Acceleration covariance in isotropic hydromagnetic turbulence

The present paper deals with the turbulent flow of an incompressible, viscous and conducting fluid which is isotropic, spatially homogeneous. The expression for acceleration covariance is derived. The obtained result shows that the defining scalars α(r, t) and β(r, t) of the acceleration covariance in MHD turbulence depend on the defining scalars of Q _ij , H _ij , Π _ij and S _{ik, j} .     

The tensor equation AX+XA=Φ(A,H), with applications to kinematics of continua

The (second-order) tensor equation AX+XA=(A,H) is studied for certain isotropic functions (A,H) which are linear in H. Qualitative properties of the solution X and relations between the solutions for various forms of are established for an inner product space of arbitrary dimension. These results, together with Rivlin's identities for tensor polynomials in two variables, are applied in three dimensions to obtain new explicit formulas for X in direct tensor notation as well as new derivations of previously known formulas. Several applications to the kinematics of continua are considered.     

Symmetry groups and the pseudo-Riemann spacetimes for mixed-hardening elastoplasticity

The constitutive postulations for mixed-hardening elastoplasticity are selected. Several homeomorphisms of irreversibility parameters are derived, among which X_a⁰ and X_c⁰ play respectively the roles of temporal components of the Minkowski and conformal spacetimes. An augmented vector X_a:=(YQ_a^t,YQ_a⁰)^t is constructed, whose governing equations in the plastic phase are found to be a linear system with a suitable rescaling proper time. The underlying structure of mixed-hardening elastoplasticity is a Minkowski spacetime $\text{M}{}^{\mathrm{n+1}}$ on which the proper orthochronous Lorentz group SO_o(n,1) left acts. Then, constructed is a Poincaré group ISO_o(n,1) on space X:=X_a+X_b, of which X_b reflects the kinematic hardening rule in the model. We also find that the space (Q_a^t,q₀^a) is a Robertson–Walker spacetime, which is conformal to X_a through a factor Y, and conformal to X_c:=(ρQ_a^t,ρQ_a⁰)^t through a factor ρ as given by ρ(q₀^a)=Y(q₀^a)/[1−2ρ₀Q_a⁰(0)+2ρ₀Y(q₀^a)Q_a⁰(q₀^a)]. In the conformal spacetime the internal symmetry is a conformal group.     

On K-BKZ and other viscoelastic models as continuum generalizations of the classical spring-dashpot models

An alternate constitutive formulation for visco-elastic materials, with particular emphasis on macromolecular viscoelastic fluids, is presented by generalizing Maxwell's idealized separation of elastic and relaxation mechanisms. The notion ofrelative rate of change of elastic stress is identified, abstracted, and formulated with the help of the established theory of finitely elastic isotropic materials. This given a local rate-type constitutive relation for an elastic mechanism in a simple material.For the simplest class of viscoelastic polymer melts, the notion of rate of change of elastic stress and its damped accumulation is identified and formulated. Under conditions of moderate strain rates, this scheme implies the reliable K-BKZ model for a class of polymer melts. An obvious extension generalizes the remaining classical spring-dashpot models. I Set of second-order tensors.A I is identified with a 3 × 3 matrix in a Cartesian co-ordinate system - I ^sym Set of symmetric second order tensors - Q Orthogonal tensor, i.e.Q ^T=Q ^–1. - Symbol for the value of the functional H:X I ^sym, whereX is the set of piecewise continuous and differentiable strain historiesF _to : [t ₀,t] I Other functionals, unless otherwise specified, should be interpreted in a similar manner.     

Nonlinear motion equations for a Non-Newtonian incompressible fluid in an orthogonal coordinate system

Summary Starting with an assumed relationship between the stress tensor, the non-Newtonian viscosity, and the strain rate tensor, the nonlinear equations of motion are developed for use in any orthogonal coordinate system. The resulting equations are written in terms of the scalar velocities, the non-Newtonian viscosity, the metric coefficients, and their derivatives.The non-Newtonian viscosity is assumed to be a scalar function of the strain rate tensor, and so depends upon the invariants of the strain rate tensor. For convenience, the necessary invariants are written out in complete form for use in any orthogonal coordinate system, in terms of the scalar velocities, the metric coefficients, and their derivatives.Using the resulting motion equations and a model of this type of viscosity, theOstwald-de Waele model, an example of time dependent flow is solved using a continuous time-discrete space method programmed on an analog computer. e _ij strain rate tensor - body force density, dynes/cm³ - F ₁,F ₂,F ₃ components of body force density, dynes/cm³ - g acceleration of gravity - H function of time - h ₁,h ₂,h ₃ metric coefficients - I ₁,I ₂,I ₃ invariants - m constant - P pressure, dynes/cm² - r radius, cm - t time, sec - velocity vector, cm/sec - v ₁,v ₂,v ₃ velocities in thex ₁,x ₂ andx ₃ directions, respectively, cm/sec - v _n (t) velocity of thenth node, cm/sec - x ₁,x ₂,x ₃ coordinate directions - z coordinate, cm - unit tensor - _ij Kronecker delta - _ij 2e _ij - nabla - _ijk alternating unit tensor - non-Newtonian viscosity, dynes/cm² - ₀, ₁ constant viscosities, dynes sec/cm², dynes sec ^m /cm² - angle, radians - v ₀,v ₁ constant kinematic viscosities, cm²/sec, cm² sec ^m-2 - density, g/cm³ - _ij stress tensor - fluid dilation With 3 figures     

On unimodular constitutive expressions

We consider constitutive expressions which the stress σ(X, t) at a particle X at time t is given by σ (X, t) = $\text{F}$[F[X, τ)] where $\text{F}$[F(X, τ)] denotes a functional of the history of the deformation gradient matrix [F(X, τ)] from time τ = 0 unti τ = t. This expression is restricted by the requirement of invariance under a superposed rotation of the physical system and by the further requirement that the constitutive expression shall be invariant under the group of unimodular transformations, i.e. $\text{F}$[F(X, τ)] = $\text{F}$[F(X, τ) H] must hold for all matrices H such that det H - 1. We employ results from the classical theory of invariants in order to determine the general form of the expression $\text{F}$[F(X, τ)] which is consistent with these restrictions. Special cases are considered where the functional is replaced by a function of the strain, rate of strain, ? matrices. The case of shear flow is briefly discussed.     

A generating function for the yield criterion of isotropic and anisotropic polycrystalline materials

Summary A yield criterion for elastic pure-plastic polycrystalline materials is generated under simplified conditions by assuming that for yielding a certain fraction Q _c of the total number of slip planes in the material has to be active. This fraction Q _c is called the critical active quantity. We suppose Q _c to be independent of the state of stress. The yield criterion is mathematically expressed as an integral, which is a function of Q _c. This criterion can also be used for anisotropic materials.For isotropic materials the ratio (r) of the yield stress in torsion to that in tension is calculated as a function of Q _c. We find 0.5r0.61.The value r=0.5 (Tresca's criterion) is obtained for Q _c=0 and Q _c=1. The value r=0.577 (von Mises criterion) is obtained for Q _c=0.34 and Q _c=0.79. The difference between two criteria with the same r is the magnitude of the yield stress. We think the value Q _c=0.79 corresponds to the experiments for f.c.c. materials, since a rough estimation gives Q _c>0.75 for yielding.The independence of Q _c on the state of stress brings on that r>0.5 is more probable. This is caused by the slower increase to Q _c in torsion compared with the case of tension.From the theory follows that in the general case (Q _c0) the middle principal stress has influence on yielding.In this paper we don't determine Q _c, but adapt its value to the experimental results. However, a rough estimation of Q _c is given for isotropic materials.     

Stability and boundedness results for certain nonlinear vector differential equations of the fourth order

We consider the equation X ⁽⁴⁾ + Φ(X″)X‴ + F(X,X′)X″ + G(X′) + H(X) = P(t,X,X′,X″,X‴) in two cases: P ≡ 0 and P ≠ 0. In the case P ≡ 0, the asymptotic stability of the zero solution X = 0 of the equation is investigated; in the case P ≠ 0, the boundedness of all solutions of the equation is proved. Published in Neliniini Kolyvannya, Vol. 9, No. 4, pp. 548–563, October–December, 2006.     

The effects of resolution and noise on kinematic features of fine-scale turbulence

The effect of spatial resolution and experimental noise on the kinematic fine-scale features in shear flow turbulence is investigated by means of comparing numerical and experimental data. A direct numerical simulation (DNS) of a nominally two-dimensional planar mixing layer is mean filtered onto a uniform Cartesian grid at four different, progressively coarser, spatial resolutions. Spatial gradients are then calculated using a simple second-order scheme that is commonly used in experimental studies in order to make direct comparisons between the numerical and previously obtained experimental data. As expected, consistent with other studies, it is found that reduction of spatial resolution greatly reduces the frequency of high magnitude velocity gradients and thereby reduces the intermittency of the scalar analogues to strain (dissipation) and rotation (enstrophy). There is also an increase in the distances over which dissipation and enstrophy are spatially coherent in physical space as the resolution is coarsened, although these distances remain a constant number of grid points, suggesting that the data follow the applied filter. This reduction of intermittency is also observed in the eigenvalues of the strain-rate tensor as spatial resolution is reduced. The quantity with which these eigenvalues is normalised is shown to be extremely important as fine-scale quantities, such as the Kolmogorov length scale, are showed to change with different spatial resolution. This leads to a slight change in the modal values for these eigenvalues when normalised by the local Kolmogorov scale, which is not observed when they are normalised by large-scale, resolution-independent quantities. The interaction between strain and rotation is examined by means of the joint probability density function (pdf) between the second and third invariants of the characteristic equation of the velocity gradient tensor, Q and R respectively and by the alignments between the eigenvectors of the strain-rate tensor and the vorticity vector. Gaussian noise is shown to increase the divergence error of a dataset and subsequently affect both the Q–R joint pdf and the magnitude of the alignment cosines. The experimental datasets are showed to behave qualitatively similarly to the numerical datasets to which Gaussian noise has been added, confirming the importance of understanding the limitations of coarsely resolved, noisy experimental data.     

Basic equations for the microscopic theory of plasticity of polycrystalline materials with given texture

An expression for the yield stress of anisotropic materials is applied to the anisotropic strength of hard rolled copper foils whose crystallographic texture is known. We assume that this crystallographic texture is the only cause of the anisotropic plastic behaviour of the material. The model used for the yield stress is also used to deduce:

1. Stress-strain relations for isotropic polycrystalline materials;
2. A formula for the fully plastic strain tensor, applied to anisotropic hard rolled copper foils.

For the anisotropic copper foils considered the calculated curves of the yield stress and of the strain tensor as a function of the angle x between rolling and tensile direction agree qualitatively with the measured values. However, the theory is not complete, since the yield stress and the plastic strain tensor are both a function of a parameter Q, the fraction of the number of available crystallographic slip planes on which the maximum shear stress has reached the critical value τ_a. We assume that for “fully” plastic deformation a certain critical fraction Q _e of the total number of slip planes has to be active. The fraction Q _e is called the critical active quantity. With the parameter Q _e we adjust the calculated curves to the measured ones. The dependence of Q _e on the properties of the material (e.g. the crystallographic texture) is discussed in Appendix I.     

On a formulation for anisotropic elastoplasticity at finite strains invariant with respect to the intermediate configuration

The paper is concerned with a formulation of anisotropic finite strain inelasticity based on the multiplicative decomposition of the deformation gradient F=F_eF_p. A major feature of the theory is its invariance with respect to rotations superimposed on the inelastic part of the deformation gradient. The paper motivates and shows how such an invariance can be achieved. At the heart of the formulation is the mixed-variant transformation of the structural tensor, defined as the tensor product of the privileged directions of the material as given in a reference configuration, under the action of F_p. Issues related to the plastic material spin are discussed in detail. It is shown that, in contrast to the isotropic case, any flow function formulated purely in terms of stress quantities, necessarily exhibits a non-vanishing plastic material spin. The possible construction of spin-free rates is discussed as well, where it is shown that the flow rule must then depend not only on the stress but on the strain as well.     

Macrodispersion by Point-Like Source Flows in Randomly Heterogeneous Porous Media

A pulse of a passive tracer is injected in a porous medium via a point-like source. The hydraulic conductivity K is regarded as a stationary isotropic random space function, and we model macrodispersion in the resulting migrating plume by means of the second-order radial spatial moment X _rr . Unlike previous results, here X _rr is analytically computed in a fairly general manner. It is shown that close to the source macrodispersion is enhanced by the large local velocities, whereas in the far field it drastically reduces since flow there behaves like a mean uniform one. In particular, it is demonstrated that X _rr is bounded between X _∞ corresponding to the short-range (far field), and X ₀ pertaining to the long-range (near-field) correlation in the conductivity field. Although our analytical results rely on the assumption of isotropic medium, they enable one to grasp in a simple manner the main features of macrodispersion mechanism, therefore providing explicit physical insights. Finally, the proposed model has potential toward the characterization of the spatial variability of K as well as testing more general numerical codes.     

Stability of some low‐order approximations for the Stokes problem

Two‐level low‐order finite element approximations are considered for the inhomogeneous Stokes equations. The elements introduced are attractive because of their simplicity and computational efficiency. In this paper, the stability of a Q₁(h)–Q₁(2h) approximation is analysed for general geometries. Using the macroelement technique, we prove the stability condition for both two‐ and three‐dimensional problems. As a result, optimal rates of convergence are found for the velocity and pressure approximations. Numerical results for three test problems are presented. We observe that for the computed examples, the accuracy of the two‐level bilinear approximation is compared favourably with some standard finite elements. Copyright © 2007 John Wiley & Sons, Ltd.     

States of classical statistical mechanical systems of infinitely many particles. I

We study a general mathematical model of a classical system of infinitely many point particles. The space X of infinite particle configurations is equipped with a natural topology as well as a measurable structure related to it. It is also connected with a family {X _A } of local spaces of finite configurations indexed by bounded open sets A in the one-particle space E. A theorem analogous to Kolmogoroff's fundamental theorem for stochastic processes is proved, according to which a consistent family { _A } of local probability measures _A defined on the X _A gives rise to a unique probability measure on X. We also study the problem of integral representation for positive linear forms defined over some linear space of real functions on X. We prove that a positive linear form F(f), defined for functions f in the class C+P, admits a uniquely determined integral representation F(f)= f () d, where is a probability measure over X.     

On the Computation of the Third Order Terms of the Series Defining the Center Manifold for a Scalar Delay Differential Equation

When computing the third order terms of the series of powers of the function whose graph is the center manifold, at an equilibrium point of a scalar delay differential equation with a single constant delay r > 0, some problems occur at the term w_2,1z²[`(z)].{w_{2,1}z^2\overline{z}.} More precisely, in order to determine the values at 0, respectively −r of the function w _2,1(.), an algebraic system of equations must be solved. We show that the two equations are dependent, hence the system has an infinity of solutions. Then we show how we can overcome this lack of uniqueness and provide a formula for w _2,1(0).     

Duality in constitutive formulation of finite-strain elastoplasticity based on F=FeFp and F=FF decompositions

A constitutive theory for large elastic–plastic deformations is presented by employing F=F^pF^e decomposition of the total deformation gradient. A duality in constitutive formulation based on this and the well-known Lee's decomposition F=F_eF_p is established for isotropic polycrystalline and single crystal plasticity.     

The fundamental equations of two-dimensional layer flows

In many studics on two-dimensional flows in field of atmosphere and ocean theequations which are extension of river-hydraulic equationsor Navier-Stokes equationsare usually used.In these equations stand forturbulent resistance.Obviously use of these equations in practice may lead to contradiction.In this paper the average of Reynolds equations over depth is taken.The motion equations,continuity equation and diffusion equation are obtained for the average physical variables.     

Symmetry and secondary bifurcation of elastic structures

We consider non-linear bifurcation problems for elastic structures modeled by the operator equation F[w;α]=0 where F:X×R^k→Y,X,Y are Banach spaces and XY. We focus attention on problems whose bifurcation equations are of the form

f_i(α₁,α₂;λ,μ)=(a_iμ+b_iλ)α_i+p_iα_i³+q_iα_i∑_j=1,j≠i^kα_j+1²+α_ih_i(λ,μ;α₁,α₂,…α_k) i=1,2,…k

which emanates from bifurcation problems for which the linearization of F is Fredholm operators of index 0. Under the assumption of F being odd we prove an important theorem of existence of secondary bifurcation. Under this same assumption we prove a symmetry condition for the reduced equations and consequently we got an existence result for secondary bifurcation. We also include a stability analysis of the bifurcating solutions.     

Global Nonexistence for Nonlinear Kirchhoff Systems

In this paper we consider the problem of non-continuation of solutions of dissipative nonlinear Kirchhoff systems, involving the p(x)-Laplacian operator and governed by nonlinear driving forces f = f (t, x, u), as well as nonlinear external damping terms Q = Q(t, x, u, u _t ), both of which could significantly dependent on the time t. The theorems are obtained through the study of the natural energy Eu associated to the solutions u of the systems. Thanks to a new approach of the classical potential well and concavity methods, we show the nonexistence of global solutions, when the initial energy is controlled above by a critical value; that is, when the initial data belong to a specific region in the phase plane. Several consequences, interesting in applications, are given in particular subcases. The results are original also for the scalar standard wave equation when p ≡ 2 and even for problems linearly damped.