocity, and (ii) an inertial drag that arises essentially by the same mechanisms that cause the lift-induced drag familiar from wing theory. Estimates of both contributions, the latter using visualizations of the wakes of the spheres, give a favorable agreement with the measured drag forces. These findings are closely related to recent numerical results of in the literature on the forces experienced by oblate ellipsoidal bubbles rising in quiescent water.