Investigation of the effect of the Coriolis force on a thin fluid film on a rotating disk

The effect of the Coriolis force on the evolution of a thin film of Newtonian fluid on a rotating disk is investigated. The thin-film approximation is made in which inertia terms in the Navier–Stokes equation are neglected. This requires that the thickness of the thin film be less than the thickness of the Ekman boundary layer in a rotating fluid of the same kinematic viscosity. A new first-order quasi-linear partial differential equation for the thickness of the thin film, which describes viscous, centrifugal and Coriolis-force effects, is derived. It extends an equation due to Emslie et al. [J. Appl. Phys. 29, 858 (1958)] which was obtained neglecting the Coriolis force. The problem is formulated as a Cauchy initial-value problem. As time increases the surface profile flattens and, if the initial profile is sufficiently negative, it develops a breaking wave. Numerical solutions of the new equation, obtained by integrating along its characteristic curves, are compared with analytical solutions of the equation of Emslie et al. to determine the effect of the Coriolis force on the surface flattening, the wave breaking and the streamlines when inertia terms are neglected.     

Wave Regimes of Viscous Film Flow down a Vertical Cylinder

The flow of a viscous liquid film down a vertical cylinder in the gravity field is considered. In the case of small Reynolds numbers for long-wave perturbations on a cylinder of radius much greater than the film thickness, the problem can be reduced to a single nonlinear equation for the evolution of the film thickness perturbation. For axially symmetric solutions, this equation coincides with the well-known Sivashinsky-Kuramoto equation. The results of a numerical analysis of this equation for three-dimensional stationary traveling solutions of the problem are reported. The effect of the problem parameters on the solution behavior is demonstrated. Soliton type solutions are presented.     

On the Linear Stability of Large Stefan Number Convection in Rotating Mushy Layers for a New Darcy Equation Formulation

The Coriolis effect on a solidifying mushy layer is considered. A near-eutectic approximation and large far-field temperature is employed in the current study for large Stefan numbers. The linear stability theory is used to investigate analytically the Coriolis effect on convection in a rotating mushy layer for a new formulation of the Darcy equation. It was found that a large Stefan number scaling allows for the presence of both the stationary and oscillatory modes of convection. In contrast to the problem of a stationary mushy layer, rotating the mushy layer has a stabilising effect on convection. It was observed that increasing the Taylor number or the Stefan number encouraged the oscillatory mode of convection.     

Numerical investigation of two-phase rotating flow

Finite difference solutions of the two-fluid equations of motion for a particle (droplet)-fluid mixture in a rotating finite axisymmetric cylinder are presented. The numerical method, which can be regarded as an extension of the Harlow & Amsden approach, employs forward time and centred space discretization and treats implicitly the pressure, Coriolis and volume flux terms. The computed flow fields are examined via a detailed comparison to previous analytic approximations, which illuminates both the physical and numerical aspects and the validity of these approximations.     

The effect of mechanical and thermal anisotropy on the stability of gravity driven convection in rotating porous media in the presence of thermal non-equilibrium

We investigate Rayleigh–Benard convection in a porous layer subjected to gravitational and Coriolis body forces, when the fluid and solid phases are not in local thermodynamic equilibrium. The Darcy model (extended to include Coriolis effects and anisotropic permeability) is used to describe the flow, whilst the two-equation model is used for the energy equation (for the solid and fluid phases separately). The linear stability theory is used to evaluate the critical Rayleigh number for the onset of convection and the effect of both thermal and mechanical anisotropy on the critical Rayleigh number is discussed.     

Application and analysis of functionally graded piezoelectrical rotating cylinder as mechanical sensor subjected to pressure and thermal loads

The exact thermoelastic analysis of a functionally graded piezoelectrical (FGP) rotating cylinder is investigated analytically. The cylinder is subjected to a combination of electrical, thermal, and mechanical loads simultaneously. The structure is a simplified model of a rotational sensor or actuator. The basic governing differential equation of the system is obtained by using the energy method. A novel term, named as the additional energy, is introduced to exact the evaluation of the energy functional. The solution to the governing differential equation is presented for two types of boundary conditions including free rotating and rotating cylinders exposed to the inner pressure. The effect of the angular velocity is investigated on the radial distribution of various components. The mentioned structure can be considered as a sensor for measuring the angular velocity of the cylinder subjected to the pressure and temperature. The obtained results indicate that the electrical potential is proportional to the angular velocity.     

轴对称横观各向同性热弹性圆柱的分解

为了得到精确的应力场、位移场、温度场,将扭转圆轴的精化理论研究方法推广到轴对称横观各向同性热弹性圆柱。利用Bessel函数以及轴对称横观各向同性热弹性圆柱的通解,给出了轴对称横观各向同性热弹性圆柱的分解定理。根据柱面齐次边界条件获得了精确的精化方程,精化方程可以分解为一阶方程、超越方程、温度方程,从而将横观各向同性热弹性圆柱的轴对称问题分解为轴向拉压问题、超越问题、热-应力耦合问题。超越部分对应端部自平衡情况,可以清晰地了解到端部应力分布对内部应力场的影响,热-应力耦合部分对应无外加应力场时圆柱内部因温度变化引起的热应力。     

Elastodynamic solution of a non-homogeneous orthotropic hollow cylinder

A theoretical method for analyzing the axisymmetric plane strain elastodynamic problem of a non-homogeneous orthotropic hollow cylinder is developed. Firstly, a new dependent variable is introduced to rewrite the governing equation, the boundary conditions and the initial conditions. Secondly, a special function is introduced to transform the inhomogeneous boundary conditions to homogeneous ones. By virtue of the orthogonal expansion technique, the equation with respect to the time variable is derived, of which the solution can be obtained. The displacement solution is finally obtained, which can be degenerated in a rather straightforward way into the solution for a homogeneous orthotropic hollow cylinder and isotropic solid cylinder as well as that for a non-homogeneous isotropic hollow cylinder. Using the present method, integral transform can be avoided and it can be used for hollow cylinders with arbitrary thickness and subjected to arbitrary dynamic loads. Numerical results are presented for a non-homogeneous orthotropic hollow cylinder subjected to dynamic internal pressure. The project supported by the National Natural Science Foundation of China (10172075 and 10002016)     

非平面应变状态下的叠层厚壁筒

抛弃任何有关位移或应力模式的人为假设，在轴对称情况下，导出正交异性厚壁筒的状态方程。在沿筒轴方向任意分布的轴对称荷载下，给出叠层厚壁筒静力问题的精确解。此解满足所有弹性力学基本方程，包含了全部弹性常数，可满足任意精度要求。数值结果和ＳＡＰ５有限元解进行了对比。     

激励Stuart-Landau方程的研究--周期解、稳定性及流动控制

解析得出了有外部激励的Stuart-Landau（S-L）方程的频率锁定周期解,对这些解与外部激励振幅和频率的依赖关系做了详细研究,并用周期系统稳定性理论确定了解的稳定性边界.还对S-L方程所描述的流动控制效果进行了研究,发现由于外部激励的作用,稳定的锁频解可能比原来的饱和解能量减少了,外部的控制最多能使扰动能量减少为原来的一半.     

Numerical simulations of miscible fluids on a rotating Hele–Shaw cell with effects of Coriolis forces

The effects of Coriolis forces to interfacial stabilities on miscible rotating Hele–Shaw flows are investigated by means of highly accurate numerical schemes. Two major influences on the interfaces, i.e. the stabilization of circumferential fingerings and the unstable body distortion of droplets, are observed. These two phenomena can be represented by two characteristic measurements, i.e. interfacial length and radius of gyration, respectively. Without an additional injection, the miscible interface is found to be more stable at higher Coriolis factors, based on both the judgments of growths in interfacial length and the radius of gyration. However, with an additional injection, a visible body distortion with suppressed interfacial fingerings at higher Coriolis factors leads to inconsistent developments in these two characteristic measurements. The stabilized effect of interfacial fingering delays and weakens the growth in interfacial length at higher Coriolis factors, while faster and greater growth in the radius of gyration is observed because of the significant body distortion. As a result, the effects of the Coriolis forces on the miscible interfacial instabilities should be addressed with great caution. Copyright © 2005 John Wiley & Sons, Ltd.     

Non-slipping adhesive contact of a rigid cylinder on an elastic power-law graded half-space

In this paper, adhesive contact of a rigid cylinder on an elastic power-law graded half-space is studied analytically with the theory of weakly singular integral equation and orthogonal polynomial method. Emphasis is placed on the coupling effect between tangential and normal directions which was often neglected in previous works. Our analysis shows that the coupling effect tends to reduce the contact area in the compressive regime. The effect of bending moment on the adhesion behavior is also examined. Like a pull-off force, there also exists a critical bending moment at which the cylinder can be bended apart from the substrate. However, unlike pull-off force, the critical bending moment is insensitive to the gradient exponent of the graded material.     

Transient thermal stresses in an orthotropic hollow cylinder for axisymmetric problems

For the thermoelastic dynamic axisymmetric problem of a finite orthotropic hollow cylinder, one comes closer to reality to involve the effect of axial strain than to consider the plane strain case only. However, additional mathematical difficulties should be encountered and a different solution procedure should be developed. By the separation of variables, the thermoelastic axisymmetric dynamic problem of an orthotropic hollow cylinder taking account of the axial strain is transformed to a Volterra integral equation of the second kind for a function of time, which can be solved efficiently and quickly by the interpolation method. The solutions of displacements and stresses are obtained. It is noted that the present method is suitable for an orthotropic hollow cylinder with an arbitrary thickness subjected to arbitrary axisymmetric thermal loads. Numerical comparison is made to show the effect of the axial strain on the displacements and stresses. The project supported by the National Natural Science Foundation of China (10172075) and China Postdoctoral Science Foundation (20040350712)     

横观各向同性圆柱体自由振动的弹性力学解

本文由横观各向同性的弹性力学方程出发，研究有限长圆柱体的自由振动问题。利用文献「１」的通解，将位移分量和应力分量分别表达成傅里叶－塞尔级数和双曲－贝塞尔级数的形式。通过边界条件和级数的正交关系，得到关于有限长圆柱自由振动频率的特征方程。利用数值方法求解特征根，从而得到圆柱体三维振动的自振频率。     

Bifurcations and exact solutions of an asymptotic rotation-Camassa–Holm equation

This paper investigates the rotation-Camassa–Holm equation, which appears in long-crested shallow-water waves propagating in the equatorial ocean regions with the Coriolis effect due to the earth’s rotation. The rotation-Camassa–Holm equation contains the famous Camassa–Holm equation and is a special case of the generalized Camassa–Holm equation. By using the approach of dynamical systems and singular traveling wave theory to its traveling wave system, in different parameter conditions of the five-parameter space, the bifurcations of phase portraits are studied. Some exact explicit parametric representations of the smooth solitary wave solutions, periodic wave solutions, peakons and anti-peakons, periodic peakons as well as compacton solutions are obtained.

     

Numerical simulations of interfacial instabilities on a rotating miscible droplet in a time‐dependent gap Hele–Shaw cell with significant Coriolis effects

Interfacial instability of a rotating miscible droplet with significant Coriolis force in a Hele–Shaw cell is simulated numerically. The influences of the relevant control parameters are first discussed qualitatively by fingering patterns. More vigorous fingerings are found at higher rotational effects, a lower viscosity contrast and a weaker effective surface tension (Korteweg constant). For a time‐dependent gap Hele–Shaw cell, a higher cell lifting rate makes the rotating droplet bear an inward straining flow, which leads to fingering enhancement. On the contrary, a higher pressing rate provides more stable effects by additional squeezing outward flow. A quantitative analysis between the Coriolis effects and tilting angles of fingers is addressed. For arbitrary combinations of all relevant control parameters, the values of tilting angles follow a nearly linear relationship with the Coriolis effects. We estimate the correlation between the relevant control parameters (dimensionless Coriolis factor Re, viscosity parameter R, cell lifting rate a) and tilting angles (θ) of fingers that can be approximated as for significant Korteweg stresses. Copyright © 2005 John Wiley & Sons, Ltd.     

THE MATHEMATICAL MODELLING OF NEAR COAST SHALLOW WATER CIRCULATION

This paper presents a finite element method to solve the shallow water circulation problem numerically. Considering the Coriolis effect, bottom friction and eddy viscosity, the continuity equation and momentum equation are integrated vertically. Using Galerkin weighted residual method, the weak variational formulation is derived for the finite element analysis. The split-time method is applied for the numerical integration instead of iteration for nonlinear terms. Moreover, an artificial smooth approach is proposed to suppress the short wavelength noise. In order to save Computer storage units, a densed storage scheme is set up, where all the zero elements in large scaled and sparse matrices are excluded.     

Berechnung der Störung der Brennkammer-Filmkühlung durch Einblasen von Luftstrahlen mit Hilfe einer Analogiebetrachtung zwischen Strahl und Zylinder

The influence of cold air jet injection upon the film cooling of combustion chamber walls is investigated theoretically and experimentally. Considering the cold air jet acting like a cylinder in the combustion gas and film cooling streams, an equation can be derived theoretically which describes the disturbance of the film cooling effectiveness downstreams of the air jet injection. The experiments, where single cold air jets are injected normal to the film cooled test chamber wall, show considerable decreases of the film cooling effectiveness downstream of the injection points. The same effect is noted in real combustion chambers of aircraft gas turbine engines. Additional experiments with cylinders simulating the disturbance effect of the air jets prove the analogy consideration between jet and cylinder. The theoretically predicted cooling effectiveness is in agreement with own experiments as well as other test results.     

旋转薄壁圆柱壳振型进动的非线性振动特性

选取在工程上常用的悬臂旋转薄壁圆柱壳为研究模型,首先推导出考虑阻尼的振型进动因子,然后根据Donnell's简化壳理论建立考虑科氏力,阻尼与几何大变形的非线性波动方程,采用Galerkin方法对波动方程进行离散化,得到模态坐标中相互耦合的三阶非线性微分方程组.应用Runge-Kutta法求解获得非线性幅频特性曲线,分析了不同模态组合下系统主模态(m=1,n=6,k=1)的共振响应.应用谐波平衡法对系统三阶非线性微分方程组解析分析,与数值解比较验证了解析解的正确性和有效性.最后分析了动力系统的运动稳定性.结果表明,节径数n和频率倍数k对于主模态共振响应的影响很小,而轴向半波数m对主共振的影响则相对较大,因此只需选取相邻的两个轴向模态(M=2)即可较为简洁,准确的描述主共振响应;谐波平衡法可以很好的解决三阶微分方程组的非线性问题,并且能够达到较为满意的精度.