Stability of advective flow in an inclined plane fluid layer bounded by solid planes with a longitudinal temperature gradient. 1. Unstable stratification

The stability of steady convective flow in an inclined plane fluid layer bounded by ideally heat conducting solid planes is studied in the presence of a homogeneous longitudinal temperature gradient under unstable stratification conditions where the layer is inclined so that the temperature is higher in the lower part than in the upper part. It is shown that the inclination leads to the transition from critical perturbations to long-wavelength helical perturbations. Flow stability maps are given for the entire range of Prandtl numbers and inclination angles corresponding to unstable stratification.     

Stability of Thermocapillary Flow in a Flat Layer with Allowance for the Soret Effect

The stability of thermocapillary two-component liquid flow is studied taking into account thermal diffusion. An explicit expression is obtained to construct neutral Marangoni numbers under the assumption of monotonicity of perturbations. The thermocapillary and hydrodynamic instability mechanisms are considered. It is shown that plane perturbations are the greatest hazard to the stability of return flow.__________Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 46, No. 5, pp. 86–92, September–October, 2005.     

Long-wave instability of an advective flow in an inclined fluid layer with perfectly heat-conducting boundaries

Stability of a steady convective flow in a plane inclined layer with perfectly heat-conducting solid boundaries in the presence of a uniform longitudinal temperature gradient to long-wave perturbations is studied. The boundaries of the domain of stability to long-wave perturbations are found, and the critical Grashof numbers for the most dangerous even helical perturbations are determined.     

The role of divergent terms in the Frank energy of nematic liquid crystals

The effect of divergent terms in the Frank orientation energy of nematic liquid crystals on the equilibrium state of the director field is studied. Such terms have no effect on the equations of motion or on the equilibrium of the medium under consideration; however, they should be taken into account in the derivation of boundary conditions. It is shown that, in the case of boundary perturbations or in the case of polar orientation angle perturbations, the divergent terms can be considered as a surface energy for the azimuth angle (this energy is similar to the Rapini-Papoular energy). In addition, these terms may cause a deviation of the director in the plane parallel to the boundary. The equilibrium problem for a nematic liquid crystal is considered as an example in the case of small periodic boundary perturbations.     

Instability of an Unbounded Elastic Plate in a Gas Flow

The stability of an unbounded plane elastic plate in gas moving on one side of the plate and at rest on the other is analyzed. The gases are inviscid and in general different. The plate is under tension and has flexural stiffness. It is shown that the system is always unstable to plane sinusoidal perturbations with wave vector parallel to the velocity. As limiting cases, a tangential discontinuity between the two gases and unilateral flow past a plate with constant pressure on the opposite side are considered. In these cases, the conditions of stability to plane perturbations are non-trivial and are investigated below.     

Effect of high-frequency vibrations on the stability of advective flow

The stability of plane convective flow in a horizontal layer with a longitudinal temperature gradient under the action of longitudinal vibrations is considered. The behavior of small normal plane and spiral perturbations is investigated. It is shown that the vibrations enhance the stability with respect to almost all types of perturbations. The sole exception is plane thermal waves whose existence domain extends toward low Prandtl numbers. Perm’. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 1, pp. 16–22, January–February, 1998. The work was supported by the Russian Foundation for Fundamental Research (project No. 94-01-01730).     

Simulation of the instability of an accelerating gaseous envelope

The development of perturbations of the parameters of a dense gaseous envelope traveling with an acceleration driven by a difference in the pressures on either side is investigated numerically. Plane and axisymmetric time-dependent flows of a compressible medium are considered. The effect of both the density of the envelope and the form of the initial perturbations of its shape and motion on the mass cumulation in the compactions formed is studied. The evolutions of the perturbations of a layer and the surface of a contact discontinuity accelerated by an impinging plane shock wave are compared.     

USEFUL RELATIVE MOTION DESCRIPTION METHOD FOR PERTURBATIONS ANALYSIS IN SATELLITE FORMATION FLYING

Nomenclature OXYZEarth’sequatorialinertialreferenceframeωArgumentofperigee SlxyzLeadingsatelliteorbitframeMMeananomaly SfxyzFollowingsatelliteorbitframefTrueanomalyaSemi majoraxisθ=ω fArgumentoflatitude eEccentricitynMeanmotion iOrbitinclinationrSatel…     

Double Transient Phase-Frequency Resonance in Vibratory Systems

Estimations are made of how an elastic structure or a pendulum system affects the accelerated motion of platforms. The platforms move at some angle to the horizontal plane. The perturbed motion of the platforms is represented as the sum of two damped harmonics. An analysis is made of how the initial phases of the harmonic perturbations affect the dynamic amplification factor of vibratory systems for certain angles between the perturbation direction and the horizontal plane. With some combinations of the frequency and dissipation parameters, the motion takes new features called the double transient resonance and antiresonance. They occur under the concurrent and partial actions of the parametric and external perturbations.     

Algorithm for transient growth of perturbations in channel Poiseuille flow

This study develops a direct optimal growth algorithm for three-dimensional transient growth analysis of perturbations in channel flows which are globally stable but locally unstable. Different from traditional non-modal methods based on the OrrSommerfeld and Squire(OSS) equations that assume simple base flows, this algorithm can be applied to arbitrarily complex base flows. In the proposed algorithm, a reorthogonalization Arnoldi method is used to improve orthogonality of the orthogonal basis of the Krylov subspace generated by solving the linearized forward and adjoint Navier-Stokes(N-S) equations. The linearized adjoint N-S equations with the specific boundary conditions for the channel are derived, and a new convergence criterion is proposed. The algorithm is then applied to a one-dimensional base flow(the plane Poiseuille flow) and a two-dimensional base flow(the plane Poiseuille flow with a low-speed streak)in a channel. For one-dimensional cases, the effects of the spanwise width of the channel and the Reynolds number on the transient growth of perturbations are studied. For two-dimensional cases, the effect of strength of initial low-speed streak is discussed. The presence of the streak in the plane Poiseuille flow leads to a larger and quicker growth of the perturbations than that in the one-dimensional case. For both cases, the results show that an optimal flow field leading to the largest growth of perturbations is characterized by high-and low-speed streaks and the corresponding streamwise vortical structures.The lift-up mechanism that induces the transient growth of perturbations is discussed.The performance of the re-orthogonalization Arnoldi technique in the algorithm for both one-and two-dimensional base flows is demonstrated, and the algorithm is validated by comparing the results with those obtained from the OSS equations method and the crosscheck method.     

Some Global Stability Results for Shear Flows of Viscoelastic Fluids

We consider plane shear flows of viscoelastic fluids. For a number of constitutive models, we prove stability of the rest state for perturbations of arbitrary size. We also consider stability of plane Poiseuille flow in a few special cases. This research was supported by the National Science Foundation under Grant DMS-0405810.     

Propagation of unsteady bulk perturbations in an elastic half-plane

At present, the problems of unsteady waves initiated by surface perturbations in an elastic half-space have been studied sufficiently well (see, e.g., [1–5]; a detailed bibliography on this problem can be found in [6]). At the same time, the analytical solutions of the corresponding unsteady problems of bulk perturbations are practically absent. It is these questions as applied to the plane problem that are considered in this paper.     

Instability of Streaming Viscoelastic Fluids in the Presence of ��Effective Interfacial Tension�� Through Porous Medium

The ??effective interfacial tension?? effect on the instability of the plane interface between two uniform, superposed, and streaming Rivlin?CEricksen viscoelastic fluids through a porous medium is considered. The case of two uniform streaming Rivlin?CEricksen viscoelastic fluids separated by a horizontal boundary is studied. In the absence of ??effective interfacial tension??, stability/instability of the system as well as perturbations transverse to the direction of streaming are found to be unaffected by the presence of streaming if the perturbations in the direction of streaming are ignored, whereas for perturbations in all other directions, there exists instability for a certain wave number range. ??Effective interfacial tension?? is able to suppress this Kelvin?CHelmholtz instability for small wavelength perturbations, the medium porosity reduces the stability range given in terms of a difference in streaming velocities.     

Waves generated on an ice cover by a source pulsating in fluid

A point source of variable intensity located at rest in a plane infinitely deep fluid layer under an ice cover is considered. The general expression for perturbations of the fluid-ice interface is obtained. In the case of the long operation in the pulsating regime the wave established on the ice cover is found.     

Exponential estimates of perturbations of rigid-plastic spreading-sink of an annulus

The time evolution of the plane picture of small perturbations imposed on the radial spreading or sink of an annulus made of incompressible ideally rigid-plastic material obeying the Mises–Hencky plasticity criterion is studied. The adhesion conditions are posed on the extending (contracting) boundaries of the annulus in both the ground and perturbed processes. The method of integral relations, which is based on variational inequalities in the corresponding complex Hilbert space, is used to reduce the linearized problem in perturbations to a single relation for quadratic functionals, which permits deriving new exponential upper bounds for the growth or decay of kinematic perturbations. It is shown that the evolution of angular harmonics with distinct numbers is qualitatively distinct.     

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.     

Wave features of a hyperbolic reaction–diffusion model for Chemotaxis

The Extended Thermodynamic theory is used to derive a hyperbolic reaction–diffusion model for Chemotaxis. Linear stability analysis is performed to study the nature of the equilibrium states against uniform and nonuniform perturbations. A particular emphasis is given to the occurrence of the Turing bifurcation. The existence of traveling wave solutions connecting the two steady states is investigated and the governing equations are numerically integrated to validate the analytical results. The propagation of plane harmonic waves is analyzed and the stability regions in terms of the model parameters are shown. The frequency dependence of the phase velocity and of the attenuation is also illustrated. Finally, in order to have a measure of the non linear stability, the propagation of acceleration waves is studied, the wave amplitude is derived and the critical time is evaluated.     

Stability of Rheometric Viscoelastic Fluid Flow with Respect to Shear Perturbations

Flow between two plates is considered for a fluid obeying the DeWitt rheological equation of state with the Jaumann derivative. It is found analytically that the steady-state Couette flow is stable or unstable with respect to plane shear perturbations when the Weissenberg numbers are less or greater than unity, respectively. The flow acceleration stage is studied analytically and numerically, a comparison with the case of an Oldroyd fluid is carried out, and the neutral stability curves are constructed. The fundamental role of perturbations of the type considered among the set of instability types which can act on the fluid in such a flow is noted.     

Vibrational Convection in an Inclined Fluid Layer Heated from Below

The stability of mechanical equilibrium of an inclined fluid layer with respect to three-dimensional perturbations under the action of high-frequency vibration is studied. It is shown that under heating from below the spiral perturbations are always the most dangerous for vibration transverse to the layer. For vertical vibration the stability limit is determined by three-dimensional perturbations whose shape depends in a complicated way on the angle of inclination of the layer and the vibrational Rayleigh number. In the limiting case of a thin vertical layer supercritical vibrational-convective motions are calculated numerically and analytically and scenarios of transition from quasi-equilibrium to irregular motions are studied.     

The spectrum of small perturbations in plane couette-poiseuille flow

The spectrum of small perturbations of plane Couette-Poiseuille flow is studied. The perturbations are classified according to their behavior at large wave numbers. The changes in the spectrum are traced as the transition is made from Poiseuille to Couette flow at fixed Reynolds number. The behavior of the perturbations is considered as a function of the Reynolds number.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 2, pp. 63–67, March–April, 1971.The author wishes to thank M. A. Gol'dshtik for his attention to the paper, V. A. Sapozhnikov for useful discussions, and V. N. Shtern for his great assistance and help with the paper and for useful discussions.