Wall slip induced by a micropolar fluid

Lubrication theory is devoted to the study of thin-film flows, More often, the fluid can be considered as a Newtonian one and no-slip boundary conditions can be retained for the velocity at the fluid solid interface. With these assumptions it is possible to deduce from the (Navier) Stokes system a simplified equation describing the flow: the Reynolds equation. It allows to compute the pressure distribution inside the film and to obtain overall performances of a lubricated device such as load and friction coefficient. For very thin films, however, surface effects at the fluid solid interface become very important and no-slip conditions cannot be retained. Solid surfaces exert some influence on the liquid molecules and the effective shear viscosity along the boundary differs from the classical bulk shear viscosity. Moreover, the microstructure of the fluid cannot be ignored, especially the effects of solid-particle additives in the lubricant. Micropolar theory for fluids is often adopted to account of such microstructure and microrotation. In the present study, a thin micropolar fluid model with new boundary conditions at the fluid–solid interface is considered. This condition links velocity and microrotation at the interface by introducing a so-called “boundary viscosity”. By way of asymptotic analysis, a generalized micropolar Reynolds equation is obtained. Numerical results show the influence of the new boundary conditions for the load and friction coefficients. Comparisons are made with other works that retain the no-slip boundary conditions.     

Singularities of viscous flow

Summary The force and torque on a slender body in Stokes flow is obtained by using a distribution of singularities along the centreline and the required image system to satisfy the no-slip condition and the plane boundary. It is found that the force on a slender body increases rapidly as it approaches within a body length of the boundary, but not the torque. To rule out “wall effects”, experiments on slender bodies, such as flagellated or ciliated micro-organisms, should be carried out a distance of many body lengths (not radii!) from all boundaries. The ratio of normal to tangential resistance coefficients is found to be greater than 2 (the maximum in an infinite fluid) in the presence of walls.     

Asymptotic analysis of the viscous micro/nano pump at low Reynolds number

The steady viscous parabolic flow past an eccentrically placed rotating cylinder is studied in the asymptotic limit of small Reynolds number. It is assumed that the flow around the rotating cylinder undergoes boundary slip described by the Navier boundary condition. This involves a single parameter to account for the slip, referred to as the slip length ?, and replaces the standard no-slip boundary condition at solid boundaries. The streamlines for ? > 0 are closer to the body than for ? = 0, and it is discovered that the loss of symmetry due to the rotation of the cylinder is significantly reduced by the inclusion of slip. This arises as a result of a balance between the rotation velocity and the slip velocity on that portion of the cylinder which rotates opposite to the free-stream flow. Streamline patterns for nonzero eccentricity partially agree with Navier–Stokes simulations of the viscous pump; the small discrepancy is primarily due to the fact that here wall effects are not explicitly considered. Expressions for the frictional drag and the torque on the cylinder are obtained. The expression for the torque agrees well with the lubrication solution for the flow past a rotating cylinder placed symmetrically in a fully developed channel flow. The results presented here may be used to validate numerical schemes developed to study the viscous pump.     

Finite element analysis of the transient motion of stratified viscous fluids under gravity

The present paper deals with the finite element analysis of two-dimensional two-layer density flows in a gravitational field. A fluid in each layer is replaced with a large number of discrete particles, and the motion and deformation of each layer is represented by moving those particles in a Lagrangian manner. The velocity distribution in the whole fluid region is given as the finite element solution of the Navier-Stokes equations and the equation of continuity. In the finite element calculation, free-slip conditions are used on solid wall boundaries because no-slip conditions may cause sticking of some particles to walls. Then, a new technique for the implementation of free-slip conditions on arbitrary curved boundaries is presented. As numerical examples, density flows in a rectangular closed container and Rayleigh-Taylor instability in the container with a circular cross-section have been computed.     

A variational principle for magnetohydrodynamics with high Hartmann number flow

By the semi-inverse method proposed by He, a variational principle is established for three-dimensional MHD equations with high Hartmann number. In order to incorporate the no-slip condition and far distance boundary condition as natural boundary conditions, a special technique is proposed in this paper. Lagrange crisis are also illustrated.     

模拟复杂流动的一种隐式直接力浸入边界方法

为避免复杂贴体网格的生成,该文采用一种隐式直接力浸入边界法模拟复杂边界流动问题。借助求解不可压缩N-S方程组的分步投影方法的思想,来求解基于浸入边界法的耦合系统方程。其中固体边界离散点的作用力密度通过强制满足固体边界的无滑移条件导出,进而通过δ光滑函数对固体壁面附近速度场进行二次修正。在空间离散上,对流项采用QUICK迎风格式,扩散项采用中心差分格式,采用二阶显式Adams-Bashforth法离散时间项。以雷诺数为25、40和300的圆柱绕流为基准数值算例,通过与实验结果和其他文献数值结果的对比,验证数值计算方法的可靠性。     

Aiding mixed convection through a vertical anisotropic porous channel with oblique principal axes

In this paper, an analytical study is carried out on the mixed convection in parallel-plate vertical porous channels with an anisotropic permeability whose principal axes are oriented in a direction which is oblique to the gravity vector. The channel walls are assumed to be isothermal and the flow fully developed at the entrance is upward, so that natural convection aids the forced flow. In the formulation of the problem, use is made of the generalized Brinkman-extended Darcy model which allows the no-slip boundary condition on solid wall, to be satisfied. The flow reversal and the limiting cases of low and high porosity media for natural and forced convection are considered. The effects of varying the anisotropic permeability ratio and the orientation angle of the principal axes on the flow and the heat transfer are investigated.     

Numerical Solutions for Heat Transfer of An Unsteady Cavity with Viscous Heating

The mechanism of viscous heating of a Newtonian fluid filled inside a cavity under the effect of an external applied force on the top lid is evaluated numerically in this exploration. The investigation is carried out by assuming a two-dimensional laminar in-compressible fluid flow subject to Neumann boundary conditions throughout the numerical iterations in a transient analysis. All the walls of the square cavity are perfectly insulated and the top moving lid produces a constant finite heat flux even though the fluid flow attains the steady-state condition. The objective is to examine the effects of viscous heating in the fully insulated lid-driven cavity under no-slip and free-slip Neumann boundary conditions coupled with variations in Reynolds and Prandtl numbers. The partial differential equations of time-dependent vorticity-stream function and thermal energy are discretized and solved using a self-developed finite difference code in MATLAB^® environment. Time dependence of fluid thermodynamics is envisaged through contour and image plots. A commercial simulation software, Ansys Fluent^® utilizing a finite element code is employed to verify the finite difference results produced. Although the effect of viscous heating is very minimal, Neumann no-slip and free-slip boundary conditions are able to trap the heat inside the fully insulated cavity as the heat flux is constantly supplied at the top lid. A lower Reynolds number and a greater Prandtl number with free-slip effects reduce temperature distribution in the cavity with a faster velocity than in the no-slip condition as the free-slip behaves as a lubricant.     

Involving the Navier-Stokes equations in the derivation of boundary conditions for the lattice Boltzmann method

By means of the continuity equation of the incompressible Navier-Stokes equations, additional physical arguments for the derivation of a formulation of the no-slip boundary condition for the lattice Boltzmann method for straight walls at rest are obtained. This leads to a boundary condition that is second-order accurate with respect to the grid spacing and conserves mass. In addition, the boundary condition is stable for relaxation frequencies close to two.     

Non‐reflecting artificial boundaries for transient scalar wave propagation in a two‐dimensional infinite homogeneous layer

This paper presents an exact non‐reflecting boundary condition for dealing with transient scalar wave propagation problems in a two‐dimensional infinite homogeneous layer. In order to model the complicated geometry and material properties in the near field, two vertical artificial boundaries are considered in the infinite layer so as to truncate the infinite domain into a finite domain. This treatment requires the appropriate boundary conditions, which are often referred to as the artificial boundary conditions, to be applied on the truncated boundaries. Since the infinite extension direction is different for these two truncated vertical boundaries, namely one extends toward x →∞ and another extends toward x→‐ ∞, the non‐reflecting boundary condition needs to be derived on these two boundaries. Applying the variable separation method to the wave equation results in a reduction in spatial variables by one. The reduced wave equation, which is a time‐dependent partial differential equation with only one spatial variable, can be further changed into a linear first‐order ordinary differential equation by using both the operator splitting method and the modal radiation function concept simultaneously. As a result, the non‐reflecting artificial boundary condition can be obtained by solving the ordinary differential equation whose stability is ensured. Some numerical examples have demonstrated that the non‐reflecting boundary condition is of high accuracy in dealing with scalar wave propagation problems in infinite and semi‐infinite media. Copyright © 2003 John Wiley & Sons, Ltd.     

Slip flow in converging and diverging channels

The classical problem of Jeffery-Hamel flow is considered in which the fluid is allowed to slip along the walls of the channel. The problem is solved analytically and the volumetric flow rate is computed and compared with that of the corresponding no-slip flow. In the converging channel case, it is found that the slip boundary condition enhances flow rates through the channel, although the effect is minimal when the Reynolds number is large.In the case of the diverging channel, the slip boundary condition in some instances actually lowers the flow rate from its no-slip value. In other instances, a stable velocity profile does not even appear to exist. These cases aside, when the mean pressure in the channel is adverse, slip flow solutions exist and increase the flow rate through the channel by at most 15.7%.     

下卧刚性基岩条件下场地土-结构体系地震反应分析方法研究

下卧刚性基岩条件下的土-结构体系地震反应分析模型是一个能量“半开放-半封闭”系统。它与完全的能量“开放”系统土-结构体系地震反应分析模型的不同之处是其采用了域内惯性力的地震动输入方式,这与刚性基岩-结构体系地震反应分析模型的地震动输入方式是相同的。不同于刚性基岩上的结构地震反应分析模型,下卧刚性基岩条件下的土-结构体系地震反应分析模型需要考虑半无限土层对截断边界处的影响。就笔者-所见,目前有许多研究者和工程技术人员对这一问题有错误的理解,认为截断边界的影响只是一个人工边界处理问题,分析中仅采用人工边界条件模拟了截断边界对辐射能量的影响,而忽略了一个更重要的影响因素--下卧刚性基岩条件下场地土的自由场效应。为澄清这一问题,基于土-结构体系地震反应的直接分析模型,严格推导了下卧刚性基岩条件下的土-结构体系地震反应分析方法,指出在通常的工程经验尺度内遗漏自由场运动效应将得不到正确的计算结果。     

Phase Field Modelling Allotropic Transformation of Solid Solution

Based on multiphase field conception and integrated with the idea of vectorvalued phase field, a phase field model for typical allotropic transformation of solid solution is proposed. The model takes the non-uniform distribution of grain boundaries of parent phase and crystal orientation into account in proper way, as being illustrated by the simulation of austenite to ferrite transformation in low carbon steel. It is found that the misorientation dependent grain boundary mobility shows strong influence on the formation of ferrite morphology comparing with the weak effect exerted by misorientation dependent grain boundary energy. The evolution of various types of grain boundaries are quantitatively characterized in terms of its respective grain boundary energy dissipation. The simulated ferrite fraction agrees well with the expectation from phase diagram, which verifies this model.     

Effects of interface slip and viscoelasticity on the dynamic response of droplet quartz crystal microbalances

In the present paper we first present a derivation based on the time-dependent perturbation theory to develop the dynamical equations which can be applied to model the response of a droplet quartz crystal microbalance (QCM) in contact with a single viscoelastic media. Moreover, the no-slip boundary condition across the device-viscoelastic media interface has been relaxed in the present model by using the Ellis-Hayward slip length approach. The model is then used to illustrate the characteristic changes in the frequency and attenuation of the QCM with and without the boundary slippage due to the changes in viscoelasticity as the coated media varies from Newtonian liquid to solid. To complement the theory, experiments have been conducted with microliter droplets of aqueous glycerol solutions and silicone oils with a viscosity in the range of 50 approximately 10,000 cS. The results have confirmed the Newtonian characteristics of the glycerol solutions. In contrast, the acoustic properties of the silicones oils as reflected in the impedance analysis are different from the glycerol solutions. More importantly, it was found that for the silicone oils the frequency steadily increased for several hours and even exceeded the initial value of the unloaded crystal as reflected in the positive frequency shift. Collaborative effects of interfacial slippage and viscoelasticity have been introduced to qualitatively interpret the measured frequency up-shifts for the silicone oils. The present work shows the potential importance of the combined effects of viscoelasticity and interfacial slippage when using the droplet QCM to investigate the rheological behavior of more complex fluids.     

Continuously deforming finite elements for the solution of parabolic problems,with and without phase change

A number of transport problems are complicated by the presence of physically important transition zones where quantities exhibit steep gradients and special numerical care is required. When the location of such a transition zone changes as the solution evolves through time, use of a deforming numerical mesh is appropriate in order to preserve the proper numerical features both within the transition zone and at its boundaries. A general finite element solution method is described wherein the elements are allowed to deform continuously, and the effects of this deformation are accounted for exactly. The method is based on the Galerkin approximation in space, and uses finite difference approximations for the time derivatives. In the absence of element deformation, the method reduces to the conventional Galerkin formulation. The method is applied to the two-phase Stefan problem associated with the melting and solidification of A substance. The interface between the solid and liquid phase form an internal moving boundary, and latent heat effects are accounted for in the associated boundary condition. By allowing continuous mesh deformation, as dictated by this boundary condition, the moving boundary always lies on element boundaries. This circumvents the difficulties inherent in interpolation of parameters and dependent variables across regions where those quantities change abruptly. Basis functions based on Hermite polynomials are used, to allow exact specification of the flux-latent heat balance condition at the phase boundary. Analytic solutions for special cases provide tests of the method.     

Purely mechanical constitutive theory for fluid-filled porous materials with broken outer boundaries

Summary The purely mechanical constitutive theory for a fluid-filled porous material is reexamined from a point of view that the outer boundary of an actual representative volume element consists of solid and fluid. Volume average kinematical and kinetical quantities for porous materials with broken outer boundaries are introduced. If a pointwise stress distribution on the inner boundary of the porous solid material is replaced by a constant hydrostatic fluid pressure, the form of constitutive equations for the broken outer boundary case is shown to be the same as that for the closed outer boundary case.     

Solution of three-dimensional viscous flows using integral velocity–vorticity formulation

In this paper, a numerical approach is presented to solve the velocity–vorticity integro-differential formulations for three-dimensional incompressible viscous flow. Both the velocity and pressure are solved in integral formulations and the general numerical method is based on standard finite volume scheme. The velocities needed at the vertexes of each control volume are calculated by a so-called generalized Biot–Savart formula combined with a fast multipole algorithm, which makes the velocity boundary conditions implicitly satisfied by maintaining the kinematic compatibility of the velocity and vorticity fields. The well-known fractional step approaches are used to solve the vorticity transport equation. No-flux boundary conditions on solid objects are satisfied as vorticity Helmholtz equation is solved. The diffusion term in the transport equation is treated implicitly using a conservative finite update. The diffusive fluxes of vorticity into flow domain from solid boundaries are determined by an iterative process in order to satisfy the no tangential-flow boundary condition. As an application example, the impulsively started flow through a sphere with different Reynolds numbers is computed using the method. The calculated results are compared with the experimental data and other numerical results and show good agreement.     

Warum hat die aufsteigende Blase einen Widerstand?

An understanding of the drag of a body relies heavily on the no-slip wall condition of boundary layer theory. When this condition does not apply, as for a rising bubble with mobile surface, rethinking is required going beyond assignment of an appropriate drag coefficient. The ultimate cause of the drag and the role of the scaling parameters is to be resolved. The Reynolds number can be interpreted as a flow variable while the Archimedes number is the key parameter. It is shown that the existence of the mobile surface does not lead automatically to potential flow as one is tempted to guess. Besides a few experimental data the present work strives for an appropriate interpretation of the drag’s origin and a consistent view on bubbles with mobile or immobile surface.     

Lubrication Models with Small to Large Slip Lengths

A set of lubrication models for the thin film flow of incompressible fluids on solid substrates is derived and studied. The models are obtained as asymptotic limits of the Navier-Stokes equations with the Navier-slip boundary condition for different orders of magnitude for the slip-length parameter. Specifically, the influence of slip on the dewetting behavior of fluids on hydrophobic substrates is investigated here. Matched asymptotics are used to describe the dynamic profiles for dewetting films and comparison is given with computational simulations. The motion of the dewetting front shows transitions from being nearly linear in time for no-slip to t ^2/3 as the slip is increased. For much larger slip lengths the front motion appears to become linear again. Correspondingly, the dewetting profiles undergo a transition from oscillatory to monotone decay into the uniform film layer for large slip. Increasing the slip further, to very large values, is associated with an increasing degree of asymmetry in the structure of the dewetting ridge profile     

Texture competition during thin film deposition – effects of grain boundary migration

In this paper, we describe an implementation of grain boundary migration in the atomistic simulator of thin film deposition (ADEPT), and apply the simulator to study effects of the grain boundary migration on texture evolution. In the implementation, atoms are classified into two categories: those belong to a single grain and those at grain boundaries. An atom is defined as one at a grain boundary if it has more than half of its neighbors occupied and not all of the neighboring atoms are in the same grain. The grain boundary atom is attempted to re-align with neighboring grains to represent the grain boundary migration; the attempt probability is defined by the grain boundary migration coefficient. Our studies show that grain boundary migration does not always assist formation of texture with a top surface of the lowest energy. At the nucleation stage of thin film deposition, high migration coefficient of grain boundaries may enhance the formation of grain nuclei with top surfaces of higher energy, and therefore effectively may suppress formation of textures with a top surface of the lowest energy. This effect may provide an extra dimension to engineer textures of thin films.