Problems of Hydrodynamics for a Triaxial Ellipsoid

Problems of motion of a triaxial ellipsoid in an ideal liquid and in a viscous liquid in the Stokes approximation and also equilibrium shapes of the rotating gravitating liquid mass are considered. Solutions of these problems expressed via four quadratures depending on four parameters are significantly simplified because they are expressed via the only function of two arguments. The efficiency of the proposed approach is demonstrated by means of analyzing the velocity and pressure fields in an ideal liquid, calculating the added mass of the ellipsoid, determining the viscous friction, and studying the equilibrium shapes and stability of the rotating gravitating capillary liquid. The pressure on the triaxial ellipsoid surface is expressed via the projection of the normal to the impinging flow velocity. The shape of an ellipsoid that ensures the minimum viscous drag at a constant volume is determined analytically. A simple equation in elementary functions is derived for determining the boundary of the domains of the secular stability of the Maclaurin ellipsoids. An approximate solution of the problem of equilibrium and stability of a rotating droplet is presented in elementary functions. A bifurcation point with non-axisymmetric equilibrium shapes branching from this point is found.     

Stability of gas-bubble equilibrium shape in uniform flow of an ideal fluid

Steady-state motion of a bubble in the shape of an ellipsoid of revolution has been studied [1, 2]. Steady-state motion and small oscillations of an ellipsoid of revolution around the equilibrium state were studied with the help of Lagrangian equations [3]. In this paper, possible equilibrium shapes of a bubble in the form of a triaxial ellipsoid are studied. The dependence of the pressure difference at the stagnation point and within the gas bubble on deformation is determined for steady-state motion. The stability of the equilibrium shape with respect to small perturbations of the axes of the ellipsoid is investigated through analysis of potential energy in the neighborhood of the extremum.     

Internal and inertial waves in a viscous rotating stratified fluid

The effects of viscosity on the propagation of a St. Andrew's cross wave which is generated by a simple-harmonic localized disturbance in a rotating stratified fluid are considered. A similarity solution of the linearised equations shows that the velocities decay and that the wave width increases away from the disturbance. Previous solutions in a stratified non-rotating fluid are recovered by letting the rotation tend to zero. The solutions are also valid in the limit of a homogeneous rotating fluid. Further solutions for waves in a realistic ocean and in an isothermal atmosphere on a rotating Earth are also included.     

Some properties of multi-fluid stokes flows

The solution of Stokes' equations for a rotating axisymmetric body which possesses reflection symmetry about a planar interface between two infinite immiscible quiescent viscous fluids is shown to be independent of the viscosities of the fluids and identical with the solution when the fluids have the same viscosity. The result is generalized to a rotating axisymmetric system of bodies which possesses reflection symmetry about each interface of a plane stratified system of fluids. An analogous result for two-fluid systems with a nonplanar static interface is also derived. The effect on torque reduction produced by the presence of a second fluid layer adjacent to a rotating axisymmetric body is considered and explicit calculations are given for the case of a sphere. A proof of uniqueness for unbounded multi-fluid Stokes' flow is given and the asymptotic far field structure of the velocity field is determined for axisymmetric flow caused by the rotation of axisymmetric bodies.     

Solution of the Reynolds Equation for Viscous-Fluid Flow in the Clearance between a Rotating Ellipsoid and a Surface

Slow viscous-fluid flows in the narrow clearance (i) between a moving ellipsoid and a straight tube of elliptic cross section and (ii) between a rotating ellipsoid and a toroidal tube, including the case of an ellipsoid near a plane, are considered. A solution of the boundary-value problem for the Reynolds equation describing the flow in the clearance is found. The similarity of the pressure profiles in the “ellipsoid-plane” and “ cylinder-plane” systems is indicated.     

An orthotropic piezoelectric ellipsoid in a centrifugal field

A rotating ellipsoid composed of an orthotropic piezoelectric material (2mm) are considered, and the stress and electric displacement fields in this rotating ellipsoid are obtained exactly and completely. The solutions of the same problem for transversely isotropic piezoelectric material (6 mm) are also given by degenerating above results. At last, numerical examples for four kinds of media are illustrated in figures for comparison. Supported by the National Natural Science Foundation of China (No. 19872060).     

Geometric shapes of the interface surface of bicomponent flows between two concentric rotating cylinders

In this paper, the shape problem of interface of bicomponent flows between two concentric rotating cylinders is investigated. With tensor analysis, the problem is reduced to an energy functional isoperimetric problem when neglecting the effects of the dissipative energy caused by viscosity. We derive the associated Euler-Lagrangian equation, which is a nonlinear elliptic boundary value problem of the second order. Moreover, by considering the effects of the dissipative energy, we propose another total energy functional to characterize the geometric shape of the interface, and obtain the corresponding Euler-Lagrangian equation, which is also a nonlinear elliptic boundary value problem of the second order. Thus, the problem of the geometric shape is converted into a nonlinear boundary value problem of the second order in both cases.     

An empirically validated model of the pressure within a droplet confined between plates at equilibrium for low Bond numbers

An analytical model is presented that describes the equilibrium pressure within a confined droplet for small Bond numbers without prior knowledge of the interface shape. An explicit equation for the pressure was developed as a function of the gap height, surface tension, and contact angle. This equation was verified empirically. The shape of the interface was found based on the pressure predicted by both the proposed model and a model commonly used in electrowetting on dielectric (EWOD) investigations. These shapes were compared against experimentally observed interfaces for aspect ratios between 3.5 and 18. The pressures and shapes predicted by the proposed model were at least an order of magnitude more accurate than those predicted with a more commonly used model. At an aspect ratio of 3.5, the average error in the predicted shape was almost 4%, but decreased below the experimental error at an aspect ratio of 6. An aspect ratio of 15 is required for an EWOD device to split water droplets in air. The error in the model pressure and its predicted interface in this case were approximately 0.3%. The analytical pressure model proposed here can be used to increase the accuracy of models of practical EWOD devices. Better accuracy can be attained for small aspect ratios by iteratively calculating pressure using the model proposed here.     

A DOMAIN DECOMPOSITION ALGORITHM WITH FINITE ELEMENT-BOUNDARY ELEMENT COUPLING

A domain decomposition algorithm coupling the finite element and the boundary element was presented. It essentially involves subdivision of the analyzed domain into sub-regions being independently modeled by two methods, i.e., the finite element method (FEM) and the boundary element method (BEM). The original problem was restored with continuity and equilibrium conditions being satisfied on the interface of the two sub-regions using an iterative algorithm. To speed up the convergence rate of the iterative algorithm, a dynamically changing relaxation parameter during iteration was introduced. An advantage of the proposed algorithm is that the locations of the nodes on the interface of the two sub-domains can be inconsistent. The validity of the algorithm is demonstrated by the consistence of the results of a numerical example obtained by the proposed method and those by the FEM, the BEM and a present finite element-boundary element (FE-BE) coupling method.     

Droplet relaxation in blends with one viscoelastic component: bulk and confined conditions

Using a counter rotating parallel plate shear flow cell, the shape relaxation of deformed droplets in a quiescent matrix is studied microscopically. Both the effects of geometrical confinement and component viscoelasticity are systematically explored at viscosity ratios of 0.45 and 1.5. The flow conditions are varied from a rather low to a nearly critical Ca number. Under all conditions investigated, viscoelasticity of the droplet phase has no influence on shape relaxation, whereas matrix viscoelasticity and geometrical confinement result in a slower droplet retraction. Up to high confinement ratios, the relaxation curves for ellipsoidal droplets can be superposed onto a master curve. Confined droplets with a sigmoidal shape relax in two stages: the first consists of a shape change to an ellipsoid with a limited amount of retraction, and the second is the retraction of this ellipsoid. The latter stage can be described by means of one single relaxation time that can be obtained from the relaxation of initially ellipsoidal droplets. The experimental results are compared to the predictions of a recently published phenomenological model for droplet dynamics in confined systems with viscoelastic components (Minale et al., Langmuir 26:126–132, 2010). However, whereas the model predicts additive effects of geometrical confinement and component viscoelasticity, the experimental data reveal more complex interactions.     

Influence of the Outer Bath on the Eigenfrequencies of Rotating Axisymmetric Liquid Bridges

In experiments with liquid bridges, the neutral buoyancy technique has frequently been used to simulate microgravity conditions. In this technique the liquid bridge is surrounded by an outer liquid with similar density to compensate partially for the effect of the hydrostatic pressure over the interface. The outer bath is expected to play a relevant role not only in the static problem, but also in the dynamical behaviour of this fluid configuration. In the present contribution the eigenfrequencies characterizing the oscillation modes of a rotating axisymmetric liquid bridge are calculated. The analysis focuses on the influence of the outer bath and the liquid bridge equilibrium shape on those quantities. The results are obtained numerically by solving the inviscid two-dimensional model by means of a finite difference scheme. The method provides accurate results as is shown by comparison with the analytical solution for the cylindrical configuration. The comparison between the theoretical predictions for the first eigenfrequency and the experimental data obtained by Perales and Meseguer (1992) shows the capability of the model to describe the dynamics of real liquid bridges.     

Validation of the tangent formulation for the solution of the non-linear Eshelby inclusion problem

An extension of the Eshelby problem for non-linear viscous materials is considered. An ellipsoidal heterogeneity is embedded in an infinite matrix. The material properties are assumed to be uniform within the ellipsoid and in the matrix. The problem of determining the average strain rate in the ellipsoid in terms of the overall applied strain rate is solved in an approximate way. The method is based on the non-incremental tangent formulation of the non-linear matrix behavior [Acta Metall. 35 (1987) 2983]. In the present work this approximate solution is verified with a good agreement by comparing to finite element calculations for various inclusion shapes and loading conditions.     

非平整运动海底上n层流动中波浪传播的Hamilton逼近

在海洋水域，界面波对大尺度变化流的作用是一种典型的分层流动现象．考虑一不可压缩、无黏的分层势流运动，建立了一个在非平整运动海底上的n层流体演化系统，并对其进行了Hamilton描述．每层流体具有各自的常密度、均匀流水平速度，其厚度由未扰动和扰动部分构成．相对于顶层流体的自由表面，刚性、运动的海底具有一般地形变化特征．在明确指出n层流体运动的控制方程和各层交界面上的运动学、动力学边界条件(包含各层交界面上张力效应)后，对该分层流动力系统进行了Hamilton构造，即给出其正则方程和其下述的正则变量：各交界面位移和各交界面上的动量势密度差。     

An Adaptive Cartesian Mesh Based Method to Simulate Turbulent Flows of Multiple Rotating Surfaces

An efficient Cartesian cut-cell/level-set method based on a multiple grid approach to simulate turbulent turbomachinery flows is presented. The finite-volume approach in an unstructured hierarchical Cartesian setup with a sharp representation of the complex moving boundaries embedded into the computational domain, which are described by multiple level-sets, ensures a strict conservation of mass, momentum, and energy. Furthermore, an efficient kinematic motion level-set interface method for the rotation of embedded boundaries described by multiple level-set fields on a computational domain distributed over several processors is introduced. This method allows the simulation of multiple boundaries rotating relatively to each other in a fixed frame of reference. To demonstrate the efficiency of the numerical method and the quality of the computed findings the generic test problem of a rotating cylinder surrounded by a stationary hull and the flow over a ducted rotating axial fan with a stationary turbulence generating grid at the inflow are simulated. The computational results of the axial fan show a good agreement with the experimental data.     

The influence of a magnetic field on the mechanical behavior of a fluid interface

This work focuses on a theoretical investigation of the shape and equilibrium height of a magnetic liquid–liquid interface formed between two vertical flat plates in response to vertical magnetic fields. The formulation is based on an extension of the so called Young–Laplace equation for an incompressible magnetic fluid forming a two-dimensional free interface. A first order dependence of the fluid susceptibility with respect to the magnetic field is considered. The formulation results in a hydrodynamic-magnetic coupled problem governed by a nonlinear second order differential equation that describes the liquid–liquid meniscus shape. According to this formulation, five relevant physical parameters are revealed in this fluid static problem. The standard gravitational Bond number, the contact angle and three new parameters related to magnetic effects in the present study: the magnetic Bond number, the magnetic susceptibility and its derivative with respect to the field. The nonlinear governing equation is integrated numerically using a fourth order Runge-Kutta method with a Newton–Raphson scheme, in order to accelerate the convergence of the solution. The influence of the relevant parameters on the rise and shape of the liquid–liquid interface is examined. The interface shape response in the presence of a magnetic field varying with characteristic wavenumbers is also explored. The numerical results are compared with asymptotic predictions also derived here for small values of the magnetic Bond number and constant susceptibility. A very good agreement is observed. In addition, all the parameters are varied in order to understand how the scales influence the meniscus shape. Finally, we discuss how to control the shape of the meniscus by applying a magnetic field.     

Asymptotic form of the free surface of a uniformly rotating liquid for large bond numbers

In [1–5] boundary-layer methods were used to solve problems concerned with the equilibrium and motion of a liquid with surface tension in a strong gravitational field (for large Bond numbers Bo). In the present paper we apply these methods to problems involving the equilibrium shape of a uniformly rotating liquid, contained in a cylindrical container of arbitrary cross section or in a container which is a surface of revolution about the z axis. Both of these problems reduce to the asymptotic integration of an equation with a small parameter involving a quasilinear elliptic operator with a nonlinear boundary condition. In the second case, owing to radial symmetry, the equation for the problem goes over into an ordinary equation; however, the wetted boundary is not known beforehand. This boundary, together with the equilibrium shape, is also determined asymptotically.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 3–12, November–December, 1973.The authors thank L. A. Slobozhanin for his help in the preparation of this paper.     

Evolution of the interface in a stratified anisotropic porous material

It is shown that the boundary-value problem describing the evolution of the interface during impregnation of a stratified inhomogeneous anisotropic porous material with a viscous fluid can be reduced to a similar problem for a stratified inhomogeneous isotropic material by nonorthogonal transformation of the coordinates. As a result, the well-known estimates of the problem parameters determining the interface configuration for impregnation of an isotropic material can be extended to the anisotropic case.     

Quasi-cylindrical figures of equilibrium of a rotating fluid

The problem of the equilibrium shape of a steady rotating rectilinear infinite cord of ideal self-gravitating homogeneous fluid is considered. The question whether, apart from the obvious solution, namely, an infinite circular cylinder, noncylindrical equilibrium figures can exist is investigated. A search is carried out among axisymmetric figures with periodic surface structure (“wavy” cylinders). The period of the wave structure and, in the first approximation, the shape of the surface are found as functions of the angular velocity of rotation. Sankt-Peterburg. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 3, pp. 22–30, May–June, 2000.     

Poincaré-Chetaev bifurcation diagrams in the problem of motion of an inhomogeneous dynamically and geometrically symmetric ellipsoid on a smooth plane

The existence, stability, and bifurcation of steady motions of an inhomogeneous dynamically and geometrically symmetric ellipsoid is considered. The mass center of the ellipsoid is shifted and located on its symmetry axis. The ellipsoid moves on a perfectly smooth horizontal plane.     

A thermodynamic viscous interface theory and associated stability problems

Constitutive equations are developed for a theory of viscous interface where the interface itself is a two-dimensional continuum. An analysis of the problem of stability for a fluid in an inverted tube of arbitrary shape, the fluid remaining in the tube due to surface tension forces, is also considered.