Non-axisymmetric instability of polar orthotropic annular plates

The non-axisymmetric instability of polar orthotropic annular plates under inplane uniform radial pressure is studied by use of the shooting method. The characteristic equations and eigenvalues under a variety of edge conditions are given. Under two appropriate hypotheses, we prove that all eigenvalues are bifurcation points. Hence, it is possible that non-axisymmetric buckled and post-buckled states branch from axisymmetric unbuckled states of an annular plate. Asymptotic formulae for buckled states are obtained and curves for the deflection and stress are shown.     

Transfer matrix solutions to axisymmetric and non-axisymmetric consolidation of multilayered soils

Transfer matrix solutions are presented in this paper to study the axisymmetric and non-axisymmetric consolidation of a multilayered soil system under an arbitrary loading. Starting with the governing equations for consolidation problems of saturated soils, the relationship of displacements, stresses, excess pore water pressure, and flux between the points at the depth z, and on the ground surface (z = 0) is established in a transformed domain by introducing the displacement functions and using the integral transform technique. Then the transfer matrix method is used with the boundary conditions to obtain the analytical solutions in the transformed domain for the multilayered soil system. Numerical inversion of the integral transform of these analytical solutions results in the solutions for the actual problems. The numerical results for axisymmetric and non-axisymmetric Biot’s consolidation problems of a single layer and a multi-layered soil system are obtained and compared with existing results by others.     

Air flow in cavities of labyrinth seals

The leakage flow rate through a sequence of labyrinth seal cavities, and the associated pressure and the circumferential velocity distributions are calculated for seals used in turbomachinery. Computational Fluid Dynamics is used to justify the use of bulk cavity variables, and to analyze the details of the flow in a single cavity under steady state, axisymmetric conditions. Periodic, analytic solutions of the continuity and circumferential momentum equations are obtained for the case of time dependent flow generated by a non-axisymmetric rotation of the shaft. The dynamic stiffness and damping coefficients necessary for the lateral stability analysis of the rotor are then calculated. The results compare reasonably well to experimentally obtained values.     

Circular edge singularities for the Laplace equation and the elasticity system in 3-D domains

Asymptotics of solutions to the Laplace equation with Neumann or Dirichlet conditions in the vicinity of a circular singular edge in a three-dimensional domain are derived and provided in an explicit form. These asymptotic solutions are represented by a family of eigen-functions with their shadows, and the associated edge flux intensity functions (EFIFs), which are functions along the circular edge. We provide explicit formulas for a penny-shaped crack for an axisymmetric case as well as a case in which the loading is non-axisymmetric. Explicit formulas for other singular circular edges such as a circumferential crack, an external crack and a 3π/2 reentrant corner are also derived. The mathematical machinery developed in the framework of the Laplace operator is extended to derive the asymptotic solution (three-component displacement vector) for the elasticity system in the vicinity of a circular edge in a three-dimensional domain. As a particular case we present explicitly the series expansion for a traction free or clamped penny-shaped crack in an axisymmetric or a non-axisymmetric situation. The precise representation of the asymptotic series is required for constructing benchmark problems with analytical solutions against which numerical methods can be assessed, and to develop new extraction techniques for the edge flux/intensity functions which are of practical engineering importance in predicting crack propagation.     

A Boubaker polynomials expansion scheme (BPES) related analytical solution to the Williams–Brinkmann stagnation point flow equation at a blunt body

An analytic solution for the problem of Williams–Brinkmann axisymmetric steady flow in the vicinity of a stagnation point at a blunt body is proposed. The boundary conditions are embedded in the main system of equations by means of the Boubaker polynomials expansion scheme (BPES). These differential equations are solved analytically and yield continuous and differentiable solutions compared to some published ones.     

Simple dynamical models for vortex breakdown of the B-type

Summary A simple, exact, locally convergent series solution of the Navier-Stokes and continuity equations for steady, incompressible, axisymmetric fluid flow is derived which exhibits the qualitative features of vortex breakdown of the bubble or B-type. This solution, which converges in a neighborhood of the axis of symmetry of the flow, is shown to produce vortex breakdown flows which are in good qualitative agreement with both experimentally and numerically observed B-type vortex breakdown phenomena.     

Free convection from a torsionally oscillating vertical plane

Free convective flow of a viscous incompressible fluid from a uniformly heated vertical plane lamina, undergoing small-amplitude sinosoidal torsional oscillations, is investigated. The non-axisymmetric fluid motion consists of the primary Rosenblat flow and the secondary buoyancyinduced cross-flow. Numerical-analytical and asymptotic solutions of the energy equation, covering the whole range of the values of the Prandtl number of the fluid, are derived for their subsequent use in the analysis of the unsteady cross-flow. The mean cross-flow is found to dominate the steady components of the primary flow.     

Flows induced by a plate moving normal to stagnation-point flow

The flow generated by an infinite flat plate advancing toward or receding from a normal stagnation-point flow is obtained as an exact reduction of the Navier?CStokes equations for the case when the plate moves at constant velocity V. Both Hiemenz (planar) and Homann (axisymmetric) stagnation flows are considered. In each case, the problem is governed by a Reynolds number R proportional to V. Small and large R behaviors of the shear stress parameters are found for both advancing and receding plates. Numerical solutions determined over an intermediate range of R accurately match onto the small and large R asymptotic behaviors. As a side note, we report an interesting exact solution for plates advancing toward or receding from an exact rotational stagnation-point flow discovered by Agrawal (1957).     

Spin-up and spin-down of a viscous fluid over a heated disk rotating in a vertical plane in the presence of a magnetic field and a buoyancy force

Summary The hydromagnetic spin-up and spin-down of an incompressible electrically conducting fluid on a heated infinite disk rotating in a vertical plane in the presence of a magnetic field and a buoyancy force have been studied. The flow is non-axisymmetric due to the imposition of the buoyancy force. We have considered the situation where there is an initial steady state which is perturbed by suddenly changing the angular velocity of the disk. By using suitable transformations the Navier-Stokes and energy equations with four independent variables (x, y, z, t) are reduced to a system of partial differential equations with two independent variables (,t ^*). Also, these transformations uncouple the momentum and energy equations, resulting in a primary axisymmetric flow with an axial magnetic field, in an energy equation dependent on the primary flow and in a buoyancy induced secondary cross flow dependent on both primary flow and energy.The results indicate that the effect of the step-change in the angular velocity of the disk is more pronounced on the primary flow than on the secondary flow and the temperature field. For both spin-up and spin-down cases the surface shear stress in the non-axial direction normal to gravity for the primary flow and the surface shear stresses for the secondary flow increase with the magnetic parameter, whilst the surface shear stress in the vertical direction and the heat transfer at the surface decrease as the magnetic parameter increases. Also, the secondary flow near the disk dominates the primary flow. We have also developed an asymptotic analysis for large magnetic parameters which complements well the numerical results obtained in the lower magnetic parameter range.     

Midplane-symmetry breaking in the flow between two counter-rotating disks

This paper considers the axisymmetric steady flow driven by exact counter rotation of two co-axial disks of finite radius. At the edges of the rotating disks one of three conditions is (typically) imposed: (i) zero velocity, corresponding to a stationary, impermeable, cylindrical shroud (ii) zero normal velocity and zero tangential fluid traction, corresponding to a (confined) free surface and (iii) an edge constraint that is consistent with a similarity solution of von Kármán form. The similarity solution is valid in an infinite geometry and possesses a pitchfork bifurcation that breaks the midplane symmetry at a critical Reynolds number. In this paper, similar bifurcations of the global (finite-domain) flow are sought and comparisons are made between the resulting bifurcation structure and that found for the similarity solution. The aim is to assess the validity of the nonlinear similarity solutions in finite domains and to explore the sensitivity of the solution structure to edge conditions that are implicitly neglected when assuming a self-similar flow. It is found that, whilst the symmetric similarity solution can be quantitatively useful for a range of boundary conditions, the bifurcated structure of the finite-domain flow is rather different for each boundary condition and bears little resemblance to the self-similar flow.     

Stability of ferrofluid flow between concentric rotating cylinders with an axial magnetic field

A linear stability analysis for the ferrofluid flow between two concentric rotating cylinders in the presence of an axial magnetic field is implemented in this study. Both of the wide-gap and small-gap cases are considered and the governing equations with respect to three-dimensional disturbances including axisymmetric and non-axisymmetric modes are derived and solved by a direct numerical procedure. A parametric study covering wide ranges of ?, the volume fraction of colloidal particles; ξ, the strength of axial magnetic field; μ, the ratio of angular velocity of the outer cylinder to that of the inner cylinder; and ε, the ratio of radius of the inner cylinder to that of the outer cylinder, is conducted. Results show that the stability characteristics depend heavily on these factors. It is found that the increases of ? and ξ, and decrease of ε tend to stabilize the basic flow for an assigned value of μ. The variations of the onset mode with these parameters are discussed in detail. An example for the practical application of present results is given to help the understanding of stability behaviour of this flow.     

Galerkin boundary integral analysis for the axisymmetric Laplace equation

The boundary integral equation for the axisymmetric Laplace equation is solved by employing modified Galerkin weight functions. The alternative weights smooth out the singularity of the Green's function at the symmetry axis, and restore symmetry to the formulation. As a consequence, special treatment of the axis equations is avoided, and a symmetric‐Galerkin formulation would be possible. For the singular integration, the integrals containing a logarithmic singularity are converted to a non‐singular form and evaluated partially analytically and partially numerically. The modified weight functions, together with a boundary limit definition, also result in a simple algorithm for the post‐processing of the surface gradient. Published in 2005 by John Wiley & Sons, Ltd.     

Stability of Non-axisymmetric Electrolyte Jet in High-frequency AC Electric Field

In the present work linear instability of capillary non-axisymmetric micro-jets of electrolyte solutions in a high-frequency alternating axial electric field is investigated theoretically. The gravity affects are neglected. The problem is described by strongly coupled nonlinear system of PDEs for ion transport, electrical field and fluid flow. Viscous liquid is taken. The problem can be divided into outer and inner ones. Solution for the unsteady double ion layer is obtained in Debye-Huckel approximation provided that the oscillation frequency is sufficiently high while Pecklet number based on the Debye layer thickness is sufficiently small. The unsteady double ion layer produces additional normal and tangential stresses on the liquid–gas interface; the latter can either stabilize or destabilize the flow. It is shown that only axisymmetric mode is unstable while non-axisymmetric perturbations are always stable. It is also shown that in unstable case there is an essential dependence of the main stability characteristics on the parameter proportional to the frequency of external field. There are two threshold values of the parameter at which a bifurcation of stability parameters occurs. In particular, the size of the formed drops suffers a jump at increase of amplitude of fluctuation of an electric field. The problem is solved in a broad region of its parameters. There is a qualitative agreement of the theory developed with the available experimental data.     

叶栅内非定常不可压流动的数值模拟

基于SMAC(SimplifiedMarkerandCell)方法推导出直接求解二维非定常、不可压N-S方程的隐式数值方法。求解的基本方程是任意曲线坐标系中以逆变速度为变量的N-S方程和椭圆型的压力Poisson方程。采用该方法,对二维叶栅非定常分离流场进行了数值模拟,叶栅表面压力的计算结果与试验结果相比比较吻合,从而验证了这种方法的可靠性。同时对叶栅非定常流场的流场结构和流动机理做了初步的探讨。在均匀来流和定常边界条件下,叶栅内部流动表现出强烈的非定常性;在小冲角和高雷诺数时,叶栅尾部产生类似卡门涡街的周期性流动。