The dynamic performance of the Weissenberg rheogoniometer III. Large amplitude oscillatory shearing — harmonic analysis

The harmonic content of the nonlinear dynamic behaviour of 1% polyacrylamide in 50% glycerol/water was studied using a standard Model R 18 Weissenberg Rheogoniometer. The Fourier analysis of the Oscillation Input and Torsion Head motions was performed using a Digital Transfer Function Analyser.In the absence of fluid inertia effects and when the amplitude of the (fundamental) Oscillation Input motion _I is much greater than the amplitudes of the Fourier components of the Torsion Head motion _Tn empirical nonlinear dynamic rheological propertiesG _n (, ⁰),G _n (, ⁰) and/or _n (, ⁰), _n (, ⁰) may be evaluated without a-priori-knowledge of a rheological constitutive equation. A detailed derivation of the basic equations involved is presented.Cone and plate data for the third harmonic storage modulus (dynamic rigidity)G ₃ (, ⁰), loss modulusG ₃ (, ⁰) and loss angle ₃ (, ⁰) are presented for the frequency range 3.14 × 10^–2 1.25 × 10² rad/s at two strain amplitudes, _CP ⁰ = 2.27 and 4.03. Composite cone and plate and parallel plates data for both the third and fifth harmonic dynamic viscosities ₃ (, ⁰), _S (, ⁰) and dynamic rigiditiesG ₃ (, ⁰),G ₅ (, ⁰) are presented for strain amplitudes in the ranges 1.10 _CP ⁰ 4.03 and 1.80 _PP ⁰ 36 for a single frequency, = 3.14 × 10^–1 rad/s. Good agreement was obtained between the results from both geometries and the absence of significant fluid inertia effects was confirmed by the superposition of the data for different gap widths.     

Investigation of stationary flow of viscous gas in a thin three-dimensional shock layer

A three-dimensional shock layer near the blunt surface of a fairly smooth body is analyzed asymptotically. Equations of the first approximation are obtained and justified in various cases of the limit 1, 0, ( – 1)^–1M ^-2 0. These equations are simplified for the flow near the stagnation point of a body with double curvature and near the blunt leading edge of a sweptback wing. The results of some calculations are given and compared with the results of [17, 18] in the case of axisymmetric flow.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 115–126, September–October, 1980.     

Some problems of nonisothermal steady flow of a viscous fluid

The paper presents solutions to the problems of plane Couette flow, axial flow in an annulus between two infinite cylinders, and flow between two rotating cylinders. Taking into account energy dissipation and the temperature dependence of viscosity, as given by Reynolds's relation =₀ exp (–T) (₀, =const). Two types of boundary conditions are considered: a) the two surfaces are held at constant (but in general not equal) temperatures; b) one surface is held at a constant temperature, the other surface is insulated.Nonisothermal steady flow in simple conduits with dissipation of energy and temperature-dependent viscosity has been studied by several authors [1–11]. In most of these papers [1–6] viscosity was assumed to be a hyperbolic function of temperature, viz. =_m 1/1+²(T–T_m.Under this assumption the energy equation is linear in temperature and can he easily integrated. Couette flow with an exponential viscosity-temperature relation. =₀ e ^–T (₀, =const), (0.1) was studied in [7, 8]. Couette flow with a general (T) relation was studied in (9).Forced flow in a plane conduit and in a circular tube with a general (T) relation was studied in [10]. In particular, it has been shown in [10] that in the case of sufficiently strong dependence of viscosity on temperature there can exist a critical value of the pressure gradient, such that a steady flow is possible only for pressure gradients below this critical value.In a previous work [11] the authors studied Polseuille flow in a circular tube with an exponential (T) relation. This thermohydrodynamic problem was reduced to the problem of a thermal explosion in a cylindrical domain, which led to the existence of a critical regime. The critical conditions for the hydrodynamic thermal explosion and the temperature and velocity profiles were calculated.In this paper we treat the problems of Couette flow, pressureless axial flow in an annulus, and flow between two rotating cylinders taking into account dissipation and the variation of viscosity with temperature according to Reynolds's law (0.1). The treatment of the Couette flow problem differs from that given in [8] in that the constants of integration are found by elementary methods, whereas in [8] this step involved considerable difficulties. The solution to the two other problems is then based on the Couette problem.     

Unsteady supersonic flow around blunt bodies with detached shock wave

The numerical method of calculating the supersonic three-dimensional flow about blunt bodies with detached shock wave presented in [1–3] is applied to the case of unsteady flow. The formulation of the unsteady problem is analogous to that of [4], which assumes smallness of the unsteady disturbances.The paper presents some results of a study of the unsteady flow about blunt bodies over a wide range of variation of the Mach number M=1.50– and dimensionless oscillation frequency l/V=0–1.0. A comparison is made with the results obtained from the Newton theory.     

Method of mass-average quantities for the boundary layer in an outer flow with transverse nonuniformity

An engineering method is proposed for calculating the friction and heat transfer through a boundary layer in which a nonuniform distribution of the velocity, total enthalpy, and static enthalpy is specified across the streamlines at the initial section x₀. Such problems arise in the vortical interaction of the boundary layer with the high-entropy layer on slender blunt bodies, with sudden change of the boundary conditions for an already developed boundary layer (temperature jump, surface discontinuity), and in wake flow past a body, etc.Notation x, y longitudinal and transverse coordinates - u,, H, h gas velocity, stream function, total and static enthalpy - p,,, pressure, density, viscosity, Prandtl number - , q friction and thermal flux at the body surface - r(x), (x) body surface shape and boundary layer thickness - V, M freestream velocity and Mach number - u⁽⁰⁾(x₀,), H⁽⁰⁾(x₀,), h⁽⁰⁾(x₀,) parameter distributions at initial section - u⁽⁰⁾(x,), h⁽⁰⁾(x,), h⁽⁰⁾(x,) profiles of quantities in outer flow in absence of friction and heat transfer at the surface of the body The indices v=0, 1 relate to plane and axisymmetric flows - , w, b, relate to quantities at the outer edge of the inner boundary layer, at the body surface in viscid and nonviscous flows, and in the freestream, respectively. The author wishes to thank O. I. Gubanov, V. A. Kaprov, I. N. Murzinov, and A. N, Rumynskii for discussions and assistance in this study.     

Optimum lifting wings with specified longitudinal trim characteristics

The case of an infinitely slender wing that slightly disturbs a supersonic ideal gas flow is considered. The plan form and the free-stream Mach number M are given. The optimum surface of the wing y=g(x, z) is determined as a result of finding a bounded function of the local angles of attack _M=g(x, z)/x that minimizes the drag coefficient c_x for given values of the lift coefficient c_y and the pitching moment coefficient m_z. The problem is solved in the class of piecewise-constant functions for wings of complex geometry [1].Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 185–189, July–August, 1987.     

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.     

Anwendung des Übertragungsverfahrens bei stark abklingenden Lösungsfunktionen

Übersicht Bei stark abklingenden Funktionen wird die Übertragungsmatrix U() aufgespalten in die Anteilc U ₁() e^– und U ₂() e. Der zweite Term spielt am Rand = 0 keinc Rolle. Die unbekannten Anfangswerte sind über die Matrix U ₁(0) an die bekannten gebunden und eindeutig bestimmbar.

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Summary For strongly decaying solution functions the transfer matrix U() is splitted into the parts U ₁() e^– and U ₂() e. The second term does not influence at the boundary = 0. The unknown initial values are related by the matrix U ₁(0) to the known values and they can be uniquely determined.

     

Calculation of the pressure on the side surface of bodies consisting of a spherical segment joined to an inverted cone in a supersonic stream

The results of investigations of inviscid flow over inverted cones with nose consisting of a spherical segment were published for the first time in Soviet literature in [1–4]. In the present paper, a numerical solution to this problem is obtained using the improved algorithms of [5, 6], which have proved themselves well in problems of exterior flow over surfaces with positive angles of inclination to the oncoming flow. It is shown that the Mach number 2 M , equilibrium and nonequilibrium physicochemical transformations in air (H = 60 km, V = 7.4 km/sec, R₀ = 1 m), and the angle of attack 0 40° influence the investigated pressure distributions. A comparison of the results of the calculations with drainage experiments for M = 6, = 0-25° confirms the extended region of applicability of the developed numerical methods. Also proposed is a simple correlation of the dependence on the Mach number in the range 1.5 M of the shape of the shock wave near a sphere in a stream of ideal gas with adiabatic exponent = 1.4.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 178–183, January–February, 1981.     

On Perturbative Expansions to the Stochastic Flow Problem

When analyzing stochastic steady flow, the hydraulic conductivity naturally appears logarithmically. Often the log conductivity is represented as the sum of an average plus a stochastic fluctuation. To make the problem tractable, the log conductivity fluctuation, f, about the mean log conductivity, lnK _G, is assumed to have finite variance, _f ². Historically, perturbation schemes have involved the assumption that _f ²<1. Here it is shown that _f may not be the most judicious choice of perturbation parameters for steady flow. Instead, we posit that the variance of the gradient of the conductivity fluctuation, _f ², is more appropriate hoice. By solving the problem withthis parameter and studying the solution, this conjecture can be refined and an even more appropriate perturbation parameter, , defined. Since the processes f and f can often be considered independent, further assumptions on f are necessary. In particular, when the two point correlation function for the conductivity is assumed to be exponential or Gaussian, it is possible to estimate the magnitude of f in terms of _f and various length scales. The ratio of the integral scale in the main direction of flow ( _x ) to the total domain length (L^*), _x ²=_x/L^*, plays an important role in the convergence of the perturbation scheme. For _x smaller than a critical value _c, _x < _c, the scheme's perturbation parameter is =_f/_x for one- dimensional flow, and =_f/_x ² for two-dimensional flow with mean flow in the x direction. For _x > _c, the parameter =_f/_x ³ may be thought as the perturbation parameter for two-dimensional flow. The shape of the log conductivity fluctuation two point correlation function, and boundary conditions influence the convergence of the perturbation scheme.     

Stability and Transition on a Swept Cylinder in a Supersonic Flow

Results of experimental investigations of the evolution of natural disturbances and laminar–turbulent transition in a supersonic boundary layer on the attachment line of a circular cylinder with a sweep angle of 68° and a freestream Mach number M = 2 are presented. The experimental studies are supplemented by calculations of the mean flow and stability characteristics. Flow regimes in the boundary layer on the attachment line are determined by a hotwire technique as functions of the Reynolds number and height of twodimensional roughness elements. The results are compared with NASA (Ames) experiments.     

About the Regularized Navier–Stokes Equations

The first goal of this paper is to study the large time behavior of solutions to the Cauchy problem for the 3-dimensional incompressible Navier–Stokes system. The Marcinkiewicz space L^3, is used to prove some asymptotic stability results for solutions with infinite energy. Next, this approach is applied to the analysis of two classical regularized Navier–Stokes systems. The first one was introduced by J. Leray and consists in mollifying the nonlinearity. The second one was proposed by J.-L. Lions, who added the artificial hyper-viscosity (–)^{/ 2}, > 2 to the model. It is shown in the present paper that, in the whole space, solutions to those modified models converge as t toward solutions of the original Navier–Stokes system.     

On the Ellipticity of the Middle Surface of a Shell and its Application to the Asymptotic Analysis of ‘Membrane’ Shells

We consider a surface S = (), where ² is a bounded, connected, open set with a smooth boundary and : ³ is a smooth map; let () denote the components of the two-dimensional linearized strain tensor of S and let 0 with length 0 > 0. We assume the the norm ,|| ()||0, in the space V0() = { H¹() × H¹() × L²(); = 0 on 0 } is equivalent to the usual product norm on this space. We then establish that this assumption implies that the surface S is uniformly elliptic and that we necessarily have 0 = .     

Versuche zu einem „shear-rate-blockage“-Effekt am Protoplasma zentrifugierter Gewebe

Ohne ZusammenfassungI thank Mr.Scott Blair for his answer reserving the term dilatancy for cases of real dilatation. Now it's only to wish that this terminology is used according to his Report on the principles of rheological nomenclature (Amsterdam 1949).     

Investigation of the flow past wings of infinite span with a blunt leading edge in the vorticity interaction regime

An asymptotic analysis of the Navier-Stokes equations is carried out for the case of hypersonic flow past wings of infinite span with a blunt leading edge when 0, Re , and M . Analytic solutions are obtained for an inviscid shock layer and inviscid boundary layer. The results of a numerical solution of the problems of vorticity interaction at the blunt edge and on the lateral surface of the wing are presented. These solutions are compared with the solution of the equations of a thin viscous shock layer and on the basis of this comparison the boundaries of the asymptotic regions are estimated.deceasedTranslated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 120–127, November–December, 1987.     

Calculation of heat transfer accompanying hypersonic three-dimensional flow of air over slender blunt-nosed cones

The effective length method [1, 2] has been used to make systematic calculations of the heat transfer for laminar and turbulent boundary layers on slender blunt-nosed cones at small angles of attack ( + 5° in a separationless hypersonic air stream dissociating in equilibrium (half-angles of the cones 0 20°, angles of attack 0 15°, Mach numbers 5 M 25). The parameters of the gas at the outer edge of the boundary layer were taken equal to the inviscid parameters on the surface of the cones. Analysis of the results leads to simple approximate dependences for the heat transfer coefficients.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 173–177, September–October, 1981.     

Hypersonic flow past blunt-edged delta wings

A method is proposed for calculating hypersonic ideal-gas flow past blunt-edged delta wings with aspect ratios = 100–200. Systematic wing flow calculations are carried out on the intervals 6 M 20, 0 20, 60 80; the results are analyzed in terms of hypersonic similarity parameters.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 175–179, September–October, 1990.     

Asymptotic Modeling of Nonlinear Wave Processes in Shock‐Loaded Elastoplastic Materials

Nonlinear wave processes in shockloaded elastoplastic materials are modeled. A comparison of the results obtained with experimental data and numerical solutions of exact systems of dynamic equations shows that the model equations proposed qualitatively describe the stressdistribution evolution in both the elasticflow and plasticflow regions and can be used to solve one and twodimensional problems of pulsed deformation and fracture of elastoplastic media.     

Flow Structure in the Separation Zone in the Interaction between Shock Waves and the Boundary Layer on a Plate with Slip

A model which makes it possible to calculate the reverse-flow parameters in the separation zone is constructed on the basis of the results of an integrated experimental study of the characteristics of the separated flow developed in the transition from free to non-free interaction between plane shock waves and the boundary layer on a plate with slip. The effect of the Mach number of the reverse flow in the separation zone on the properties of inner boundary layer separation is analyzed. Features of the interference flow due to boundary layer transition are described. The present study is a continuation of investigations [1–3] devoted to the study of a new steady-state type of interaction between shock waves and the boundary layer on a plate with slip in which the separation line formed would propagate upstream beyond the sharp leading edge if no leading edge was present, i.e., so-called non-free interaction.     

“Thermal layer” effect in MHD: Interaction of a parallel shock with a layer of decreased density

Interaction of a parallel fast MHD shock with a layer of decreased density is discussed using ideal MHD approach. This is an extrapolation of gas dynamic thermal layer effect on ideal MHD. Computer simulations show that a magnetic field of a moderate intensity ( 1) may change the character of the flow for intermediate Mach numbers (M 5) and a new raking regime may occur which is not observed in the absence of a magnetic field. Self similar precursor analogous to that in gas dynamics may develop in the case of highM and low density in the layer but magnetic forces essentially decrease its growth rate. This problem appears in connection with cosmical shock propagation where planetary magnetic tails play the role of the thermal layer, and it may also be observed in the laboratory when the shock is strong enough to heat the walls ahead of it.This article was processed using Springer-Verlag T_EX Shock Waves macro package 1.0 and the AMS fonts, developed by the American Mathematical Society.