On a convective flow of a binary mixture in a vertical layer

Under study is an invariant solution of the equations of thermal diffusive convection which describes a stationary process of a binary mixture flow in a vertical layer under the action of the pressure gradient and the buoyancy force that depends nonlinearly on temperature and concentration. Some general properties of this solution are established and an existence theorem is proved. Analysis of the numerical solution of the problem is carried out in the cases of a power-law and exponential dependence of the buoyancy force on its argument.     

On the stability of space-periodic convective flows in a vertical layer with sinuous boundaries : PMM vol. 43, no. 6, 1979, pp. 998–1007

The problem of convection in a vertical layer with harmonically distorted boundaries is examined by perturbation theory methods for a small amplitude of sinuosity. The solutions obtained are applicable both in the stability region as well as in the supercritical region of the plane-parallel flow. The stability of the solutions found is investigated with respect to a certain class of space-bounded perturbations that are not necessarily space-periodic. The method of amplitude functions [1], generalized to the case of curved boundaries, is used. The Grashof critical number is found as a function of the period of sinuosity and the form of the neutral curve for the space-periodic motions and their stability region are obtained. It is established that if the deformation period of the boundaries is close to the wavelength of the critical perturbation for the plane-parallel flow or is twice as great, then as the Grashof number grows stability loss does not occur and the motion's amplitude changes continuously (cf. [2 — 4]). A comparison is made with the results of the numerical calculation in [5], An attempt was made in [6] to construct a stationary periodic motion in a layer with weakly-deformed boundaries, in the form of series in powers of a small sinuosity amplitude. However, the solution obtained diverges in a neighborhood of the neutral curve of the plane-parallel flow and approximates unstable motion in the supercritical region of the unperturbed problem. Flows under a finite sinuosity amplitude are calculated by the net method in [5] wherein the stability of the flows was investigated as well, but only with respect to perturbations with wave numbers that are multiples of 2π/l, where l is the length of the calculated region.     

On the stability of fluid motions

The fundamental ideas concerning the linear and nonlinear stability of fluid motions are presented. The thermal convection in a layer of a dielectric fluid subject to an alternative current is studied.     

Unsteady convective boundary layer flow of a viscous fluid at a vertical surface with variable fluid properties

In this paper we present numerical solutions to the unsteady convective boundary layer flow of a viscous fluid at a vertical stretching surface with variable transport properties and thermal radiation. Both assisting and opposing buoyant flow situations are considered. Using a similarity transformation, the governing time-dependent partial differential equations are first transformed into coupled, non-linear ordinary differential equations with variable coefficients. Numerical solutions to these equations subject to appropriate boundary conditions are obtained by a second order finite difference scheme known as the Keller-Box method. The numerical results thus obtained are analyzed for the effects of the pertinent parameters namely, the unsteady parameter, the free convection parameter, the suction/injection parameter, the Prandtl number, the thermal conductivity parameter and the thermal radiation parameter on the flow and heat transfer characteristics. It is worth mentioning that the momentum and thermal boundary layer thicknesses decrease with an increase in the unsteady parameter.     

On thermal stability of a reactive third-grade fluid in a channel with convective cooling the walls

In this paper, the thermal stability of a reactive third-grade liquid flowing steadily between two parallel plates with symmetrical convective cooling at the walls is investigated. The system is assumed to exchange heat with the ambient following Newton’s cooling law and the reaction is exothermic under Arrhenius kinetics, neglecting the consumption of the material. Approximate solutions are constructed for the governing nonlinear boundary value problem using regular perturbation techniques together with a special type of Hermite-Padé approximants and important properties of the velocity and temperature fields including bifurcations and thermal criticality conditions are discussed. It is observed that a combined increase in non-Newtonian parameter and convective cooling enhances the thermal stability of the material.     

On the stability of steady motions of systems with cyclic coordinates

A method of solving stabilization problems by isolating a controlled subsystem of possibly smaller dimension [1, 2] is developed further. The stabilizing action is determined by the solution of an optimal stabilization problem [3] for a linear controlled subsystem. The control that is found is implemented in the form of a feedback loop that uses an estimate [4] of the state vector (or part of it) constructed by measuring the perturbations of the positional coordinates. The stability of the unperturbed motion in a complete closed system is established by reducing the problem to a special case of the theory of critical cases [5, 6] or to the problem of stability under constantly acting perturbations [6].     

On stability boundaries of conservative systems

Stability boundaries of linear conservative systems smoothly dependent on several parameters are studied. Generic singularities appearing on the stability boundaries are classified. Explicit formulae for the approximations to the stability domain at regular and singular points of the boundary are derived. These formulae use information on the system only at the point under consideration (eigenvectors and derivatives of the stiffness matrix with respect to parameters). As an example a buckling problem of a column loaded by an axial force is considered and discussed in detail.     

On the stability of the steady-state motions of systems with quasicyclic coordinates

The stability of the steady-state motions of a system with quasicyclic coordinates under the action of potential and dissipative forces and also forces which depend on the quasicyclic velocities is investigated. The results are applied to the problem of the stability of the steadystate plane-parallel motions of a rotor on a shaft which is set up in elasticated bearings with a non-linear reaction /1/.

The stability of the stationary motions and relative equilibria of systems with a single cyclic (quasicyclic) coordinate has previously been investigated /2/ from a common point of view. The question of the stability of the stationary motions of systems with quasicyclic coordinates under the action of constant and dissipative forces has been considered in /3/. The results obtained in /2/ have been generalized /4/ to systems with several cyclic (quasicyclic) coordinates and, additionally, a third regime of uniform motions, which includes the regime considered in /3/, has also been investigated.     

Numerical study of mixed convection flow of a micropolar fluid towards permeable vertical plate with convective boundary condition

In this article, the mixed convective flow of a micropolar fluid along a permeable vertical plate under the convective boundary condition is analyzed. The scaling group of transformations is applied to get the similarity representation of the system of partial differential equations of the problem and then the resulting equations are solved by using Spectral Quasi-Linearisation Method. This study reveals that the dual solutions exists for certain values of mixed convection parameter. The outcomes are analyzed with dual solutions in detail. Effects of micropolar parameter, Biot number and suction/injection parameters on different flow profiles are discussed and depicted graphically.     

Magnetohydrodynamic stationary and oscillatory convective stability in a mushy layer during binary alloy solidification

For the case of solidification of a bottom cooled binary alloy, the magnetohydrodynamic stationary and oscillatory convective stability in the mushy layer is investigated analytically using normal mode linear stability analysis. In the limit of large Stefan number (S_t), a near–eutectic approximation with large far field temperature is considered in the present research. To ascertain the instability in the mushy layer, the strength of the superimposed magnetic field is so chosen that it corresponds to a given mush Hartmann number (Ha_m) of the problem. The results are presented for various values of mush Hartmann numbers in the range, 0 ≤ Ha_m ≤ 50. The critical Rayleigh number for stationary convection shows a linear relationship with increasing Ha_m. The magnetohydrodynamic effect imparts a stabilizing influence during stationary convection. In comparison to that of the stationary convective mode, the oscillatory mode appears to be critically susceptible at higher values of β (β = S_t/℘² ϒ², ℘ is the compositional ratio, ϒ = 1 + S_t/℘), and vice versa for lower β values. Analogous to the behavior for stationary convection, the magnetic field also offers a stabilizing effect in oscillatory convection and thus influences global stability of the mushy layer. Increasing magnetic strength shows reduction in the wavenumber and in the number of rolls formed in the mushy layer.     

On weakly nonlinear evolution of convective flow in a passive mushy layer

The problem of weakly nonlinear convective flow in a mushy layer, with a permeable mush–liquid interface and constant permeability, is studied under operating conditions for an experiment. A Landau type nonlinear evolution equation for the amplitude of the secondary solutions, which is based on the Landau theory and formulation for the Rayleigh, $R$, number close to its critical value, $R_c$, is developed. Using numerical and analytical methods, the solutions to the evolution equation are calculated for both supercritical and subcritical conditions. We found, in particular, that for $R<R_c$, the amplitude of the secondary solutions decays with time. For $R>R_c$, the tendency for chimney formation in the mushy layer increases with time. In addition, in such a supercritical regime, the basic flow is linearly unstable and we see the presence of steady flow for large values of time. These results suggest a possible slow transition to turbulence in such a flow system.     

Convection in a rotating layer with nearly insulating boundaries

Summary Finite amplitude steady convection in a horizontal layer of fluid heated from below with nearly insulating boundaries and rotating about a vertical axis is investigated. The main result is that the only steady stable convective motion in the form of square pattern becomes unstable once the rotation parameter exceeds the value 7.45. The disturbances which have the highest growth rates are in the form of rolls inclined at an angle of 45° to the basic wave vectors of the steady motion.

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Zusammenfassung Das Problem der stationären Konvektion endlicher Amplitude wird untersucht für eine horizontale von unten erhitzten Flüssigkeitsschicht, die um eine vertikale Achse rotiert. Das Hauptresultat ist, daß die einzige stabile stationäre Konvektionsströmung in der Form von Quadratzellen instabil wird, wenn der Rotationsparameter den kritischen Wert 7.45 übersteigt. Die Strömungen höchster Anwachsrate haben die Form von Rollen, die einen Winkel von 45° mit den Wellenzahlvektoren der stationären Lösung einschließen.

     

On the solution of mixed problems for a fiber with a vertical circular cylindrical cavity

A method of solving paired integral equations that appear in considerations of mixed problems of elasticity and thermoelasticity theory is given, with the help of generalized integral Weber transforms. The paired equations are reduced to an integral Fredholm equation of the second kind on the semiaxis, which have a discontinuous kernel, or to Fredholm equations of the second kind on a finite interval and infinite systems of linear algebraic equations, which are normal in the sense of Poincare-Koch. As an example, contact problems for an inhomogeneous fiber with a cavity are considered. If the fiber is bonded with the elastic half-space, then a second appproach is realized, which is based on a reduction to an equation with a self-adjoint operator, for which some method of sequential iteractions and the Bubnov-Galerkin method are justified.Translated from Dinamicheskie Sistemy, No. 7, pp. 95–102, 1988.     

Wave motions in a spatial boundary layer

The propagation of perturbations in a boundary layer under conditions when the velocity of the approaching stream may be both subsonic and supersonic is considered. With regard to the initial flow in the boundary layer it is assumed that it is stationary and possesses a spatial character which is caused by the external pressure gradient and not by the curvature of the body around which the flow occurs (boundary layers of this kind are extensively used in experiments at the present time). The linearized equations describing waves of vanishingly small amplitude are studied in detail. An analysis of the dispersion relation which links the frequency of the free oscillations with the components of the wave vector reveals a number of special features which are only present in motions with a three-dimensional velocity field. In particular, it is established that the Cauchy problem for the system of linear equations is ill-posed.     

On the orbital stability of pendulum-like motions of a rigid body in the Bobylev-Steklov case

We deal with the problem of orbital stability of pendulum-like periodic motions of a heavy rigid body with a fixed point. We suppose that the geometry of the mass of the body corresponds to the Bobylev-Steklov case. Unperturbed motion represents oscillations or rotations of the body around a principal axis, occupying a fixed horizontal position. The problem of the orbital stability is considered on the basis of a nonlinear analysis.