A model for fluid flow in non-fully filled tribological interfaces – Part 1: Basics and Numerics

In literature, most contributions on starved lubrication focus on the occurring pressures in macroscopic devices. Hereby, usually the Reynolds equation is modified in different ways. In contrast to this proceeding, this paper's intention is the general investigation of this tribological regime to get a fundamental comprehension on the transition from boundary lubrication to mixed lubrication. The respective model describes the flow of the fluid through two rough surfaces moving relative to each other. The lack of fluid is regarded by the fact that elements may not be fully filled with the fluid. Only elements where the fluid fully fills the gap, generate a pressure. This effect is considered by a type of unilateral constraint in combination with a penalty function. The fluid flow is computed according to the Navier-Stokes equation. In combination with the continuity equation, a set of implicit nonlinear equations has to be solved. Its potential and basic application fields are finally discussed. A further paper will show applications of the algorithm towards different scenarios. (© 2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Influence of thermal dispersion on forced convection in a composite parallel-plate channel

This paper concentrates on the analytical study of the effect of thermal dispersion on fully developed forced convection in a parallel-plate channel partly filled with a fluid saturated porous medium. The walls of the channel are subject to a constant heat flux. The central part of the channel is occupied by a homogeneous fluid, while peripheral parts of the channel are occupied by a fluid saturated porous medium of uniform porosity. It is assumed that the momentum flow in the porous region is described by the Brinkman-Forchheimer-extended Darcy equation. Since thermal dispersion becomes appreciable in high speed flows, that is, for the same situation when accounting for the Forchheimer term in the momentum equation is essential, the effect of thermal dispersion should be taken into account simultaneously with accounting for the Forchheimer term in the momentum equation. The objective of the present research is to determine in which situations accounting for thermal dispersion can significantly influence the solution.     

Investigation of the Free-Surface Flow in Partially Filled Extruders

In this work we present the free-surface flow of high viscous Newtonian liquids in a simplified extruder model, namely the flat-plate model. Due to high density and viscosity ratios, a free-surface approach based on the volume of fluid method is used. The material distribution for different degree of fillings and the resulting power consumption is investigated. In addition, the obtained power characteristic is compared against analytical consideration and numerical simulation of a fully filled case. The investigation shows that the power characteristic of a partially filled extruder can be estimated by a fully filled computation. (© 2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

On the dynamics of a fluid-particle interaction model: The bubbling regime

This article deals with the issues of global-in-time existence and asymptotic analysis of a fluid-particle interaction model in the so-called bubbling regime. The mixture occupies the physical space Ω⊂R³ which may be unbounded. The system under investigation describes the evolution of particles dispersed in a viscous compressible fluid and is expressed through the conservation of fluid mass, the balance of momentum and the balance of particle density often referred as the Smoluchowski equation. The coupling between the dispersed and dense phases is obtained through the drag forces that the fluid and the particles exert mutually by the action-reaction principle. We show that solutions exist globally in time under reasonable physical assumptions on the initial data, the physical domain, and the external potential. Furthermore, we prove the large-time stabilization of the system towards a unique stationary state fully determined by the masses of the initial density of particles and fluid and the external potential.     

A porous media approach for plantar tissue during gait

Recently a new computational model, based on the thermodynamically constrained averaging theory, has been proposed to predict tumor initiation and proliferation and afterwards to study plantar tissue mechanics. The foot tissue is modeled as an elastic porous medium, in large strain regime and completely filled by a fluid phase. By considering the interstitial fluid, it is possible to mimic the viscoelastic behavior of the plantar tissue observed experimentally. Being the global response of the bi-phase system viscoelastic, it is shown that the duration of stance as well as of each of gait cycle has an influence on tissue stress field. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

混合微通道中不稳定的广义Couette流

在一个平行板通道中,部分充满了均匀的多孔介质,部分为纯流体的流动区,对其微通道中完全发展的不稳定层流进行了数值分析,流动由其中一块板的运动和压力梯度所引起.多孔介质区域的流动,采用扩展的Brinkman模型,即Darcy模型,纯净流动区域的流动,采用Stokes方程.还对稳定的完全发展流进行了理论分析,给出了分界面速度、边界板处的速度和表面摩擦的闭式解.通过数值计算发现,稳定完全发展流的闭式解,和不稳定流动的数值解,在所有时间点上得到很好地吻合.     

On the oscillations of a thin-walled structure containing a fluid, in the presence of a hydrodynamic damper : PMM vol. 43, no. 6, 1979, pp. 1095–1101

A model of a hydrodynamic oscillation damper is proposed. The model is used to obtain the equations describing longitudinal oscillations of a structure which includes a shell partially filled with fluid, and contains a hydrodynamic damper. It is shown that the use of the damper leads to considerable increase in the damping of the oscillations of specified frequencies within the structure.

In modern technology one encounters various types of problems connected with restricting the amplitudes of the axisymmetric vibrations of shells and of the longitudinal oscillations of structures consisting of shells partially filled with fluid. Various devices have been proposed [1] for solving these problems. All these devices have a common feature, namely an elastic shell filled with gas and placed in the fluid. The natural frequency of oscillations of such a shell in a fluid can be tuned to required frequency. The effect of such a device is analogous to the effect of a dynamic vibration damper in mechanical systems [2]. A part of the fluid contained in the shell serves as the active mass of the dynamic damper, and for this reason we shall call such devices the hydrodynamic vibration dampers.     

Synchronization effects in rotors partly filled with fluid

In fluid-filled rotors self-excited vibrations occur induced by a surface wave of the fluid. A characteristic property is the instability over the full range of angular velocity above the Eigenfrequency of the system. A possible explanation is the occurrence of synchronization effects between fluid and rotor. The behaviour of rotors partly filled with fluid was mostly studied under the aspect of stability in steady-state conditions. For non-steady-state investigations, discrete models with reduced number of degrees of freedom and reasonable ability to model the system behaviour are desirable due to the complexity of fluid modelling. This paper analyses a simple minimal model and shows synchronization effects between fluid and rotor model. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Pseudo-cyclic policies for multi-queue single server systems

In this paper we evaluate the waiting time performance of cycle-time guided algorithms in semi-dynamic polling models. In these models knowledge of the system state is used by the server in the process of on-line determination of the visit pattern. Our main focus is on pseudo-cyclic algorithms which achievefairness by visiting every station exactly once in each cycle.It is shown that in fully symmetric systems the performance ofevery pseudocyclic policy is bounded between the performance of the cycle maximization and the cycle minimization strategies. In particular, stochastic dominance is shown with regard to system workload, implying dominance of mean waiting times. In a fluid approximation model the performance ratio between the two bounding policies is derived and shown to be always between 1 and 3/4 under the gated service regime and between 1 and 1/2 under the exhaustive service regime. Under both regimes, the ratio approaches 3/4 when the number of stations grows to infinity. Simulation results suggest that a similar performance ratio holds for the general (non-symmetric) stochastic model.Further, we study strategies which are guided by the cycle maximization (minimization) criteria, but which do not constrain themselves to pseudo-cyclic orders. It is shown that depending on the switch-over parameters these more dynamic policies may perform much worse than the pseudo-cyclic schemes.     

Fluid-structure interaction of an elastic pipe immersed in a coaxial rigid pipe: a model for the Tullio Phenomenon

An axisymmetric, elastic pipe is filled with an incompressible fluid and is immersed in a second, coaxial rigid pipe which contains the same fluid. A pressure pulse in the outer fluid annulus deforms the elastic pipe which invokes a fluid motion in the fluid core. It is the aim of this study to investigate streaming phenomena in the core which may originate from such a fluid-structure interaction. This work presents a numerical solver for such a configuration. It was developed in the OpenFOAM software environment and is based on the Arbitrary Lagrangian Eulerian (ALE) approach for moving meshes. The solver features a monolithic integration of the one-dimensional, coupled system between the elastic structure and the outer fluid annulus into a dynamic boundary condition for the moving surface of the fluid core. Results indicate that our configuration may serve as a mechanical model of the Tullio Phenomenon (sound-induced vertigo). (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Synchronization effects in rotors partially filled with fluid-influence on propulsion

In fluid-filled Rotors occur self-excited vibrations induced by surface waves of the fluid. A characteristic property is the instability over an interval of angular velocity above the natural frequency of the system. One explanation is the occurrence of synchronization effects between fluid waves and the critical rotor speed. The behaviour of rotors partly filled with fluid was mostly studied under the aspect of stability in steady-state conditions. For non-steady-state investigations, discrete models with reduced number of degrees of freedom and reasonable ability to model the system behaviour are desirable for observer-based real-time control. This paper analyses a model based on a laval rotor and shows synchronization effects between fluid waves and rotor model and its influence on the rotor propulsion. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Vibrations of a cylinder in a concentric vessel filled with a two-layer fluid

A hybrid vibrational system containing a solid (a cylinder) with an elastic connection to a coaxial cylindrical cavity, completely filled with a heavy ideal stably stratified two-layer fluid, is considered. The combined self-consistent vibrations of the body and the fluid (of the internal waves) are studied. An explicit solution of the internal boundary value problem of an inhomogeneous liquid in an annular domain for a specified motion of the body is obtained. An integrodifferential equation of the Newton type is constructed on the basis of this. This equation describes the self-consistent oscillations of the cylinder. In the case of weak coupling of the interaction between the solid and the medium, an approximate solution is obtained using asymptotic methods and an analysis is carried out. Qualitative effects of the mutual effect of the motions of the cylinder and the fluid are found.     

Stability of a Generalized Sobolev System

The linear stability problem of the rotational motion of a rigid body around a fixed point containing an inner cavity filled up with an ideal fluid is considered. In this paper, we also assume that the fluid is rotating. The effect of the angular velocities of the rigid body and the fluid in the stability problem is studied. The case of a cavity ellipsoidal is presented in detail.     

多孔介质平板通道传热模型的两种求解方法

基于Brinkman Darcy扩展模型和非局部热平衡模型,考虑液相和固相含有内热源的情况,建立了多孔介质平板通道传热的一般模型.分别采用直接法和间接法将液相与固相能量方程解耦,进而求得充分发展传热条件下的多孔介质温度场.与直接解耦法相比,间接解耦法可在原始边界条件下求解二阶微分方程,更加简单易行.通过对无量纲温度表达式系数以及温度分布的比较,验证了两种求解方法的等价性.在两种极限情形下,间接法所得温度分布解析解与现有文献结果相当吻合,这也在一定程度上证明了所建模型更具一般性.参数分析表明,液固两相温差随着Biot数或有效导热系数比的增大而减小,Nusselt数随着内热源比的增大而减小.     

Vlasov limit and discreteness effects in cosmological N-body simulations

We present the problematic of controlling the discreteness effects in cosmological N-body simulations. We describe a perturbative treatment which gives an approximation describing the evolution under self-gravity of a lattice perturbed from its equilibrium, which allows to trace the evolution of the fully discrete distribution until the time when particles approach one another (“shell-crossing”). Perturbed lattices are typical initial conditions for cosmological N-body simulations and thus we can describe precisely the early time evolution of these simulations. A quantitative comparison with fluid Lagrangian theory permits to study discreteness effects in the linear regime of the simulations. We show finally some work in progress about quantifying discreteness effects in the non-perturbative (highly non-linear) regime of cosmological N-body simulations by evolving different discretizations of the same continuous density field.     

Lubrication of porous solids in reference to human joints

A simple mechanical model which has some features in common with load bearing human joints is described. The normal approach of two plane surfaces, one of which is covered with porous material is analysed. The gap between the two surfaces is filled with micropolar fluid to represent a particulate suspension (i.e, synovial fluid) as lubricant. The poresize diameter is so small that only the suspending medium,i.e, the viscous fluid enters into the porous matrix due to the filtration action. The problem has been solved separately in two regions; flow of viscous fluid in the porous matrix and the squeeze film lubrication with micropolar fluid as lubricant in between the two approaching surfaces along with suitable matching conditions at the porous boundary. Several interesting results have been brought out. Agreement with available experimental results and the computational results presented, herein, is quite good.     

On the Generation of Mean Flows by the Interaction of Görtler Vortices and Tollmien-Schlichting Waves in Curved Channel Flows

There are many fluid flows where the onset of transition can be caused by different instability mechanisms which compete in the nonlinear regime. Here the interaction of a centrifugal instability mechanism with the viscous mechanism which causes Tollmien-Schlichting waves is discussed. The interaction between these modes can be strong enough to drive the mean state; here the interaction is investigated in the context of curved channel flows so as to avoid difficulties associated with boundary layer growth. Essentially it is found that the mean state adjusts itself so that any modes present are neutrally stable even at finite amplitude. In the first instance the mean state driven by a vortex of short wavelength in the absence of a Tollmien-Schlichting wave is considered. It is shown that for a given channel curvature and vortex wavelength there is an upper limit to the mass flow rate which the channel can support as the pressure gradient is increased. When Tollmien-Schlichting waves are present then the nonlinear differential equation to determine the mean state is modified. At sufficiently high Tollmien-Schlichting amplitudes it is found that the vortex flows are destroyed, but there is a range of amplitudes where a fully nonlinear mixed vortex-wave state exists and indeed drives a mean state having little similarity with the flow which occurs without the instability modes. The vortex and Tollmien-Schlichting wave structure in the nonlinear regime has viscous wall layers and internal shear layers; the thickness of the internal layers is found to be a function of the Tollmien-Schlichting wave amplitude.     

A Quasi‐Planar Model for Gravity‐Capillary Interfacial Waves in Deep Water

A dynamical model equation for interfacial gravity‐capillary (GC) waves between two semi‐infinite fluid layers, with a lighter fluid lying above a heavier one, is derived. The model proposed is based on the fourth‐order truncation of the kinetic energy in the Hamiltonian of the full problem, and on weak transverse variations, in the spirit of the Kadomtsev‐Petviashvilli equation. It is well known that for the interfacial GC waves in deep water, there is a critical density ratio where the associated cubic nonlinear Schrödinger equations changes type. Our numerical results reveal that, when the density ratio is below the critical value, the bifurcation diagram of plane solitary waves behaves in a way similar to that of the free‐surface GC waves on deep water. However, the bifurcation mechanism in the vicinity of the minimum of the phase speed is essentially similar to that of free‐surface gravity‐flexural waves on deep water, when the density ratio is in the supercritical regime. Different types of lump solitary waves, which are fully localized in both transverse and longitudinal directions, are also computed using our model equation. Some dynamical experiments are carried out via a marching‐in‐time algorithm.     

A new spectral-homotopy analysis method for solving a nonlinear second order BVP

A modification of the homotopy analysis method (HAM) for solving nonlinear second-order boundary value problems (BVPs) is proposed. The implementation of the new approach is demonstrated by solving the Darcy–Brinkman–Forchheimer equation for steady fully developed fluid flow in a horizontal channel filled with a porous medium. The model equation is solved concurrently using the standard HAM approach and numerically using a shooting method based on the fourth order Runge–Kutta scheme. The results demonstrate that the new spectral homotopy analysis method is more efficient and converges faster than the standard homotopy analysis method.     

Convergence of a non-stiff boundary integral method for interfacial flows with surface tension

Boundary integral methods to simulate interfacial flows are very sensitive to numerical instabilities. In addition, surface tension introduces nonlinear terms with high order spatial derivatives into the interface dynamics. This makes the spatial discretization even more difficult and, at the same time, imposes a severe time step constraint for stable explicit time integration methods.

A proof of the convergence of a reformulated boundary integral method for two-density fluid interfaces with surface tension is presented. The method is based on a scheme introduced by Hou, Lowengrub and Shelley [ J. Comp. Phys. 114 (1994), pp. 312-338] to remove the high order stability constraint or stiffness. Some numerical filtering is applied carefully at certain places in the discretization to guarantee stability. The key of the proof is to identify the most singular terms of the method and to show, through energy estimates, that these terms balance one another.

The analysis is at a time continuous-space discrete level but a fully discrete case for a simple Hele-Shaw interface is also studied. The time discrete analysis shows that the high order stiffness is removed and also provides an estimate of how the CFL constraint depends on the curvature and regularity of the solution.

The robustness of the method is illustrated with several numerical examples. A numerical simulation of an unstably stratified two-density interfacial flow shows the roll-up of the interface; the computations proceed up to a time where the interface is about to pinch off and trapped bubbles of fluid are formed. The method remains stable even in the full nonlinear regime of motion. Another application of the method shows the process of drop formation in a falling single fluid.