Nonlinear analysis of the forced response of a beam with three mode interaction

An analysis is presented for the primary resonance of a clamped-hinged beam, which occurs when the frequency of excitation is near one of the natural frequencies,_n . Three mode interaction (₂ 3₁ and _{3 1} + 2₂) is considered and its influence on the response is studied. The case of two mode interaction (₂ 3₁) is also considered to compare it with the case of three mode interaction. The straight beam experiencing mid-plane stretching is governed by a nonlinear partial differential equation. By using Galerkin's method the governing equation is reduced to a system of nonautonomous ordinary differential equations. The method of multiple scales is applied to solve the system. Steady-state responses and their stability are examined. Results of numerical investigations show that there exists no significant difference between both modal interactions' influences on the responses.     

Impacts of local dispersion and first-order decay on solute transport in randomly heterogeneous porous media

Stochastic subsurface transport theories either disregard local dispersion or take it to be constant. We offer an alternative Eulerian-Lagrangian formalism to account for both local dispersion and first-order mass removal (due to radioactive decay or biodegradation). It rests on a decomposition of the velocityv into a field-scale componentv , which is defined on the scale of measurement support, and a zero mean sub-field-scale componentv _s , which fluctuates randomly on scales smaller than. Without loss of generality, we work formally with unconditional statistics ofv _s and conditional statistics ofv . We then require that, within this (or other selected) working framework,v _s andv be mutually uncorrelated. This holds whenever the correlation scale ofv is large in comparison to that ofv _s . The formalism leads to an integro-differential equation for the conditional mean total concentration c which includes two dispersion terms, one field-scale and one sub-field-scale. It also leads to explicit expressions for conditional second moments of concentration cc. We solve the former, and evaluate the latter, for mildly fluctuatingv by means of an analytical-numerical method developed earlier by Zhang and Neuman. We present results in two-dimensional flow fields of unconditional (prior) mean uniformv . These show that the relative effect of local dispersion on first and second moments of concentration dies out locally as the corresponding dispersion tensor tends to zero. The effect also diminishes with time and source size. Our results thus do not support claims in the literature that local dispersion must always be accounted for, no matter how small it is. First-order decay reduces dispersion. This effect increases with time. However, these concentration moments c and cc of total concentrationc, which are associated with the scale below, cannot be used to estimate the field-scale concentrationc directly. To do so, a spatial average over the field measurement scale is needed. Nevertheless, our numerical results show that differences between the ensemble moments ofc and those ofc are negligible, especially for nonpoint sources, because the ensemble moments ofc are already smooth enough.     

Evaluation of the dynamic-mechanical properties of solids from a cooperative model for stress relaxation

For many solid materials the stress relaxation process obeys the universal relationF = – (d/d lnt)_max = (0.1 ± 0.01) ( ₀ — _i ), regardless of the structure of the material. Here denotes the stress,t the time, ₀ the initial stress of the experiment and _i the internal stress. A cooperative model accounting for the similarity in relaxation behaviour between different materials was developed earlier. Since this model has a spectral character, the concepts of linear viscoelasticity are used here to evaluate the corresponding prediction of the dynamic mechanical properties, i.e. the frequency dependence of the storageE () and lossE () moduli. Useful numerical approximations ofE () andE () are also evaluated. It is noted that the universal relation in stress relaxation had a counterpart in the frequency dependence ofE (). The theoretical prediction of the loss factor for high-density polyethylene is compared with experimental results. The agreement is good.     

Numerical calculation of creep compliance from dynamic data for linear viscoelastic materials

Summary Numerical formulae are given for calculation of creep compliance from the known course of the storage and loss compliance with frequency for linear viscoelastic materials. These formulae involve values of the storage compliance and/or loss compliance at frequencies which are equally spaced on a logarithmic frequency scale. The ratio between successive frequencies corresponds to a factor of two.A method is introduced by which bounds for the relative error of those formulae can be derived. These bounds depend on the value of the damping, tan, at the angular frequency, ₀, at which the calculation is performed. The lower this damping, the easier is the calculation of the creep compliance. This calculation involves either the value of the storage compliance at a frequency ₀ = 1/t, and the values of the loss compliance in a rather narrow frequency region around ₀; or the value of the storage compliance at frequency ₀, the value of the loss compliance at frequency ₀/2, and the derivative of the storage compliance with respect to the logarithm of frequency in a frequency region around ₀.

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Zusammenfassung Numerische Formeln werden gegeben, die die Berechnung der Kriechfunktion aus der dynamischen Nachgiebigkeit ermöglichen. In diesen Formeln treten Werte der Speicher- bzw. Verlustkomponente der dynamischen Nachgiebigkeit auf, die bei logarithmisch äquidistanten Frequenzen gemessen wurden. Das Verhältnis zweier aufeinanderfolgender Frequenzen entspricht stets einem Faktor 2.Für alle Formeln werden obere und untere Schranken für den relativen Fehler abgeleitet. Diese Schranken hängen vom Werte der Dämpfung (tan) ab, die bei der Kreisfrequenz ₀ auftritt, für die die Berechnung erfolgt. Die Berechnung der Kriechfunktion ist desto leichter, je niedriger der Wert der Dämpfung ist. Zu dieser Berechnung benötigt man entweder den Wert der Speicherkomponente der dynamischen Nachgiebigkeit bei der Kreisfrequenz ₀ = 1/t und die Werte der Verlustkomponente der dynamischen Nachgiebigkeit in einem ziemlich engen Frequenzintervall um ₀; oder den Wert der Speicherkomponente bei der Kreisfrequenz ₀, den Wert der Verlustkomponente bei der Kreisfrequenz ₀/2 und den Wert der logarithmischen Frequenzableitung der Speicherkomponente in einem Frequenzintervall um ₀.

     

Approximate First Integrals of a Galaxy Model

First-order approximate first integrals (conserved quantities)of a Hamiltonian dynamical system with two degrees of freedomwhich arises in the modeling of central part of a deformed galaxy [1] havebeen obtained based on the approximate Noether symmetries for resonances₁=₂, ₁=2₂ and 2₁=3₂. Furthermore,KAM curves have been obtained analytically and they have been compared with thenumerical ones on the Poincaré surface of section.     

Flow in porous media II: The governing equations for immiscible,two-phase flow

The Stokes flow of two immiscible fluids through a rigid porous medium is analyzed using the method of volume averaging. The volume-averaged momentum equations, in terms of averaged quantities and spatial deviations, are identical in form to that obtained for single phase flow; however, the solution of the closure problem gives rise to additional terms not found in the traditional treatment of two-phase flow. Qualitative arguments suggest that the nontraditional terms may be important when / is of order one, and order of magnitude analysis indicates that they may be significant in terms of the motion of a fluid at very low volume fractions. The theory contains features that could give rise to hysteresis effects, but in the present form it is restricted to static contact line phenomena.Roman Letters (, = , , and ) A interfacial area of the- interface contained within the macroscopic system, m² - A _e area of entrances and exits for the -phase contained within the macroscopic system, m² - A interfacial area of the- interface contained within the averaging volume, m² - A ^* interfacial area of the- interface contained within a unit cell, m² - A _e ^* area of entrances and exits for the-phase contained within a unit cell, m² - g gravity vector, m²/s - H mean curvature of the- interface, m^–1 - H area average of the mean curvature, m^–1 - H – H , deviation of the mean curvature, m^–1 - I unit tensor - K Darcy's law permeability tensor, m² - K permeability tensor for the-phase, m² - K viscous drag tensor for the-phase equation of motion - K viscous drag tensor for the-phase equation of motion - L characteristic length scale for volume averaged quantities, m - characteristic length scale for the-phase, m - n unit normal vector pointing from the-phase toward the-phase (n = –n ) - p _c p – P , capillary pressure, N/m² - p pressure in the-phase, N/m² - p intrinsic phase average pressure for the-phase, N/m² - p – p , spatial deviation of the pressure in the-phase, N/m² - r ₀ radius of the averaging volume, m - t time, s - v velocity vector for the-phase, m/s - v phase average velocity vector for the-phase, m/s - v intrinsic phase average velocity vector for the-phase, m/s - v – v , spatial deviation of the velocity vector for the-phase, m/s - V averaging volume, m³ - V volume of the-phase contained within the averaging volume, m³ Greek Letters V /V, volume fraction of the-phase - mass density of the-phase, kg/m³ - viscosity of the-phase, Nt/m² - surface tension of the- interface, N/m - viscous stress tensor for the-phase, N/m² - / kinematic viscosity, m²/s     

Normal forms for random diffeomorphisms

Given a dynamical system (,, ,) and a random diffeomorphism (): ^{d d} with fixed point at x=0. The normal form problem is to construct a smooth near-identity nonlinear random coordinate transformation h() to make the random diffeomorphism ()=h()^–1() h() as simple as possible, preferably linear. The linearization D(, 0)=:A() generates a matrix cocycle for which the multiplicative ergodic theorem holds, providing us with stochastic analogues of eigenvalues (Lyapunov exponents) and eigenspaces. Now the development runs pretty much parallel to the deterministic one, the difference being that the appearance of turns all problems into infinite-dimensional ones. In particular, the range of the homological operator is in general not closed, making the conceptof-normal form necessary. The stochastic versions of resonance and averaging are developed. The case of simple Lyapunov spectrum is treated in detail.     

Symplectic approach to the theory of quantized fields. II

We prove that the set D of vector fields on the configuration space B of a field whose 1-parameter groups locally associated are groups of fibre-preserving transformations of B that leave invariant that field in the sense of variational theory, is a Lie algebra with respect to ordinary addition, multiplication by real numbers and Lie brackets. We see that this Lie algebra structure can be carried over to the corresponding set of Noether invariants, which then becomes a Lie algebra in a natural way.Further, we define the n-form of Poincaré-Cartan of a field, and we use it to generalize the Lie algebras D and in a reasonable way. The algebras D and are subalgebras of the new Lie algebras D and introduced. A main result in this connection is the following: the differential d of the n-form of Poincaré-Cartan is –(d+f), where (, d+f) are the field equations on the vertical bundle B.The symplectic manifold of solutions associated with a field is introduced in a formal way and the former Lie algebras D, , D, are interpreted on this manifold. In imitation of the case of analytical dynamics, the main results in this direction are: a) Every vector field of the Lie algebra D defines, in a canonical way, a vector field on the manifold of solutions such that its polar 1-form with respect to the symplectic metric ₂ is the differential of its corresponding Noether invariant, and b) the Lie bracket [, ] of two Noether invariants , is the Noether invariant given by ₂(D, D), where D, D are the vector fields on the manifold of solutions defined, in the sense a), by two infinitesimal generators of , , respectively. This will allow us to regard the Lie algebra as the analogous object in field theory to the Poisson algebra of analytic dynamics.We apply the general formalism to the relativistic theory of non-linear scalar fields, and we compare our results with the formalism developed by I. Segal for this case.     

Oscillatory measurements of silicone oils —Loss and storage modulus master curves

The work describes a way to obtain loss modulus and storage modulus master curves from oscillatory measurements of silicone oils.The loss modulus master curve represents the dependence of the viscous flow behavior on · ₀ ^* and the storage modulus master curve — the dependence of the elastic flow behavior on · ₀ ^* .The relation between the values of the loss modulus and storage modulus master curves (at a certain frequency) is a measurement of the viscoelastic behavior of a system. The G/G-ratio depends on · ₀ ^* which leads to a viscoelastic master curve. The viscoelastic master curve represents the relation between the elastic and viscous oscillatory flow behavior.     

Harmonic solutions of micropolar elastodynamics

The equations of micropolar elastodynamics are considered for an unbounded continuum subjected to a body force and a body couple. These act harmonically with the same real frequency , but with individual arbitrary spatial distributions. Over a harmonic state, the displacement and microrotation are related to two radiation conditioned harmonic vectors, each acquiring three eigenvalue contributions, assuming a noncritical -frequency. Altogether, four distinct eigenvalues are admissible. If ²<2² ₀, ₀ being a frequency parameter of the continuum, two of these are real while two are purely imaginary. But if ²<2² ₀, then all admissible eigenvalues are real. Each eigenvalue contribution resolves into a series of Hankel and Bessel functions coupled to Hankel type transforms of: (i) spherical integrals which, in turn, can be expanded via spherical harmonics for the 3-dimensional problem, (ii) circular integrals for the 2-dimensional problem. Axisymmetric and spherically symmetric results are deduced in 3-dimensions. Asymptotic solutions are also established; they disclose long-range formation of radially attenuated spherical (or circular) waves propagating with, generally, anisotropic amplitudes but, invariably, isotropic eikonals.If, in the absence of a body couple, a body force acts radially in 3-dimensions with a spherically symmetric strength, then the elastic displacement behaves likewise while the microrotation vanishes identically. Another application is made to a 2-dimensional problem for a 1 × 3 source system of body force plus body couple without longitudinal variation but with magnitudes symmetric about a longitudinal axis.As approaches a certain critical frequency , dependent solely on the continuum, at least two eigenvalues approach the same value. The phenomenon is explored for a continuum consistent with ²<2² ₀ and under the hypothesis ²<2² ₀. All admissible eigenvalues are then real throughout an -neighbourhood of . Here, two associated eigenvalue contributions behave singularly. Nevertheless, their essential singularities cancel out within the relevant combination. Examination of a far-field suggests that critical frequency attainment sets off a slow instability in the 2-dimensional configuration. In the 3-dimensional configuration, however, it preserves stability and eliminates radial attenuation; an exact solution is formulated for this case.     

Hausdorff Dimension of Invariant Sets for Random Dynamical Systems

Suppose X() is a compact random set, invariant with respect to a continuously differentiable random dynamical system (RDS) on a separable Hilbert space. It is shown that the Hausdorff dimension dim _H (X()) is an invariant random variable, and it is bounded by d, provided the RDS contracts d-dimensional volumes exponentially fast. Both exponential decrease of d-volumes as well as the approximation of the RDS by its linearization are assumed to hold uniformly in . The results are applied to reaction diffusion equations with additive noise and to two-dimensional Navier–Stokes equations with bounded real noise.     

Vorticity characteristics of the turbulent intermediate wake

Measurements of the lateral components _j (j=2 and 3) of the vorticity fluctuation vector have been made, using a vorticity probe consisting of two X-wires, in the intermediate wake of a circular cylinder. The effect of the spatial resolution of the probe on the measurement of _j has been studied. As the spatial resolution impairs, the variance and flatness factor of _j decrease whereas the skewness of _j increases. Reasonably accurate values of _j ² can be obtained by applying spectral corrections for the spatial resolution effect.Near the beginning of the intermediate wake, the variance of ₂ is larger than that of ₃ due to the significant contribution from ribs which connect consecutive spanwise roll vortices. This difference decreases with downstream distance. Also, the presence of the rolls is reflected by a local extremum in the skewness of ₃ on each side of the wake centerline. The magnitude of the extremum decreases with downstream distance.The support of the Australian Research Council is gratefully acknowledged.     

Emissivities and spectral absorptivities of combustion products

This paper is concerned with the determination of the spectral absorptivities K and emissivities of a thermodynamic-equilibrium mixture of combustion products consisting of CO₂, H₂O, CO, OH, HCl, HF, H₂, and NO molecules on the temperature interval 2000–5000° K. The proposed calculation method, based on the use of a graph of the molecular absorptivities reduced to atmospheric pressure, enables K and to be calculated for any gas mixture composed of the above-mentioned molecules.     

A non-affine network model for polymer melts

In this paper, a network model of polymer melts is proposed in which network junction points move non-affinely. In this non-affine motion, junction points follow particle paths as seen by an observer rotating at the fluid element's net rigid-rotation rate. The speed at which junction points move is reduced as the network segments near their maximum extensions. In order to maintain a frame invariant model, it is necessary that the vorticity be decomposed into two portions, such that, = _R + _D . The deformational vorticity, _D , arises from shear deformation and is frame invariant while the rigid vorticity, _R , is frame dependent. A constitutive equation based on this finitely extensible network strand (FENS) motion is developed. The model illustrates how rotations that cause changes in the relative orientation of a fluid element with its surroundings can be incorporated into a constitutive equation using the deformational vorticity. The FENS model predicts a shear-thinning viscosity, and the Trouton viscosity predicted by the model is finite for all elongation rates. Finally, stochastic simulation results are presented to justify a mathematical approximation used in deriving the constitutive equation.     

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     

Flow visualization and analysis of aerodynamically detuned supersonic rotors

The effect of aerodynamic detuning on the supersonic steady and unsteady blade passage flow field is experimentally investigated on a free surface water table by means of color Schlieren and shadowgraph flow visualization techniques. Two aerodynamic detuning mechanisms are considered: (1) alternate circumferential spacing of adjacent airfoils; (2) the replacement of alternate airfoils with splitters. The steady flow visualization demonstrates the significant effect of aerodynamic detuning on the passage flow field and, in particular, the shock wave-airfoil surface intersection locations. The unsteady flow visualization studies show the importance of the interblade phase angle. A mathematical model is also described and utilized to demonstrate the enhanced aeroelastic stability associated with the altered cascade passage shock wave structure due to these aerodynamic detuning mechanisms.List of symbols a dimensionless perturbation sonic velocity - C airfoil chord - I _x mass moment of inertia - k reduced frequency, k = C/2 - K spring constant - M _R dimensionless unsteady aerodynamic moment - M Mach number - P _split split splitter circumferential spacing - P _start splitter leading edge location - R reference full chord airfoil - R _s reference splitter - Sp _chord splitter chord length - u dimensionless perturbation chordwise velocity - v dimensionless perturbation normal velocity - amplitude of oscillation - interblade phase angle - level of aerodynamic detuning - undamped natural torsional frequency - ₀ reference frequency - flutter frequency     

Relations between steady flow and oscillatory shear measurements

Summary Results are given of a comparison between dynamic oscillatory and steady shear flow measurements with some polymer melts. Comparison of the steady shear flow viscosity,, with the absolute value of the dynamic viscosity, ¦¦, at equal values of the shear rate,q, and the circular frequency,, has shown the relation thatCox andHerz had found empirically to be substantially correct.Further, the coefficients of the normal stress differences obtained by streaming birefringence techniques have been compared with 2G () · ^{– 2} in the same range of shear rates as covered by the viscosity measurements (G is the real part of the dynamic shear modulus). Two polystyrenes with narrow molecular weight distribution showed the same shift factor along the orq axis for the normal stress coefficients with respect to 2G () · ^{– 2} and the steady shear flow viscosities with respect to the real part of the dynamic viscosity,. For two polyethylenes the results are not so conclusive owing to the smallness of the shift factor found. An empirical equation is proposed predicting the main normal stress difference from dynamic measurements only.

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Zusammenfassung Die Ergebnisse von Messungen unter erzwungenen Schwingungen und stationärer Scherströmung an einigen Polymerschmelzen werden miteinander verglichen. Der Vergleich der stationären Viskosität mit der absoluten dynamischen Viskosität ¦¦ bei gleichen Werten des Strömungsgradientenq und der Kreisfrequenz zeigt die Gültigkeit der empirischen Beziehung vonCox undHerz.Weiter wurden die Koeffizienten der Normalspannungsdifferenzen, welche durch Messung der Strömungsdoppelbrechung erhalten wurden, mit 2G() · ^–2 verglichen, und zwar wiederum bei gleichen Werten vonq und, wobeiG die Speicherkomponente des dynamischen Schubmoduls ist. Zwei Polystyrole mit enger Molekulargewichtsverteilung zeigen die gleiche Verschiebung entlang der-oderq-Achse für die Normalspannungskoeffizienten in bezug auf2G()· ^–2 und für die stationären Scherviskositäten in bezug auf den Realteil der dynamischen Viskosität. Für zwei Polyäthylene sind die Ergebnisse weniger signifikant, da die entsprechenden Verschiebungen zu klein waren. Eine empirische Beziehung zwischen den Hauptnormalspannungsdifferenzen und den dynamischen Meßwerten wird vorgeschlagen.

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Paper presented at the British Society of Rheology Conference, held at Shrivenham, from 9th–12th September, 1968.     

The extension of Beltrami's ellipsoid to anisotropic bodies

Summary The spectral decomposition of the compliance, stiffness, and failure tensors for transversely isotropic materials was studied and their characteristic values were calculated using the components of these fourth-rank tensors in a Cartesian frame defining the principal material directions. The spectrally decomposed compliance and stiffness or failure tensors for a transversely isotropic body (fiber-reinforced composite), and the eigenvalues derived from them define in a simple and efficient way the respective elastic eigenstates of the loading of the material. It has been shown that, for the general orthotropic or transversely isotropic body, these eigenstates consist of two double components, ₁ and ₂ which are shears (₂ being a simple shear and ₁, a superposition of simple and pure shears), and that they are associated with distortional components of energy. The remaining two eigenstates, with stress components ₃, and ₄, are the orthogonal supplements to the shear subspace of ₁ and ₂ and consist of an equilateral stress in the plane of isotropy, on which is superimposed a prescribed tension or compression along the symmetry axis of the material. The relationship between these superimposed loading modes is governed by another eigenquantity, the eigenangle .The spectral type of decomposition of the elastic stiffness or compliance tensors in elementary fourth-rank tensors thus serves as a means for the energy-orthogonal decomposition of the energy function. The advantage of this type of decomposition is that the elementary idempotent tensors to which the fourth-rank tensors are decomposed have the interesting property of defining energy-orthogonal stress states. That is, the stress-idempotent tensors are mutually orthogonal and at the same time collinear with their respective strain tensors, and therefore correspond to energy-orthogonal stress states, which are therefore independent of each other. Since the failure tensor is the limiting case for the respective _x, which are eigenstates of the compliance tensor S, this tensor also possesses the same remarkable property.An interesting geometric interpretation arises for the energy-orthogonal stress states if we consider the projections of _x in the principal₃D stress space. Then, the characteristic state ₂ vanishes, whereas stress states ₁, ₃ and ₄ are represented by three mutually orthogonal vectors, oriented as follows: The ₃ and ₄ lie on the principal diagonal plane (₃₁₂) with subtending angles equaling (–/2) and (-), respectively. On the positive principal ₃-axis, is the eigenangle of the orthotropic material, whereas the ₁-vector is normal to the (₃₁₂)-plane and lies on the deviatoric -plane. Vector ₂ is equal to zero.It was additionally conclusively proved that the four eigenvalues of the compliance, stiffness, and failure tensors for a transversely isotropic body, together with value of the eigenangle , constitute the five necessary and simplest parameters with which invariantly to describe either the elastic or the failure behavior of the body. The expressions for the _x-vector thus established represent an ellipsoid centered at the origin of the Cartesian frame, whose principal axes are the directions of the ₁-, ₃- and ₄-vectors. This ellipsoid is a generalization of the Beltrami ellipsoid for isotropic materials.Furthermore, in combination with extensive experimental evidence, this theory indicates that the eigenangle alone monoparametrically characterizes the degree of anisotropy for each transversely isotropic material. Thus, while the angle for isotropic materials is always equal to _i = 125.26° and constitutes a minimum, the angle || progressively increases within the interval 90–180° as the anisotropy of the material is increased. The anisotropy of the various materials, exemplified by their ratiosE _L/2G_L of the longitudinal elastic modulus to the double of the longitudinal shear modulus, increases rapidly tending asymptotically to very high values as the angle approaches its limits of 90 or 180°.     

Primary and secondary bifurcation in baroclinic instability

In modelling atmospheric flows the baroclinic instability of the flow in a differentially heated rotating annulus plays a central role. This paper deals with an experimental study using LDV and flow visualization techniques. Usually the temperature difference, T, was kept fixed while the angular velocity, , was varied. On crossing the stability boundary, the primary bifurcation, the basic flow gives way to a baroclinic wave flow. For a given annulus geometry the wave number, m, of the first wave pattern was found to be uniquely defined by T. The measured critical values of , _crit, agree reasonably well with those obtained by other authors. On increasing above _crit the wave number changed, this process showing hysteresis. The situation might indicate secondary bifurcation phenomena. Flow visualization using aluminium particles shows surface flow details.This paper is dedicated to Prof. Dr. K. Gersten on the occasion of his 60th birthday     

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.