Investigation of the stability problems of elastic bodies using the method of mathematical theory of elasticity

In this paper, using the equilibrium equations and boundary conditions of elastic stability problem of and the method of mathematical theory of elasticity, we solve some elastic stability problems, which were studied by Ишлынский^[2] and Войцеховская^[3,4],and obtained more reasonable results than theirs.     

Experimental investigations of the stability limit of the helical flow of pseudoplastic liquids

The stability of the laminar helical flow of pseudoplastic liquids has been investigated with an indirect method consisting in the measurement of the rate of mass transfer at the surface of the inner rotating cylinder. The experiments have been carried out for different values of the geometric parameter = R ₁/R ₂ (the radius ratio) in the range of small values of the Reynolds number,Re < 200. Water solutions of CMC and MC have been used as pseudoplastic liquids obeying the power law model. The results have been correlated with the Taylor and Reynolds numbers defined with the aid of the mean viscosity value. The stability limit of the Couette flow is described by a functional dependence of the modified critical Taylor number (including geometric factor) on the flow indexn. This dependence, general for pseudoplastic liquids obeying the power law model, is close to the previous theoretical predictions and displays destabilizing influence of pseudoplasticity on the rotational motion. Beyond the initial range of the Reynolds numbers values (Re>20), the stability of the helical flow is not affected considerably by the pseudoplastic properties of liquids. In the range of the monotonic stabilization of the helical flow the stability limit is described by a general dependence of the modified Taylor number on the Reynolds number. The dependence is general for pseudoplastic as well as Newtonian liquids.Nomenclature C _i concentration of reaction ions, kmol/m³ - d = R ₂ –R ₁ gap width, m - F _M () Meksyn's geometric factor (Eq. (1)) - F ₀ Faraday constant, C/kmol - i _l density of limit current, A/m³ - k _c mass transfer coefficient, m/s - n flow index - R ₁,R ₂ inner, outer radius of the gap, m - Re = V _m ·2d·/µ _m Reynolds number - Ta _c = _c ·d^3/2·R ₁ ^1/2 ·/µ _m Taylor number - Z _i number of electrons involved in electrochemical reaction - = R ₁/R ₂ radius ratio - µ apparent viscosity (local), Ns/m² - µ _m mean apparent viscosity value (Eq. (3)), Ns/m² - µ _i apparent viscosity value at a surface of the inner cylinder, Ns/m² - density, kg/m³ - _c angular velocity of the inner cylinder (critical value), 1/s     

Explicit integration of one problem of motion of the generalized Kowalevski top

In the problem of motion of the Kowalevski top in a double force field the four-dimensional invariant submanifold of the phase space was pointed out by [Kharlamov, M.P., 2002. Mekh. Tverd. Tela 32, 33–38]. We show that the equations of motion on this manifold can be separated by the appropriate change of variables, the new variables s₁, s₂ being elliptic functions of time. The natural phase variables (components of the angular velocity and the direction vectors of the forces with respect to the movable basis) are expressed via s₁, s₂ explicitly in elementary algebraic functions.     

Stability of Couette flow of liquids with power law viscosity

The stability of the Couette flow of the liquid with the power law viscosity in a wide annular gap has been investigated theoretically in this work with the aid of the method of small disturbances. The Taylor number, being a criterion of the stability, has been defined using the mean apparent viscosity value in the main flow. In the whole range of the radius ratio, R _i /R _o and the flow index, n, considered (R _i /R _o 0.5, n = 0.25–1.75 ), the critical value of the Taylor number Ta _c is an increasing function of the flow index, i.e., shear thinning has destabilizing influence on the rotational flow, and dilatancy exhibits an opposite tendency.In the wide ranges of the flow index, n > 0.5, and the radius ratio, R _i /R _o > 0.5, the wide-gap effect on the stability limit is predicted to be almost the same for non-Newtonian fluids as for Newtonian ones. The ratio on the critical Taylor numbers for non-Newtonian and Newtonian fluids: Ta _c (n) and Ta _c (n = 1) obey a generalized functional dependence: Ta _c (n)/Ta _c (n = 1) = g(n), where g(n) is a function corresponding to the solution for the narrow gap approximation.Theoretical predictions have been compared with experimental results for pseudoplastic liquids. In the range of the radius ratio R _i /R _o > 0.6 the theoretical stability limit is in good agreement with the experiments, however, for R _i /R _o < 0.6, the critical Taylor number is considerably lower than predicted by theory.     

Influence of M w of LDPE and vinyl acetate content of EVA on the rheology of polymer modified asphalt

Asphalt binder was modified by low-density polyethylene (LDPE) and ethyl vinyl acetate (EVA) polymers to investigate the structure–property relationships of polymer-modified asphalt (PMA). The PMA was prepared in a high-shear blender at 160 °C. The optimum blending time (OBT) for each polymer was determined following a separate investigation. OBT was influenced by M_w, MWD, and polymer structure. The influence of M_w of LDPE and vinyl acetate (VA) content of EVA on PMAs was studied by rheological tools. Polymer modification improved the rheological properties of base asphalt. EVA–PMAs were found to be less temperature sensitive than LDPE-modified asphalts. LDPE modification increased flow activation energy (E_a) but EVA modification decreased E_a. Both VA content and M_w of LDPE have influenced the storage stability of PMAs. The low-temperature properties of PMAs and short ageing tests were not influenced by polymer type. On the other hand, the high-temperature properties of PMAs were strongly influenced by M_w of LDPE and VA content of EVA. Overall, EVA with low VA content showed the best temperature resistance to high- temperature deformations, the highest upper service temperature as well as the best storage stability.     

Degree of branching of polypropylene measured from Fourier-transform rheology

In this study, linear and branched polypropylenes (PP) were compared under medium strain amplitude oscillatory shear (usually strain amplitude range from 10 to 100%) with Fourier-transform rheology (FT rheology). On a log–log diagram, the third relative intensity (I ₃/I ₁), which is a parameter to represent nonlinearity, shows a linear relationship with the strain amplitude in the range of medium strain amplitude. The slope of I ₃/I ₁ of linear PP with various molecular weight and molecular weight distribution was 2 as most constitutive equations predict, while that of branched PP was 1.64, which is lower than that of linear PP. When the linear and branch PP were blended, the slope of I ₃/I ₁ was proportional to the composition of the branch PP. Therefore, it is suggested that the degree of branching can be defined in terms of the slope of I ₃/I ₁ under medium amplitude oscillatory shear.     

Slow Eigenvalues of Self-similar Solutions of the Dafermos Regularization of a System of Conservation Laws: an Analytic Approach

The Dafermos regularization of a system of n hyperbolic conservation laws in one space dimension has, near a Riemann solution consisting of n Lax shock waves, a self-similar solution u = u _ε(X/T). In Lin and Schecter (2003, SIAM J. Math. Anal. 35, 884–921) it is shown that the linearized Dafermos operator at such a solution may have two kinds of eigenvalues: fast eigenvalues of order 1/ε and slow eigenvalues of order one. The fast eigenvalues represent motion in an initial time layer, where near the shock waves solutions quickly converge to traveling-wave-like motion. The slow eigenvalues represent motion after the initial time layer, where motion between the shock waves is dominant. In this paper we use tools from dynamical systems and singular perturbation theory to study the slow eigenvalues. We show how to construct asymptotic expansions of eigenvalue-eigenfunction pairs to any order in ε. We also prove the existence of true eigenvalue-eigenfunction pairs near the asymptotic expansions.     

A formulation of PANS capable of mimicking IDDES

The partially averaged Navier–Stokes (PANS) model, proposed in Girimaji (2006), allows to simulate turbulent flows either in RANS, LES or DNS mode. The PANS model includes $f_k$ which denotes the ratio of modeled to total kinetic energy. In RANS, $f_k=1$ while in DNS it tends to zero. In the present study we propose an improved formulation for $f_k$ based on the H-equivalence introduced by Friess et al. (2015). In this formulation the expression of $f_k$ is derived to mimic Improved Delayed Detached Eddy Simulation (IDDES). This new formulation behaves in a very similar way as IDDES, even though the two formulations use different mechanisms to separate modeled and resolved scales. They show very similar performance in separated flows as well as in attached boundary layers. In particular, the novel formulation is able to (i) treat attached boundary layers as properly as IDDES, and (ii) “detect” laminar initial/boundary conditions, in which case it enforces RANS mode. Furthermore, it is found that the new formulation is numerically more stable than IDDES.     

Generators of the product of two Schoenflies motion groups

Schoenflies motion is often termed X-motion for conciseness. A set of X-motions with a given direction of its axes of rotations has the algebraic properties of a Lie group for the composition product of rigid-body motions or displacements. The product of two X-subgroups, which is the mathematical model of a serial concatenation of two kinematic chains generating two distinct X-motions, characterizes a noteworthy type of 5-dimensional (5D) displacement set called double Schoenflies motion or X–X motion for brevity. This X–X motion set is a 5D submanifold of the displacement 6D Lie group. Such a motion type includes any spatial translation (3T) and any two sequential rotations (2R) provided that the axes of rotation are parallel to two fixed independent vectors. This motion set also contains the rotations that are products of the foregoing two rotations. In the paper, some preliminary fundamentals on the 4D X-motion are recalled; the 5D set of X–X motions is emphasized. Then implementing serial arrays of one-dof Reuleaux pairs and hinged parallelograms, we enumerate all serial mechanical generators of X–X motion, which have no redundant internal mobility. Based on the group-theoretic concepts, one can differentiate two families of irreducible representations of an X–X motion. One family is realized by twenty-one open chains including the doubly planar motion generators as special cases. The other is generally classified into eight major categories in which one hundred and six distinct open chains generating X–X motion are revealed and nineteen more ones having at least one parallelogram are derived from them. Meanwhile, these kinematic chains are graphically displayed for a possible use in the structural synthesis of parallel manipulators.     

On Asymptotics of Solutions of Nonlinear Differential Equations of the Second Order

We study the problem of asymptotics of unbounded solutions of differential equations of the form y″ = α₀ p(t)ϕ(y), where α₀ ∈ {−1, 1}, p: [a, ω[→]0, +∞[, −∞ < a < ω ≤ +∞, is a continuous function, and ϕ: [y ₀, +∞[→]0, +∞[ is a twice continuously differentiable function close to a power function in a certain sense.__________Translated from Neliniini Kolyvannya, Vol. 8, No. 1, pp. 18–28, January–March, 2005.     

Influence of NaClO3 on the rheological behaviour of a micellar solution of CPCl

It is now well know that a small addition of salt to a micellar solution often increases the zero-shear viscosity η₀ of the solution, the understanding of the behaviour at high salt content is more questionable. In this situation, addition of more salt induces a decrease of η₀. In this experimental work we investigate the linear and non-linear rheological behaviour of a new micellar system: CPCl (surfactant)/NaClO₃ (salt). Studies of the evolution of η₀ as well as G₀ (the elastic modulus) or τ_R (the relaxation time) are in agreement with the hypothesis of a diminution of the mean micellar length when, after the maximum η₀, the salt content increases. In the non-linear behaviour (non-Newtonian viscosity) the evolution of γ˙ _c, (which defines the occurrence of the shear thinning) with salt concentration C_S is also in agreement with such a hypothesis. Received: 29 March 1999/Accepted: 20 March 2000     

Eigenvalues of Self-Similar Solutions of the Dafermos Regularization of a System of Conservation Laws via Geometric Singular Perturbation Theory

The Dafermos regularization of a system of n conservation laws in one space dimension admits smooth self-similar solutions of the form u=u(X/T). In particular, there are such solutions near a Riemann solution consisting of n possibly large Lax shocks. In Lin and Schecter (2004, SIAM. J. Math. Anal. 35, 884–921), eigenvalues and eigenfunctions of the linearized Dafermos operator at such a solution were studied using asymptotic expansions. Here we show that the asymptotic expansions correspond to true eigenvalue–eigenfunction pairs. The proofs use geometric singular perturbation theory, in particular an extension of the Exchange Lemma.     

Uniaxial deformation of a random packing of particles

Summary Mechanical behavior of dense packing spheres with small irregularities is investigated in this paper. A generalization of the hertzian contact model for surfaces of the form x ^k yields a normal contact force F _n , which is proportional to ζ^{1+1/ k} , with the normal displacement ζ. For oblique forces, the frictional force can be calculated, [10]. Different load cases are explained in detail. It is shown that the stress-strain curve during initial loading of the packing is identical with the force-displacement relation at the contact point, using an appropriate constant. The results for uniaxial loading, unloading and reloading are illustrated. As experimentally observed, the axial pressure in unloading is smaller than during loading, while the lateral pressure increases. The stress-strain relation is compared with well-known empirical relations of rock and soil mechanics, and the wave velocity for spherical irregularities agrees with earlier geomechanical theories for random packing of smooth spheres. Received 19 July 1998; accepted for publication 19 October 1998     

Higher-order correction of effective permeability of heterogeneous isotropic formations of lognormal conductivity distribution

Steady flow of an incompressible fluid takes place in a porous formation of spatially variable hydraulic conductivityK. The latter is regarded as a lognormal stationary random space function and Y=ln(K/K _G ), whereK _G is the geometric mean ofK, is characterized by its variance ² and correlation scale I. Exact results are known for the effective conductivityK _eff in one- and two-dimensional flows. In contrast, only a first-order term in a perturbation expansion in ² has been derived exactly for the three-dimensional flow. A conjecture has been made in the past onK _eff for any ², but it was not yet proved exactly. This study derived the exact nonlinear correction, i.e. the termO(⁴) ofK _eff, which is found to be the one resulting from the conjecture, strengthening the confidence in it. It is also shown that the self-consistent approximation leads to the exact results for one-dimensional and two-dimensional flows, but underestimates the nonlinear correction ofK _eff for in the three-dimensional case.     

Twist viscosity of mixtures of low molar mass nematics

Measurements of the twist viscosity, γ₁(DLS) and twist elastic coefficient, K₂₂(DLS) by electric-field-dependent dynamic light scattering (EFDLS) are reported for low molar mass nematics (LMMNs) 4′-heptyl-4-cyanobiphenyl (7CB) and 4′-octyl-4-cyanobiphenyl (8CB), and their binary mixtures at several temperatures in the nematic state. The results are compared with values (γ₁(Rheol)=α₃–α₂) computed from rheological measurements of the Leslie viscosities α₂ and α₃. For the binary mixtures, at each temperature, the measured twist viscosity γ₁(DLS) and corresponding twist elastic constant K₂₂(DLS) show approximately a linear additive dependence on concentration. The calculated twist viscosity, γ₁(Rheol), agrees with γ₁(DLS) for the pure components, but is significantly smaller for the binary mixtures. Our observations appear to be consistent with a recent report of a discrepancy between values of the tumbling parameter λ, determined using a small-strain oscillatory optical technique, vs those measured by a rheological method. These results suggest that, in the rheological measurements at large strains, the rate of director rotation for mixtures may be affected by a flow-induced change in structure, e.g., shear-induced biaxiality. Received: 17 March 2000 Accepted: 17 July 2000     

On the structure of continua and the mathematical properties of algebraic elastodynamic of a triclinic structural system

This paper is neither laudatory nor derogatory but it simply contrasts with what might be called elastostatic (or static topology), a proposition of the famous six equations. The extension strains and the shearing strains which were derived by A.L. Cauchy, are linearly expressed in terms of nine partial derivatives of the displacement function (u _i ,u _j ,u _h )=u(x ⁱ ,x ^j ,x ^k ) and it is impossible for the inverse proposition to sep up a system of the above six equations in expressing the nine components of matrix (∂(u _i ,u _j ,u _h )/∂(x ⁱ ,x ^j ,x ^k )). This is due to the fact that our geometrical representations of deformation at a given point are as yet incomplete^[1]. On the other hand, in more geometrical language this theorem is not true to any triangle, except orthogonal, for “squared length” in space^[2]. The purpose of this paper is to describe some mathematic laws of algebraic elastodynamics and the relationships between the above-mentioned important questions.     

The contribution of internal degrees of freedom to the non-Newtonian rheology of model polymer fluids

We report non-equilibrium molecular dynamics simulations of rigid and non-rigid dumbbell fluids to determine the contribution of internal degrees of freedom to strain-rate-dependent shear viscosity. The model adopted for non-rigid molecules is a modification of the finitely extensible nonlinear elastic (FENE) dumbbell commonly used in kinetic theories of polymer solutions. We consider model polymer melts — that is, fluids composed of rigid dumbbells and of FENE dumbbells. We report the steady-state stress tensor and the transient stress response to an applied Couerte strain field for several strain rates. We find that the rheological properties of the rigid and FENE dumbbells are qualitatively and quantitatively similar. (The only exception to this is the zero strain rate shear viscosity.) Except at high strain rates, the average conformation of the FENE dumbbells in a Couette strain field is found to be very similar to that of FENE dumbbells in the absence of strain. The theological properties of the two dumbbell fluids are compared to those of a corresponding fluid of spheres which is shown to be the most non-Newtonian of the three fluids considered.Symbol Definition b dimensionless time constant relating vibration to other forms of motion - F force on center of mass of dumbbell - F _i force on bead i of dumbbell - F force between center of masses of dumbbells and - F _ij force between beads i and j - h vector connecting bead to center of mass of dumbbell - H dimensionless spring constant for dumbbells, in units of / ² - I moment of inertia of dumbbell - J general current induced by applied field - k _B Boltzmann's constant - L angular momentum - m mass of bead, (= m/2) - M mass of dumbbell, g - N number of dumbbells in simulation cell - P translational momentum of center of mass of dumbbell - P pressure tensor - P _xy xy component of pressure tensor - Q separation of beads in dumbbell - Q _eq equilibrium extension of FENE dumbbell and fixed extension of rigid dumbbell - Q ₀ maximum extension of dumbbell - r _ij vector connecting beads i and j - r position vector of center of mass dumbbell - R vector connecting centers of mass of two dumbbells - t time - t ^* dimensionless time, in units of m/ - T ^* dimensionless temperature, in units of /k - u potential energy - u velocity vector of flow field - u _x x component of velocity vector - V volume of simulation cell - X general applied field - strain rate, s^–1 - ^* dimensionless shear rate, in units of /m ² - general transport property - Lennard-Jones potential well depth - friction factor for Gaussian thermostat - shear viscosity, g/cms - ^* dimensionless shear viscosity, in units of m/ ² - ^* dimensionless number density, in units of ^–3 - Lennard-Jones separation of minimum energy - relaxation time of a fluid - angular velocity of dumbbell - orientation angle of dumbbell      

Regularity and asymptotic behavior of solutions of nonautonomous differential equations

The existence of a nonautonomous approximate inertial manifold is shown for problems of the formu + Au + N(t,u)=0, in whichA is a self-adjoint operator with compact resolvent in a Hilbert spaceH. The operatorN(t, u) = G(u) + F(t, u) is nonlinear withG a monotone gradient that is locally Lipschitz fromD(A ^1/2) intoH, andF:⁺×HH a Lipschitz perturbation that is Hölder continuous int. Weak solutions are shown to be uniformly locally Hölder continuous intoD(A) with equicontinuity in families of solutions with ¦u(0)¦ r.A priori estimates of ¦Au(t)¦ are also verified and used in a skew-product flow to show there is a global attractor whose component elements form a equicontinuous family of solutions.     

Optimization of the deflagration to detonation transition: reduction of length and time of transition

The aim of this experimental investigation is the study of Deflagration to Detonation Transition (DDT) in tubes in order to (i) reduce both run-up distance and time of transition (L _DDT and t _DDT) in connection with Pulsed Detonation Engine applications and to (ii) attempt to scale L _DDT with λ_CJ (the detonation cellular structure width). In DDT, the production of turbulence during the long flame run-up can lead to L _DDT values of several meters. To shorten L _DDT, an experimental set-up is designed to quickly induce highly turbulent initial flow. It consists of a double chamber terminated with a perforated plate of high Blockage Ratio (BR) positioned at the beginning of a 26 mm inner diameter tube containing a “Shchelkin spiral” of BR ≈ 0.5. The study involves stoichiometric reactive mixtures of H₂, CH₄, C₃H₈, and C₂H₄ with oxygen and diluted with N₂ in order to obtain the same cell width λ_CJ≈10 mm at standard conditions. The results show that a shock-flame system propagating with nearly the isobaric speed of sound of combustion products, called the choking regime, is rapidly obtained. This experimental set-up allows a L _DDT below 40 cm for the mixtures used and a ratio L _DDT/λ_CJ ranging from 23 to 37. The transition distance seems to depend on the reduced activation energy (E _a/RT _c) and on the normalized heat of reaction (Q/a ₀ ²). The higher these quantities are, the shorter the ratio L _DDT/λ_CJ is. PACS 47.40.Rs · 47.60.+i · 47.70.Pq · 47.80.CbThis paper was based on the work that was presented at the 19th International Colloquium on the Dynamics of Explosions and Reactive Systems, Hakone, Japan, July 27–August 1, 2003.