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.     

Electrical charging of bodies as a consequence of charged-particle extraction by a hydrodynamic flow

We investigate the problem of electrical charging of bodies as a result of charged-particle extraction by a hydrodynamic flow. The analysis is performed in view of the application to the problem of motion electrification of aircraft caused by a stream of charged particles into the surrounding space. We formulate the appropriate system of nonstationary electrohydrodynamic equations. It is shown that in many applications the charging of electrically insulated bodies consists of two successive intermediate processes. The first process is the formation of charge Q on the body in time T₁ The second process consists of a change of the body potential (with a constant charge Q) as a consequence of the stream of charged particles into the outside space noted above. At the end of the second process (with duration T₂) the body potential is at . We also investigate the problem of charging a spherical source of neutral and charged particles. Using the analytical solution we find the quantities Q and and the characteristic times T₁ and T₂. It is shown that the time T₂ can exceed T₁ by several orders of magnitude. We formulate the problems of nonstationary electric fields during the extraction of several types of particles.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 94–103, September–October, 1977.     

Numerical investigation of laminar separation in the case of supersonic flow of viscous gas past spiked bodies

The complete Navier-Stokes equations for a compressible viscous perfect heat conducting gas have been used in a numerical investigation of laminar separation in the case of supersymmetric axisymmetric flow past cylinders with a conical nose and a spike at the front of finite thickness. The flow structure has been studied in its dependence on the length of the spike and the half-angle of the conical tip. For the considered free-stream parameters (2 M 6, 100 Re 500) and spike lengths, which do not exceed the diameter of the cylinder, the existence of steady flow regimes has been established and it has been shown that the spike in front of the body reduces its total drag and the heat flux to its surface.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 126–131, March–April, 1984.     

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.     

Existence theorem for a minimum problem with free discontinuity set

We study the variational problem Where is an open set in ⁿ ,n2gL ^q () L (), 1q<+, O<, <+ andH _n–1 is the (n–1)-dimensional Hausdorff Measure.     

Green's functions,bi-linear forms,and completeness of the eigenfunctions for the elastostatic strip and wedge

The bi-harmonic Green's functionG(r,r) for the infinite strip region -1y1, -<x<, with the boundary conditionsG=G/y ony=±1, is obtained in integral form. It is shown thatG has an elegant bi-linear series representation in terms of the (Papkovich-Fadle) eigenfunctions for the strip. This representation is then used to show that any function bi-harmonic in arectangle, and satisfying the same boundary conditions asG, has a unique representation in the rectangle as an infinite sum of these eigenfunctions. For the case of the semi-infinite strip, we investigate conditions on sufficient to ensure that is exponentially small asx. In particular it is proved that this is so, solely under the condition that be bounded asx.A corresponding pattern of results is established for the wedge of general angle. The Green's function is obtained in integral form and expressed as a bilinear series of the (Williams) eigenfunctions. These eigenfunctions are proved to be complete for all functions bi-harmonic in anannular sector (and satisfying the same boundary conditions as the Green's function). As an application it is proved that if an elastostatic field exists in a corner region with free-free boundaries, and with either (i) the total strain energy bounded, or (ii) the displacement field bounded, then this field has a unique representation as a sum of those Williams eigenfunctions whichindividually posess the properties (i), (ii).The methods used here extend to all other linear homogeneous boundary conditions for these geometries.On leave of absence at the University of British Columbia, Vancouver, B.C. Canada, during 1977–79. This work was supported in part by N.R.C. grants Nos. A9259 and A9117.     

Lp-estimates for the nonlinear spatially homogeneous Boltzmann equation

This paper studies L^p-estimates for solutions of the nonlinear, spatially homogeneous Boltzmann equation. The molecular forces considered include inverse k^th-power forces with k > 5 and angular cut-off.The main conclusions are the following. Let f be the unique solution of the Boltzmann equation with f(v,t)(1 + ¦v²¦)^(s ₁ ^{+ /p)/2} L¹, when the initial value f ₀ satisfies f ₀(v) 0, f ₀(v) (1 + ¦v¦²)^(s ₁ ^{+ /p)/2} L¹, for some s₁ 2 + /p, and f ₀(v) (1 + ¦v¦²)^s/2 L^p. If s 2/p and 1 < p < , then f(v, t)(1 + ¦v¦²)^{(s s} ₁)^/2 L^p, t > 0. If s >2 and 3/(1+ ) < p < , thenf(v,t) (1 + ¦v¦²)^(s(s ₁ ^{+ 3/p))/2} L^p, t > 0. If s >2 + 2C₀/C₁ and 3/(l + ) < p < , then f(v,t)(1 + ¦v¦²)^s/2 L^p, t > 0. Here 1/p + 1/p = 1, x y = min (x, y), and C₀, C₁, 0 < 1, are positive constants related to the molecular forces under consideration; = (k – 5)/ (k – 1) for k^th-power forces.Some weaker conclusions follow when 1 < p 3/ (1 + ).In the proofs some previously known L-estimates are extended. The results for L^p, 1 < p < , are based on these L-estimates coupled with nonlinear interpolation.     

On the Asymptotic Properties of Solutions of Linear Functional Differential Equations with Constant Coefficients and Linearly Transformed Argument

We establish new properties of C ¹(0, +)-solutions of the linear functional differential equation in the neighborhood of the singular point t = +.     

Rheology of molten polymers

Summary TheCross equation describes the flow of pseudoplastic liquids in terms of an upper and a lower Newtonian viscosity corresponding to infinite and zero shear, and ₀, and of a third material constant related to the mechanism of rupture of linkages between particles in the intermediate, non-Newtonian flow regime, Calculation of of bulk polymers is important, since it cannot be determined experimentally. The equation was applied to the melt flow data of two low density polyethylenes at three temperatures.Using data in the non-Newtonian region covering 3 decades of shear rate to extrapolate to the zero-shear viscosity resulted in errors amounting to about onethird of the measured ₀ values. The extrapolated upper Newtonian viscosity was found to be independent of temperature within the precision of the data, indicating that it has a small activation energy.The ₀ values were from 100 to 1,400 times larger than the values at the corresponding temperatures.The values of were large compared to the values found for colloidal dispersions and polymer solutions, but decreased with increasing temperature. This shows that shear is the main factor in reducing chain entanglements, but that the contribution of Brownian motion becomes greater at higher temperatures.

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Zusammenfassung Die Gleichung vonCross beschreibt das Fließverhalten von pseudoplastischen Flüssigkeiten durch drei Konstante: Die obereNewtonsche Viskosität (bei sehr hohen Schergeschwindigkeiten), die untereNewtonsche Viskosität ₀ (bei Scherspannung Null), und eine Materialkonstante, die vom Brechen der Bindungen zwischen Partikeln im nicht-Newtonschen Fließbereich abhängt. Die Berechnung von ist wichtig für unverdünnte Polymere, wo man sie nicht messen kann.Die Gleichung wurde auf das Fließverhalten der Schmelzen von zwei handelsüblichen Hochdruckpolyäthylenen bei drei Temperaturen angewandt. Die Werte von ₀, durch Extrapolation von gemessenen scheinbaren Viskositäten im Schergeschwindigkeitsbereich von 10 bis 4000 sec^–1 errechnet, wichen bis 30% von den gemessenen ₀-Werten ab. Die Aktivierungsenergie der war so klein, daß die-Werte bei den drei Temperaturen innerhalb der Genauigkeit der Extrapolation anscheinend gleich waren. Die ₀-Werte waren 100 bis 1400 mal größer als die-Werte.Im Verhältnis zu kolloidalen Dispersionen und verdünnten Polymerlösungen war das der Schmelzen groß, nahm aber mit steigender Temperatur ab. Deshalb wird die Verhakung der Molekülketten hauptsächlich durch Scherbeanspruchung vermindert, aber der Beitrag derBrownschen Bewegung nimmt mit steigender Temperatur zu.

     

Hodograph design of lifting airfoils with high critical mach numbers

We wish to construct airfoils that have the highest free-stream Mach number M for a given set of geometric constraints for which the flow is nowhere supersonic. Nonlifting airfoils which maximize M for a given thickness ratio are known to possess long sonic segments at their critical speed. To construct lifting airfoils, we proceed under the conjecture that the optimal airfoil satisfying a given set of constraints is the one possessing the longest possible arc length of sonic velocity. A boundary-value problem is formulated in the hodograph plane using transonic small-disturbance theory whose solution determines an airfoil with long sonic arcs. For small lift coefficients, the hodograph domain covers two Riemann sheets and a finite-difference method is used to solve the boundary-value problem on this domain. A numerical integration of the solution around the boundary yields an airfoil shape, and three examples are discussed. The performance of these airfoils is compared with standard airfoils having the same lift coefficient and , and it is shown that the calculated airfoils have a 6%–10% increase in critical M .This research was supported by the Air Force Office of Scientific Research under Grant F49620-93-1-0022DEF and by the National Science Foundation under Grant DMS-9157546. Research support for M.C.A.K. was given by the IBM Corporation under a Graduate Research Fellowship.     

The behavior of a spherical bubble near a solid wall in a compressible liquid

Summary The behavior of a spherical bubble near a solid wall is analysed by considering the liquid compressibility. The equation of motion of the bubble with first order correction for the effects of liquid compressibility and solid wall is derived. The equation obtained here coincides with the known result in case of L or C . Further experimental study is made on the motion of bubbles produced by a spark discharge in water. The theoretical results are in good agreement with the experiments.

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Das Verhalten einer kugelförmigen Blase in einer kompressiblen Flüssigkeit in der Nähe einer festen Wand

Übersicht Bei Berücksichtigung der Flüssigkeitskompressibilität wird das Verhalten einer kugelförmigen Blase in der Nähe einer festen Wand analysiert. Die Gleichung der Bewegung der Blase wird mit der Korrektur erster Ordnung für den Einfluß der Flüssigkeitskompressibilität und der festen Wand angegeben. Aus der erhaltenen Gleichung wird für L oder C das bekannte Ergebnis hergeleitet. Darüber hinaus wird eine experimentelle Untersuchung der Blasenbewegung durchgeführt. Die Blase wird mit Hilfe von Funkendurchschlägen zwischen Elektroden in Wasser erzeugt. Die theoretischen Ergebnisse stimmen gut mit den Experimenten überein.

     

Asymptotic behavior of solutions of nonlinear elliptic equations

We study and obtain formulas for the asymptotic behavior as ¦x¦ of C ² solutions of the semilinear equation u=f(x, u), x (*) where is the complement of some ball in ⁿ and f is continuous and nonlinear in u. If, for large x, f is nearly radially symmetric in x, we give conditions under which each positive solution of (*) is asymptotic, as ¦x¦, to some radially symmetric function. Our results can also be useful when f is only bounded above or below by a function which is radially symmetric in x or when the solution oscillates in sign. Examples when f has power-like growth or exponential growth in the variables x and u usefully illustrate our results.     

Contribution to the asymptotic theory of quasisteady viscous flows

The aim of this paper is to obtain equations which, as Re , describe quasisteady viscous incompressible flows with stream surfaces that are closed only in the limit as Re , and also to examine the simpler consequences of these equations.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 46–51, July–August, 1985.The author wishes to thank V. Ya. Neiland for taking an interest in his work and discussing the results.     

Evolution of the Flow Field around a Circular Cylinder and a Sphere upon Instantaneous Start with a Supersonic Velocity

Evolution of the flow field around a circular cylinder and a sphere instantaneously starting with a constant supersonic velocity (M = 5 and Re = 10⁵) from the state at rest is studied by means of numerical integration of unsteady twodimensional Navier–Stokes equations.     

Linearized dynamics for step jumps of velocity and displacement of shearing flows of a simple fluid

We consider linearized dynamics associated with step jumps in the velocity or displacement of the boundary of a fluid in a shearing motion. The discontinuity will propagate into the interior with a speed ( is the density) if the initial valuesG(0) andG(0) of the fading memory kernels are bounded, 0 <G(0) < , – <G (0) < 0. IfG(0) butG(0) = – , then the boundary of the support of the solution still propagates with the speedC. However, the solutions on both sides of the boundary match together in aC -fashion. IfG(0) butG(0) = 0, the amplitude of the discontinuity will not damp as in a purely elastic fluid. IfG(0) = , the step change is felt immediately throughout the fluid, without shocks, as in Navier-Stokes fluids. This same type of parabolic behavior can be achieved by a small Newtonian contribution added to the integral form of the stress but if this contribution is small, a smooth transition layer around the shock will propagate with the speedC. In the case of step displacement, from rest to rest, singular surfaces of infinite velocity can propagate into the interior with speed of propagationC. The singular surfaces undergo multiple reflections off bounding walls, but the final steady state reached asymptotically is in universal form independent of material.     

Chaos and integrability in a nonlinear wave equation

We consider the parametrized family of equations _tt ,u- _xx u-au+u ₂ ² u=O,x(0,L), with Dirichlet boundary conditions. This equation has finite-dimensional invariant manifolds of solutions. Studying the reduced equation to a four-dimensional manifold, we prove the existence of transversal homoclinic orbits to periodic solutions and of invariant sets with chaotic dynamics, provided that =2, 3, 4,.... For =1 we prove the existence of infinitely many first integrals pairwise in involution.     

Spectral and correlation characteristics of the turbulent boundary layer on an extended flexible cylinder

In marine geophysical seismological prospecting extensive use is made of towed receiving systems consisting of extended flexible cylinders containing acoustic sensors over which the water flows in the longitudinal direction. The boundary layer pressure fluctuations on the cylinder surface are picked up by the sensors as hydrodynamic noise. This paper is concerned with the study of the energy and spacetime characteristics of the pressure fluctuations in the turbulent boundary layer on an extended flexible cylinder in a longitudinal flow. The pressure fluctuations on the cylinder surface have been investigated experimentally for Re_X=(2–4)·10⁷, a dimensionless diameter of the pressure fluctuation sensors d _p ⁺ =d_pu^*/=500, where d_p is the sensor diameter, u^* the dynamic viscosity, and the viscosity coefficient, and frequencies 0.02311.259 (=^*/U). The spectral and correlation characteristics of the pressure fluctuations on the surface of the flexible cylinder are found to differ from the corresponding characteristics for a rigid cylinder [1–4].Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i aza, No, 5, pp. 49–54, September–October, 1989.     

Interaction between a distributed source and the free surface of a viscous fluid

A self-similar solution of the Navier-Stokes equations describing steady-state axisymmetric viscous incompressible fluid flow in a half-space is investigated. The motion is induced by sources or sinks distributed over a vertical axis with a constant density. The horizontal plane bounding the fluid is a free surface. It is found that in the presence of sources a solution of the above type exists and is unique for any value of the Reynolds numberR > 0, but in the case of sinks only on the interval –2 R < 0. The results of calculating the self-similar solutions are presented. The asymptotics of the solutions are found asR 0 andR .Novosibirsk. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 2, pp. 53–65, March–April, 1996.     

Hyperbolic phenomena in a strongly degenerate parabolic equation

We consider the equation u _t =((u) (u _x )) _x , where >0 and where is a strictly increasing function with lim _s = <. We solve the associated Cauchy problem for an increasing initial function, and discuss to what extent the solution behaves qualitatively like solutions of the first-order conservation law u _t = ((u)) _x . Equations of this type arise, for example, in the theory of phase transitions where the corresponding free-energy functional has a linear growth rate with respect to the gradient.     

Injection of ideal fluid from a slot into a free stream

A cold gas is injected from a slot into a free stream of hot gas. In a simple model this leads to a two-fluid free boundary problem with the jump relation |u^-|²–|u⁺|² = ( constant) on the free boundary {u=0}, where u is the stream function. We prove that for any (–1, ) there exists a unique solution (Q, u) where Q is the flux of the injected fluid. Various properties of the solution u and of the free boundary are established.