Stability of pipe flow with blowing

The stability of self-similar flows in a porous circular pipe [1–4] with respect to classical and self-similar perturbations of axisymmetric and nonaxisymmetric form is investigated. The case of blowing through the porous lateral surface is examined. Two formulations of the linear stability problem are considered and stability in the sense of self-similar evolution is also investigated. The limiting stability situations are analyzed. Relations for the critical values of the blowing rate parameters are presented for all the types of perturbations investigated. It is shown that nonaxisymmetric classical perturbations are the most dangerous.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 63–71, January–February, 1989.     

About a stability conjecture concerning unilateral contact with friction

A new notion of stability specially adapted to systems with unilateral contact and Coulomb friction is introduced. Whereas classical stability results in mechanics concern perturbations of the initial data in a classical phase space, we establish here results concerning the trajectories issued from a perturbation of the external forces. In such a context we state a conjecture concerning stability with respect to external forces that we back up by analytical computations in the case of simple models and then by numerical computations for more complex systems.     

Effect of Rotation on the Stability of Advective Flow in a Horizontal Fluid Layer at a Small Prandtl Number

The stability of advective flow in a rotating infinite horizontal fluid layer with rigid bound-aries is investigated for a small Prandtl number Pr = 0.1 and various Taylor numbers for perturbations of the hydrodynamic type. Within the framework of the linear theory of stability, neutral curves describing the dependence of the critical Grashof number on the wave number are obtained. The behavior of finite-amplitude perturbations beyond the stability threshold is studied numerically.__________Translated from Izvestiya Rossiiskoi Academii Nauk, Mekhanika Zhidkosti i Gaza, No. 2, 2005, pp. 29–38.Original Russian Text Copyright © 2005 by Schwarz.     

Stability of plane magnetohydrodynamic couette flow with asymmetrical velocity profile

The hydrodynamic stability of plane magnetohydrodynamic Couette flow with asymmetrical velocity profile formed by a transverse magnetic field is investigated within the framework of the linear theory. The complete spectrum of the small perturbations is studied for the characteristic Hartmann numbers. The perturbations are classified in accordance with their phase velocity at large wave numbers. It is established that the stability of the flow is controlled by only one type of perturbations. The critical parameters of the problem are determined. The instability in question recalls the instability of Hartmann flow against asymmetrical perturbations.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 3, pp. 12–18, May–June, 1971.The author thanks M. A. Gol'dshtik for interest in the work and V. A. Sapozhnikov and V. N. Shtern for useful discussions.     

Propagation of Perturbations in Plane Poiseuille Flow between Walls of Nonuniform Compliance

The evolution of small perturbations in longitudinally nonuniform flows is studied with reference to the problem of the propagation of flow perturbations in a plane channel with walls of variable elasticity. Using the solution of the problem of the receptivity of the flow to local vibrations of the walls, the problem considered can be reduced to the solution of an integral equation for a single function, namely, the complex vibration amplitude of the walls. A numerical method for solving this equation on the basis of a piecewise-linear approximation of the unknown function is proposed. It is shown that the instability wave amplitude changes discontinuously at the junction of the rigid and elastic channel sections. A second method of investigating the process of propagation of perturbations in the flow considered is proposed. This method is based on laws of evolution of perturbations in nonuniform flows and an analytic solution of the problem of perturbation scattering on the junction of walls with different compliance. On the basis of this method the classical stability theory is generalized to include the case of nonuniform flows.     

Stability of poiseuille flow of a viscoplastic fluid with respect to perturbations of finite amplitude

The study made in [1] revealed that the Poiseuille flow of a viscoplastic fluid is stable with respect to infinitely small perturbations. At the same time, it is a known fact that at large Reynolds numbers a turbulent-flow regime of a viscoplastic fluid has been observed experimentally (see [2]). The divergence in the results from the linear theory of hydrodynamic stability of the experimental data indicates the need for investigating the stability of the Poiseuille flow of a viscoplastic fluid with respect to finite amplitude perturbations; this forms the main content of the present paper.     

弹性体表面分层压缩失稳问题的数学弹性力学解

采用数学弹性力学的稳定平衡方程并结合富氏积分变换的方法研究了含表面平行裂纹的弹性体在压缩载荷下的表面分层失稳问题。导出了一级显式的精确齐次奇异积分方程组,然后.通过Gauss-Chebyshev积分公式,得到一组齐次代数方程组,从而求出临界压缩载荷。并将结果与经典的材料力学梁板稳定的研究方法所得结果进行了比较,指出经典方法误差太大而不适于求解此问题。最后,利用数学弹性力学解求出的等效弹性支承常数给出一个简单精确的临界压缩载荷计算公式。     

The nonlinear evolution of two-dimensional and three-dimensional waves in mixing layers

The authors consider problems connected with stability [1–3] and the nonlinear development of perturbations in a plane mixing layer [4–7]. Attention is principally given to the problem of the nonlinear interaction of two-dimensional and three-dimensional perturbations [6, 7], and also to developing the corresponding method of numerical analysis based on the application to problems in the theory of hydrodynamic stability of the Bubnov—Galerkin method [8–14].Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhldkosti i Gaza, No. 1, pp. 10–18, January–February, 1985.     

Stability of elliptical vortices from “Imperfect–Velocity–Impulse” diagrams

In 1875, Lord Kelvin proposed an energy-based argument for determining the stability of vortical flows. While the ideas underlying Kelvin’s argument are well established, their practical use has been the subject of extensive debate. In a forthcoming paper, the authors present a methodology, based on the construction of “Imperfect–Velocity–Impulse” (IVI) diagrams, which represents a rigorous and practical implementation of Kelvin’s argument for determining the stability of inviscid flows. In this work, we describe in detail the use of the theory by considering an example involving a well-studied classical flow, namely the family of elliptical vortices discovered by Kirchhoff. By constructing the IVI diagram for this family of vortices, we detect the first three bifurcations (which are found to be associated with perturbations of azimuthal wavenumber m = 3, 4 and 5). Examination of the IVI diagram indicates that each of these bifurcations contributes an additional unstable mode to the original family; the stability properties of the bifurcated branches are also determined. By using a novel numerical approach, we proceed to explore each of the bifurcated branches in its entirety. While the locations of the changes of stability obtained from the IVI diagram approach turn out to match precisely classical results from linear analysis, the stability properties of the bifurcated branches are presented here for the first time. In addition, it appears that the m = 3, 5 branches had not been computed in their entirety before. In summary, the work presented here outlines a new approach representing a rigorous implementation of Kelvin’s argument. With reference to the Kirchhoff elliptical vortices, this method is shown to be effective and reliable.     

Effect of an electric current on necking in a tensile rod

Necking conditions in a tensile thermoviscoplastic rod with passage through it of an alternating electric current are studied. Modeling is performed with allowance for the complex constitutive relations for the rod material, heat transfer in the rod, and the current distribution across the section of the rod as a function of the current frequency (skin effect). The stability of uniform tension is examined by linear analysis of perturbations using the Routh-Hurwitz theory. The results were refined by nonlinear analysis taking into account the effect of the amplitude curve of perturbations on the stability of plastic deformation. Institute of Mechanics and Applied Mathematics, Rostov State University, Rostov-on-Don 344090. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 40, No. 5, pp. 173–178, September–October, 1999.     

Generation of the tollmien–schlichting wave in a supersonic boundary layer by two sinusoidal acoustic waves

The problem is solved using parabolized equations of stability for threedimensional perturbations of a compressible boundary layer on a flat plate. Nonlinearity is taken into account by quadratic terms that are most significant in estimates of the viscous critical layer of the stability theory. These terms are determined by the total field of two acoustic perturbations, and the equations become linear and inhomogeneous. The calculations are performed for one acoustic wave being stationary in the reference system fitted to the plate for Mach numbers M=2 and 5. Solutions are presented, which are identified very accurately with Tollmien–Schlichting waves at a rather large distance from the plate edge.     

Rayleigh stability of shear flow in relaxing media

The stability of steady-state flow is considered in a medium with a nonlocal coupling between pressure and density. The equations for perturbations in such a medium are derived in the linear approximation. The results of numerical integration are given for shear motion. The stability of parallel layered flow in an inviscid homogeneous fluid has been studied for a hundred years. The mathematics for investigating an inviscid instability has been developed, and it has been given a physical interpretation. The first important results in flow stability of an incompressible fluid were obtained in the papers of Helmholtz, Rayleigh, and Kelvin [1] in the last century. Heisenberg [2] worked on this problem in the 1920's, and a series of interesting papers by Tollmien [3] appeared subsequently. Apparently one of the first problems in the stability of a compressible fluid was solved by Landau [4]. The first investigations on the boundary-layer stability of an ideal gas were carried out by Lees and Lin [5], and Dunn and Lin [6]. Mention should be made of a series of papers which have appeared quite recently [7–9]. In all the papers mentioned flow stability is investigated in the framework of classical single-phase hydrodynamics. Meanwhile, in recent years, the processes by which perturbations propagate in media with relaxation have been intensively studied [10–12].Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 3, pp. 87–93, May–June, 1976.     

About the Linear Stability of the Spherically Symmetric Solution for the Equations of a Barotropic Viscous Fluid under the Influence of Self-Gravitation

This paper is concerned with the question of linear stability of motionless, spherically symmetric equilibrium states of viscous, barotropic, self-gravitating fluids. We prove the linear asymptotic stability of such equilibria with respect to perturbations which leave the angular momentum, momentum, mass and the position of the center of gravity unchanged. We also give some decay estimates for such perturbations, which we derive from resolvent estimates by means of analytic semigroup theory.     

Finite-amplitude convection in mixtures with heat sources proportional to the concentration of one of the components

A numerical investigation has been made into the equilibrium stability with respect to finite perturbations of a mixture with heat sources proportional to the concentration of an active component. The convective motions that develop after the loss of stability were also studied. The equations of thermoconcentration convection were solved by the grid method for a planar region of rectangular shape simulating a convective cell in the horizontal layer. Neutral curves for finite-amplitude perturbations are constructed, the regions of existence of subcritical motions are found, and a comparison with the results of linear theory is made.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 10–16, November–December, 1982.We thank E. M. Zhukovitskii for discussing the results of the paper.     

On the question of the stability of powder combustion in a half-closed space

The influence of gas temperature perturbations on the stability of powder combustion in a rocket chamber is investigated theoretically on the basis of the Zel'dovich-Novozhilov theory of powder combustion. The influence of the bow space adjacent to the burning channel is also examined.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 6, pp. 74–79, November–December, 1971.     

Effects of Surface Tension on the Stability of Dynamical Liquid-Vapor Interfaces

The persistence of sharp and flat interfaces under multidimensional perturbations is investigated from a viewpoint analogous to Majdas treatment of classical shocks. The main novelty here is that jump conditions for the free interfaces take into account surface tension. This means that, unlike classical jump conditions, they are non-homogeneous, containing first-order and second-order differentials of the front. A normal modes analysis shows that neutral modes may propagate along the front. In the standard setting, this would imply a weak stability result, involving energy estimates with loss of derivatives. In our case the lack of homogeneity of the underlying boundary value problem implies that neutral modes can only be of large enough wave length. Suitable frequency cut-offs then yield energy estimates without loss of derivatives – for the constant-coefficients linearized problem, as in the case of uniformly stable classical shocks.     

Construction of the solution of the linear problem of shock wave stability

Within the framework of the linear theory a solution is obtained in explicit form for a solitary plane shock using Fourier and Laplace transforms and assuming only the finiteness of the small perturbations. In the case of three-dimensional flows the small deformations of the shock wave surface are represented in the form of integral functionals, with Poisson kernels, of the initial perturbations of both the shape of the shock wave and the parameters of the flow field beyond it. The solution for plane flows is then constructed by the method of descent. From the equations obtained it follows that: for the region of stability and the intermediate region the solution has a finite domain of dependence on the initial perturbations; despite the fact that the structure of the domain of dependence in these regions is different, at large times the damping of the perturbations proceeds in accordance with a single law at a rate that depends on the dimensionality of the shock front.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. A, pp. 130–138, July–August, 1988.     

Certain types of vibrationally convective instability of a two-dimensional fluid layer in zero gravity

The stability of a new equilibrium configuration possible in a two-dimensional layer of nonisothermal fluid executing high-frequency vibrations in zero gravity is investigated in the framework of the linear theory. A study is made on the basis of the equations of vibrational convection. Instability with respect to one-dimensional and two-dimensional perturbations is studied. An elementary exact solution is obtained for the one-dimensional perturbations. Vibrationally connective instability of a fluid in zero gravity has been studied in a number of papers [1-3].Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 4–7, September–October, 1987.The author expresses his gratitude to G. Z. Gershuni for his constant interest in my work.