Spontaneous rotation in the exact solution of magnetohydrodynamic equations for flow between two stationary impermeable disks

Magnetohydrodynamic (MHD) flow of a viscous electrically conducting incompressible fluid between two stationary impermeable disks is considered. A homogeneous electric current density vector normal to the surface is specified on the upper disk, and the lower disk is nonconducting. The exact von Karman solution of the complete system of MHD equations is studied in which the axial velocity and the magnetic field depend only on the axial coordinate. The problem contains two dimensionless parameters: the electric current density on the upper plate Y and the Batchelor number (magnetic Prandtl number). It is assumed that there is no external source that produces an axial magnetic field. The problem is solved for a Batchelor number of 0–2. Fluid flow is caused by the electric current. It is shown that for small values of Y, the fluid velocity vector has only axial and radial components. The velocity of motion increases with increasing Y, and at a critical value of Y, there is a bifurcation of the new steady flow regime with fluid rotation, while the flow without rotation becomes unstable. A feature of the obtained new exact solution is the absence of an axial magnetic field necessary for the occurrence of an azimuthal component of the ponderomotive force, as is the case in the MHD dynamo. A new mechanism for the bifurcation of rotation in MHD flow is found.     

Effects of Entropic Wave in Vibroacoustic Problem Using Boundary Element Analysis

A variational formulation for a vibroacoustic problem of a membrane and a viscothermal fluid is investigated in this paper. The formulation combines a variational formulation by integral equations of the fluid, that takes into account the acoustic and entropic waves coupling, with a variational formulation of the membrane. The formulation has been implemented numerically for the problems with axisymmetric geometry. The numerical results are compared to the analytical solution for a circular membrane coupled to a cylindrical cavity filled with air. These results show the validity of numerical implementation and illustrate the thermal effects of air on the membrane-cavity system modes in the micro cavities cases.     

On the flow between two counter-rotating infinite plane disks

This paper studies the boundary-value problem arising from the behaviour of a fluid occupying the region -1≦x≦1 between two rotating disks, rotating about a common axis perpendicular to their planes when the disks are rotating with the same speed Ω₀ but in the opposite sense. The equations which describe the axially symmetric similarity solutions of this problem are $$\varepsilon H^{iv} + HH''' + GG' = 0$$ $$\varepsilon G'' + HG' - H'G = 0$$ with the boundary conditions $$H( \pm 1) = H'( \pm 1) = 0$$ $$G( - 1) = - 1,{\text{ }}G(1) = 1$$ where ?=v/2Ω₀ and v is the kinematic viscosity. The existence of an odd solution is established. This particular solution satisfies many special conditions, for example, G′ (x, ?)>0. Moreover, precise estimates are obtained on the size and behaviour of the solution as ? ↓ 0.     

Pressure pulse on the boundary of an elastic inhomogeneous half-plane

The problem about the motion of a pressure pulse at constant velocity along the boundary of an elastic homogeneous half-plane has been examined in [1–3]. The problem was considered as stationary in [1, 2], while in [3] it was solved by using a Laplace time transformation. An analogous problem is considered in this paper for an elastic half-plane with variable Lamé parameters and density of the medium.     

Secondary thermocapillary motions of soliton type

An exact analytic solution is obtained for the problem of the stability of the axisymmetric thermocapillary motion due to a point heat source of constant power located on the horizontal free surface of a viscous fluid. Analytic expressions are found for monotonic neutral disturbances of hydrodynamic and thermal type. The critical values of the dimensionless source power for disturbances with arbitrary quantum numbersl andm are determined, together with the secondary motions near the stability threshold. An exact solution of the problem of the axisymmetric thermocapillary motion due to a spherical heat source is presented and its stability is investigated. It is shown that it is always possible to select physical heater properties such that for arbitrarily small source power, the axisymmetric motion is unstable relative to the vortex motion. A comparison is made with experiment.Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No.4, pp. 20–27, July–August, 1992.     

Dynamics of a flat cavity in a low-viscosity liquid

The problem of the motion of a cavity in a plane-parallel flow of an ideal liquid, taking account of surface tension, was first discussed in [1], in which an exact equation was obtained describing the equilibrium form of the cavity. In [2] an analysis was made of this equation, and, in a particular case, the existence of an analytical solution was demonstrated. Articles [3, 4] give the results of numerical solutions. In the present article, the cavity is defined by an infinite set of generalized coordinates, and Lagrange equations determining the dynamics of the cavity are given in explicit form. The problem discussed in [1–4] is reduced to the problem of seeking a minimum of a function of an infinite number of variables. The explicit form of this function is found. In distinction from [1–4], on the basis of the Lagrauge equations, a study is also made of the unsteady-state motion of the cavity. The dynamic equations are generalized for the case of a cavity moving in a heavy viscous liquid with surface tension at large Reynolds numbers. Under these circumstances, the steady-state motion of the cavity is determined from an infinite system of algebraic equations written in explicit form. An exact solution of the dynamic equations is obtained for an elliptical cavity in the case of an ideal liquid. An approximation of the cavity by an ellipse is used to find the approximate analytical dependence of the Weber number on the deformation, and a comparison is made with numerical calculations [3, 4]. The problem of the motion of an elliptical cavity is considered in a manner analogous to the problem of an ellipsoidal cavity for an axisymmetric flow [5, 6]. In distinction from [6], the equilibrium form of a flat cavity in a heavy viscous liquid becomes unstable if the ratio of the axes of the cavity is greater than 2.06.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 15–23, September–October, 1973.The author thanks G. Yu. Stepanov for his useful observations.     

Onset of Cavitation in Compressible,Isotropic, Hyperelastic Solids

In this work, we derive a closed-form criterion for the onset of cavitation in compressible, isotropic, hyperelastic solids subjected to non-symmetric loading conditions. The criterion is based on the solution of a boundary value problem where a hyperelastic solid, which is infinite in extent and contains a single vacuous inhomogeneity, is subjected to uniform displacement boundary conditions. By making use of the “linear-comparison” variational procedure of Lopez-Pamies and Ponte Castañeda (J. Mech. Phys. Solids 54:807–830, 2006), we solve this problem approximately and generate variational estimates for the critical stretches applied on the boundary at which the cavity suddenly starts growing. The accuracy of the proposed analytical result is assessed by comparisons with exact solutions available from the literature for radially symmetric cavitation, as well as with finite element simulations. In addition, applications are presented for a variety of materials of practical and theoretical interest, including the harmonic, Blatz-Ko, and compressible Neo-Hookean materials.     

Propagation of a stress pulse in a viscoelastic rod

Experimental work is reported on the propagation of a stress pulse in a viscoelastic waveguide. The data obtained are compared with results of analysis using one-dimensional wave-propagation theory. The waveguide used in this work is a low-density polyethylene rod 1/2 in. in diameter and 30-in. long. Stress input to the waveguide and the resulting particle velocity at three stations are measured using a crystal stress transducer, two Faraday-principle velocity transducers and a capacitor transducer. The experiment is described mathematically as a boundary-value problem formulated in terms of the one-dimensional equation of motion, the strain-displacement relationship, a hereditary constitutive equation and the stress-boundary condition. Fourier transform and inversion yield an integral expression for velocity which is evaluated numerically at three stations using measured values for the stress-boundary condition, material attenuation and phase velocity. The analytical results compare favorably with the experimental data. The one-dimensional theory appears adequate to describe pulse propagation of this type. The attenuation and phase velocity used here are found to be a linear function and a logarithmic increasing function of frequency respectively.     

On the local stability analysis of the approximate harmonic balance solutions

Periodic response of nonlinear oscillators is usually determined by approximate methods. In the "steady state" type methods, first an approximate solution for the steady state periodic response is determined, and then the local stability of this solution is determined by analyzing the equation of motion linearized about this predicted "solution". An exact stability analysis of this linear variational equation can provide erroneous stability type information about the approximate solutions. It is shown that a consistent stability type information about these solutions can be obtained only when the linearized variational equation is analyzed by approximate methods, and the level of accuracy of this analysis is consistent with that of the approximate solutions. It is demonstrated that these consistent stability results do not imply that the approximate solution is qualitatively correct. It is also shown that the difference between an approximate and the next higher order stability analysis can be used to "guess" the role of higher harmonics in the periodic response. This trial and error procedure can be used to ensure the qualitatively correct and numerically accurate nature of the approximate solutions and the corresponding stability analysis.     

The problem of a gas bubble in plane ideal fluid flow

We study the problem of two-dimensional fluid flow past a gas bubble adjacent to an infinite rectilinear solid wall.Two-dimensional ideal fluid flow past a gas bubble on whose boundary surface-tension forces act (or a gas bubble bounded by an elastic film) has been studied by several authors. Zhukovskii, who first studied jet flows with consideration of the capillary forces, constructed an exact solution of the problem of symmetric flow past a gas bubble in a rectilinear channel [1]. However, Zhukovskii's solution is not the general solution of the problem; in particular, we cannot obtain the flow past an isolated bubble from his solution. Slezkin [2] reduced the problem of symmetric flow of an infinite fluid stream past a bubble to the study of a nonlinear integral equation. The numerical solution of this problem has recently been found by Petrova [3]. McLeod [4] obtained an exact solution under the assumption that the gas pressure p₁ in the bubble equals the flow stagnation pressure p₀. Beyer [5] proved the existence of a solution to the problem of flow of a stream having a given velocity circulation provided p₁p₀.We examine the problem of two-dimensional ideal fluid flow past a gas bubble adjacent to an infinite rectilinear solid wall. The solution depends on the value of the contact angle . The existence of a solution is proved in some range of variation of the parameters, and a technique for finding this solution is given. The situation in which =1/2 is studied in detail.     

Rotating disk heat transfer in a fluid swirling as a forced vortex

The paper represents results of an exact solution of a laminar heat transfer problem for a rotating disk in a fluid co-rotating with the disk as a solid body. The angular speed of the fluid is less than the angular speed of the disk. Disks surface temperature varies radially accordingly to a power law. Results for the laminar regime are compared with computations for turbulent heat transfer obtained using an integral method developed earlier. On the basis of the exact solution for laminar flow and basic ideas of the integral methods solution for turbulent flow, an integral method for laminar regime is designed and an approximate analytical solution of the considered problem is derived. Inaccuracies of the laminar approximate solution over the main range of variation of the influencing parameters and Prandtl numbers from 0.71 to 1 do not exceed 2.5%. It is shown that the dependence of the Nusselt number on the ratio of the angular speeds of disk and fluid varying from 0 to 0.3 is weak and has a point of maximum within this region for laminar flow. The obtained results are important in predictions of fluid flow and heat transfer in different types of rotating machinery.     

Simulation of two‐phase flow with moving immersed boundaries

A two‐dimensional multi‐phase model for immiscible binary fluid flow including moving immersed objects is presented. The fluid motion is described by the incompressible Navier–Stokes equation coupled with a phase‐field model based on van der Waals' free energy density and the Cahn–Hilliard equation. A new phase‐field boundary condition was implemented with minimization of the free energy in a direct way, to specifically improve the physical behavior of the contact line dynamics for moving immersed objects. Numerical stability and execution time were significantly improved by the use of the new boundary condition. Convergence toward the analytical solution was demonstrated for equilibrium contact angle, the Lucas–Washburn theory and Stefan's problem. The proposed model may be used for multi‐phase flow problems with moving boundaries of complex geometry, such as the penetration of fluid into a deformable, porous medium. Copyright © 2010 John Wiley & Sons, Ltd.     

On the unsteady motion of a biplane cascade

Many studies have been devoted to the questions of unsteady flow about a moving cascade of profiles which are located at the same distance t from one another. However, the unsteady motion in a fluid of a single-row multiplane cascade consisting of a finite or infinite number of profile systems (groups) has received little study. It is assumed that in such a cascade each i-th system (group) of profiles has its own pitch t_i (i=1, 2,, s) and constant phase shift _i of the oscillations between neighboring profiles. The distance h_i between the multiplane cascade profile groups is different; h_it_i. Such cascades arise, for example, in the solution of the problem of the motion of one or two profiles under the free surface of a weightless liquid at a distance h from a solid wall. In the present paper we consider as an example the unsteady motion of a biplane cascade in an incompressible inviscid fluid. We obtain an expression for the complex flow velocity outside the cascade profiles and outside its vortex trails. The corresponding integral equation is obtained for the unknown function u ()-the discontinuity of the tangential component of the velocity along the vortex trail. The solution of this equation is written out for harmonic oscillations of the profiles.     

管内上随体Maxwell流体非定常流动

本文研究了上随体Maxwell流体在圆管内非定常流动规律,对于上随体Maxwell流体模型,导出了特殊的运动方程,分别应用隐式差分格式和Kantorovich变分法,求得数值解,对两类方法的结果进行比较,揭示了粘弹流效应对管内非定常流动规津的影响,根据上述研究认为,以上的特殊的变分方法适应于研究非定常流动。     

Stability of unsteady rotational motion of a plane layer of ideal fluid with free boundaries

The exact solution of the equations of an ideal incompressible fluid describing the unsteady rotational motion of a plane layer with free boundaries is obtained. For constant vorticity the stability problem is studied in the linear approximation. The asymptotic behavior of the free boundaries of the layer as t is calculated. It is shown that the vorticity of the basic motion stabilizes the boundaries of the layer.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 15–21, March–April, 1986.     

Flow of viscoelastic fluids through a porous channel—I

The flow of viscoelastic fluids through a porous channel with one impermeable wall is computed. The flow is characterized by a boundary value problem in which the order of the differential equation exceeds the number of boundary conditions. Three solutions are developed: (i) an exact numerical solution, (ii) a perturbation solution for small R, the cross-flow Reynold's number and (iii) an asymptotic solution for large R. The results from exact numerical integration reveal that the solutions for a non-Newtonian fluid are possible only up to a critical value of the viscoelastic fluid parameter, which decreases with an increase in R. It is further demonstrated that the perturbation solution gives acceptable results only if the viscoelastic fluid parameter is also small. Two more related problems are considered: fluid dynamics of a long porous slider, and injection of fluid through one side of a long vertical porous channel. For both the problems, exact numerical and other solutions are derived and appropriate conclusions drawn.     

Motion of a solidcontaininga spherical damper

For the purpose of modeling the motion of a solid with a cavity filled with a viscous fluid, M. A. Lavrent'ev [1] has proposed a model in the form of a solid with a spherical cavity in which another solid, spherical in shape, is enclosed. The sphere is separated from the cavity walls by a small, clearance in which viscous forces act (a lubricating film). This simple model with a finite number of degrees of freedom possesses certain mechanical properties of a solid with a cavity containing a viscous fluid. Study of this model is therefore of interest.The present paper examines certain properties of the model, which will be termed a solid with a damper. It is shown that for a highviscosity lubricant the motion of a solid with a damper can be described by the same equations which pertain to the motion of a solid with a spherical cavity filled with a high-viscosity fluid. Expressions relating the parameters of the systems are obtained. If these relations are fulfilled, the systems become mechanically equivalent.The steady motions of a free solid with a damper and their stability conditions are determined.These motions and stability conditions hold for a body with a cavity filled with a viscous fluid [2].     

Nonlinear problem of a thin jet impinging on the surface of a heavy fluid

In a previous note by the author [1] the problem of symmetric forms of contact with oblique incidence of a free jet on a liquid was posed as a problem of the eigenfunctions of a nonlinear integral equation.Here we consider a more general flow scheme-a model of the jet curtain of an air cushion vehicle above the water surface (Fig. 1); the jet of inviscid, incompressible, weightless fluid of density ₁ impinges from a nozzle on the surface of a stationary liquid of density ₂, where, generally speaking, the pressures p₀ and p₁ are different. The problem is two-dimensional. We derive nonlinear integral equations, one of which is analogous to the Nekrasov equation for exact wave theory [2], In the limiting case of a thin jet we obtain a simple differential equation and exact solutions of the problem are constructed.Some data from the numerical calculations for the nonlinear problem of a thin jet curtain are presented in [3]; the problem has been solved in linearized form in [4],The author wishes to thank M. I. Gurevich and G. Yu. Stepanov, to whom he is indebted for his interest in the problem on jet impingement on a liquid and whose advice has been of assistance in improving the present note.     

Motion of an ideal fluid of constant density in the presence of sinks

The problem under consideration is the unsteady motion of an ideal fluid with constant density in an unbounded volume when the velocity divergence is nonzero and is specified by the sink density a which depends on the coordinates r and the time t. It is well known that the introduction of such idealized hydrodynamic objects as a point vortex, a source, or a sink and the related studies of fluid flows are useful in solving a number of specific hydrodynamic problems [1, 2]. There have been many studies of point vortices, and some of the earliest are reviewed in [3], whereas the motion of free point sinks or sources has not been studied. The reason for this situation is that it is hard to find the appropriate hydrodynamic counterparts. The aim of the present paper is to study the basic laws governing the motion of a system of sinks and sources, both point and distributed, and then apply the results obtained to a simulation of thermal convection in a plane horizontal fluid layer consisting, for example, of periodic convective cells. Special attention is given to the asymptotic behavior of as t. Conservation laws for a system of N point sinks are derived and discussed. The qualitative behavior of the system for large t is investigated. Under the assumption of a frozen sink density in the velocity field of the fluid, an evolution equation for is obtained for an arbitrary initial distribution of the velocity divergence. In the case of a finite integrated intensity of the sink density in an unbounded volume, an exact solution of the evolution equation is given for a cylindrically symmetric initial distribution. The asymptotic behavior of this solution as t is studied in three qualitatively different cases. Finally, a steady-state solution of the evolution equation is obtained.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 21–27, July–August, 1976.The author thanks A. A. Zaitsev for his interest in the work, valuable advice, and discussion of the results.     

Motion of a small sphere in a nonhomogeneous flow of an incompressible liquid

We consider the motion of a small sphere in an arbitrary potential flow of an ideal liquid. For the general case we obtain an integral of the equations of motion and a particular solution. We find flows in which the force acting on the sphere is central. We also obtain exact solutions of the equations of motion of the sphere for the cases of stationary flows around a cylinder and around a body of revolution when the forces are noncentral. N. E. Zhukovskii [1] calculated the force acting on a fixed sphere in an arbitrary nonstationary flow. Kelvin [2] obtained the equations of motion of a sphere in a stationary flow of a liquid circulating through a hole in a solid. A formula for the force F, acting on a fixed small body of volume V in a stationary flow with speed v, was obtained by Taylor [3]: F = (T ₀ / v)Vv + 1/2_V v ² Here T₀ is the kinetic energy of an unbounded liquid in which a body moves with velocity v. This problem was solved in [3] through a direct integration of the pressure forces over the surface of the body in a flow defined by multipoles of the first and second orders at infinity.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 5, pp. 57–61, September–October, 1973.