Group Classification of Equations of the Form y″ = f(x,y)

The problem of classification of ordinary differential equations of the form y = f(x,y) by admissible local Lie groups of transformations is solved. Standard equations are listed on the basis of the equivalence concept. The classes of equations admitting a oneparameter group and obtained from the standard equations by invariant extension are described.     

On Lyapunov Exponents for Non-Smooth Dynamical Systems with an Application to a Pendulum with Dry Friction

We consider dynamical systems from mechanics for which, due to some non-smooth friction effects, Oseledets' Multiplicative Ergodic Theorem cannot be applied canonically to define Lyapunov exponents. For general non-smooth systems which fit into a natural formal framework, we construct a suitable cocycle which lives on a good invariant set of full Lebesgue measure. Afterwards, this construction is applied to investigate a pendulum with dry friction, described through the equation . The Lyapunov exponents obtained by our construction show a good agreement with the dynamical behaviour of the system, and since we will prove that these Lyapunov exponents are always non-positive, we conclude that the system does not show chaotic behaviour.     

The Gibbs-Thompson relation within the gradient theory of phase transitions

This paper discusses the asymptotic behavior as 0+ of the chemical potentials associated with solutions of variational problems within the Van der Waals-Cahn-Hilliard theory of phase transitions in a fluid with free energy, per unit volume, given by ²¦¦²+ W(), where is the density. The main result is that is asymptotically equal to E/d+o(), with E the interfacial energy, per unit surface area, of the interface between phases, the (constant) sum of principal curvatures of the interface, and d the density jump across the interface. This result is in agreement with a formula conjectured by M. Gurtin and corresponds to the Gibbs-Thompson relation for surface tension, proved by G. Caginalp within the context of the phase field model of free boundaries arising from phase transitions.     

Solution of regular beam equations in arbitrary emission conditions on a curvilinear surface

The regular beam equations are solved analytically for the case of emission from an arbitrary surface in conditions of total space charge (-mode) and in a given external magnetic field H (§2) for temperature-limited emission (T-mode), in an external magnetic field H (§3); and for emission with nonzero initial velocity (§4). The emitter is taken as the coordinate surface x¹=0 in an orthogonal system x¹ (i = =1,2,3), while the current density J and field on it are given functions j(x², x³), (x², x³. The solution is written as series in (x¹) with coefficients dependent on x², x³, determined from recurrence relations. For emission in the -mode and H 0, =1/3; for temperature-limited emission, =1/2; with nonzero initial velocity, =1. The results are extended to the case of a beam in the presence of a moving background of uniform density (5).     

Magnetohydrodynamic natural convection heat transfer from radiate vertical porous surfaces

Magnetohydrodynamic natural convection heat transfer from radiate vertical surfaces with fluid suction or injection is considered. The nonsimilarity parameter is found to be the conductive fluid injection or suction along the streamwise coordinate = V{4x/² g(T _w – T )}^1/4. Three dimensionless parameters had been found to describe the problem: the magnetic influence number N = B ² _y /V ², the radiation-conduction parameter R _d = k _R /4aT ³ , and the Gebhart number Ge _x = gx/cp to represent the effect of the viscous dissipation. It is found that increasing the magnetic field strength causes the velocity and the heat transfer rates inside the boundary layer to decrease. Its apparent that increasing the radiation-conduction parameter decreases the velocity and enhances the heat transfer rates. The Gebhart number, i.e, the viscous dissipation had no effect on the present problem.Nomenclature a Stefan-Boltzmann constant - B _y Magnetic field flux density Wb/m² - Cf _x Local skin friction factor - c _p Specific heat capacity - f Dimensionless stream function - Ge _x Gebhart number, gx/cp - g Gravitational acceleration - k Thermal Conductivity - L Length of the plate - N Magnetic influence number, B ² _y /V ² - p Pressure - Pr Prandtl number - q _r Radiative heat flux - q _w (x) Local surface heat flux - Q _w (x) Dimensionless Local surface heat flux - R _d Planck number (Radiation-Conduction parameter), k _R /4aT ³ - T Temperature - T Free stream temperature - T _w Wall temperature - u, v Velocity components in x- and y-directions - V Porous wall suction or injection velocity - V _w Porous wall suction or injection velocity - x, y Axial and normal coordinates - Thermal diffusivity Greek symbols _R Roseland mean absorption coefficient, 4/3R _d - Coefficient of thermal expansion - Nonsimilarity parameter, V{4x/² g(T _w – T )}^1/4 - Peseudo-similarity variable - Dimensionless temperature - _w Ratio of surface temperature to the ambient temperature, T _w /T - Dynamice viscosity - Kinemtic viscosity - Fluid density - Electrical conductivity - _w Local wall shear stress - Dimensional stream function     

An application of nonsymmetric Lax-Milgram lemma to nonassociated plasticity

Based on a general assumption for plastic potential and yield surface, some properties of the nonassociated plasticity are studied, and the existence and uniqueness of the distribution of incremental stress and displacement for work-hardening materials are proved by using nonsymmetric Lax-Milgram lemma, when the work-hardening parameter A>F/Q/–F/, Q/.     

Reservoir transport equations by compositional approach

The objective of this paper is to present an overview of the fundamental equations governing transport phenomena in compressible reservoirs. A general mathematical model is presented for important thermo-mechanical processes operative in a reservoir. Such a formulation includes equations governing multiphase fluid (gas-water-hydrocarbon) flow, energy transport, and reservoir skeleton deformation. The model allows phase changes due to gas solubility. Furthermore, Terzaghi's concept of effective stress and stress-strain relations are incorporated into the general model. The functional relations among various model parameters which cause the nonlinearity of the system of equations are explained within the context of reservoir engineering principles. Simplified equations and appropriate boundary conditions have also been presented for various cases. It has been demonstrated that various well-known equations such as Jacob, Terzaghi, Buckley-Leverett, Richards, solute transport, black-oil, and Biot equations are simplifications of the compositional model.Notation List B reservoir thickness - B formation volume factor of phase - Cⁱ mass of component i dissolved per total volume of solution - C ⁱ mass fraction of component i in phase - C heat capacity of phase at constant volume - C_p heat capacity of phase at constant pressure - D ⁱ hydrodynamic dispersion coefficient of component i in phase - D_MTf thermal liquid diffusivity for fluid f - F = F(x, y, z, t) defines the boundary surface - f_p fractional flow of phase - g gravitational acceleration - H_p enthalpy per unit mass of phase - J_p volumetric flux of phase - k_rf relative permeability to fluid f - k₀ absolute permeability of the medium - M_p ⁱ mass of component i in phase - n porosity - N rate of accretion - P_f pressure in fluid f - p_ca capillary pressure between phases and =p-p - Rⁱ rate of mass transfer of component i from phase to phase - Rⁱ _source source rate of component i within phase - S saturation of phase - s gas solubility - T temperature - t time - U displacement vector - u velocity in the x-direction - v velocity in the y-direction - V volume of phase - V_s velocity of soil solids - W_i body force in coordinate direction i - x horizontal coordinate - z vertical coordinate Greek Letters _p volumetric coefficient of compressibility - _T volumetric coefficient of thermal expansion - _ij Kronecker delta - volumetric strain - _m thermal conductivity of the whole matrix - internal energy per unit mass of phase - gf suction head - density of phase - _ij tensor of total stresses - _ij tensor of effective stresses - volumetric content of phase - f viscosity of fluid f     

Theory of plane flows of a readily conducting plasma in a pipe

Simplified equations are obtained describing slowly changing plane flows of a readily conducting quasineutral inviscid plasma in a pipe. The practically interesting case of flow in a channel with solid metal ideally conducting walls (electrodes) is analyzed. When the gas pressure is large by comparison with the magnetic pressure ( 1), the field and current distribution is determined by gas dynamic factors, and the solid electrodes perturb the longitudinal electric field in a skin of the flow, symmetrically on the two sides of the flow, leading to attenuation of the longitudinal electric field near the input to the pipe; we also consider problems in the motion of the plasma under ideal and under poor conductivity. In the converse limiting case ( 1), it is shown that as the motion of the plasma in the pipe accelerates near the anode, there is observed an increase in the intensity of the electric field which is sharply inhomogeneous in the transverse direction. The possibility of the plasma breaking away from the anode (the limiting regime) is indicated, this being accompanied by a divergence between the electron velocity and the velocity of the ions. A criterion is obtained for the breakaway of the plasma, and its possible connection with the occurrence of pre-anode explosions is noted. It is shown that for 1, Joule losses are small by comparison with the power in the charge and the magnitude of the losses is independent of the conductivity of the plasma.Translated from Zhurnal Prikladnoi Mekhaniki i Teknicheskoi Fiziki, No. 4, pp. 9–19, July–August, 1970.     

The dynamics of solid-solid phase transitions 2. Incoherent interfaces

Incoherent phase transitions are more difficult to treat than their coherent counterparts. The interface, which appears as a single surface in the deformed configuration, is represented in its undeformed state by a separate surface in each phase. This leads to a rich but detailed kinematics, one in which defects such as vacancies and dislocations are generated by the moving interface. In this paper we develop a complete theory of incoherent phase transitions in the presence of deformation and mass transport, with phase interface structured by energy and stress. The final results are a complete set of interface conditions for an evolving incoherent interface.Frequently used symbols A_i,C_i generic subsurface of S_t - B_i undeformed phase-i region - C configurational bulk stress, Eshelby tensor - F deformation gradient - G inverse deformation gradient - H relative deformation gradient - J bulk Jacobian of the deformation - ¯K, K_i total (twice the mean) curvature of and S_i - Lin (U, V) linear transformations from U into V - Lin⁺ linear transformations of ³ with positive determinant - Orth⁺ rotations of ³ - Q^a external bulk mass supply of species a - ¯S bulk Cauchy stress tensor - S bulk Piola-Kirchhoff stress tensor - S_i undeformed phase i interface - U_i relative velocity of S_i - Unim⁺ linear transformations of ³ with unit determinant - ¯V, V_i normal velocity of and S_i - intrinsic edge velocity of S and A _i S - W_i volume flow across the phase-i interface - X material point - b external body force - e internal bulk configurational force - f_i external interfacial force (configurational) - ¯g external interfacial force (deformational) - grad, div spatial gradient and divergence - gradient and divergence on - h relative deformation - h^a, diffusive mass flux of species a and list of mass fluxes - ¯m outward unit normal to a spatial control volume - ¯n, n_i unit normal to and S_i - n subspace of ³ orthogonal to n - ¯q^a external interfacial mass supply of species a - s ......... - ¯v, v_i compatible velocity fields of and S_i - ¯w, w_i compatible edge velocity fields for and A_i - x spatial point - y_i deformation or motion of phase i - y^. material velocity - generic subsurfaces of - , _i deformed body and deformed phase-i region - () energy supplied to by mass transport - symmetry group of the lattice - _i, surface jacobians - lattice - () power expended on - spatial control volume - S deformed phase interface - lattice point density - interfacial power density - , A total surface stress - C configurational surface stress for phase 1 (material) - ¯C_i configurational surface stress (spatial) - F_i tangential deformation gradient - G_i inverse tangential deformation gradient - H incoherency tensor - ¯1(x), 1_i(X) inclusions of ¯n(x) and n _i (X) into ³ - K configurational surface stress for phase 2 (material) - ¯L, l_i curvature tensor of and S_i - ¯P(x), P_i(X) projections of ³ onto ¯n(x) and n_i (X) - ¯S, S deformational surface stress (spatial and material) - ¯a, a normal part of total surface stress - c normal part of configurational surface stress for phase 1 (material) - e_i internal interfacial configurational force - ¯v, v_i unit normal to and A _i - (x),_i(X) projections of ³ onto ¯n(x) and n _i (X) - _i normal internal force (material) - bulk free energy - slip velocity - _i=(–1)ⁱ _i ......... - ^a, chemical potential of species a and list of potentials - ^a, bulk molar density of species a and list of molar densities - _i normal internal force (spatial) - surface tension - , _i effective shear - referential-to-spatial transform of field - interfacial energy - grand canonical potential - l unit tensor in ³ - x, vector and tensor product in ³ - (...)^., _t(...) material and spatial time derivative - , Div material gradient and divergence - gradient and divergence on S_i - (...), (...) normal time derivative following and S_i - (...) limit of a bulk field asx ,x_i - [...],...> jump and average of a bulk field across the interface - (...)_ext extension of a surface tensor to ³ - tangential part of a vector (tensor) on and S_i     

Study of initial segment of turbulent jets of various gases in a parallel air stream

Results are presented of a study of the gasdynamic parameters and the geometric characteristics of the mixing zone of axisymmetric jets of gases of differing density (Freon-12, air, and helium) propagating in a parallel air stream, within the limits of the initial segment (0x/R3–30). Experimental data are presented on the effect of different densities (0. 27 n8.2) and velocities (0m1.7) of the gas jet and the parallel stream on the mixing process.     

Infinite Memory Expectations in a Dynamic Model with Hyperbolic Demand

In this study we will research the dynamics shown by a cobweb-type model with hyperbolic demand, sigmoidal supply and with backward-looking mechanism of expectation creation, whereby the new state of the system is obtained from all the previous states observed by weighted arithmetical mean with exponentially decreasing weights in the region. The study herewith presented aims at confirming the existence of a stabilising effect due to the presence of infinite memory since, with all the other conditions begin the same, a memory rate > exists at which market equilibrium is a sink. An unstable system, therefore, becomes stable in the presence of sufficiently resistant expectations with infinite historical memory, although this transition to stability is accompanied by the onset of chaos. The resulting effect, therefore is one of qualitative destabilisation, that is with reference to the qualitative dynamic performance produced, associated to a quantitative stabilisation, that is to say with reference to the decreasing width of the invariant sets within which relevant dynamics occur.     

Some remarks on differential equations of quadratic type

In this paper we study differential equations of the formx(t) + x(t)=f(x(t)), x(0)=x ₀ C HereC is a closed, bounded convex subset of a Banach spaceX,f(C) C, and it is often assumed thatf(x) is a quadratic map. We study the differential equation by using the general theory of nonexpansive maps and nonexpansive, non-linear semigroups, and we obtain sharp results in a number of cases of interest. We give a formula for the Lipschitz constant off: C C, and we derive a precise explicit formula for the Lipschitz constant whenf is quadratic,C is the unit simplex inR ⁿ, and thel ₁ norm is used. We give a new proof of a theorem about nonexpansive semigroups; and we show that if the Lipschitz constant off: CC is less than or equal to one, then lim_tf(x(t))–x(t)=0 and, if {x(t):t 0} is precompact, then lim_tx(t) exists. Iff¦C=L¦C, whereL is a bounded linear operator, we apply the nonlinear theory to prove that (under mild further conditions on C) lim_t f(x(t))–x(t)=0 and that lim_t x(t) exists if {x(t):t 0} is precompact. However, forn 3 we give examples of quadratic mapsf of the unit simplex ofR ⁿ into itself such that lim_t x(t) fails to exist for mostx ₀ C andx(t) may be periodic. Our theorems answer several questions recently raised by J. Herod in connection with so-called model Boltzmann equations.     

Some self-similar solutions of the Leibenzon equation

Self-similar one-dimensional solutions of the Leibenzon equation c²_t= _zz ^k (z 0, k 2) are considered. Approximate solutions are constructed for the two cases in which the initial value = ₁ = const > 0 and on the boundary either a constant value = ₂ < ₁ is maintained or the flow (directed outwards) is given. In the first problem the dependence of the boundary flow on the governing parameters is determined. A characteristic property of the types of motion in question is the existence near the boundary of a region, expanding with time, in which the flow is almost independent of the coordinate.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 145–150, September–October, 1991.     

Effects of temperature and pressure on the rheological behavior of montmorillonite sols

Summary The effects of high temperatures up to 180 C and high pressures up to 560 kg/cm² on the rheological properties of pure montmorillonite suspensions with well-defined base-exchange cations have been investigated. The suspensions behave very much asBingham plastics according to the equation=+, in which is the shear stress,D the shear rate, the plastic viscosity, and the yield stress which is largely a measure of residual flocculation.The observed effects depend strongly on the interparticle forces that govern the colloidal stability and the rheological behavior of the suspensions. One can distinguish between two categories of suspensions:P-type sols in which the clay particles are associated through Coulombic attraction between positive edges and negative faces and are located in a primary potential energy minimum, andS-type sols in which the particles are associated edge-to-edge and are located in a weaker secondary potential minimum obtained by the summation ofvan der Waals attraction and double layer repulsion.Both theBingham yield stress and the plastic viscosity of theP-type sols decrease with increasing temperature. The temperature dependence of follows theArrhenius equation. TheP-type suspensions are either weakly or not at all thixotropic at room temperature and are definitely non-thixotropic at higher temperatures. Pressure slightly increases both the plastic viscosity and the yield stress.TheS-type sols, on the other hand, display an increase in yield stress and degree of thixotropy with increasing temperature and generally a decrease in the plastic viscosity. This behavior is modified in the case of Ca-montmorillonite suspensions, in which both and pass through a maximum at 150C, followed by a decline. The maximum can be explained by disaggregation of face-to-face aggregated clay packets.Pressure causes a decrease both in and the degree of thixotropy in theS-type suspensions, while it causes a slight increase in the plastic viscosity. This behavior is a consequence of the destruction of the hydration shell caused by high pressure.

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Zusammenfassung Der Einfluß von hohen Temperaturen (bis 180 C) und hohen Drucken (bis zu 560 kp/cm²) auf die rheologischen Eigenschaften reiner Suspensionen von Montmorillonit mit definierten Austauschionen wurde untersucht. Die Suspensionen folgen dem Binghamschen Flie\gesetz gemÄ\ der Gleichung=+, worin die Schubspannung,D das GeschwindigkeitsgefÄlle, dieBingham- ViskositÄt und die Flie\spannung darstellen. Letztere zeigt in erster Linie das Ausma\ der Koagulation an.Die beobachteten Effekte hÄngen stark von den KrÄften zwischen den Teilchen ab, welche die StabilitÄt der Kolloide und das rheologische Verhalten der Suspensionen bestimmen. Man kann zwei Kategorien von Suspensionen unterscheiden: Sole vom sog.P-Typ, in denen die Tonteilchen durch Coulombsche AnziehungskrÄfte zwischen positiven Kanten und negativen FlÄchen assoziiert sind und sich in einem primÄren Potentialminimum befinden, und Sole vomS-Typ, in denen die Teilchen Kante-zu-Kante assoziiert sind und sich in einem flacheren sekundÄren Potentialminimum befinden, welches durch das Zusammenwirken vonvan der Waalsschen AnziehungskrÄften und von Absto\ungskrÄften infolge der Wechselwirkung der elektrischen Doppelschichten entsteht.Sowohl die Flie\grenze als auch dieBingham-ViskositÄt der Sole vomP-Typ nehmen mit wachsender Temperatur ab. Die TemperaturabhÄngigkeit von folgt derArrhenius-Gleichung. Die Suspensionen vomP-Typ sind bei Zimmertemperatur entweder gar nicht oder nur schwach thixotrop, wÄhrend sie bei hoher Temperatur in keinem Fall thixotrop sind. Mit wachsendem Druck erhöht sich sowohl dieBingham-ViskositÄt als auch die Flie\grenze ein wenig.Bei den Solen vomS-Typ andererseits steigt die Flie\spannung und der Grad der Thixotropie mit steigender Temperatur, wÄhrend dieBingham-ViskositÄt im allgemeinen abnimmt. Bei Ca-Montmorillonit-Suspensionen ist das Verhalten etwas anders: Sowohl als auch erreichen bei 150 C ein Maximum und fallen dann wieder ab. Das Maximum kann durch Desaggregation flÄchenhaft aggregierter Tonpartikel erklÄrt werden. Bei steigendem Druck fÄllt sowohl als auch der Grad der Thixotropie in den Suspensionen vomS-Typ ab, wÄhrend dieBingham-YiskositÄt leicht ansteigt. Dieses Verhalten ist eine Folge der dann eintretenden Zerstörung der Solvathüllen.

     

Optical co-ordinates in general relativity 2 — the problem of distance

Summary Let denote the congruence of null geodesics associated with a given optical observer inV ₄. We prove that determines a unique collection of vector fieldsM₍₎ ( =1, 2, 3) and ₍₀₎ overV ₄, satisfying a weak version of Killing's conditions.This allows a natural interpretation of these fields as the infinitesimal generators of spatial rotations and temporal translation relative to the given observer. We prove also that the definition of the fieldsM₍₎ and ₍₀₎ is mathematically equivalent to the choice of a distinguished affine parameter f along the curves of, playing the role of a retarded distance from the observer.The relation between f and other possible definitions of distance is discussed.

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Sommario Sia la congruenza di geodetiche nulle associata ad un osservatore ottico assegnato nello spazio-tempoV ₄. Dimostriamo che determina un'unica collezione di campi vettorialiM₍₎ ( =1, 2, 3) e ₍₀₎ inV ₄ che soddisfano una versione in forma debole delle equazioni di Killing. Ciò suggerisce una naturale interpretazione di questi campi come generatori infinitesimi di rotazioni spaziali e traslazioni temporali relative all'osservatore assegnato. Dimostriamo anche che la definizione dei campiM₍₎, ₍₀₎ è matematicamente equivalente alla scelta di un parametro affine privilegiato f lungo le curve di, che gioca il ruolo di distanza ritardata dall'osservatore. Successivamente si esaminano i legami tra f ed altre possibili definizioni di distanza in grande.

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Work performed in the sphere of activity of: Gruppo Nazionale per la Fisica Matematica del CNR.     

A New Optimal Growth Constant for the Hele–Shaw Instability

We study the immiscible displacement of the oil from a homogeneous porous medium by using a less viscous fluid (water). We use the Hele–Shaw model, then a sharp interface exists between the fluids. The fingering phenomenon appears, first studied by Saffman and Taylor (1958). Gorell and Homsy (1983) consider an intermediate region (I. R.) between water and oil, containing a polymer mixture. The unknown viscosity in I. R. is a parameter which can improve the stability of the I. R.–oil interface. A numerical optimal viscosity profile in I. R. is given. Carasso and Paa (1998) obtain an explicit formula for an optimal viscosity profile in I. R. An upper estimation of the growth constant is given. In this paper, a very slow viscosity profile in I. R. is defined and an optimal formula for the growth constant is obtained, less than the previous estimation of Carasso and Paa. Moreover, this formula is similar with the Saffman–Taylor result, only the water viscosity is replaced by the limit value of viscosity in I. R. on the interface with the oil. We explain the apparent contradiction between the previous results of Gorell and Homsy (1983) and Paa and Polisevski (1992).     

Transport in ordered and disordered porous media V: Geometrical results for two-dimensional systems

In this paper we continue the geometrical studies of computer generated two-phase systems that were presented in Part IV. In order to reduce the computational time associated with the previous three-dimensional studies, the calculations presented in this work are restricted to two dimensions. This allows us to explore more thoroughly the influence of the size of the averaging volume and to learn something about the use of anon-representative region in the determination of averaged quantities.

Nomenclature

Roman Letters A interfacial area of the– interface associated with the local closure problem, m² - a _i i=1, 2, gaussian probability distribution used to locate the position of particles - l unit tensor - characteristic length for the-phase particles, m - ⁰ reference characteristic length for the-phase particles, m - characteristic length for the-phase, m - _i i=1,2,3 lattice vectors, m - m convolution product weighting function - m _V special convolution product weighting function associated with a unit cell - n _i i=1, 2 integers used to locate the position of particles - n unit normal vector pointing from the-phase toward the-phase - r _p position vector locating the centroid of a particle, m - r gaussian probability distribution used to determine the size of a particle, m - r ₀ characteristic length of an averaging region, m - V averaging volume, m³ - V volume of the-phase contained in the averaging volume,V, m³ - x position of the centroid of an averaging area, m - x ₀ reference position of the centroid of an averaging area, m - y position vector locating points in the-phase relative to the centroid, m Greek Letters V /V, volume average porosity - _{a i} standard deviation ofa _i - _r standard deviation ofr - intrinsic phase average of      

Effect of Transitional Nonequilibrium on the Molecular–Dissociation Rate in a Hypersonic Shock Wave

The problem of the influence of a nonequilibrium (non–Maxwellian( distribution of translational energy over the degrees of freedom of molecules on the rate of their dissociation in a hypersonic shock wave is considered. An approximate beam—continuous medium model, which was previously applied to describe a hypersonic flow of a perfect gas, was used to study translational nonequilibrium. The degree of dissociation of diatomic molecules inside the shock–wave front, which is caused by the nonequilibrium distribution over the translational degrees of freedom, is evaluated. It is shown that the efficiency of the first inelastic collisions is determined by the dissociation rate exponentially depending on the difference in the kinetic energy of beam molecules and dissociation barrier.     

Ši’lnikov Chaos in the Generalized Lorenz Canonical Form of Dynamical Systems

This paper studies the generalized Lorenz canonical form of dynamical systems introduced by elikovský and Chen [International Journal of Bifurcation and Chaos 12(8), 2002, 1789]. It proves the existence of a heteroclinic orbit of the canonical form and the convergence of the corresponding series expansion. The ilnikov criterion along with some technical conditions guarantee that the canonical form has Smale horseshoes and horseshoe chaos. As a consequence, it also proves that both the classical Lorenz system and the Chen system have ilnikov chaos. When the system is changed into another ordinary differential equation through a nonsingular one-parameter linear transformation, the exact range of existence of ilnikov chaos with respect to the parameter can be specified. Numerical simulation verifies the theoretical results and analysis.     

Nonsimilar Solutions for Mixed Convection in Non‐Newtonian Fluids Along a Vertical Plate in a Porous Medium

A nonsimilar boundary layer analysis is presented for the problem of mixed convection in powerlaw type nonNewtonian fluids along a vertical plate with powerlaw wall temperature distribution. The mixed convection regime is divided into two regions, namely,the forced convection dominated regime and the free convection dominated regime. The two solutions are matched. Numerical results are presented for the details of the velocity and temperature fields. A discussion is provided for the effect of viscosity index on the surface heat transfer rate.