Effect of permeability on the instability of a non-Newtonian film down a porous inclined plane

A thin film of a power–law fluid flowing down a porous inclined plane is considered. It is assumed that the flow through the porous medium is governed by the modified Darcy’s law together with Beavers–Joseph boundary condition for a general power–law fluid. Under the assumption of small permeability relative to the thickness of the overlying fluid layer, the flow is decoupled from the filtration flow through the porous medium and a slip condition at the bottom is used to incorporate the effects of the permeability of the porous substrate. Applying the long-wave theory, a nonlinear evolution equation for the thickness of the film is obtained. A linear stability analysis of the base flow is performed and the critical condition for the onset of instability is obtained. The results show that the substrate porosity in general destabilizes the film flow system and the shear-thinning rheology enhances this destabilizing effect. A weakly nonlinear stability analysis reveals the existence of supercritical stable and subcritical unstable regions in the wave number versus Reynolds number parameter space. The numerical solution of the nonlinear evolution equation in a periodic domain shows that the fully developed nonlinear solutions are either time-dependent modes that oscillate slightly in the amplitude or time independent stable two-dimensional nonlinear waves with large amplitude referred to as ‘permanent waves’. The results show that the shape and the amplitude of the nonlinear waves are strongly influenced by the permeability of the porous medium and the shear-thinning rheology.     

Galloping instability and control of a rigid pendulum in a flowing soap film

A pendulum suspended in a fast flowing soap film may show sustained oscillations. The conditions necessary for self-excited motion to occur are outlined: a flow velocity above a threshold value along with geometrical constraints. The role of vortex shedding is discussed, and the observed instability is shown to be well-described by the galloping instability. Experimental results are supported by numerical simulations. Furthermore, we observe that the instability may be suppressed by attaching a long enough filament to the rear of the pendulum.     

THE OSCILLATION MODE OF INCOMPRESSIBLE CAVITY FLOW

采用二维大涡模拟方法进行了空腔水流场的数值计算, 考察空腔前缘动量损失厚度及来流速度等因素如何影响空腔流的振荡, 同时考察了空腔长深比与空腔流振荡模式的关系. 用空腔水流场的粒子图像测速测量结果验证了数值计算的可信性.结果表明, 空腔水流是否发生振荡取决于壁面摩擦速度.瞬时涡结构和空腔阻力系数2个方面的特征显示空腔水流场有2种典型的振荡模式, 剪切层模式与尾流模式, 确定振荡模式的关键因素是空腔长深比.     

Flow induced oscillation of two rigid rectangular plates in a side-by-side configuration

The present work is an experimental study of two oscillating rigid plates placed in side-by-side configuration, hinged at their leading edges, subjected to low subsonic flow. This problem is investigated using smoke-wire flow visualization, hot-wire anemometry, and time resolved particle image velocimetry. It is found that beyond a critical Reynolds number, the plates set into oscillatory motion. This critical Reynolds number depends on the gap between the plates. It is also seen that this value of Reynolds number, at lower values of gap to thickness ratio (<7) is significantly higher than that of the single plate configuration value. The frequency and amplitude of the oscillating plates at various gaps and Reynolds numbers have been studied and compared with the characteristics of an oscillating single plate. It is also found that depending on the gap and acceleration of the free-stream, there exist two modes of oscillation - (i) in-phase and (ii) out-of-phase. For gap to thickness ratio less than 10, only in-phase oscillations take place for all values of free-stream velocity considered in the present work, whereas, when this ratio is greater than 10, the mode of oscillation depends on the initial conditions up to a certain free-stream velocity, beyond which the plates switch to in-phase mode. Smoke wire flow visualization technique along with time resolved particle image velocimetry reveal that the vorticity distributions around the plates are responsible for the initiation of the two modes of oscillations.     

Modal analysis of cantilever plate flutter

This study illustrates the mechanism of modal coupling in cantilever plate flutter using the full Theodorsen airfoil theory within the linear framework. An accurate, pseudo-spectral method is employed to calculate the fluid loading and the eigenvalue problem is solved numerically following the Galerkin procedure. For plates with a structure-to-fluid mass ratio around unity, the first two in vacuo modes are dominant and the Kutta condition at the trailing edge plays a central role in the flutter mechanism. The fluid loading induced by the first mode excites significant second and higher order modes. The fluid loading on the second mode is coupled strongly with the structural vibration velocity of the first mode, which is identified as the main mechanism of energy transfer from flow to plate. It is demonstrated that the response of the second mode is suppressed and the plate is stabilized when a concentrated mass is added near the middle of the plate length. Theoretical prediction is supported by experimental data although the latter is affected by many practical factors that are difficult to model precisely.     

Experimental study of thermoacoustic effects on a single plate Part I: Temperature fields

 The thermal interaction between a heated solid plate and the acoustically driven working fluid was investigated by visualizing and quantifying the temperature fields in the neighbourhood of the solid plate. A combination of holographic interferometry and high-speed cinematography was applied in the measurements. A better knowledge of these temperature fields is essential to develop systematic design methodologies for heat exchangers in oscillatory flows. The difference between heat transfer in oscillatory flows with zero mean velocity and steady-state flows is demonstrated in the paper. Instead of heat transfer from a heated solid surface to the colder bulk fluid, the visualized temperature fields indicated that heat was transferred from the working fluid into the stack plate at the edge of the plate. In the experiments, the thermoacoustic effect was visualized through the temperature measurements. A novel evaluation procedure that accounts for the influence of the acoustic pressure variations on the refractive index was applied to accurately reconstruct the high-speed, two-dimensional oscillating temperature distributions. Received on 22 March 1999     

Numerical simulation of a drop undergoing large amplitude oscillatory shear

Direct numerical simulations are conducted for a Newtonian drop in a Newtonian matrix subjected to large amplitude oscillatory shear flows. In the experimental study of Guido et al. (in Rheol Acta 43:575–583, 2004), the drop shape is found to oscillate at higher harmonics of the forcing frequency when the capillary number is increased. Their phenomenological model requires a much smaller capillary number for predicting the harmonic nature of the experimental data. In this paper, computational results on the evolution of drop length and inclination angle are obtained at the same fluid and flow properties as the experiments, and are shown to reasonably reproduce the experimental data. In particular, the computed velocity fields around the drop are shown to elucidate the over-rotation, which is a mechanism for the experimentally observed harmonics.     

The viscous curtain: General formulation and finite-element solution for the stability of flowing viscous sheets

The stability of thin viscous sheets has been studied so far in the special case where the base flow possesses a direction of invariance: the linear stability is then governed by an ordinary differential equation. We propose a mathematical formulation and a numerical method of solution that are applicable to the linear stability analysis of viscous sheets possessing no particular symmetry. The linear stability problem is formulated as a non-Hermitian eigenvalue problem in a 2D domain and is solved numerically using the finite-element method. Specifically, we consider the case of a viscous sheet in an open flow, which falls in a bath of fluid; the sheet is mildly stretched by gravity and the flow can become unstable by ‘curtain’ modes. The growth rates of these modes are calculated as a function of the fluid parameters and of the geometry, and a phase diagram is obtained. A transition is reported between a buckling mode (static bifurcation) and an oscillatory mode (Hopf bifurcation). The effect of surface tension is discussed.     

Classification of periodic orbits of two-dimensional homogeneous granular crystals with no pre-compression

In the present study we classify the periodic orbits of a squarely packed, uncompressed and undamped, homogeneous granular crystal, assuming that all elastic granules oscillate with the same frequency (i.e., under condition of 1:1 resonance); this type of Hamiltonian periodic orbits have been labeled as nonlinear normal modes. To this end we formulate an auxiliary system which consists of a two-dimensional, vibro-impact lattice composed of non-uniform “effective particles” oscillating in an anti-phase fashion. The analysis is based on the idea of balancing linear momentum in both horizontal and vertical directions for separate, groups of particles, whereby each such a group is represented by the single effective particle of the auxiliary system. It is important to emphasize that the auxiliary model can be defined for general finite, squarely packed granular crystals composed of n rows and m columns. The auxiliary model is successful in predicting the total number of such periodic orbits, as well as the amplitude ratios for different periodic regimes including strongly localized ones. In fact this methodology enables one to systematically study the generation of mode localization in these strongly nonlinear, highly degenerate dynamical systems. Good correspondence between the results of the theoretical model and direct numerical simulations is observed. The results presented herein can be further extended to study the intrinsic dynamics of the more complex granular materials, such as heterogeneous two-dimensional and three-dimensional granular crystals and multi-layered structures.     

Asynchronous free oscillations of linear mechanical systems: A general appraisal and a digression on a column with a follower force

The concept of asynchronous oscillatory response of mechanical systems in free vibrations is addressed as an introduction to a definition of asynchronous modes of vibration. Connection of the asynchronous modes with the localization and other dynamical phenomena is discussed. Then, conditions for the occurrence of asynchronous modes are investigated in depth. It is seen that a non-conservative system, with an asymmetric stiffness matrix – caused, for instance, by circulatory loads, may have an asynchronous mode. Yet, it is also seen that non-conservativeness is not actually mandatory. Two- to four-degree-of-freedom models of a column with a follower force are addressed, allowing for a mathematical modelling and phenomenological discussion.     

Exact solutions of starting flows for second grade fluid in a porous medium

This paper concentrates on the unsteady flows of a magnetohydrodynamic (MHD) second grade fluid filling a porous medium. The flow modeling involves modified Darcy's law. Three problems are considered. They are (i) starting flow due to an oscillating edge, (ii) starting flow in a duct of rectangular cross-section oscillating parallel to its length, and (iii) starting flow due to an oscillating pressure gradient. Analytical expressions of velocity field and corresponding tangential stresses are developed. These expressions are found to be significantly affected by the applied magnetic field, permeability of the porous medium and the material parameter of the fluid. Moreover, the influence of pertinent parameters on the flows is delineated and appropriate conclusions are drawn. Finally, a comparison is also made with the existing results in the literature.     

Thermal instability in a rotating anisotropic porous layer saturated by a viscoelastic fluid

Linear and non-linear thermal instability in a rotating anisotropic porous medium, saturated with viscoelastic fluid, has been investigated for free-free surfaces. The linear theory is being related to the normal mode method and non-linear analysis is based on minimal representation of the truncated Fourier series analysis containing only two terms. The extended Darcy model, which includes the time derivative and Coriolis terms has been employed in the momentum equation. The criteria for both stationary and oscillatory convection is derived analytically. The rotation inhibits the onset of convection in both stationary and oscillatory modes. A weak non-linear theory based on the truncated representation of Fourier series method is used to find the thermal Nusselt number. The transient behaviour of the Nusselt number is also investigated by solving the finite amplitude equations using a numerical method. The results obtained during the analysis have been presented graphically.     

流向强迫振荡圆柱绕流的涡脱落模态分析

通过求解采用ALE方法描述的运动坐标系Navier-Stokes方程组，分析均匀来流下雷诺 数为150的静止和流向振荡的圆柱绕流. 主要研究了强迫振荡频率和较大振幅比 (A/D=0.3-1.2)对圆柱升力、阻力变化特性以及涡脱落模态的影响. 研究表 明，流向振荡圆柱绕流存在多种涡脱落模态，如对称S以及反对称A-I, A-III, A-IV等多种形式；比较研究结果，拓展了各模态下对应的锁定区域，并将其分为5个 子区；A-I模态中圆柱受力较以前所知更复杂；通过分析计算结果，发现最大加速度 比Af_{c}^{2}/Df_{s0}^{2}可能是涡脱落模态(尤其是对称S模态)最有效的控制参数.     

Instability of sloshing motion in a vessel undergoing pivoted oscillations

Suspending a rectangular vessel partially filled with an inviscid fluid from a single rigid pivoting rod produces an interesting physical model for investigating the dynamic coupling between the fluid and vessel motion. The fluid motion is governed by the Euler equations relative to the moving frame of the vessel, and the vessel motion is given by a modified forced pendulum equation. The fully nonlinear, two-dimensional, equations of motion are derived and linearised for small-amplitude vessel and free-surface motions, and the natural frequencies of the system analysed. It is found that the linear problem exhibits an unstable solution if the rod length is shorter than a critical length which depends on the length of the vessel, the fluid height and the ratio of the fluid and vessel masses. In addition, we identify the existence of 1:1 resonances in the system where the symmetric sloshing modes oscillate with the same frequency as the coupled fluid/vessel motion. The implications of instability and resonance on the nonlinear problem are also briefly discussed.     

Implicit–explicit finite‐difference lattice Boltzmann method with viscid compressible model for gas oscillating patterns in a resonator

Difficulties for the conventional computational fluid dynamics and the standard lattice Boltzmann method (LBM) to study the gas oscillating patterns in a resonator have been discussed. In light of the recent progresses in the LBM world, we are now able to deal with the compressibility and non‐linear shock wave effects in the resonator. A lattice Boltzmann model for viscid compressible flows is introduced firstly. Then, the Boltzmann equation with the Bhatnagar–Gross–Krook approximation is solved by the finite‐difference method with a third‐order implicit–explicit (IMEX) Runge–Kutta scheme for time discretization, and a fifth‐order weighted essentially non‐oscillatory (WENO) scheme for space discretization. Numerical results obtained in this study agree quantitatively with both experimental data available and those using conventional numerical methods. Moreover, with the IMEX finite‐difference LBM (FDLBM), the computational convergence rate can be significantly improved compared with the previous FDLBM and standard LBM. This study can also be applied for simulating some more complex phenomena in a thermoacoustics engine. Copyright © 2008 John Wiley & Sons, Ltd.     

The Stokes boundary layer for a power-law fluid

We develop semi-analytical, self-similar solutions for the oscillatory boundary layer (‘Stokes layer’) in a semi-infinite power-law fluid bounded by an oscillating wall (the so-called Stokes problem). These solutions differ significantly from the classical solution for a Newtonian fluid, both in the non-sinusoidal form of the velocity oscillations and in the manner at which their amplitude decays with distance from the wall. In particular, for shear-thickening fluids the velocity reaches zero at a finite distance from the wall, and for shear-thinning fluids it decays algebraically with distance, in contrast to the exponential decay for a Newtonian fluid. We demonstrate numerically that these semi-analytical, self-similar solutions provide a good approximation to the flow driven by a sinusoidally oscillating wall.     

Oscillatory flow due to eccentrically rotating porous disk and a fluid at infinity embedded in a porous medium

An oscillatory flow due to non-coaxial rotations of an oscillating porous disk and a fluid at infinity rotating about an axis parallel to the axis of rotation of the disk through a fixed point has been investigated. An analytical solution of the unsteady Navier-Stokes equations is obtained for three cases when the frequency parameter is less than, equal to or greater than the rotation parameter. The influences of the physical parameters acting on the flow are explained with the help of the figures. It is found that the depth of the penetration or the wave length of the layers decreases with an increase in porosity parameter.     

On unsteady hydromagnetic flows of a dusty fluid between two oscillating plates

An initial value investigation is made of the motion of an incompressible viscous conducting fluid with embedded small spherical particles bounded by two infinite rigid non-conducting plates. The flow is generated in the fluid-particle system due to rectilinear oscillations of given frequencies superimposed on the plates in presence of an external transverse magnetic field. The operational method is used to derive exact solutions for the fluid and the particle velocities and the wall shear stress. It is shown that the effect of the dust particles on the fluid velocity depends on the time periods of the oscillating plates. When the time-periods are small, i.e., when the plates oscillate with high frequency, the fluid motion is found to be retarded by the particles. However, when the plates oscillate with larger time periods (smaller frequencies), the fluid velocity is increased by the presence of the particles at the early stage of the motion, and this effect persists until the equilibrium is reached when the particles exert their influence to resist the flow.     

Mechanics of mixed-mode ductile material removal with a conical tool and the size dependence of the specific energy

The mechanics of material removal during a single-grit rotating scratch has been investigated both analytically and experimentally. The models for cutting, plowing and mixed modes of material removal are analyzed based on the pressure and the frictional resistance. The mixed-mode model takes into account the contribution of built-up edge (BUE) ahead of the tool. To validate the model, single-grit rotating scratch experiments were conducted with a conical diamond tool on pure titanium. It was noticed that the adhesion between the tool and the deformed material, and the hardening properties of material play active roles in the scratching process and provide a driving force to the formation of the BUE. The overall frictional coefficient was found to oscillate strongly on both ends of the scratch but increases steadily over the central span of the scratch length. It is shown that the mixed-mode model captures the salient features of material removal and the size dependence of specific energy during the formation of a rotating scratch. The size dependence of specific energy may be attributed to the size effect of the yield pressure in titanium.