On the squeeze flow of a power-law fluid between rigid spheres

The lubrication solution for the squeeze flow of a power-law fluid between two rigid spherical particles has been investigated. It is shown that the radial pressure distribution converges to zero within the gap between the particles for any value of the flow index, n, provided that the gap separation distance is sufficiently small. However, in the case of the viscous force, it is useful to consider that there are two contributions. The first is developed in the inner region of the gap and corresponds to the lubrication limit. The second is due to an integration of the pressure in the adjacent outer region of the gap. The relative contribution to the force in this outer region increases as n decreases and the separation distance increases. In particular, for flow indices in the range n>1/3, the contribution in the outer region is negligible if the separation distance is sufficiently small. For n1/3, this is the dominant term and an accurate prediction of the viscous force is possible only for discrete liquid bridges.Based on “zero” pressure and lubrication criteria for the upper limits of integration, two closed-form solutions have been derived for the viscous force. Both are accurate for n>0.5 and are in close agreement with a previously published asymptotic solution in the range n>0.6. For smaller values of n, the asymptotic solution over-estimates the viscous force and predicts a singularity when n approaches 1/3. The two closed-form solutions show continuous and monotonic behaviour for all values of n. Moreover, the solution satisfying the lubrication limit is valid in the range n<1/3 provided that it is restricted to liquid bridges.     

The squeeze flow of a bi-viscosity fluid between two rigid spheres with wall slip

In this paper, the squeeze flow between two rigid spheres with a bi-viscosity fluid is examined. Based on lubrication theory, the squeeze force is calculated by deriving the pressure and velocity expressions. The results of the normal squeeze force are discussed, and fitting functions of the squeeze and correction coefficients are given. The squeeze force between the rigid spheres increases linearly or logarithmically with the velocity when most or part of the boundary fluid reaches the yield state, respectively. Furthermore, the slip correction coefficient decreases with the increase in the velocity. The investigation may contribute to the further study of bi-viscosity fluids between rigid spheres with wall slip.     

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.     

Flow of a viscoelastic fluid over a stretching sheet

This paper presents a study of the flow of an incompressible second-order fluid past a stretching sheet. The problem has a bearing on some polymer processing application such as the continuous extrusion of a polymer sheet from a die.     

Dissipation in a dilute suspension of spheres in a second-order fluid

We examine the effective medium properties of a dilute suspension of spheres in a second-order fluid under linear shear. Since the second-order fluid is the first step toward the general viscoelastic fluid, the results obtained may provide a qualitative feel for the problem in which the suspending fluid obeys a more complicated (and realistic) constitutive relation.The dissipation in the medium is calculated by determining the rate of working by surface forces; this is compared to the dissipation in a homogeneous fluid to give the effective properties. The results show that the term linear in volume fraction increases the corresponding rheological coefficient, just as in the Newtonian case. It is to be noted that the second-order dissipation is zero for simple shear and other weak flows, whereas for strong flows the small correction may increase or decrease the overall dissipation.     

Interaction of spheres in a viscoelastic fluid

Summary The interaction between two identical spheres of radiusa in a second-order fluid is studied, if the undisturbed flow is a general homogeneous flow. WithR the (instantaneous) distance between the sphere centers only the situationa/R 1 is considered. It turns out that it is not sufficient to know thea/R-term of the perturbation velocity, since certain contributions of the (a/R)²-terms are also needed. For two spheres sedimenting in a quiescent fluid a change of the relative position vector is predicted: the distance decreases and so does the orientation, i.e. the spheres tend to fall along their line of centers. If the motion of the individual sphere is restrained via a rigid connection (rigid dumbbell) this change of orientation implies that the dumbbell rotates until its axis is parallel to the direction of the applied force (stable orientation). In simple shear the first-order dumbbell (a/R-terms due to interaction) ultimately ends up in the plane normal to the gradient direction, independent of the rate of shear. This contrasts the behavior of a second-order dumbbell: if the symmetry axis lies in the plane of flow it will rotate around the vorticity axis at small rates of shear. Increasing the shear rate this dumbbell reaches a spinfree terminal state in which the angle between the symmetry axis and the flow direction is non-zero (although it is small). It is conjectured that for arbitrary initial orientations (not in the flow plane) the axis of the second-order dumbbell will not rotate in the Jeffrey orbits but rather show a systematic drift to become oriented parallel to the vorticity axis.

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Zusammenfassung Die Wechselwirkung zweier identischer Kugeln vom Radiusa in einer beliebigen homogenen Strömung einer Flüssigkeit zweiter Ordnung wird untersucht. MitR dem (augenblicklichen) Abstand der Kugelzentren beschränken wir uns auf die Situationa/R 1. Es zeigt sich, daß es nicht genügt, diea/R-Glieder der Störungsgeschwindigkeit zu kennen, da einige Beiträge der (a/R)²-Terme ebenso benötigt werden. Für zwei in einer ruhenden Flüssigkeit sedimentierende Kugeln wird eine Änderung der relativen Position vorausgesagt: der Abstand verkleinert sich, und das gleiche gilt für die Orientierung, d. h. die Kugeln streben die Situation, hintereinander zu fallen, an. Schränkt man die Bewegung der individuellen Kugeln durch eine starre Verbindung ein (starre Hantel), so zieht diese Orientierungsänderung eine Rotation nach sich, die die Hantelachse parallel zur Richtung der angreifenden Kraft ausrichtet (stabile Orientierung). Bei einer einfachen Scherung wandert unabhängig von der Schergeschwindigkeit die Achse einer Hantel erster Ordnung (die nur die Wechselwirkungsgliedera/R enthält) in der Ebene, deren Normale in Gradientenrichtung zeigt. Damit verhält sie sich völlig anders als eine Hantel zweiter Ordnung: Liegt bei letzterer die Symmetrieachse in der Strömungsebene, so rotiert diese bei kleinen Schergeschwindigkeiten um eine Achse senkrecht zur Strömungsebene. Bei einer Vergrößerung der Schergeschwindigkeit wird dagegen eine rotationsfreie Lage erreicht, bei der die Hantelachse unter einem kleinen Winkel zur Strömungsrichtung steht. Bei einer beliebigen Anfangsorientierung (außer in der Strömungsebene) schließen wir auf eine Wanderung der Achse einer Hantel zweiter Ordnung, bis diese parallel zur indifferenten Richtung steht.

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With 5 figures     

On unsteady motions of a second-order fluid over a plane wall

In this paper, three types of unsteady flows of second-order fluids are considered, namely, flow caused by impulsive motion of a flat plate, flow induced by a constantly accelerating plane and flow imposed by a flat plate that applies a constant tangential stress to the fluid. The previous attempts made regarding these problems, by using the Laplace transform, have failed. In this paper, the sine and the cosine transforms are used to solve these problems and exact solutions for the velocity distributions are found in terms of definite integrals. It is shown that these exact solutions satisfy the initial and the boundary conditions and the governing equation.     

Corner flow of a suspension of rigid rods

Stress singularities in the neighbourhood of sharp corners can be a source of severe problems in the numerical simulation of non-Newtonian flows leading to loss of convergence with grid refinement (G.G. Lipscombe, R. Kennings and M.M. Denn, J. Non-Newtonian Fluid Mech., 24 (1987) 85 [1]). For Newtonian flows the nature of this singularity is given by the analysis of Dean and Montagnon (W.R. Dean and P.E. Montagnon, Phil. Trans. R. Soc. London, Ser. A., 308 (1949) 199 [2]) in terms of similarity solutions. In this paper we extend this similarity analysis to a suspension of rigid rods. In the limit of nearly full extension the FENE constitutive model has the same behaviour as such a suspension. Our analysis predicts the possibility of lip vortices but their behaviour is somewhat inconsistent with those observed experimentally.     

The viscous-slip, diffusion-slip, and thermal-creep problems for a binary mixture of rigid spheres described by the linearized Boltzmann equation

An analytical version of the discrete-ordinates method (the ADO method) is used with recently established analytical expressions for the rigid-sphere scattering kernels to develop concise and particularly accurate solutions to the viscous-slip, the diffusion-slip, and the half-space thermal-creep problems for a binary gas mixture described by the linearized Boltzmann equation. In addition to a computation of the viscous-slip, the diffusion-slip, and the thermal-slip coefficients, for the case of Maxwell boundary conditions for each of the two species, the velocity, heat-flow, and shear-stress profiles are established for each species of particles. Numerical results are reported for two binary mixtures (Ne–Ar and He–Xe) with various molar concentrations.     

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.     

Squeeze flow of a power-law fluid between two rigid spheres with wall slip

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Diffusion of a line vortex in a second-order fluid

In this paper, the diffusion of a line vortex in a second-order fluid is considered. The Hankel transform is used to solve this problem and an exact solution for the velocity distribution is found in terms of a definite integral. The integrand is an oscillatory function and the integration is performed by a numerical technique. It is found that there are pronounced effects of viscoelastic properties on the velocity distribution with respect to that of the Newtonian fluid.     

2.5D scattering of waves by rigid inclusions buried under a fluid channel via BEM

A 2.5D Boundary Element Method (BEM) formulation, applied in the frequency domain, is developed to compute the scattering of waves by rigid inclusions buried in a semi-infinite solid under a fluid layer, when this system is excited by a spatially-sinusoidal harmonic load.The BEM algorithm includes Green's functions for a horizontal fluid layer over a semi-infinite solid, which avoids the discretrization of the horizontal surfaces, and thus only the rigid inclusion needs to be discretized by boundary elements. The model uses complex frequencies with a small imaginary part to avoid aliasing phenomena. Time domain responses are obtained by applying an inverse Fourier Transform to the frequency results. The source is modeled as a Ricker pulse. The simulations are performed for three different properties of the solid medium: a fast formation, a slow formation and a sediment formation.     

Flow of a viscoelastic fluid between eccentric cylinders

The velocity field in the annular region between two eccentric cylinders for a second-order fluid is determined. Second-order velocity terms appear only as a result of the interaction between imposed axial and planar motions. When only one of the two motions is imposed by the boundary conditions, the velocity field coincides with that of a Navier-Stokes fluid.The results of the present study are to be used in a future investigation of the stresses, forces, and torques acting on the walls of the cylinders to enable a rheometer based on the present boundary value problem to be suggested.     

On the effects of dilute polymers on driven cavity turbulent flows

Effects of dilute polymer solutions on a lid-driven cubical cavity turbulent flow are studied via particle image velocimetry (PIV). This canonical flow is a combination of a bounded shear flow, driven at constant velocity and vortices that change their spatial distribution as a function of the lid velocity. From the two-dimensional PIV data we estimate the time averaged spatial fields of key turbulent quantities. We evaluate a component of the vorticity–velocity correlation, namely 〈ω₃v〉, which shows much weaker correlation, along with the reduced correlation of the fluctuating velocity components, u and v. There are two contributions to the reduced turbulent kinetic energy production −〈u v〉S_uv, namely the reduced Reynolds stresses, −〈u v〉, and strongly modified pointwise correlation of the Reynolds stress and the mean rate-of-strain field, S_uv. The Reynolds stresses are shown to be affected because of the derivatives of the Reynolds stresses, ∂〈u v〉/∂y that are strongly reduced in the same regions as the vorticity–velocity correlation. The results, combined with the existing evidence, support the phenomenological model of polymer effects propagating from the polymer scale to the velocity derivatives and through the mixed-type correlations and Reynolds stress derivatives up to the turbulent velocity fields. The effects are shown to be qualitatively similar in different flows regardless of forcing type, homogeneity or presence of liquid–solid boundaries.     

Run-up and spin-up in a viscoelastic fluid. IV

Three problems are discussed — run-up when the fluid is contained between infinite parallel plates and longitudinal run-up and spin-up when it is contained in an infinitely long circular cylinder. The procedure adopted for solving these problems differs from that employed in Part I of this series, where these three problems were previously discussed, and yields results for the velocity fields in quite different forms. It is similar to that used in Part III in the context of the problem of run-up between parallel plates.