Parameter identification in a soil with constant diffusivity

We consider infiltration into a soil that is assumed to have hydraulic conductivity of the form K = K = K_se^h and water content of the form = K – _r. Here h denotes capillary pressure head while K_s, , and _r represent soil specific parameters. These assumptions linearize the flow equation and permit a closed form solution that displays the roles of all the parameters appearing in the hydraulic function K and . We assume K_s and _r to be known. A measurement of diffusivity fixes the product of and resulting in a parameter identification problem for one parameter. We show that this parameter identification problem, in some cases, has a unique solution. We also show that, in some cases, this parameter identification problem can have multiple solutions, or no solution. In addition it is shown that solutions to the parameter identification problem can be very sensitive to small changes in the problem data.     

Stress analysis of plates with a circular hole reinforced by flange reinforcing member

This paper deals with the problem of stress analysis of plates with a circular hole reinforced by flange reinforcing member. The so called flange reinforcing member here means that the reinforcing member is built up by setting shapes or bars with any section shape on both sides of the plates along the edge of the hole. Two cases of external loads are considered. In one case the external loads are stressesσ_X^(∞),σ_Y^(∞),and τ_XY^(∞) acting at infinite point of the plate, and in the other the external loads are linear distributed normal stresses. The procedure of solving the problems mentioned above consists of three steps. Firstly, the reinforcing member is taken out from the plates and considered to be a circular bar being solved to determine its deformation under the action of radial force q₀(θ) and tangential force t₀(θ) which are forces acting upon each other between reinforcing member and plate. Secondly, the displacements of plate with a circular hole under the action of q₀(θ) and t₀(θ) and external loads are determined. Finally, forces q₀(θ) and t₀(θ) are obtained by the compatibility of deformations between reinforcing member and plate. Then the internal forces and displacements of reinforcing member and plate are deduced from q₀(θ) and t₀(θ) obtained.     

Numerical solution of two-dimensional problem of gas flow from a chamber with flat walls

A study is made of the classical problem of steady flow of ideal gas from an infinite two-dimensional chamber with straight wall generator making angle _w with the symmetry plane of the flow. Specification of _w, the pressure ratio _a-P_a/P_o (P_o is the stagnation pressure of the gas in the chamber, P is the pressure in the ambient medium), and the specific-heat ratio completely determines the flow of gas from the chamber.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 177–181, July–August, 1988.I thank A. N. Kraiko for his interest in the work and helpful discussions of the results.     

The three dimensional incompressible laminar boundary layer on a spinning cone

Summary The motion of an incompressible viscous fluid induced by a spinning cone is analytically studied and similar solutions of the relevant steady state boundary equations are obtained. Some of the numerical results are shown to be obtainable from the Karman-Cochran solution for the infinite disc.Symbols and Notation p Pressure - p Pressure at infinity - p ₀ Pressure at the wall - Density - Transverse component of velocity - Normal component of velocity - Radial component of velocity - Angular velocity - Semi-vertex angle - Re Reynolds number with respect to _o - _o Transverse component of velocity at the cone surface - Kinematic viscosity This research is sponsored by the Air Force Office of Scientific Research, Fluid Mechanics Division, under Contract Number AF 18(600)-498.     

Temperament Development Modeled as a Nonlinear Complex Adaptive System

Using approach-withdrawal (AW) as a specific instance of temperament, a theoretical model of temperament as a complex dynamic system is proposed. Developmental contextualism (Lerner, 1998) serves as a guiding theory in determining the structural components of the system and Kauffman's (1993) Boolean models of self-organization are adapted to estimate the parameter functions. In this model P(AW) = f(, ) where P(AW) is the probability density function of an approach or a withdrawal response, ( is a standardized parameter estimate of the biological sensitivity to stimulation, and is a standardized parameter estimate of the contextual response to an approach or withdrawal response. It is theorized that the functions of ( and follow a Hill function of the forms: d /dt = (²/c² + ²) – K₁ d /dt = ( ²/c² + ²) – K₂, where K₁, K₂, and c are system constants. This results in a double sigmoid function in which at extreme values of and the system stabilizes on a steady state of either approach or withdrawal response patterns. At intermediate parameter values the probability density functions of approach and withdrawal responses are wider. Thus, AW can be modeled as representing two basins of attraction. In addition, considerations are given to the systems sensitivity to initial conditions.     

Elastico-viscous properties of deflocculated china clay suspensions — concentration effects

Summary The physical properties of deflocculated china clay suspensions are studied in a combined steady and low-amplitude oscillatory shear flow. Concentration effects are examined and it is shown that, with increasing concentration, an initial shear thinning region is followed by a shear thickening one. Qualitative agreement is obtained between theory and experiment for a range of concentrations of suspensions, all of which exhibit marked elastic properties. The experimental results were obtained using a Weissenberg Rheogoniometer.

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Zusammenfassung Es werden die physikalischen Eigenschaften deflockulierter Suspensionen von Porzellanerde in einer kombinierten stationären und oszillatorischen Scherströmung mit niedriger Amplitude studiert. Der Einfluß der Konzentration wird untersucht, und es wird gezeigt, daß mit wachsender Konzentration sich an den anfänglich allein vorhandenen Bereich mit Scherentzähung ein Bereich mit Scherverzähung anschließt. Zwischen Theorie und Experiment wird eine qualitative Übereinstimmung in einem Konzentrationsbereich gefunden, in dem ausgeprägte viskoelastische Eigenschaften vorhanden sind. Die experimentellen Ergebnisse werden mit Hilfe eines Weissenberg-Rheogoniometers erhalten.

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c phase lag in oscillatory testing - D(t – t) deformation history - F, G non-dimensional complex functions of - complex conjugate ofF - G dynamic rigidity - i - I % increase in mean couple under superposed shear rates - I ₁ moment of inertia of the top platen (i.e. cone) - J amplitude ratio, ₁/ ₁ - K ₁ restoring constant of the torsion bar - q steady shear rate - r, , spherical polar coordinates - t current time - v _i velocity vector - w/w concentration by weight - W a function of andt - ₁ angular amplitude of the motion of the plate - shear rate - /q - apparent viscosity - dynamic viscosity - ^* complex dynamic viscosity - ₀ limiting viscosity at small rates of shear - ₀ gap angle in cone and plate system - ₁, ₂, ₃, ₄,µ ₀ relaxation time constants - shear stress - ₀ unperturbed shear stress - ₁, ₂ kernel functions - angular frequency of oscillation - steady angular velocity of the plate With 16 figures     

The secondary flow induced around a sphere in an oscillating stream of elastico-viscous liquid

The present paper is devoted to the theoretical study of the secondary flow induced around a sphere in an oscillating stream of an elastico-viscous liquid. The boundary layer equations are derived following Wang's method and solved by the method of successive approximations. The effect of elasticity of the liquid is to produce a reverse flow in the region close to the surface of the sphere and to shift the entire flow pattern towards the main flow. The resistance on the surface of the sphere and the steady secondary inflow increase with the elasticity of the liquid.Nomenclature a radius of the sphere - b ^ik contravariant components of a tensor - e contravariant components of the rate of strain tensor - F() see (47) - G total nondimensional resistance on the surface of the sphere - g _ik covariant components of the metric tensor - f, g, h secondary flow components introduced in (34) - k ₀ measure of relaxation time minus retardation time (elastico-viscous parameter) - K =k ₀ ²/V ₀ ² , nondimensional parameter characterizing the elasticity of the liquid - n measure of the ratio of the boundary layer thickness and the oscillation amplitude - N, T defined in (44) - p arbitrary isotropic pressure - p _ik covariant components of the stress tensor - p ^ik contravariant components of the stress tensor associated with the change of shape of the material - R =V ₀ a/v, the Reynolds number - S =a/V ₀, the Strouhall number - r, , spherical polar coordinates - u, v, w r, , component of velocity - t time - V(, t) potential velocity distribution around the sphere - V ₀ characteristic velocity - u, v, t, y, P nondimensional quantities defined in (15) - reciprocal of s - density - defined in (32) - defined in (42) - ₀ limiting viscosity for very small changes in deformation velocity - complex conjugate of - oscillation frequency - = ₀/, the kinematic coefficient of viscosity - , defined in (52) - (, y) stream function defined in (45) - =(NT/2n)^1/2 y - /t convective time derivative (1) _ik      

Injection moulding of thermoplastics: Freezing during mould filling

The injection moulding of thermoplastic polymers involves, during mould filling, flows of hot melts into mould networks, the walls of which are so cold that frozen layers form on them. Theoretical analyses of such flows are presented here. Br Brinkman number - c _L polymer melt specific heat capacity - c _S frozen polymer specific heat capacity - e exponential function - erf() error function - Gz Graetz number in thermal entrance region - Gz ^* modified Graetz number in thermal entrance region - Gz overall Graetz number - h channel half-height - h ^* half-height of polymer melt region - H mean heat transfer coefficient - k _L polymer melt thermal conductivity - k _S frozen polymer thermal conductivity - ln( ) natural logarithm function - L length of thermal entrance region in pipe or channel - m viscosity shear rate exponent - M(,,) Kummer function - Nu Nusselt number - p pressure - P pressure drop in thermal entrance region - P _f pressure drop in melt front region - Pe Péclet number - Pr Prandtl number - Q volumetric flow rate - r radial coordinate in pipe - R pipe radius - R ^* radius of polymer melt region - Re Reynolds number - Sf Stefan number - t time - T temperature - T _i inlet polymer melt temperature - T _m melting temperature of polymer - T _w pipe or channel wall temperature - U(,,) Kummer function - u _r radial velocity in pipe - u _x axial velocity in channel - u _y cross-channel velocity - u _z axial velocity in pipe - V melt front velocity - w channel width - x axial coordinate in channel - x _f melt front position in channel - y cross-channel coordinate - z axial coordinate in pipe - z _f melt front position in pipe - () gamma function - dimensionless thickness of frozen polymer layer - _i i-th term (i = 1,2,3) in power series expansion of - dimensionless axial coordinate in pipe - _f dimensionless melt front position in pipe - dimensionless cross-channel coordinate - ^* dimensionless half-height of polymer melt region - dimensionless temperature - _i i-th term (i = 0, 1, 2, 3) in power series expansion of - _i first derivative of _i with respect toø - _i second derivative of _i with respect toø - ^* dimensionless wall temperature - thermal diffusivity ratio - - latent heat of fusion - µ viscosity - µ ^* unit shear rate viscosity - dimensionless axial coordinate in channel - _f dimensionless melt front position in channel - dimensionless pressure drop in thermal entrance region - _f dimensionless pressure drop in melt front region - _L polymer melt density - _s frozen polymer density - dimensionless radial coordinate in pipe - ^* dimensionless radius of polymer melt region - ø dimensionless similarity variable in thermal entrance region - dummy variable - dimensionless contracted axial coordinate in thermal entrance region - dimensionless similarity variable in melt front region - ^* constant     

An exact numerical method of calculating inviscid heated flows in two dimensions with an example of duct flow

Summary A twodimensional flow problem with heat addition can be expressed in terms of five parameters (pressure p, density , flow speed u, flow direction , rate of heating q) which must satisfy four equations (continuity, two components of momentum, and energy). It is shown how the equations become particularly simple, being linear and hyperbolic, if is specified and solutions obtained for the other four variables. An example is given of the flow through a supersonic combustion chamber.

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Übersicht Zweidimensionale Strömungsprobleme mit Wärmezufuhr können mit Hilfe von 5 Größen (Druck p, Dichte , Strömungsgeschwindigkeit u, Strömungsneigung , Wärmezufuhr q) formuliert werden, die 4 Gleichungen erfüllen müssen (Erhaltungssätze für Masse, Energie und zwei Komponenten des Impulses). Es wird gezeigt, daß die Gleichungen besonders einfach werden, nämlich linear und hyperbolisch, wenn vorgegeben wird und Lösungen für die andern 4 Veränderlichen bestimmt werden. Als Beispiel wird die Überschallströmung in einer Brennkammer behandelt.

     

Forced oscillations of a rotating shaft with nonlinear spring characteristics and internal damping (1/2 order subharmonic oscillations and entrainment)

Nonlinear forced oscillations of a rotating shaft with nonlinear spring characteristics and internal damping are studied. In particular, entrainment phenomena at the critical speeds of 1/2 order subharmonic oscillations of forward and backward whirling modes are investigated. A self-excited oscillation appears in the wide range above the major critical speed. The amplitude of this oscillation reaches a limit value and then a self-sustained oscillation occurs. In the vicinity of a 1/2 order subharmonic oscillation of a forward whirling mode, a self-excited oscillation is entrained by a subharmonic oscillation. In the vicinity of a 1/2 order subharmonic oscillation of a backward whirling mode, either a self-excited oscillation or a subharmonic oscillation occurs.Experiments were made by an elastic rotating shaft with a disc. Nonlinearity in its restoring force was due to an angular clearance of a bearing and internal damping was due to friction between the shaft and an inner ring of the bearing. A self-excited oscillation was observed in the range above the major critical speed and this self-excited oscillation was entrained by a 1/2 order subharmonic oscillation of a forward whirling mode.Nomenclature O–xyz rectangular coordinate system - , _x, _y inclination angle of a shaft and its projections on the xz- and yz-planes - _x, _y inclination angles in rotating coordinates - , polar coordinates - I _p polar moment of inertia of a rotor - I diametral moment of inertia of a rotor - i _p ratio of I _p to I - dynamic unbalance of a rotor - rotating speed (angular velocity) - F magnitude of a dynamic unbalance force, F = (1 – i _p )² - c external damping coefficient - h internal damping coefficient - t time - D _x , D _y internal damping terms in stationary coordinates - D _x , D _y internal damping terms in rotating coordinates - N _x , N _y nonlinear terms in restoring forces     

Numerical analysis of the transient conjugated heat transfer in a circular duct with a power-law fluid

We treat numerically in this paper, the transient analysis of a conjugated heat transfer process in the thermal entrance region of a circular tube with a fully developed laminar power-law fluid flow. We apply the quasi-steady approximation for the power-law fluid, identifying the suitable time scales of the process. Thus, the energy equation in the fluids is solved analytically using the well-known integral boundary layer technique. This solution is coupled to the transient energy equation for the solid where the transverse and longitudinal heat conduction effects are taken into account. The numerical results for the temporal evolution of the average temperature of the tube wall, _av, is plotted for different nondimensional parameters such as conduction parameter, , the aspect ratios of the tube, and ₀ and the index of power-law fluid, n.     

On the thermodynamics of elastic-plastic materials with temperature-dependent moduli and yield stresses

Summary The subject of this article is the thermodynamics of perfect elastic-plastic materials undergoing unidimensional, but not necessarily isothermal, deformations. The first and second laws of thermodynamics are employed in a form in which only the following quantities appear: the temperature , the elastic strain _e, the plastic strain _p, the elastic modulus (gq), the yield strain (gq), the heat capacity (_e, _p,), the latent elastic heat _e(_e, _p, ), and the latent plastic heat _p(_e, _p, ). Relations among the response functions , , , _e, and _p are derived, and it is shown that a set of these relations gives a necessary and sufficient condition for compliance with the laws of thermodynamics. Some observations are made about the existence and uniqueness of energy and entropy as functions of state.Dedicated to Clifford Truesdell on the occasion of his 60th birthdayThis research was supported by the U.S. National Science Foundation.     

Gas flow from plane vessels with large angles of inclination of the walls

Numerical solutions are obtained for problems of steady supersonic gas flow from infinite plane vessels with angles of inclination of the walls to the plane of symmetry _c on the interval 90° < _c 180°. The problems are posed and solved in the hodograph plane. It is shown that starting from a certain _c* the flow choking mechanism is determined not by the arrival of the limiting characteristic at the edge of the opening (classical choking mechanism) but by the interaction of the jet with the outside surface of the vessel wall. The effect of _c and the ambient pressure on the local and integral flow characteristics is investigated.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 144–151, January–February, 1991.The author wishes to thank A. N. Kraiko for discussing the results and N. I. Tillyaev for assisting with the work.     

Unsteady effects in a thin viscous shock layer near a three-dimensional stagnation point of a body moving with given acceleration and deceleration

An unsteady viscous shock layer near a stagnation point is studied. The Navier-Stokes equations are analyzed in the limit 1, Re₀ , df/dt = ^n-mF(t/^m). The Reynolds number Re₀ is defined in the paper by Eq. (1.3) (df/dt is the velocity of the body with respect to an inertial frame of reference moving with the original steady velocity –V'_t8, ₂ = ( – 1)/( + 1)). Various flow regimes in the case 1, l, n max(2m, m + 1), m 0, where ² = 1/Re₀ are analyzed. Equations are derived that generalize the asymptotic analysis to the case of a viscous unsteady flow of gas in a thin three-dimensional shock layer. The problem of a thin unsteady viscous shock layer near the stagnation point of a body with two curvatures is formulated. Examples of numerical solution are given for different ratios of the principal curvatures of the body, the wall temperature, the parameters of the original steady flow, and the acceleration and deceleration regimes.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 100–111, March–April, 1981.I thank Yu. D. Shevelev for a fruitful discussion of the work.     

Heat conduction in a cylinder crossed by an electric current with skin effect

The steady periodic temperature distribution in an infinitely long solid cylinder crossed by an alternating current is evaluated. First, the time dependent and non-uniform power generated per unit volume by Joule effect within the cylinder is determined. Then, the dimensionless temperature distribution is obtained by analytical methods in steady periodic regime. Dimensionless tables which yield the amplitude and the phase of temperature oscillations both on the axis and on the surface of copper or nichrome cylindrical electric resistors are presented.

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Wärmeleitung in einem stromdurchflossenen Zylinder unter Berücksichtigung des Skin-Effektes

Zusammenfassung Es wird die periodische Temperaturverteilung für den eingeschwungenen Zustand in einem unendlich langen, von Wechselstrom durchflossenen Vollzylinder ermittelt. Zuerst erfolgt die Bestimmung der zeitabhängigen, nichgleichförmigen Energiefreisetzung pro Volumeneinheit des Zylinders infolge Joulescher Wärmeentwicklung und anschließend die Ermittlung der quasistationären Temperaturverteilung auf analytischem Wege. Amplitude und Phasenverzögerung der Temperaturschwingungen werden für die Achse und die Oberfläche eines Kupfer- oder Nickelchromzylinders tabellarisch in dimensionsloser Form mitgeteilt.

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Nomenclature A integration constant introduced in Eq. (2) - ber, bei Thomson functions of order zero - Bi Biot numberhr ₀/ - c speed of light in empty space - c ₁,c ₂ integration constants introduced in Eq. (46) - c _p specific heat at constant pressure - E electric field - E _z component ofE alongz - E time independent part ofE, defined in Eq. (1) - f function ofs and defined in Eq. (11) - g function ofs and defined in Eq. (37) - h convection heat transfer coefficient - H magnetic field - i imaginary uniti=(–1)^1/2 - I electric current - I _eff effective electric currentI _eff=I/2^1/2 - Im imaginary part of a complex number - J _n Bessel function of first kind and ordern - J electric current density - q _g power generated per unit volume - time average of the power generated per unit volume - time averaged power per unit length - r radial coordinate - R electric resistance per unit length - r ₀ radius of the cylinder - Re real part of a complex number - s dimensionless radial coordinates=r/r ₀ - s, s integration variables - t time - T temperature - time averaged temperature - T _f fluid temperature outside the boundary layer - time average of the surface temperature of the cylinder - u, functions ofs, and defined in Eqs. (47) and (48) - W Wronskian - x position vector - x real variable - Y _n Bessel function of second kind and ordern - z unit vector parallel to the axis of the cylinder - z axial coordinate - · modulus of a complex number - equal by definition Greek symbols amplitude of the dimensionless temperature oscillations - electric permittivity - dimensionless temperature defined in Eq. (16) - ₀, ₁, ₂ functions ofs defined in Eq. (22) - thermal conductivity - dimensionless parameter=(2)^1/2 - magnetic permeability - ₀ magnetic permeability of free space - function of defined in Eq. (59) - dimensionless parameter=c _p/() - mass density - electric conductivity - dimensionless time=t - phase of the dimensionless temperature oscillations - function ofs:= ₁+i ₂ - angular frequency - dimensionless parameter=()^1/2 r ₀     

A rotating hot-wire technique for spatial sampling of disturbed and manipulated duct flows

The documentation and control of flow disturbances downstream of various open inlet contractions was the primary focus with which to evaluate a spatial sampling technique. An X-wire probe was rotated about the center of a cylindrical test section at a radius equal to one-half that of the test section. This provided quasi-instantaneous multi-point measurements of the streamwise and azimuthal components of the velocity to investigate the temporal and spatial characteristics of the flowfield downstream of various contractions. The extent to which a particular contraction is effective in controlling ingested flow disturbances was investigated by artificially introducing disturbances upstream of the contractions. Spatial as well as temporal mappings of various quantities are presented for the streamwise and azimuthal components of the velocity. It was found that the control of upstream disturbances is highly dependent on the inlet contraction; for example, reduction of blade passing frequency noise in the ground testing of jet engines should be achieved with the proper choice of inlet configurations.List of symbols K _uv correlation coefficient= - P percentage of time that an azimuthal fluctuating velocity derivative dv/d is found - U streamwise velocity component U=U (, t) - V azimuthal or tangential velocity component due to flow and probe rotation V=V (, t) - mean value of streamwise velocity component - U _m resultant velocity from and - mean value of azimuthal velocity component induced by rotation - u fluctuating streamwise component of velocity u=u(, t) - v fluctuating azimuthal component of velocity v = v (, t) - u phase-averaged fluctuating streamwise component of velocity u=u(0) - v phase-averaged fluctuating azimuthal component of velocity v=v() - û average of phase-averaged fluctuating streamwise component of velocity (u()) over cases I-1, II-1 and III-1 û = û() - average of phase-averaged fluctuating azimuthal component of velocity (v()) over cases I-1, II-1 and III-1 - u fluctuating streamwise component of velocity corrected for non-uniformity of probe rotation and/or phase-related vibration u = u(0, t) - v fluctuating azimuthal component of velocity corrected for non-uniformity or probe rotation and/or phase-related vibration v=v (, t) - u ² rms value of corrected fluctuating streamwise component of velocity - rms value of corrected fluctuating azimuthal component of velocity - phase or azimuthal position of X-probe     

On the effects of uniform high suction on the rotationally symmetric flow of a conducting liquid near a stationary disc in the presence of a transverse magnetic field

In the present paper an attempt has been made to find out effects of uniform high suction in the presence of a transverse magnetic field, on the motion near a stationary plate when the fluid at a large distance above it rotates with a constant angular velocity. Series solutions for velocity components, displacement thickness and momentum thickness are obtained in the descending powers of the suction parameter a. The solutions obtained are valid for small values of the non-dimensional magnetic parameter m (= 4 _e ² H ₀ ² /) and large values of a (a2).Nomenclature a suction parameter - E electric field - E _r , E , E _z radial, azimuthal and axial components of electric field - F, G, H reduced radial, azimuthal and axial velocity components - H magnetic field - H _r , H , H _z radial, azimuthal and axial components of magnetic field - H ₀ uniform magnetic field - H* displacement thickness and momentum thickness ratio, */ - h induced magnetic field - h _r , h , h _z radial, azimuthal and axial components of induced magnetic field - J current density - m nondimensional magnetic parameter - p pressure - P reduced pressure - R Reynolds number - U ₀ representative velocity - V velocity - V _r , V , V _z radial, azimuthal and axial velocity components - w ₀ uniform suction through the disc. - density - electrical conductivity - kinematic viscosity - _e magnetic permeability - a parameter, (/)^1/2 z - a parameter, a - * displacement thickness - momentum thickness - angular velocity     

Displacement type unified equation for bending, buckling and vibration of conical shells and its applications

By using Donnell's simplication and starting from the displacement type equations of conical shells, and introducing a displacement functionU(s,,) (In the limit case, it will be reduced to cylindrical shell displacement function introduced by V. S. Vlasov) and a generalized loadq,(s,,),the equations of conical shells are changed into an eighth—order solvable partial differential equation about the displacement functionU(s,,). As a special case, the general bending problem of conical shells on Winkler foundation has been studied. Detailed numerical results and boundary coefficients for edge unit loads are obtained.The project supported by the National Natural Science Foundation of China.     

Heteroclinic orbits in singular systems: A unifying approach

We consider singularly perturbed systems , such that=f(, _o, 0). _o ^m , has a heteroclinic orbitu(t). We construct a bifurcation functionG(, ) such that the singular system has a heteroclinic orbit if and only ifG(, )=0 has a solution=(). We also apply this result to recover some theorems that have been proved using different approaches.     

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.