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.     

Dynamically Equilibrium Shape of Intrusive Vortex Formations in the Ocean

The problem of finding the dynamically equilibrium shape of a rotating mass of liquid with homogeneous density (lens) submerged in a stratified ocean at rest on the rotating Earth is formulated. An equation for the shape of the interface between water masses is derived. An exact solution of the problem for an anticyclonically rotating lens in a linearly stratified ocean in the neighborhood of the lens depth shows that the dynamically equilibrium shape of the interface is a triaxial ellipsoid inclined with respect to the horizon and similar to an ellipsoid of revolution for real parameters of the phenomenon. The limiting values of the latitudes at which these formations can exist are determined. Degeneration of the shape with decrease in the intrinsic lens angular rate is investigated.     

Dynamics and Stability of a Liquid Triaxial Ellipsoid

The problem of the dynamics of a liquid triaxial ellipsoid confined inside a uniformly rotating rigid shell is formulated and solved within the framework of the Poincare-Joukovski problem. The system of three non-linear first-order differential equations has kinetic energy and circulation integrals representing two ellipsoids with displaced centers in parameter space. All the steady-state and time-dependent solutions are studied; time-dependent solutions exist in four zones, where they are represented by elliptic Jacobi functions, and on three zone boundary interval (represented in terms of elementary functions). On the three other boundary intervals there exist liquid steady-state ellipsoids for which the vortex vector either coincides with one of the principal axes of the ellipsoid or lies in one of its principal planes of symmetry.     

粘性液体中浮体的非线性稳定理论

本文应用连续系统稳定性理论,处理粘性不可压缩流体所支持的平衡浮体的非线性稳定问题,其中考虑到浮体的方位角和质心位置以及流体速度的非线性扰动,给出了各种尺度的非线性稳定、渐近稳定判据及不稳定判据。     

The Integral Formula for Pressure Field in the Nonstationary Barotropic Flows of Viscous Fluid

An integral expression of pressure via dynamical characteristics of the vortex and velocity fields for viscous fluid is presented. This expression may be considered as an analog (or generalization) of the Bernoulli equation for nonstationary vortex flows of ideal or viscous fluids, including the case of an external nonconservative mass force. The presented formulas are useful for calculating the pressure field when meshless vortex methods are applied for the flow simulation.     

微粗糙平面上翻身陀螺的稳定性

本文讨论微粗糙平面上翻身陀螺绕直立的极轴永久转动的稳定性。应用关于部分变量的稳定性定理,导出在粘性和库仑两种摩擦条件下的解析形式稳定性判据。该判据用永久转动角速度的大小而不是进动角速度的大小作为判别准则,较为合理。     

Drag of a rotating sphere

We consider the problem of steady incompressible viscous fluid flow about a rotating sphere, with the flow specified on a sphere of finite radius, which reduces to the solution of the complete Navier-Stokes equations.The dimensionless stream functions and circulai velocity are sought in the form of series in powers of the Reynolds numbers, which converge for small values of this number. Recurrence formulas are derived for determining the coefficients of these series. The pressure, rotational resistance torque, and drag are determined. It is established that the rotating sphere has higher drag than a stationary sphere. The leading term of the series in powers of the Reynolds number for the drag and resistive torque is calculated.     

Ellipsoidal model of the rise of a Taylor bubble in a round tube

The rise velocity of long gas bubbles (Taylor bubbles) in round tubes is modeled by an ovary ellipsoidal cap bubble rising in an irrotational flow of a viscous liquid. The analysis leads to an expression for the rise velocity which depends on the aspect ratio of the model ellipsoid and the Reynolds and Eötvös numbers. The aspect ratio of the best ellipsoid is selected to give the same rise velocity as the Taylor bubble at given values of the Eötvös and Reynolds numbers. The analysis leads to a prediction of the shape of the ovary ellipsoid which rises with same velocity as the Taylor bubble.     

Flow field determination on the surface of some bodies in an incompressible fluid stream

Simple relationships of local type are established for the velocity and pressure distributions on the surface of an elliptical cylinder, an ellipsoid of revolution, and an arbitrary triaxial ellipsoid in an incompressible fluid stream. It is shown that in an exact formulation the flow parameters at a given point of the body surface depend only on the local angle of stream impact with a surface element. In particular, Newtonian impact theory, which is used extensively in hypersonic gas flow computations, turns out to be valid for the velocity distribution.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 187–190, March–April, 1976.The author is grateful to G. Yu. Stepanov and A. G. Petrov for useful discussions of this paper.     

Plane vortex flow induced by a mass source (sink) in a rotating viscous fluid

A solution of the self-similar type, describing the development with time of a plane vortex flow excited by an axisymmetric mass source (sink) in a rotating viscous fluid, is obtained. Sources of two kinds — impulsive and of constant strength — are considered. The solutions for the velocity and vorticity fields are expressed in the form of functions similar to incomplete gamma functions and are presented in the form of graphs for various flow Reynolds numbers.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 172–175, July–August, 1991.     

Equilibrium shapes and stability of rotating drops

Direct variational methods are used to obtain nonaxisymmetric equilibrium shapes of a rotating drop. The results are given of experiments on the interaction of freely floating drops of viscous fluid. It is shown that if the dimensionless angular momentum of the system is > 3.4, then a sequence of three-dimensional shapes similar to ellipsoids bifurcates from the sequence of axisymmetric shapes The stability of these shapes is studied in Poincaré's scheme.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 13–20, July–August, 1982.     

Hydrostatics in weak force fields. On annular figures of equilibrium of a rotating liquid and their stability

We consider the problem of annular equilibrium figures of a rotating weightless liquid, having surface tension, and their stability. This question has been studied by Charreaux (for a discussion of his results see [1]), who examined the evolution of the forms of annular equilibrium figures and showed that there exists a family of stable equilibrium shapes. However, these studies of Charreaux are incomplete, and the conclusion on the existence of stable forms is valid only for axisymmetric disturbances.In the following we examine the properties of annular equilibrium figures of a rotating liquid. From the results of numerical integration on a digital computer we construct a family of equilibrium forms and present data which permit finding the corresponding equilibrium form from the ensemble of physical parameters which define the equilibrium state. In studying the stability we use the technique of [2, 3].The results of the numerical calculation and the asymptotic representat ons show that stable annular equilibrium figures of a rotating liquid do not exist.The author wishes to thank M. A. Belyaev for compiling the program for the numerical calculation, and also N. D. Kopachevskii, A. D. Myshkis, and A. D. Tyuptsov for discussions of the results and helpful remarks.     

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.     

Stability of a liquid jet into incompressible gases and liquids: Part 2. Effects of the irrotational viscous pressure

In this paper we investigate the effects of an irrotational, viscous pressure on the stability of a liquid jet into gases and liquids. The analysis extends our earlier work (part 1) in which the stability of the viscous jet was studied assuming that the motion and pressure are irrotational and the viscosity enters through the jump in the viscous normal stress in the normal stress balance at the interface. The liquid jet is always unstable; at high Weber numbers the instability is dominated by capillary instability; at low W the instability is dominated by Kelvin–Helmholtz (KH) waves generated by pressures driven by the discontinuous velocity. In the irrotational analysis the viscosity is important but the effects of shear are neglected. In fact a discontinuous velocity is not compatible with the continuity of the tangential components of velocity and shear stress so that KH instability is not properly posed for exact study using the no-slip condition but some of the effects of viscosity can be ascertained using viscous potential flow. The theory is called viscous potential flow (VPF). Here we develop another irrotational theory in which the discontinuities in the irrotational tangential velocity and shear stress are eliminated in the global energy balance by selecting viscous contributions to the irrotational pressure. These pressures generate a hierarchy of potential flows in powers of the viscosity, but only the first one, linear in viscosity, in the irrotational viscous stress, is thought to have physical significance. The tangential velocity and shear stress in an irrotational study cannot be made continuous, but the effects of the discontinuous velocity and stress in the mechanical energy balance can be removed “in the mean.” This theory with the additional viscous pressure is called VCVPF, viscous correction of VPF. VCVPF is VPF with the additional pressures. The theory here cannot be compared with an exact solution, which would not allow the discontinuous velocity and stress. In other problems, like capillary instability, in which VCVPF can be compared with an exact solution, the agreements are uniformly excellent in the wave number when one of the fluids is gas and in good but not uniform, agreement when both fluids are liquids.     

Steady and periodic modes in the problem of motion of a heavy material point on a rotating sphere (the viscous friction case)

The problem of motion of a heavy material point on a sphere uniformly rotating about a fixed axis is considered in the case of viscous friction. The angle of inclination between the axis and the horizon is constant. The existence, bifurcation, and stability of the equilibrium positions are discussed for such a mechanical system. The existence of periodic motions is also studied. An approach is proposed to find such motions in the case of low viscosity.     

Bleustein–Gulyaev waves in 6 mm piezoelectric materials loaded with a viscous liquid layer of finite thickness

In this paper, we analyze the propagation of Bleustein–Gulyaev waves in an unbounded piezoelectric half-space loaded with a viscous liquid layer of finite thickness within the linear elastic theories. Exact solutions of the phase velocity equations are obtained in the cases of both electrically open circuit and short circuit by solving the equilibrium equations of piezoelectric materials and the diffusion equation of viscous liquid. A PZT-5H/Glycerin system is selected to perform the numerical calculation. The results show that the mass density and the viscous coefficient have different effects on the propagation attenuation and phase velocity under different electrical boundary conditions. In particular, the penetration depth of the waves is of the same order as the wavelength in the case of electrically short circuit. These effects can be used to manipulate the behavior of the waves and have implications in the application of acoustic wave devices.     

Stability of motion of the plane boundary separating two fluids

The effect of viscosity on the disintegration of liquid jets can be considered in two ways. First, viscous forces alter the basic flow: they form a boundary layer whose presence necessarily alters wave formation. Second, viscous forces can have a direct effect on the development of perturbations for a given velocity profile of the basic flow. In this case the study of stability must be based on the Navier-Stokes equations instead of the equations of an ideal fluid. This complicates analysis considerably. Available data [1] indicate that this influence is very minor in the case of moderately viscous fluids. It appears, therefore, that the principal role is played by changes in the velocity profile alone, and that the behavior of the perturbations is described by the equations of an ideal fluid.In the present study we investigated the stability of motion and wave formation at the boundary between two fluids in order to determine the effect of viscosity on the drop formation mechanism. The simplest case of oscillation of the boundary is chosen in order to keep the analysis as simple as possible.     

圆环旋转黏性液体射流破碎液滴粒径与速度数量密度分布相关性研究

压力旋流喷嘴被广泛应用于航空发动机、船用发动机、车用汽油缸内直喷发动机、燃气轮机等动力机械的燃油喷射系统中.以压力旋流喷嘴射流为研究对象,开展了圆环旋转黏性液体射流破碎液滴粒径与速度数量密度分布相关性问题研究.对于液体射流,以往的研究往往对破碎液滴粒径数量密度分布或速度数量密度分布进行单独研究,对于这两种数量密度分布之间关系的研究较少;从相关性的角度对圆环旋转黏性液体射流破碎液滴粒径与速度数量密度分布之间的关系进行研究.采用最大熵原理方法建立了圆环旋转黏性液体射流破碎液滴粒径与速度联合概率密度函数.对圆环旋转黏性液体射流破碎液滴粒径与速度联合概率密度函数进行了讨论,对圆环旋转黏性液体射流破碎液滴粒径数量密度分布与速度数量密度分布的相关性问题进行了研究.研究结果表明,为了给出正确的圆环旋转黏性液体射流破碎液滴粒径与速度联合概率密度函数,射流守恒约束条件中必须同时包括质量守恒定律、动量守恒定律以及能量守恒定律;破碎液滴粒径的数量密度分布与速度数量密度分布密切相关;射流旋转强度对破碎液滴粒径数量密度与速度数量密度分布结构影响不大,对破碎液滴粒径数量密度和速度数量密度的分布区域影响较大.      

Multicomponent three-dimensional viscous shock layer on blunt bodies with a catalytic surface at angles of attack and yaw

The three-dimensional supersonic flow of nonequilibrium dissociating air past smooth blunt bodies on whose surface heterogeneous chemical reactions are taking place is investigated within the framework of thin viscous shock layer theory. An economical numerical method of solving the equations with an improved order of approximation with respect to the normal coordinate is employed. This method does not require the preliminary solution of the Stefan-Maxwell relations for the diffusion fluxes and makes it possible to calculate flows that do not possess a plane of symmetry. The effect of the angles of attack and yaw, the catalytic reaction model and a number of other parameters of the problem on the pressure, heat flux and equilibrium surface temperature distributions is analyzed with reference to the example of flow past a triaxial ellipsoid.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 143–150, January–February, 1990.Tha authors are grateful to G. A. Tirskii for useful discussions of their results.     

Oscillations of a vessel containing a viscous fluid

The oscillations of a rigid body having a cavity partially filled with an ideal fluid have been studied in numerous reports, for example, [1–6]. Certain analogous problems in the case of a viscous fluid for particular shapes of the cavity were considered in [6, 7]. The general equations of motion of a rigid body having a cavity partially filled with a viscous liquid were derived in [8]. These equations were obtained for a cavity of arbitrary form under the following assumptions: 1) the body and the liquid perform small oscillations (linear approximation applicable); 2) the Reynolds number is large (viscosity is small). In the case of an ideal liquid the equations of [8] become the previously known equations of [2–6]. In the present paper, on the basis of the equations of [8], we study the free and the forced oscillations of a body with a cavity (vessel) which is partially filled with a viscous liquid. For simplicity we consider translational oscillations of a body with a liquid, since even in this case the characteristic mechanical properties of the system resulting from the viscosity of the liquid and the presence of a free surface manifest themselves.The solutions are obtained for a cavity of arbitrary shape. We then consider some specific cavity shapes.