The effect of the slip condition on Stokes and Couette flows due to an oscillating wall: exact solutions

Stokes and Couette flows produced by an oscillatory motion of a wall are analyzed under conditions where the no-slip assumption between the wall and the fluid is no longer valid. The motion of the wall is assumed to have a generic sinusoidal behavior. The exact solutions include both steady periodic and transient velocity profiles. It is found that slip conditions between the wall and the fluid produces lower amplitudes of oscillations in the flow near the oscillating wall than when no-slip assumption is utilized. Further, the relative velocity between the fluid layer at the wall and the speed of the wall is found to overshoot at a specific oscillating slip parameter or vibrational Reynolds number at certain times. In addition, it is found that wall slip reduces the transient velocity for Stokes flow while minimum transient effects for Couette flow is achieved only for large and small values of the wall slip coefficient and the gap thickness, respectively. The time needed to reach to steady periodic Stokes flow due to sine oscillations is greater than that for cosine oscillations with both wall slip and no-slip conditions.     

Analysis of flow channeling near the wall in packed beds

The phenomenon of flow channeling in packed beds is investigated in the present analysis. A closed-form solution is obtained from the volume-averaged second-order momentum equation with the no-slip condition at the wall, using an exponential porosity distribution typical of packed beds. The predicted velocity profile shows the most important features of flow channeling: the sharp peak in the velocity near the wall, and the approach to the Darcy velocity far from the wall. The predictions are consistent with previous numerical results and with velocity measurements made downstream of a packed bed. The compact expression for the velocity given here is shown to be convenient for analyzing other packed-bed characteristics, such as the effective permeability of a finite-sized bed.     

Lyapunov exponents estimation for hysteretic systems

This article discusses the Lyapunov exponent estimation of non-linear hysteretic systems by adapting the classical algorithm by Wolf and co-workers [Wolf, A., Swift, J.B., Swinney, H.L., Vastano, J.A., 1985. Determining Lyapunov exponents from a times series. Physica D 16, 285–317.]. This algorithm evaluates the divergence of nearby orbits by monitoring a reference trajectory, evaluated from the equations of motion of the original hysteretic system, and a perturbed trajectory resulting from the integration of the linearized equations of motion. The main issue of using this algorithm for non-linear, rate-independent, hysteretic systems is related to the procedure of linearization of the equations of motion. The present work establishes a procedure of linearization performing a state space split and assuming an equivalent viscous damping in order to represent hysteretic dissipation in the linearized system. The dynamical response of a single-degree of freedom pseudoelastic shape memory alloy (SMA) oscillator is discussed as an application of the proposed algorithm. The restitution force of the oscillator is provided by an SMA element described by a rate-independent, hysteretic, thermomechanical constitutive model. Two different modeling cases are considered for isothermal and non-isothermal heat transfer conditions, and numerical simulations are performed for both cases. The evaluation of the Lyapunov exponents shows that the proposed procedure is capable of quantifying chaos capturing the non-linear dissipation of hysteretic systems.     

Slow motions of a circular cylinder experiencing slip near a plane wall

A theoretical study is presented for the two-dimensional creeping flow caused by a long circular cylindrical particle translating and rotating in a viscous fluid near a large plane wall parallel to its axis. The fluid is allowed to slip at the surface of the particle. The Stokes equations for the fluid velocity field are solved in the quasi-steady limit using cylindrical bipolar coordinates. Semi-analytical solutions for the drag force and torque acting on the particle by the fluid are obtained for various values of the slip coefficient associated with the particle surface and of the relative separation distance between the particle and the wall. The results indicate that the translation and rotation of the confined cylinder are not coupled with each other. For the motion of a no-slip cylinder near a plane wall, our hydrodynamic drag force and torque results reduce to the closed-form solutions available in the literature. The boundary-corrected drag force and torque acting on the particle decrease with an increase in the slip coefficient for an otherwise specified condition. The plane wall exerts the greatest drag on the particle when its migration occurs normal to it, and the least in the case of motion parallel to it. The enhancement in the hydrodynamic drag force and torque on a translating and rotating particle caused by a nearby plane wall is much more significant for a cylinder than for a sphere.     

Low Reynolds number hydrodynamic interaction of a solid particle with a planar wall

The slow viscous flow problem of an arbitrary solid particle in motion near a planar wall is recast into a boundary integral formulation. The present formulation employs the Green function appropriate to the planar wall problem and is developed in sufficient generality to allow calculations for arbitrary particles in any base flow which satisfies Stokes equations and no-slip on the wall. The resulting integral equations are easily discretized and solved for the particle surface tractions. Calculations are performed for axisymmetric motions of a variety of ellips?ids near the planar wall. Agreement with existing theory is excellent.     

Slip yield stress effects in start-up Newtonian Poiseuille flows

Analytical solutions are derived for various start-up Newtonian Poiseuille flows assuming that slip at the wall occurs when the wall shear stress exceeds a critical value, known as the slip yield stress. Two distinct regimes characterise the steady axisymmetric and planar flows, which are defined by a critical value of the pressure gradient. If the imposed pressure gradient is below this critical value, the classical no-slip, start-up solution holds. Otherwise, no-slip flow occurs only initially, for a finite time interval determined by a critical time, after which slip does occur. For the annular case, there is an additional intermediate (steady) flow regime where slip occurs only at the inner wall, and hence, there exist two critical values of the pressure gradient. If the applied pressure gradient exceeds both critical values, the velocity evolves initially with no-slip at both walls up to the first critical time, then with slip only along the inner wall up to the second critical time and finally with slip at both walls.     

在椭圆截面管道中的磁流体动力学管流

对于在外加横向均匀磁场下,不可压缩、粘滞、导电液体流经椭圆截面管道的一维定常流动问题,本文第一次给出其解析解,文中假设流体在绝缘管壁上没有滑动。     

基于虚拟区域法的黏弹性流体中微生物游动的数值模拟和应用

微生物是自然生态系统的重要组成部分,掌握微生物在复杂流体中的运动特性可以为微型器件的设计制造提供理论指导.壁面效应是微生物游动研究中的重要问题之一,已有研究表明微生物在壁面附近存在复杂的行为特征.然而已有研究大多集中于微生物在牛顿流体中的游动模拟,仅有少数涉及黏弹性流体等非牛顿流体.本文采用直接力虚拟区域法与乔列斯基分解相结合的数值方法,引入Squirmer微生物游动模型,研究了微生物在黏弹性流体中的游动问题.首先给出求解黏弹性流体本构方程的数值格式;并将该方法应用于研究微生物游动中的壁面效应.研究结果表明,游动方向是影响微生物颗粒壁面效应的重要因素.流体弹性应力会对微生物产生一个反向转矩,影响微生物的游动方向,从而阻碍微生物逃离壁面.微生物颗粒在黏弹性流体中与壁面作用时间较长,几乎达到牛顿流体的两倍以上.     

High-speed imaging of an ultrasound-driven bubble in contact with a wall: “Narcissus” effect and resolved acoustic streaming

We report microscopic observations of the primary flow oscillation of an acoustically driven bubble in contact with a wall, captured with the ultra high-speed camera Brandaris 128 (Chin et al. 2003). The driving frequency is up to 200 kHz, and the imaging frequency is up to 25 MHz. The details of the bubble motion during an ultrasound cycle are thus resolved, showing a combination of two modes of oscillations: a radius oscillation and a translation oscillation, perpendicular to the wall. This motion is interpreted using the theory of acoustic images to account for the presence of the wall. We conclude that the bubble is subjected to a periodic succession of attractive and repulsive forces, exerted by its own image. Fast-framing recordings of a tracer particle embedded in the liquid around the particle are performed. They fully resolve the acoustic streaming flow induced by the bubble oscillations. This non-linear secondary flow appears as a tiny drift of the particle position cycle after cycle, on top of the primary back and forth oscillation. The high oscillation frequency accounts for a fast average particle velocity, with characteristic timescales in the millisecond range at the lengthscale of the bubble. The features of the bubble motion being resolved, we can apply the acoustic streaming theory near a wall, which provides predictions in agreement with the observed streaming velocity.     

Measurement of velocity profiles in a rectangular microchannel with aspect ratio <Emphasis Type="Italic">α</Emphasis> = 0.35

In this work, we measured 14 horizontal velocity profiles along the vertical direction of a rectangular microchannel with aspect ratio α = h/w = 0.35 (h is the height of the channel and w is the width of the channel) using microPIV at Re = 1.8 and 3.6. The experimental velocity profiles are compared with the full 3D theoretical solution, and also with a Poiseuille parabolic profile. It is shown that the experimental velocity profiles in the horizontal and vertical planes are in agreement with the theoretical profiles, except for the planes close to the wall. The discrepancies between the experimental data and 3D theoretical results in the center vertical plane are less than 3.6%. But the deviations between experimental data and Poiseuille’s results approaches 5%. It indicates that 2D Poiseuille profile is no longer a perfect theoretical approximation since α = 0.35. The experiments also reveal that, very near the hydrophilic wall (z = 0.5–1 μm), the measured velocities are significantly larger than the theoretical velocity based on the no-slip assumption. A proper discussion on some physical effects influencing the near wall velocity measurement is given.     

The effect of an additional boundary condition on the plane creeping flow of a second-order fluid

A similarity solution is obtained for the two-dimensional creeping flow of a second-order fluid with non-parallel porous walls. The resulting ordinary differential equation is fifth order. Thus, an additional velocity boundary condition is needed, the other four being due to the usual no-slip conditions. Having chosen to prescribe the rate of shear at the wall, the problem is solved by a standard numerical routine. A singular perturbation analysis is developed for small values of the Deborah number. A type of boundary layer forms for which the viscous Newtonian case is the outer solution.     

Motion of a circular cylinder near a vertical wall

We consider the arbitrary motion of a circular cylinder in an ideal fluid near a vertical wall. This problem is usually solved in the approximate formulation with a degree of error which is difficult to assess, increasing with approach of the cylinder to the wall [1, 2], The exact solution has previously been carried out only for the case of purely circulatory flow about the cylinder [3].     

Modelling the flow of power law fluids in a packed bed using a volume-averaged equation of motion

The development of a theoretical model for the prediction of velocity and pressure drop for the flow of a viscous power law fluid through a bed packed with uniform spherical particles is presented. The model is developed by volume averaging the equation of motion. A porous microstructure model based on a cell model is used. Numerical solution of the resulting equation is effected using a penalty Galerkin finite element method. Experimental pressure drop values for dilute solutions of carboxymethylcellulose flowing in narrow tubes packed with uniformly sized spherical particles are compared to theoretical predictions over a range of operating conditions. Overall agreement between experimental and theoretical values is within 15%. The extra pressure drop due to the presence of the wall is incorporated directly into the model through the application of the no-slip boundary condition at the container wall. The extra pressure drop reaches a maximum of about 10% of the bed pressure drop without wall effect. The wall effect increases as the ratio of tube diameter to particle diameter decreases, as the Reynolds number decreases and as the power law index increases.     

粗糙度动力学特性的初步研究

建立了一个描写粗糙壁面附近流场的理论模型，并进行了相应的数值分析，发现粗糙度具有随机边界层外部风速的增大而减小的动力学特性，这与最新的实验结果相符合，说明粗糙度不但是一个与壁面形态有关的几何参数，而且是一个与粗糙壁面附近流场流场情况有关的动力学参数，从理论上打破了粗糙度仅依赖于粗糙壁面几何形态的传统看法。     

Slip effects on shearing flows in a porous medium

This paper deals with the magnetohydrodynamic （MHD） flow of an Oldroyd 8-constant fluid in a porous medium when no-slip condition is no longer valid. Modified Darcy＇s law is used in the flow modelling. The non-linear differential equation with non-linear boundary conditions is solved numerically using finite difference scheme in combination with an iterative technique. Numerical results are obtained for the Couette, Poiseuille and generalized Couette flows. The effects of slip parameters on the velocity profile are discussed.     

Large free vibrations of a beam carrying a moving mass

The focus of this work is to develop a technique to obtain numerical solution over a long range of time for non-linear multi-body dynamic systems undergoing large amplitude motion. The system considered is an idealization of an important class of problems characterized by non-linear interaction between continuously distributed mass and stiffness and lumped mass and stiffness. This characteristic results in some distinctive features in the system response and also poses significant challenges in obtaining a solution.

In this paper, equations of motion are developed for large amplitude motion of a beam carrying a moving spring–mass. The equations of motion are solved using a new approach that uses average acceleration method to reduce non-linear ordinary differential equations to non-linear algebraic equations. The resulting non-linear algebraic equations are solved using an iterative method developed in this paper. Dynamics of the system is investigated using a time-frequency analysis technique.     

On corner flows of Oldroyd-B fluids

Exact series solutions for planar creeping flows of Oldroyd-B fluids in the neighbourhood of sharp corners are presented and discussed. Both reentrant and non-reentrant sectors are considered. For reentrant sectors it is shown that more than one type of series solution can exist formally, one type exhibiting Newtonian-like asymptotic behaviour at the corner, away from walls, and another type exhibiting the same kind of asymptotics as an Upper Convected Maxwell (UCM) fluid. The solutions which are Newtonian-like away from walls are shown to develop non-integrable stress singularities at the walls when the no-slip velocity boundary condition is imposed. These mathematical solutions are therefore inadmissible from the physical viewpoint under no-slip conditions. An inadmissible solution, with stress singularities which are not everywhere integrable, is identified among the solutions of UCM-type. For a 270° reentrant sector the radial behaviour of the normal stress is everywhere r^−0.613. In the viscometric region near a wall, the radial normal stress σ^rr behaves like (rε)^−0.613, where ε is the angle made with the wall. In addition σ^rθ is infinite (not integrable) at the wall even when r is non-zero. Another UCM-type solution has a normal stress behaviour away from walls which is r^−0.985 for 270° sector. Again, this solution has a non-integrable stress singularity and is therefore inadmissible. Finally, for non-reentrant sectors it is shown that the flow is always Newtonian-like away from walls.     

Influence of the hall effect on development of rotational MHD flow in a flat channel

In an experimental study of the heat transfer from a partially ionized gas it was found that the heat flux to the wall for flow of an electrically conducting gas in a circular tube located in a magnetic field of a solenoid depends not only on the magnitude of the magnetic field but also on the field orientation [1]; with the magnetic field parallel to the velocity the heat transfer is reduced by 15%, with antiparallel orientation it is reduced by only 1% in comparison with the heat transfer without the magnetic field. No explanation for this was given either in [1] or in the subsequent discussion [2]; moreover, on the basis of the constructed equations [1] this effect cannot be obtained at all, since the solution of the equations clearly is not changed by a change of the field sign. In the following we attempt to explain this effect by the processes which take place during the development of rotational flow of an anisotropically conducting medium. The idea of the possibility of such an explanation for this effect was proposed in general form in the survey paper [3].The detailed calculation of the development of MHD flows has been made previously only for the case of a transverse magnetic field and very simple channel geometry (see, for example, the survey [3]).In all the considered problems the components of the electrical field which appeared in the motion equations were known with an accuracy to constants from symmetry considerations. Therefore, under the assumption of smallness of the induced magnetic field these problems reduced simply to the solution of the equation of motion with additional terms which are linear in the velocity. In the present paper we construct an approximate simultaneous solution of a system consisting of the motion equations and the equation for the electrical potential.     

Note on the reflexion of water waves at a nearly vertical cliff in the presence of surface tension

In this note the two-dimensional problem of incoming surface waves against an approximately vertical wall or cliff in water of infinite depth is examined, using velocity potential formulation and linearized boundary-value problem theory for time-harmonic motion. The cliff has arbitrary profile but for simplicity is taken to be vertical at the free surface. The approximate first-order solution is determined subject to a dynamical edge condition apposite to the presence of surface tension, and contains partially reflected outgoing waves. The solution is obtained by perturbation theory in a form involving known unperturbed and first-order correction potentials that is applicable also to water of finite constant depth. The motivation for the note is to point out a correction in principle to the results of a recent investigation for a specific profile, in which reflexion is ignored and another error made in obtaining a first-order solution by a method that is restricted to infinite depth.