Numerical Investigation of the Second Transition in the Problem of Plane Convective Flow through a Porous Medium

The problem of convective flow through a porous medium in a plane rectangular vessel with a linear temperature profile steadily maintained on the boundary is considered. Single-parameter families of steady-state regimes resulting from the existence of cosymmetry of the corresponding differential equations are investigated using the Galerkin method. The onset and development of instability on these families and the characteristics of convective regimes as functions of the seepage Rayleigh number and the rectangle side ratio are studied. It is shown that the number of regimes which lose stability, the instability type, the number of convective rollers developed, and the heat transfer depend significantly on the vessel geometry. Several bifurcations of single-parameter families of steady-state regimes are identified and investigated.     

Stabilization of the unstable steady-state regimes of a flow reactor

Stabilization of the unstable regime of a flow reactor with distributed parameters is considered. The stabilizing system relates the rate of supply of material to the reactor to the deviation of the output concentration from the steady-state value. An isothermal autocatalytic reaction is chosen as the simplest reaction for which nonuniqueness of the steady-state regimes is possible. It has been found [3] that there are three possible steady-state regimes. Their stability was investigated in [8]. It has been shown that the middle regime is unstable, only the first mode of the small disturbances being unstable over a broad range of the parameters.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 115–119, March–April, 1989.     

On the stability of the shear flow of a viscoelastic fluid with slip along the fixed wall

In this paper we solve the time-dependent shear flow of an Oldroyd-B fluid with slip along the fixed wall. We use a non-linear slip model relating the shear stress to the velocity at the wall and exhibiting a maximum and a minimum. We assume that the material parameters in the slip equation are such that multiple steady-state solutions do not exist. The stability of the steady-state solutions is investigated by means of a one-dimensional linear stability analysis and by numerical calculations. The instability regimes are always within or coincide with the negative-slope regime of the slip equation. As expected, the numerical results show that the instability regimes are much broader than those predicted by the linear stability analysis. Under our assumptions for the slip equation, the Newtonian solutions are stable everywhere. The interval of instability grows as one moves from the Newtonian to the upper-convected Maxwell model. Perturbing an unstable steady-state solution leads to periodic solutions. The amplitude and the period of the oscillations increase with elasticity.     

Filtration convection in an annular domain and branching of a family of steady-state regimes

A plane problem of convection of incompressible fluid saturating a porous annular domain is considered. Using the Darcy model and a finite-differencemethod, which retains the cosymmetry of the original problem, we investigate the branching of a family of steady-state regimes from the mechanical equilibrium. The evolution of convective motions with increase in the Rayleigh number is studied and the onset of instability on the family of steady-state regimes is analyzed. For narrow annular domains, the cosymmetric effect of a delay in the branching of a secondary self-oscillation mode from the family is detected, while for domains of moderate thickness and sectors the monotonic instability is typical.     

Stochasticity formation in a thick spherical layer for different methods of variation of the boundary conditions

The influence of the mode of formation of the flow in a rotating spherical layer on the parameters of steady-state chaotic regimes is studied experimentally near the limit of onset of these regimes. The case of counter-rotating boundaries of a thick spherical layer, whose thickness is equal to the inner-sphere radius, is considered. With the reference to two different flow formation conditions, namely, a sequential and a simultaneous increase in the angular velocity of the spherical boundaries, the possibility of transition to stochastic flow regimes with different turbulent-fluctuation spectra and different values of the correlation dimension is demonstrated.     

Study of the stable and unstable operating regimes of a chemical reactor

The problem of the stabilization of the unstable operating regimes of a flow reactor is considered. It is shown that for the same values of the external parameters up to five steady-state regimes may exist, and that without the use of stabilizing systems it is impossible to realize not only the unstable but also some stable regimes.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 54–59, May–June, 1988.     

Numerical Investigation of the Plane Problem of Convection of a Multicomponent Fluid in a Porous Medium

Scenarios of the development of continuous families of steady-state regimes branching off from mechanical equilibrium are investigated for the plane problem of filtrational convection of a multicomponent fluid saturating a porous block of rectangular cross-section. Convection of two- and three-component fluids is considered and unidirectional and differently directed vertical temperature and concentration gradients are analyzed. A new scenario of the formation of a continuous family of steady-state solutions realized in the case of oscillatory instability of mechanical equilibrium is studied.     

Scenarios of the onset of unsteady regimes in the problem of plane convective flow through a porous medium

The problem of plane convective flow through a porous medium in a rectangular vessel with a linear temperature profile steadily maintained on the boundary is considered. The onset of unsteady regimes is investigated numerically. It is shown that their onset scenarios depend on the vessel dimensions and the seepage Rayleigh number and may be as follows: the generation of stable and unstable periodic regimes as a result of a one-sided bifurcation, the generation of a stable periodic regime as a result of an Andronov-Hopf cosymmetric bifurcation, the formation of a chaotic attractor, the branching-out of a stable quasi-periodic regime from a point of a single-parameter family of steady-state regimes, and the generation of unstable periodic regimes as a result of disintegration of homoclinic trajectories. The specifics of most of the bifurcations mentioned above are attributable to the cosymmetry of the problem considered.     

Admissible Steady-State Regimes of Crack Propagation in a Square-Cell Lattice

In this paper, the authors return to the classical problem of crack propagation in a lattice. The authors study the problems concerned with the possible regimes of stable steady-state crack propagation in an anisotropic lattice. They show that the steady-state crack propagation is impossible for some relations between the strength and elastic properties of the lattice. The authors also discuss the possibility of stable crack propagation at low speeds.     

Investigation of thermogravitational-thermocapillary steady-state convective flow stability at low Prandtl numbers

The stability of gravitational-capillary flow in a square cavity with isothermal vertical and adiabatic horizontal boundaries is investigated. The region of stable regimes in the Grashof number-Marangoni number plane is determined for a fluid with a Prandtl number equal to 0.02. In [1] the stability of steady-state thermogravitational convection regimes in a laterally heated square cavity was numerically investigated. The Galerkin method with a system of coordinate functions constructed as proposed in [2] was used to solve the system of equations of free convection in the Oberbeck-Boussinesq approximation. Below, the variant of the Galerkin method described in [2] is used to investigate the stability of steady-state regimes of free convection flow developing under the combined influence of thermogravitational and thermocapillary forces.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 8–13, March–April, 1990.     

A model of hydraulic transport of a permeable elastic body in a pipe

A steady-state axisymmetric model of transport and transition into a plastic state of a permeable elastic body carried by an incompressible power-law fluid in a vertical pipe flow is constructed. Possible flow regimes are found and the threshold values of the parameters responsible for the change of a regime are determined. Semi-analytical expressions for the fluid velocity distribution in the annulus, the pressure difference, and the steady-state velocity of the body are obtained. The conditions of the transition of the body into a plastic state are studied depending on the governing parameters of the problem.     

Simulation of the Onset of Nonstationarity and Chaos in a Hydrodynamic System Governed by a Small Number of Degrees of Freedom

The hypothesis of the onset of nonstationarity and chaos in a hydrodynamic system as a result of the nonlinear interaction of a small number of degrees of freedom is verified experimentally with reference to fluid convection in a toroidal channel. Regimes of motion of a fluid medium which correspond qualitatively to the Lorenz model are obtained experimentally. These include steady-state regimes, their bifurcations, nonuniqueness and instability, unsteady periodic and stochastic regimes. The spectral and statistical characteristics of the and unsteady processes are investigated, the nature of the onset of chaos is analyzed, and the results are compared with calculations. The mathematical model of the problem is refined.     

Steady states of a continuous-flow adiabatic chemical reactor

Flow reactors are widely used in the chemical industry for purposes of catalytic reactions [1,2]. Calculation of reactors of this type, even in one-dimemional approximation, is complicated and possible only with the use of numerical methods [1, 3]. Such calculations make it possible to find the steady-state distribution of temperature and concentration in the chemical reactor if one exists; in general, however, there may be other steady-state regimes which may be preferable from the standpoint of obtaining a different degree of conversion of the starting product, operating stability, etc.In this connection special interest attaches to the question of the existence and number of steady-state solutions of the system of equations describing the reactor process.This problem was previously considered in [4–7]. Thus, in [4, 5] it was pointed out that in certain special cases more than one steady-state regime may exist. In [6, 7] the question of sufficient conditions of uniqueness was investigated. In [7] it was shown that the steady-state regime is unique in the ease of short reactors or a dilute mixture of reactants. In [8] the problem of the existence and uniqueness of the steady-state regime was examined for a chain reaction model with direct application of the general theorems of functional analysis.The present paper includes an analysis of a very simple mathematical model of an adiabatic chemical reactor in which an exothermic or endothermie reaction takes place. It is established that in the case of an endothermic process a unique steady-state regime always exists. In the exothermic case the problem of the steady-state regime also always has a solution which, however, may be nonunique; the possibility of the existence of several steady-state regimes, associated with the form of the temperature dependence of the heat release rate, is substantiated.The authors thank G. I. Barenblatt, A. I. Leonov, L. M. Pis'men, and Yu. I. Kharkats for discussing and commenting on the work.     

The Onset and Nonlinear Regimes of Convection in a Two-Layer System of Fluid and Porous Medium Saturated by the Fluid

The onset of convection and its nonlinear regimes in a heated from below two-layer system consisting of a horizontal pure fluid layer and porous medium saturated by the same fluid is studied under the conditions of static gravitational field. The problem is solved numerically by the finite-difference method. The competition between the long-wave and short-wave convective modes at various ratios of the porous layer to the fluid layer thicknesses is analyzed. The data on the nature of convective motion excitation and flow structure transformation are obtained for the range of the Rayleigh numbers up to quintuple supercriticality. It has been found that in the case of a thick porous layer the steady-state convective regime occurring after the establishment of the mechanical equilibrium becomes unstable and gives way to the oscillatory regime at some value of the Rayleigh number. As the Rayleigh number grows further the oscillatory regime of convection is again replaced by the steady-state convective regime.     

Condition of the onset and nonlinear regimes of convection of a three-component isothermal mixture in a porous rectangle under modulation of the concentration gradient

Convection of an isothermal three-component mixture saturating a porous block with a rectangular cross-section is investigated numerically when the concentration gradient of one of the components is modulated about a certain mean value. The shape of the neutral curves separating the growing and decaying disturbances in the amplitude-modulation frequency plane is investigated for various diffusion Rayleigh numbers. The results of calculations based on the linearized and complete systems of equations which describe convection of a mixture in a porous medium are analyzed and compared. The supercritical regimes of concentration convection are studied for steady-state boundary conditions and in the case of a periodic modulation of the concentration gradient of one of the components.     

Establishment of Surface Waves Generated by Hydrodynamic Singularities in Plane Flow

A method of direct proof of the existence of steady-state regimes in classical problems of small waves on the surface of a plane flow of ideal heavy fluid of infinite depth accompanying the uniform motion of a hydrodynamic singularity inside the flow or the operation of a pulsating immersed source is proposed.     

Multi-degree-of-freedom systems with a Coulomb friction contact: analytical boundaries of motion regimes

Nonlinear Dynamics - This paper proposes an approach for the determination of the analytical boundaries of continuous, stick-slip and no motion regimes for the steady-state response of a...     

Flow pattern recognition in heated vertical channels: Steady and transient conditions

Experimental work was carried out to determine the flow pattern map in vertical heated pipes under steady-state and transient conditions, using Freon 12 in forced convective flow as working fluid and optical probes for the measurements Existing maps are based on adiabatic tests, steady-state conditions, and fluids different from Freon 12. Signals from optical probes (whose response is based on the variations in fluid refractive index) are analyzed in terms oflocal void fraction, using either the probability density function (PDF) or the ratio between the average and maximum values of the signal. From the analysis of the experimental measurements the definition of a map for annular and intermittent flow regimes was achieved. The map turned out to be in good agreement with the Weisman and Kang map developed in adiabatic, steady-state conditions Qualitative results for the transient conditions are also presented.     

Direct numerical simulation of the laminar-turbulent transition in a thick spherical layer

The results of a numerical study of the laminar-turbulent transition in unsteady isothermal three-dimensional flows of viscous incompressible fluid in a thick spherical layer between counter-rotating spherical boundaries are presented. The calculations are performed for the governing parameters corresponding to the experimental data [1, 2]. The numerical investigations include both solving the complete system of Navier-Stokes equations and analyzing the linear stability of steady-state axisymmetric flows with respect to three-dimensional disturbances. A stochastic flow regime is calculated for the first time. The limits of existence of different flow regimes and the hysteresis regions are found. The spatial flow patterns and frequency characteristics are obtained, which makes it possible to extend and refine the existing experimental data.