Buckling of a flush-mounted plate in simple shear flow

The buckling of an elastic plate with arbitrary shape flush-mounted on a rigid wall and deforming under the action of a uniform tangential load due to an overpassing simple shear flow is considered. Working under the auspices of the theory of elastic instability of plates governed by the linear von Kármán equation, an eigenvalue problem is formulated for the buckled state resulting in a fourth-order partial differential equation with position-dependent coefficients parameterized by the Poisson ratio. The governing equation also describes the deformation of a plate clamped around the edges on a vertical wall and buckling under the action of its own weight. Solutions are computed analytically for a circular plate by applying a Fourier series expansion to derive an infinite system of coupled ordinary differential equations and then implementing orthogonal collocation, and numerically for elliptical and rectangular plates by using a finite-element method. The eigenvalues of the resulting generalized algebraic eigenvalue problem are bifurcation points in the solution space, physically representing critical thresholds of the uniform tangential load above which the plate buckles and wrinkles due to the partially compressive developing stresses. The associated eigenfunctions representing possible modes of deformation are illustrated, and the effect of the Poisson ratio and plate shape is discussed.     

Particle suspension in a simple shear flow

A hydrodynamic model describing the particle distribution over the cross-section of a finely dispersed flow is proposed. The model is constructed on the basis of notions concerning the diffusion of particles induced by their random displacements in the process of relative motion of neighboring layers at constant shear velocity. It is shown that the suspension capacity of the flow is large for small particles due to thermal fluctuations and for relatively large particles due to shear-induced particle pulsations. There are critical particle sizes for which the particles are suspended and transported by the flow less effectively than larger or smaller particles.Ekaterinburg. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 1, pp. 112–121, January–February, 1995.     

Grain shear flow in a rotating drum

 In the present paper we report on the experimental activities carried out on a rotating drum partially filled with grains or glass beads. The experiments give information about rheology through velocity profiles and through the velocity covariance tensor structure. We used a LDV system to measure the velocity of the grains at several points along three vertical sections. The data were also used to obtain the grain volume concentration, with encouraging results. Instantaneous velocity data were elaborated in order to obtain velocity and pseudotemperature profiles for all the experiments; for a subset of the experiments a large set of data were elaborated to obtain the velocity covariance. The velocity covariance is not collinear with the rate of deformation tensor. An attempt to justify the rotation of the tensor axes as a consequence of the kinetically induced anisotropy and of some free surface perturbations slowly moving upstream was partially successful. Received: 23 April 1999 / Accepted: 15 July 2001     

Nonstationary shear flow of a viscoplastic medium

We consider problems involving nonstationary shear flow of a viscoplastic medium between two parallel plates and also in a cylindrical tube under the action of a time-varying shear stress applied to the walls of the passage.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 4, pp. 133–137, July–August, 1972.     

Unsteady shear flow of a viscoplastic material

A viscoplastic, or yield-stress, liquid occupies the space between two infinite parallel plates. Initially the whole system is at rest. The lower plate is suddenly jerked into motion with given speed or shear stress, while the upper plate is kept fixed. The flow consists of two regions; (1) a lower sheared region bounded above by the yield surface, (2) an upper unyielded region bounded below by the yield surface. The yield surface propagates to the upper plate as time proceeds. We first consider the equivalent one plate problem of flow over a jerked plate, and find similarity solutions and small time asymptotic solutions for prescribed shear and speed cases respectively. These solutions are used as initial solutions for the two plate case. The motion of the yield surface and the time taken for the entire material to yield are investigated. The problems considered here are two dimensional representations of some control devices, for example the light duty clutch, which consists of two corotating, coaxial discs separated by a layer of electrorheological material. In this application it is useful to know the time taken for all the material to yield.     

Vortex shedding from a ring in shear flow

Vortex shedding from a bluff ring in a linear shear flow has been investigated experimentally. The shedding frequency, measured using a hot wire in various positions within the near wake of the ring, was found to be remarkably insensitive to a mean flow shear parameter, , defined by (d/U _o).U/y, where d is the mean ring diameter and U _o is the undisturbed upstream velocity on the ring axis; even for = 0.41, corresponding to a velocity variation of about ±24% across the outer ring diameter, the Strouhal number was only about 5% lower than in uniform flow. However, the strong axisymmetric shedding which dominates the flow for = 0 is significantly affected by shear. Indeed, spatial correlation measurements demonstrated that the shedding actually becomes anti-symmetric at the highest values of . The implication of these results is that vortex flowmeters constructed using a ring as the shedding body would be relatively unaffected by changes in the upstream mean flow profile.     

Gravitational sedimentation of aggregating particles in a shear flow

A theoretical model describing the sedimentation of aggregating particles under the influence of gravity in Couette shear flow is proposed. The relation between the parameters characterizing the dependence of the sedimentation properties on the shear rate and the parameters characterizing the aggregation of the particles is traced.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 95–98, March–April, 1989.The authors wish to thank S. A. Regirer for useful discussions.     

Direct simulation of compressible turbulence in a shear flow

The purpose of this study is to investigate compressibility effects on the turbulence in homogeneous shear flow. We find that the growth of the turbulent kinetic energy decreases with increasing Mach number—a phenomenon which is similar to the reduction of turbulent velocity intensities observed in experiments on supersonic free shear layers. An examination of the turbulent energy budget shows that both the compressible dissipation and the pressure-dilatation contribute to the decrease in the growth of kinetic energy. The pressure-dilatation is predominantly negative in homogeneous shear flow, in contrast to its predominantly positive behavior in isotropic turbulence. The different signs of the pressure-dilatation are explained by theoretical consideration of the equations for the pressure variance and density variance. We previously obtained the following results for isotropic turbulence: first, the normalized compressible dissipation is of O(M _t ² ), and, second, there is approximate equipartition between the kinetic and potential energies associated with the fluctuating compressible mode. Both these results have now been substantiated in the case of homogeneous shear. The dilatation field is significantly more skewed and intermittent than the vorticity field. Strong compressions seem to be more likely than strong expansions.Dedicated to Professor J.L. Lumley on the occasion of his 60th birthday.This research was supported by the National Aeronautics and Space Administration under NASA Contract No. NAS1-18605 while the authors were in residence at the Institute for Computer Applications in Science and Engineering (ICASE), NASA Langley Research Center, Hampton, VA 23665, U.S.A.     

Linear shear flow past a porous particle

Linear shear flow past a porous spherical particle is studied using a generalized boundary condition proposed by Jones. The torque on a porous sphere rotating in a quiescent fluid is calculated. Streamlines patterns are illustrated for the case of a particle freely suspended in a simple shear flow. These patterns are shown to differ significantly from those associated with an impermeable rigid sphere. Finally, an expression for the effective viscosity of a dilute suspension of porous spherical particles is obtained.Nomenclature A, B dimensionless flow parameter - a radius of the porous sphere - C, E, F constants of integration - d shear strength - d constant rate of deformation of ambient field - e rate of strain tensor - f, g functions of distance - k permeability of the porous medium - n unit normal vector - p pressure - p unit vector - Q coefficient of spherical harmonic - q filter velocity within the porous medium - r polar spherical coordinate - S _p surface of porous particle - S, T, T* coefficients of spherical harmonics - T torque exerted on the particle - u fluid velocity vector - x cartesian coordinates - dimensionless constant - , polar spherical coordinates - dimensionless flow parameter - viscosity of the fluid - stress tensor - rotational velocity of the particle - rotational velocity of the ambient field.     

Experiments on the flow past long circular cylinders in a shear flow

This paper describes an experimental investigation of the flow past circular cylinders, with the mean flow perpendicular to the cylinder axis, at conditions that yield a strong three-dimensional behaviour. The experiments were carried out in the subcritical regime. Long cylinders with end plates were subjected to shear flow with a linear velocity profile in the spanwise direction generated by means of a curved gauze. It was concluded that spanwise cellular structures of vortex shedding emerged in the wake, more clearly for some boundary conditions than others. These structures are characterised by a portion of spanwise length, a cell, having constant shedding frequency over a time average, which implies that there were no vortex dislocations inside that cell during that time. These features were studied using flow visualisation and hot-film anemometry. Spectra of the local shedding frequency are shown, revealing the effect of the shear parameter (=0.02 and 0.04) and aspect ratio L/D (=20.6 and 8) on the stability and geometry of the cells at several Reynolds numbers in the range of 3.13×10³Re_m1.25×10⁴.     

Vortices in time-periodic shear flow

Vortices emerging in geophysical turbulence may experience deformations due to the non-uniform ambient flow induced by neighbouring vortices. At first approximation this ambient flow is modeled by a linear shear flow. It is well known from previous studies that the vortex may be (partially) destructed through removal of weak vorticity at the vortex edge—a process referred to as ‘stripping’. While most previous studies considered a stationary external shear flow, we have examined the behaviour of the vortex embedded in a linear shear flow whose strength changes harmonically in time. Aspects of the vortex dynamics and the (chaotic) transport of tracers have been studied by both laboratory experiments and numerical simulations based on a simple kinematical model.     

Driven torsion pendulum for measuring the complex shear modulus in a steady shear flow

To investigate the viscoelastic behavior of fluid dispersions under steady shear flow conditions, an apparatus for parallel superimposed oscillations has been constructed which consists of a rotating cup containing the liquid under investigation in which a torsional pendulum is immersed. By measuring the resonance frequency and bandwidth of the resonator in both liquid and in air, the frequency and steady-shear-rate-dependent complex shear modulus can be obtained. By exchange of the resonator lumps it is possible to use the instrument at four different frequencies: 85, 284, 740, and 2440 Hz while the steady shear rate can be varied from 1 to 55 s^–1. After treatment of the theoretical background, design, and measuring procedure, the calibration with a number of Newtonian liquids is described and the accuracy of the instrument is discussed.Notation a radius of the lump - A geometrical constant - b inner radius of the sample holder - c constant - C ₁, C ₂ apparatus constants - D damping of the pendulum - e _x , e _y , e _z Cartesian basis - e _r , e , e _z orthonormal cylindrical basis - E geometrical constant - E _t , ⁰ E _t , _t relative strain tensor - f function of shear rate - F _t relative deformation tensor - G (t) memory function - G ^* complex shear modulus - G Re(G ^* ) - G Im(G ^* ) - h distance between plates - H ^* transfer function - , functional - i imaginary unit: i ²= – 1 - I moment of inertia - J _exc excitation current - J ₀ amplitude of J _exc - k ^* = k – ik complex wave number - K torsional constant - K fourth order tensor - l length of the lump - L mutual inductance - M _dr driving torque - M _liq torque exerted by the liquid - ⁰ M _liq, _liq steady state and dynamic part of Mliq - n power of the shear rate - p isotropic pressure - Q quality factor - r radial position - R,R ₀, R _c Re(Z ^*, Z ₀ ^* , Z _c ^* ) - s time - t, t time - T temperature - T, ⁰ T, stress tensor - u velocity - U lock-in output - ₀ velocity - V _det detector output voltage - V _sig, V _cr signal and cross-talk part of V _det - x Cartesian coordinate - X , X ₀, X _c Im(Z ^*, Z ₀ ^* , Z _c ^* ) - y Cartesian coordinate - z Cartesian coordinate, axial position     

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].     

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.