Magnetohydrodynamic laminar flow between porous disks

The steady two-dimensional laminar flow of an incompressible conducting fluid between two parallel circular disks in the presence of a transverse magnetic field is investigated. A solution is obtained by perturbing the creeping flow solution and it is valid only for small suction or injection Reynolds numbers. Expressions for velocity, induced magnetic field, pressure, and shear stress distribution are determined and are compared with the creeping flow and hydrodynamic solutions. It is found that the overall effect of the magnetic field on the flow is the same as that in the Hartmann flow.Nomenclature stream function - 2h channel width - z, r axial and radial coordinates - radius of the disk - U _r radial component of velocity - U _r average velocity in the radial direction, U _r d - U _z axial component of velocity - U ₀ injection or suction velocity - dimensionless axial coordinate, z/h - f() function defined in (8) - density - coefficient of kinematic viscosity - electrical conductivity - magnetic permeability - H ₀ impressed magnetic field - h _r induced magnetic field, H _r /H ₀ - M Hartmann number, H ₀ h(/)^1/2 - R Reynolds number, U ₀ h/ - R _m magnetic Reynolds number, U ₀ r - A constant defined in (15) - K constant defined in (27) - C ₂ constant defined in (26) - p pressure - C _p pressure coefficient - C _f skin friction coefficient     

Unsteady flow past a cascade of staggered thin profiles

Many studies have been made of plane flow of an incompressible inviscid fluid past a cascade of profiles with arbitrary stagger angle 0. For example, in the particular case of the motion of a cascade with the stagger angle at zero-oscillation phase-shift angle =0 Khaskind [1] determined the unsteady lift force theoretically by isolating the singularities with the Sedov method [2], applying a conformal mapping to the cascade of unstaggered flat plates. Belotserkovskii et al. [3] calculated the over-all unsteady aerodynamic characteristics of a cascade in the particular case =0 and for any on a computer by the method of discrete vortices, and for the more general case (0) Whitehead [4] has done the same using a vortex method. Gorelov and Dominas [5] calculated the over-all unsteady force and moment coefficients of a profile in a cascade with stagger angle 0 and phase shift 0.The calculation method was based on unsteady theory for a slender isolated profile whose flow pattern is known, with subsequent account for the interference of the profiles and the vortex wakes behind them.In the present study the singularity isolation method [2] is extended to slender profiles with arbitrary stagger angle 0 and arbitrary phase shift 0 of the oscillations between neighboring profiles. It is shown that the solution reduces to the solution of a Fredholm integral equation of the first kind in terms of the sum of the tangential velocity components along the profile. It is found that the relative effect of the unsteady flow due to the system of vortex trails behind the cascade with stagger may be determined without solving this integral equation. However, this solution must be found to calculate the added masses of the cascade and the total magnitudes of the unsteady forces. It is found that regularization transforms the integral equation of the first kind to an integral equation of the second kind, for which solution methods are known.Thus the expressions for the unsteady forces are determined in the form of separate terms, each of which has a physical significance: as a result we obtain finite formulas (improper integrals) for calculating the variable forces; from these formulas are derived the asymptotic expressions for the forces in the limiting cases of high and low solidities and Strouhal numbers, which as a rule are lost in numerical calculations. The proposed method may be considered as one of the techniques for improving the convergence of the numerical methods (elimination of singularities). Moreover, this method may be used to solve the problems of unsteady flow past cascades of arbitrary systems of slender profiles for various profile incidence angles relative to the x-axis and in the presence of a finite cavitation zone on the profiles.The limited practical application of this method is explained by the extreme theoretical difficulties in its applications to cascades with stagger angle. In the present studies these difficulties are examined using the example of a cascade with stagger angle.     

Mass transfer effects on flow past a vertical oscillating plate with variable temperature

An exact solution to the flow of a viscous incompressible fluid past an infinite vertical oscillating plate, in the presence of a foreign mass has been derived by the Laplace-transform technique when the plate temperature is linearly varying as time. The velocity profiles are shown on graphs and the numerical values of the skin-friction are listed in a table. It is observed that the skin-friction increases with increasingSc, t orGr but decreases with increasingGm ort, whereSc (the Schmidt number), (frequency),t (time),Gr (the Grashof number) andGm is (the modified Grashof number) andt.

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Einfluß von Stoffübergang auf die Strömung entlang einer senkrechten oszillierenden Platte veränderlicher Temperaturen

Zusammenfassung Mit Hilfe der Laplace-Transformation wird eine exakte Lösung für die Strömung einer zähen, inkompressiblen Flüssigkeit entlang einer unendlich ausgedehnten, senkrechten, oszillierenden Platte gewonnen, wobei die Einwirkung eines Feststoffs Berücksichtigung findet und die Plattentemperatur linear mit der Zeit veränderlich sein soll. Die Geschwindigkeitsprofile sind in Diagrammen dargestellt und die numerischen Werte der Reibungsschubspannung in einer Tabelle. Letztere wächst mit der Schmidt-ZahlSc, der Grashof-ZahlGr und dem Produkt aus Frequenz und Zeitt; sie nimmt ab, wennGm (die modifizierte Grashof-Zahl) undt zunehmen.

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Nomenclature C species concentration in the fluid near the plate - C ^– species concentration in the fluid away from the plate - C _W species concentration at the plate - C _p specific heat at constant pressure - D chemical molecular diffusivity - Gm modified Grashof number - Gr Grashof number - g acceleration due to gravity - K thermal conductivity - P Prandtl number - Sc Schmidt number - T Temperature of the fluid near the plate - T _W temperature of the plate - T temperature of the fluid far away from the plate - t time - u velocity of the fluid in the upward direction - U ₀ amplitude of oscillation - x coordinate axis along the plate in the vertically upward direction - y coordinate axis normal to the plate Greek symbols skin-friction - viscosity - coefficient of volume expansion - ^* coefficient of species expansion - density - frequency     

Heat Transfer on the Surface on a Spherically Blunted Cone Exposed to a Supersonic Spatial Flow with Injection of a Coolant Gas

The heattransfer processes in a supersonic spatial flow around a spherically blunted cone with allowance for heat overflow along the longitudinal and circumferential coordinates and injection of a coolant gas are studied numerically. The prospects of using highly heatconducting materials and injection of a coolant gas for reduction of the maximum temperatures at the body surface are demonstrated. The solutions of the direct and inverse problems in one, two, and threedimensional formulations for different shell materials are compared. The error of the thinwall method in determining the heat flux on the heatloaded boundary of the body is estimated.     

Stability, Paths, and Dynamic bending of a Blunt Body of Revolution Penetrating into an Elastoplastic Medium

The deep penetration of a thin body with a blunt nose and rear into a lowstrength medium is explored. The motion of the body is described by a system of autonomous integrodifferential equations using the physical model of a separated asymmetric flow over the body and the localinteraction method. An analytical calculation of the Lyapunov stability boundary for straightline motion is performed for bodies with a parabolic meridian. The dependences of the dynamic stability of the body on various parameters are studied numerically. Curved motion paths are constructed in the region of instability, and the classification of paths proposed in previous studies of the motion of pointed bodies is confirmed. It is shown that an reverse ejection is possible when a blunt impactor enters a semiinfinite target. It is established that there is a fundamental possibility of attaining a path close to a specified one and that there is a weak dependence of motion characteristics with a developed separation on the separation angle. Examples are given of calculations of the evolution of the lateral load, the transverse force and moment, and the strength margin of the body using the theory of dynamic bending of a nonuniform rod.     

Thixotropic behaviour of concentrated bauxite residue suspensions

An experimental study on the time-dependent rheology of highly concentrated and flocculated suspensions of bauxite residue (red mud) is presented. Both the thixotropic breakdown with shear and recovery at rest have been quantitatively examined using a vane-shear instrument and a capillary rheometer. It is demonstrated that both the yield stress and the apparent viscosity of the material can be drastically reduced, by orders of magnitude, by shear-induced agitation with a simple anchor impeller. The rate of thixotropic decay is a function of solids concentration, agitation time and speed. With prolonged agitation, the flow properties are brought to an equilibrium level characterized by a finite yield stress and a shear rate-dependent (shear-thinning) viscosity. In the absence of shear, the yield stress gradually increases with time but at a rate considerably slower than the rate of decay in the shear field. Even after an extended rest period of several months, only a fraction of the initial yield stress can be regained. The observed behaviour has been interpreted in terms of a time-dependent but non-reversible transformation of a network structure of flocculated particles initially developed in the red mud. Using a structural kinetic approach, models have been developed for correlating the experimental kinetic data. The paper concludes with a discussion on the practical consequences of the work. D Diameter of capillary tube - k ₁ Breakdown rate constant in eq. (2) - k ₁ Modified rate constant in eq. (7) - k ₂ Recovery rate constant in eq. (8) - N Impeller rotational speed - P Pressure - t Agitation (mixing time) - t Resting (aging) time - V Average velocity - Average mixing shear rate - Apparent viscosity ( _w/(8V/D)) - _w Wall shear stress - _y Yield stress - _y ^E Yield stress at infinite resting time - Dimensionless structural parameter - e Equilibrium condition (t ) - o Initial condition (t = 0) - Conditions att and      

Physical properties of the flow in the separation region for three-dimensional interaction of a boundary layer with a shock wave

In a supersonic stream we consider the three-dimensional flow in the plane of symmetry in the region of interaction of a boundary layer with a shock wave which arises ahead of an obstacle mounted on a plate. The principal characteristic of this flow is the penetration of a filament of the ideal fluid within the separation zone and the formation on the surface of the plate and obstacle of narrow segments with high pressures, high velocity gradients, and large heat transfer coefficients.Pressure distribution measurements were made, shadow and schlieren photos were taken, and photographs of the flow pattern on the surface were made using dye coatings and low-melting models. The basic physical characteristics of the separation flow are established. The independence of the separation zone length of the boundary layer thickness is shown. Local supersonic flows are detected in the separation region, flow regimes are identified as a function of the angle of encounter of the separating flow with the obstacles, characteristic flow zones in the interaction region are identified.Notation s coordinate of separation point on the plate - l length of separation zone - H obstacle height - d obstacle transverse dimension - u freestream velocity - velocity gradient on stagnation line of obstacle - b jet width - compression shock standoff from the body - p static pressure - p_* pressure at stagnation point on obstacle - density - viscosity coefficient - boundary-layer thickness - compression shock angle - effective angle of separation zone - setting angle of obstacle on plate - M Mach number - R Reynolds number - P Prandtl number     

An “ideal equivalent gas method” for the study of shock waves in supersonic real gas flows

Summary A method developed by the author for the systematic study of the thermodynamic and dynamic properties of the gas behind a shock wave is reported.The method is applicable to supersonic flow regimes for which the excitation, dissociation and ionization effects invalidate the usually adopted hypothesis of ideal gas.An Ideal Equivalent Gas, having the ratio of the specific heats _s dependent on Mach number and altitude of flight is postulated.On the basis of the mass, momentum and energy conservation equations, valid through the shock wave, the relations defining the thermodynamic and dynamic state of the gas behind the shock wave are derived. These relations establish an extension of the classic relations valid for the ideal gas and reduce to them identically for _s=.The dependence of the ratio of specific heats _s of the Ideal Equivalent Gas on Mach number and altitude has been established, over a wide range, on the basis of the real gas solutions derived by Huber.

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Sommario Nella presente nota viene esposto un metodo sviluppato dall'autore per lo studio sistematico dello stato termodinamico e dinamico del gas a valle di un'onda d'urto in regime supersonico, allorchè cioè gli effetti dell'eccitazione dei gradi di libertà vibrazionali delle molecole e della loro dissociazione e successiva ionizzazione invalidano l'ipotesi di gas ideale generalmente adottata.Viene definito un gas ideale equivalente avente rapporto dei calori specifici _s funzione del numero di Mach e della quota di volo ed in base alle equazioni di conservazione della massa, della quantità di moto e dell'energia, valide attraverso all'onda d'urto, vengono derivate delle relazioni definenti lo stato termodinamico e dinamico del gas a valle dell'onda d'urto. Tali relazioni costituiscono una estensione delle classiche relazioni dell'urto valide per il gas ideale alle quali si riducono per _s=.La dipendenza del rapporto dei calori specifici _s del gas ideale equivalente, dal numero di Mach e dalla quota è stata stabilita sulla base delle soluzioni ottenute da Huber per il gas reale.

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Waves in a plasma with Hall dispersion

In a magnetohydrodynamic approximation, an investigation is made of the propagation of waves in a plasma, whose characteristic frequency is much less than the collision frequency of the electrons _e ^–1. It is assumed that the magnetic field is sufficiently strong so that the equality _ee1 will be satisfied, where _e is the cyclotron frequency of the rotation of the electrons. With large magnetic Reynolds numbers (R_m1), which are characteristic for many astrophysical problems, this latter condition leads to a need to take account of dispersion effects connected with Hall currents, in the absence of Joule dissipation. The dispersion equation for the propagation of small perturbations is analyzed in the limiting cases of weak dispersion and of a wave propagating along the magnetic field. In the case of weak dispersion, an equation is derived for nonlinear waves. The solutions are found in the form of stationary solitons. The region of such solutions is analyzed. A typical example of a medium with Hall dispersion is an interplanetary plasma, in which the parameter _ee is generally great.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 108–113, May–June, 1974.     

Supersonic nonequilibrium flow past segmental bodies of a mixture simulating the atmosphere of venus

Calculations of the flow of the mixture 0.94 CO₂+0.05 N₂+0.01 Ar past the forward portion of segmentai bodies are presented. The temperature, pressure, and concentration distributions are given as a function of the pressure ahead of the shock wave and the body velocity. Analysis of the concentration distribution makes it possible to formulate a simplified model for the chemical reaction kinetics in the shock layer that reflects the primary flow characteristics. The density distributions are used to verify the validity of the binary similarity law throughout the shock layer region calculated.The flow of a CO₂+N₂+Ar gas mixture of varying composition past a spherical nose was examined in [1]. The basic flow properties in the shock layer were studied, particularly flow dependence on the free-stream CO₂ and N₂ concentration.New revised data on the properties of the Venusian atmosphere have appeared in the literature [2, 3] One is the dominant CO₂ concentration. This finding permits more rigorous formulation of the problem of blunt body motion in the Venus atmosphere, and attention can be concentrated on revising the CO₂ thermodynamic and kinetic properties that must be used in the calculation.The problem of supersonic nonequilibrium flow past a blunt body is solved within the framework of the problem formulation of [4].Notation V body velocity - shock wave standoff - universal gas constant - ratio of frozen specific heats - hRt/m enthalpy per unit mass undisturbed stream P pressure - density - T temperature - m molecular weight - c_p specific heat at constant pressure - (X) concentration of component X (number of particles in unit mass) - R body radius of curvature at the stagnation point - _j rate of j-th chemical reaction shock layer P V ² pressure - density - TT temperature - mm molecular weight Translated from Izv. AN SSSR. Mekhanika Zhidkosti i Gaza, Vol. 5, No. 2, pp. 67–72, March–April, 1970.The author thanks V. P. Stulov for guidance in this study.     

General expressions for unsteady forces in a lattice of profiles

A large number of studies have been devoted to the unsteady flow of a viscid incompressible fluid past a lattice of thin profiles and the determination of the resulting aerodynamic forces and moments. For example, in the particular case of the motion of a lattice with stagger with zero phase shift of the oscillations between neighboring profiles, Haskind [1] determined the unsteady lift force and moment. Popescu [2] suggested expressions for the force and moment in the case when =0 and =0, using the method of conformal mapping. Samoilovich [3] obtained equations for the unsteady lift force and moment by the method of the acceleration potential for phase shift =0 and = of the oscillations between neighboring profiles. Musatov [4] used an electronic digital computer to calculate the overall unsteady aerodynamic characteristics of a grid by the vortex method, taking into account the amplitude of the oscillations and the initial circulation for =m (m1). Gorelov [5] determined the coefficients of the over-all unsteady aerodynamic force and moment of a profile in a lattice with the stagger and any value of =m. He used a method based on the unsteady flow past an isolated profile with subsequent account for the interference of the profiles in the lattice.In the following we find general expressions for the unsteady lift force and moment acting on a lattice moving in an incompressible fluid with the constant velocity U. These formulas generalize the known formulas for the isolated profile [6]. The profiles of a staggered grid (Section 1) are considered to be thin and slightly curved, and perform oscillations with a phase shift of the oscillations between neighboring profiles. The method of separation of singularities is used to obtain the solution in closed form. The coefficients of the expansion of the complex velocity in a series in the derivatives of a function are calculated. An integral equation relative to the unknown tangential velocity component in the wake is derived (Section 2), and its analytic solution is given (Section 3). For =0 the solution coincides with the solution obtained earlier in [7]. Expressions are obtained for the forces and moments (Section 4) in the form of four terms. The first two terms determine the force and moment for motion with constant circulation, and the last two determine these characteristics for motion with variable circulation. The suction force acting at the leading edges of the profiles is found in a general form. Particular cases of closely and widely spaced lattices are considered. Computational results are presented.     

On natural convection from a vertical plate with a prescribed surface heat flux in porous media

This paper presents a theoretical and numerical investigation of the natural convection boundary-layer along a vertical surface, which is embedded in a porous medium, when the surface heat flux varies as (1 +x ²)), where is a constant andx is the distance along the surface. It is shown that for > -1/2 the solution develops from a similarity solution which is valid for small values ofx to one which is valid for large values ofx. However, when -1/2 no similarity solutions exist for large values ofx and it is found that there are two cases to consider, namely < -1/2 and = -1/2. The wall temperature and the velocity at large distances along the plate are determined for a range of values of .Notation g Gravitational acceleration - k Thermal conductivity of the saturated porous medium - K Permeability of the porous medium - l Typical streamwise length - q _w Uniform heat flux on the wall - Ra Rayleigh number, =gK(q _w /k)l/(v) - T Temperature - Too Temperature far from the plate - u, v Components of seepage velocity in the x and y directions - x, y Cartesian coordinates - Thermal diffusivity of the fluid saturated porous medium - The coefficient of thermal expansion - An undetermined constant - Porosity of the porous medium - Similarity variable, =y(1+x ) ^/3/x ^1/3 - A preassigned constant - Kinematic viscosity - Nondimensional temperature, =(T – T )Ra^1/3 k/qw - Similarity variable, = =y(log_e x)^1/3/x ^2/3 - Similarity variable, =y/x ^2/3 - Stream function     

Flow in vicinity of blunt body stagnation point in diverging hypersonic stream

In the hypersonic thin shock layer approximation for a small ratio k of the densities before and after the normal shock wave the solution of [1] for the vicinity of the stagnation point of a smooth blunt body is extended to the case of nonuniform outer flow. It is shown that the effect of this nonuniformity can be taken into account with the aid of the effective shock wave radius of curvature R_*, whose introduction makes it possible to reduce to universal relations the data for different nonuniform outer flows with practically the same similarity criterion k. The results of the study are compared with numerical calculations of highly underexpanded jet flow past a sphere.Notations x, y a curvilinear coordinate system with axes directed respectively along and normal to the body surface with origin at the forward stagnation point - R radius of curvature of the meridional plane of the body surface - uV, vV., , p V ² respectively the velocity projections on the x, y axes, density, and pressure - and V freestream density and velocity The indices =0 and=1 apply to plane and axisymmetric flows Izv. AN SSSR, Mekhanika Zhidkosti i Gaza, Vol. 5, No. 3, pp. 102–105, 1970.     

Equilibrium of a flexible permeable shell in a supersonic gas flow

The plane steady-state regime of uniform supersonic ideal-gas flow around a uniaxial absolutely flexible open permeable shell is investigated numerically [1]. The effect of the angle of attack and the degree of nonuniformity of the permeability distribution on the aerodynamic characteristics and the equilibrium shape of the shell is determined for various methods of shell attachment. Approximate localization relations, which take into account the dependence of the distributed load on a concave permeable screen on its degree of permeability and the free-stream parameters, are formulated. An example of the application of these relations to the solution of three-dimensional problems of the equilibrium of permeable shells of the circular dome type in a supersonic flow [2] is given.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 152–158, September–October, 1989.     

Flow over a body with development of a steady separation zone as re → ∞

This paper discusses formulation of the total problem of flow of an incompressible liquid over a body, with formation of a closed stationary separation zone as Re . The scheme used is based on the method of matched asymptotic expansions [1]. Following [1], it is postulated that the separated zone is developed (i.e., it is not infinitely fragmented and does not vanish as Re ), and the flow inside it has a definite degree of regularity with respect to Re. With these hypotheses we can use the Prandtl-Batchelor theorem [2], which states that, in the limit as Re , a region of circulating flow becomes vortex flow of an inviscid liquid with constant vorticity . Therefore, a basis for constructing matched asymptotic expansions is the vortex-potential problem (the problem of determining a stream function , satisfying the equation = 0 in the region of translational motion and the equation = in a certain region, unknowna priori, of circulating motion). In the general case the solution of the vortex-potential problem depends on two parameters: the total pressure p_o and the vorticity in the separated zone. These parameters appear in the condition for matching the solutions of the first and second boundary-layer approximations (at the boundary of the separated zone for the end Re values) with the corresponding solutions for the inviscid flow. It is shown in the present paper that the conditions for matching the cyclic boundary layer with the external translational flow are the same additional relations which allow us to close the total problem. Thus, in using the method of matched asymptotic expansions to solve the problem of flow over a body with closed stationary separation zones one must simultaneously consider no less than two approximations.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 28–37, March–April, 1978.The authors thank G. Yu. Stepanov for discussion of the paper and valuable comments.     

Experimental study of stability of strongly underexpanded laminar free and impact jets

The growth rates of density fluctuations in the mixing layer of strongly underexpanded lowdensity jets are measured by the electronbeam technique within the range of Reynolds numbers Re_L = 50 – 230. Regimes of selfinduced oscillations are determined for these flow conditions in the case of a jet incident normally onto a finitesize target. Results on the frequency of pressure oscillations on the target are obtained and compared with the spectra of the growth rates of density fluctuations in the mixing layer of free jets. It is shown that selfinduced oscillations cannot be sustained because of the development of instability in the mixing layer of the jet.     

Effect of nose shape on hypersonic flow past slender blunt bodies at an angle of attack

We examine some characteristics of hypersonic flow past slender blunt bodies of revolution at a small angle of attack 1, where is the relative body thickness. It is shown that, within the framework of hypersonic theory, for a correct-consideration of the effect of the conditions in the transitional section between the nose and the lateral surface it is necessary, in the general case, to specify the circumferential distribution of the force effect for the nose and the mass of the gas. For small , the effect of the nose, just as in two-dimensional flows [1–4], shows up only through its drag coefficient c_x, for =0. On this basis, the similarity law [1–4] for flow past such bodies, with arbitrary form of the lateral surface and differing in the shape of the nose blunting, which is valid over the entire disturbed region, with the exception of a small vicinity of the nose, is extended to the case in question.The notation r₀ and L maximum nose radius and characteristic body length - V, M, and density, velocity, Mach number, and adiabatic exponent of the gas in the approaching stream - , V²i, and V²p density, enthalpy, and pressure - x, r, and coordinate system of the cylindrical body with its center at the transitional section between the nose and the side surface - Vu, Vv, and Vw corresponding velocity components     

Model of a plastically compressible material and its application to the analysis of the compaction of a porous body

This paper considers a model of a plastically compressible porous medium with a cylindricaltype yield condition and its associated constitutive relations, which ensure independent mechanisms of shear and compaction of the porous material. This allows one to use the wellknown theorems of plastic theory to analyze plastically compressible media and obtain analytical solutions for a number of boundaryvalue problems, including those taking into account conditions on strongdiscontinuity surfaces. Results from fullscale studies of the structural periodicity of noncompact materials using wavelet analysis were employed to choose a physical model for a porous body and determine the properties and dimensions of a representative volume. The problem of extrusion of a porous material through a conical matrix was solved.     

Direct derivation of the two-shaft system balance conditions for the second order reciprocal inertia forces on the V-type eight cylinders internal combustion engines with a plane crankshaft

By employing the four shafts balance concept paper [1] has reported a balance regime for the second order reciprocal inertia forces on the V-type eight cylinder internal combustion engines with a plane crankshaft. Thereafter, paper [2] has acquired a two-shafts balance regime, but through a rather tedious roudabout degenerating manipulation. The present article has, but starting out directly from the two-shafts balance concept, successfully acquired the same results as those in paper [2]. In addition, we propose, herein, a third balance system which might be, in general, called the slipper balance regime.     

Flow stability of a Grad-model liquid layer on an inclined plane

A study is presented of the flow of stability of a Grad-model liquid layer [1, 2] flowing over an inclined plane under the influence of the gravity force.It is assumed that at every point of the considered material continuum, along with the conventional velocity vector v, there is defined an angular velocity vector , the internal moment stresses are negligibly small, and in the general case the force stress tensor _kj is asymmetric. The model is characterized by the usual Newtonian viscosity , the Newtonian rolling viscosity _r, and the relaxation time = J/4 _r, where J is a scalar constant of the medium with dimensions of moment of inertia per unit mass, is the density. It is assumed that the medium is incompressible, the coefficients , _r, J are constant [2].The exact solution of the equations of motion, corresponding to flow of a layer with a plane surface, coincides with the solution of the Navier-Stokes equations in the case of flow of a layer of Newtonian fluid. The equations for three-dimensional periodic disturbances differ considerably from the corresponding equations for the problem of the flow stability of a layer of a Newtonian medium. It is shown that the Squire theorem is valid for parallel flows of a Grad liquid.The flow stability of the layer with respect to long-wave disturbances is studied using the method of sequential approximations suggested in [3, 4].