Hydrodynamic formation of stable anisotropic structures in Couette flow of dilute suspensionFormation hydrodynamique pour structures anisotropiques stables dans un flux de Couette d'une suspension diluée

The problem of rotary motion of rigid axially symmetric elongated particles in the Couette flow of dilute suspension with anisotropic carrier fluid is solved. It is shown that the stable stationary solutions of the dynamical set of ordinary differential equations describing the particles rotary motion are possible in the case of forming the stationary anisotropy in the carrier fluid of the suspension. It allows us to detect the stationary orientation of suspended particles and formation of stable anisotropic liquid-crystalline structures in the considered suspension under the action of hydrodynamic forces. The study of rheological properties of such a structured suspension shows that it behaves as a viscoelastic quasi-Newtonian anisotropic liquid medium. To cite this article: E.Yu. Taran et al., C. R. Mecanique 332 (2004).     

Contribution to the theory of stationary separation zones

Experiment shows that the stationary flow pattern about a bluff body with closed separation zone, in the case of laminar flow about the body and in the separation zone, breaks down for a subsonic stream velocity in the Reynolds number range from 10¹ to 10². However, experiment shows that for a supersonic stream velocity a stable stationary flow pattern is observed with the existence of laminar stagnant zones adjacent to the body (the stagnant zone behind an aft-facing step on the body surface, the stagnant zone ahead of a gradual forward-facing step on the body surface, the forward separation zone formed by the tip of a spike, the stagnant zone formed when a shock impinges on a body surface) at high Reynolds numbers of the order of 10⁴–10⁶.Thus, experiments indicate that in certain ranges of variation of M and R, under certain boundary condition, stationary solutions of the viscous fluid equations of motion exist and are stable. Outside these ranges and under other boundary conditions the flow about a body with a closed separation zone has a more (Karman vortex street for M1) or less (pulsating flow in the near wake behind the body for M>1) marked unsteady nature, indicating instability of the stationary solutions of the equations of motion under these conditions. To date no theoretical justification has been presented for the existence of stable stationary flows with separation zones in the ranges indicated.In the following an attempt is made to find the region of existence of possible stationary flows with a closed separation zone in that range of Reynolds numbers in which the flow in the viscous mixing region may be described by the Prandtl equations. In so doing the boundary conditions for the flow within the separation zone are selected so that the flow pattern within the zone is significantly simplified and use of the analysis methods applicable in hydrodynamics becomes possible. In the first part (§§1–4) we study the field of possible stationary flows for the case of an incompressible fluid. It is shown that only under special boundary conditions within the separation zone (ideal dissipator) does the flow about a flat plat as R approach the Kirchhoff flow with fluid at rest within the zone. In this case the drag coefficient of the system consisting of the plate plus the ideal dissipator c_x/(^{+ +}4), i.e., it approaches a value which is half that obtained by Kirchhoff for an ideal fluid.A qualitative study of the field of possible stationary flows in the c_xR plane made it possible to discover the existence of a region, having an upper bound at R10², which degenerates into a line. In this region the stationary flows have a singular flow configuration with inviscid vortical-type attachment.The existence of a connection between the flow configuration in the inviscid vortical attachment region and the stability of the stationary solutions is investigated in the second part (§§6–7), both for the case of individual solutions obtained by the method of linear hydrodynamic stability theory and on the basis of the available experimental data obtained over a wide range of Reynolds numbers for both subsonic and supersonic flow velocities. This investigation makes it possible to formulate a rule for finding stable stationary flows with separation zones and to apply this rule to analyze separation-type flows, both laminar and in certain special cases turbulent.     

Asymptotic Stability of Landau Solutions to Navier–Stokes System

It is known that the three-dimensional Navier–Stokes system for an incompressible fluid in the whole space has a one parameter family of explicit stationary solutions that are axisymmetric and homogeneous of degree −1. We show that these solutions are asymptotically stable under any L ²-perturbation.     

Existence and Characterization of Attractors for a Nonlocal Reaction–Diffusion Equation with an Energy Functional

In this paper we study a nonlocal reaction–diffusion equation in which the diffusion depends on the gradient of the solution. Firstly, we prove the existence and uniqueness of regular and strong solutions. Secondly, we obtain the existence of global attractors in both situations under rather weak assumptions by defining a multivalued semiflow (which is a semigroup in the particular situation when uniqueness of the Cauchy problem is satisfied). Thirdly, we characterize the attractor either as the unstable manifold of the set of stationary points or as the stable one when we consider solutions only in the set of bounded complete trajectories.

     

Local discontinuous Galerkin method for a nonlocal viscous conservation laws

The purpose of this article is to investigate high‐order numerical approximations of scalar conservation laws with nonlocal viscous term. The viscous term is given in the form of convolution in space variable. With the help of the characteristic of viscous term, we design a semidiscrete local discontinuous Galerkin (LDG) method to solve the nonlocal model. We prove stability and convergence of semidiscrete LDG method in L² norm. The theoretical analysis reveals that the present numerical scheme is stable with optimal convergence order for the linear case, and it is stable with sub‐optimal convergence order for nonlinear case. To demonstrate the validity and accuracy of our scheme, we test the Burgers equation with two typical nonlocal fractional viscous terms. The numerical results show the convergence order accuracy in space for both linear and nonlinear cases. Some numerical simulations are provided to show the robustness and effectiveness of the present numerical scheme.     

Artificial Boundary Conditions of Pressure Type for Viscous Flows in a System of Pipes

In a three-dimensional domain Ω with J cylindrical outlets to infinity the problem is treated how solutions to the stationary Stokes and Navier–Stokes system with pressure conditions at infinity can be approximated by solutions on bounded subdomains. The optimal artificial boundary conditions turn out to have singular coefficients. Existence, uniqueness and asymptotically precise estimates for the truncation error are proved for the linear problem and for the nonlinear problem with small data. The results include also estimates for the so called “do-nothing” condition.     

Transition range drop tower J-R curve testing of A106 steel

Fracture-toughness properties should be measured in the laboratory at loading rates and temperatures similar to those expected in the application of interest. This is not usually the case because of the experimental difficulties involved. This report describes a method being used to obtainJ _Ic, J-R curves, andJ at cleavage for three-point-bend tests conducted at drop tower rates through the ductile to brittle transition regime of the ferritic A106 steel being tested. The major conclusion is that these tests can now be accomplished, though a high degree of expertise and considerable practical experience is necessary to obtain good test results. The steel tested here is quite rate dependent as shown both by tensile tests and fracture-toughness tests. A load elevation of 30 to 50 percent results in the drop tower 100 in./s (2.5 m/s) tests on this material in comparison with static tests when both tests are conducted on the ductile upper shelf. Nonetheless, for this materialJ _Ic andJ-R curves are not elevated by the loading rate. Looking at the elastic and plastic components ofJ one sees that theJ _EL increases with increased loading rate but also thatJ _PL decreases with loading rate. It is also demonstrated that for the high rate tests more crack extension is present at a given bend angle for the rapid tests than with the static tests. Paper was presented at the 1987 SEM Fall Conference on Experimental Mechanics held in Savannah, GA on October 25–28.     

Linear stability of stationary solutions of the Vlasov-Poisson system in three dimensions

Rigorous results on the stability of stationary solutions of the Vlasov-Poisson system are obtained in the contexts of both plasma physics and stellar dynamics. It is proved that stationary solutions in the plasma physics (stellar dynamics) case are linearly stable if they are decreasing (increasing) functions of the local, i.e., particle, energy. The main tool in the analysis is the free energy, a conserved quantity of the linearized system. In addition, an appropriate global existence result is proved for the linearized Vlasov-Poisson system and the existence of stationary solutions which satisfy the above stability condition is established.     

Stability of Gravity Driven Convection in a Cylindrical Porous Layer Subjected to Vibration

The linear stability theory is used to investigate analytically the effects of gravity modulation on convection in a homogeneous cylindrical porous layer heated from below. The linear stability results show that increasing the frequency of vibration stabilizes the convection. In addition the aspect ratio of the porous cylinder is shown to influence the stability of convection for all frequencies analysed. It was also observed that only synchronous solutions are possible in cylindrical porous layers, with no transition to subharmonic solutions as was the case in Govender (2005a) [Transport Porous Media 59(2), 227–238] for rectangular layers or cavities.     

An elastic-plastic analysis of an asymmetric cracked specimen subjected to longitudinal shear

Some recent elastic-plastic analyses of cracked specimens subjected to symmetric mode III loading are extended to include asymmetric loading and geometry. Solutions are given for arbitrary work hardening behaviour in any specimen that is amenable to a linear elastic analysis. It is shown that asymmetry has a major influence on the shape of the plastic zone, but does not affect the J-integral unil the loading is well into the large scale yielding range. In particular the “plastic zone corrected” estimate of J, obtained by elastically solving a problem for a crack longer than the actual one, is shown to remain a valid two-term asymptotic expansion in the presence of asymmetry. The general results are applied to a crack at an angle to a uniform stress field in a power law hardening material. The growth of the plastic zone is displayed graphically for various hardening exponents and crack orientations. No other asymmetric solution is available, but values of J are compared with those obtained from a fully plastic analysis in the symmetric case.     

A generalization of the method of averaging for systems with two time scales

In this paper we shall consider systems of the form x = ? f(t, ?t, x, y, ?), y = g(t,?t, x, y,?), where x and y are vectors of finite dimensions, f and g are assumed to be bounded for all t, and ? is a real parameter. Sufficient conditions are obtained for the existence of certain solutions which are bounded for all t. These solutions are shown to approach special solutions of a derived simpler averaged system of equations as ? → 0. Moreover, it is shown that there exists only one such bounded solution in the neighborhood of each special solution. In the special case when y is not present, it is shown that if a special solution is stable, solutions starting in nonlocal neighborhoods of this special solution approach the bounded solutions adjacent to it as t → ∞. These results generalize most of the existing work for systems of the type discussed here. Finally, we employ our results to study some problems of physical importance.     

Special Cases in Optimization Problems for Stationary Linear Closed-Loop Systems

Consideration is given to problems of solving the algebraic Riccati equation (ARE)—J-factorization of matrix polynomials and J-factorization of rational matrices—to which traditional solution algorithms are not applicable. In this connection, solution algorithms for these problems are discussed where the eigenvalues of the Hamiltonian matrix corresponding to the ARE and the zeros of matrix polynomials are located on the imaginary axis. Moreover, a procedure is set forth for asymptotic expansion of a stabilizing solution of the ARE in the neighborhood of a point at which the ARE has no stabilizing solution. It is shown how this expansion can be used for constructing canonical J-factorization of matrix polynomials that is nearly a noncanonical J-factorization. It is pointed out that the algorithms described can be implemented with the help of MATLAB routines     

Dynamics of nonlinear ecosystems under colored noise disturbances

Stochastic ecosystems of prey-predator type subjected to colored noises with broad-band spectra are investigated. Nonlinear models are considered for two different scenarios: one is the case of possible abundant prey supply and another is the case of possible large predator population. The stochastic averaging procedure is applied to obtain stationary probability solutions of the nonlinear systems. Two types of colored noise are considered: one is the low-pass filtered noise with the spectrum peak at zero frequency, and another is the randomized harmonic process with the spectrum peak at a nonzero frequency. For either type of the noises, the band width reflecting the level of the noise color can be adjusted using a single parameter. The analytical results are substantiated by those obtained from Monte Carlo simulations. It is found that the noise color has significant effects on the stationary state of the system. A narrower band width leads to a less stable system in the sense that the prey and predator populations deviate farther from the equilibrium point of the system without noise disturbances.     

J-integral and crack driving force in elastic–plastic materials

This paper discusses the crack driving force in elastic–plastic materials, with particular emphasis on incremental plasticity. Using the configurational forces approach we identify a “plasticity influence term” that describes crack tip shielding or anti-shielding due to plastic deformation in the body. Standard constitutive models for finite strain as well as small strain incremental plasticity are used to obtain explicit expressions for the plasticity influence term in a two-dimensional setting. The total dissipation in the body is related to the near-tip and far-field J-integrals and the plasticity influence term. In the special case of deformation plasticity the plasticity influence term vanishes identically whereas for rigid plasticity and elastic-ideal plasticity the crack driving force vanishes. For steady state crack growth in incremental elastic–plastic materials, the plasticity influence term is equal to the negative of the plastic work per unit crack extension and the total dissipation in the body due to crack propagation and plastic deformation is determined by the far-field J-integral. For non-steady state crack growth, the plasticity influence term can be evaluated by post-processing after a conventional finite element stress analysis. Theory and computations are applied to a stationary crack in a C(T)-specimen to examine the effects of contained, uncontained and general yielding. A novel method is proposed for evaluating J-integrals under incremental plasticity conditions through the configurational body force. The incremental plasticity near-tip and far-field J-integrals are compared to conventional deformational plasticity and experimental J-integrals.     

The Stress Response of Functionally Graded Isotropic Linearly Elastic Rotating Disks

The purpose of this research is to investigate the effects of material inhomogeneity on the response of linearly elastic isotropic solid circular disks or cylinders, rotating at constant angular velocity about a central axis. The work is motivated by the recent research activity on functionally graded materials (FGMs), i.e., materials with spatially varying properties tailored to satisfy particular engineering applications. The analog of the classic problem for a homogeneous isotropic rotating solid disk or cylinder is considered. The special case of a body with Young"s modulus depending on the radial coordinate only, and with constant Poisson"s ratio, is examined. For the case when the Young"s modulus has a power-law dependence on the radial coordinate, explicit exact solutions are obtained. It is shown that the stress response of the inhomogeneous disk (or cylinder) is significantly different from that of the homogeneous body. For example, the maximum radial and hoop stresses do not, in general, occur at the center as in the case for the homogeneous material. Furthermore, for the case where the Young"s modulus increases with radial distance from the center, it is shown that radially symmetric solutions exist provided the rate of growth of the Young"s modulus is, at most, cubic in the radial variable. It is also shown for the general inhomogeneous isotropic case how the material inhomogeneity may be tailored so that the radial and hoop stress are identical throughout the disk. This revised version was published online in August 2006 with corrections to the Cover Date.     

Limiting Solutions of Nonlocal Dispersal Problem in Inhomogeneous Media

This paper is concerned with the nonlocal dispersal problem in inhomogeneous media. Our goal is to show the limiting behavior of perturbation equation with parameters. By analyzing the asymptotic behavior of solutions when the parameter is small, we find that convection appears in inhomogeneous media. Moreover, if the effect of inhomogeneous media changes, then we prove a convergence result that convection disappears in nonlocal dispersal problems.

     

The stress field near the front of an arbitrarily shaped crack in a three-dimensional elastic body

The problem stated in the title is investigated with special emphasis on the first three terms of the stress expansion, proportional to r ^-1/2, r ⁰=1 and r ^1/2 respectively, where r denotes the distance to the crack front. The particular case of a plane crack with a straight front and of stresses independent of the distance along the latter is studied first. It is shown that the classical plane strain and antiplane solutions must be supplemented by a few additional particular solutions to obtain the full stress expansion. The general case is then considered. The stress expansion is studied by writing the field equations (equilibrium, strain compatibility and boundary conditions) in a system of suitable curvilinear coordinates. It is shown that the number of independent constants in the stress expansion is the same as in the particular case considered previously but that the curvatures of the crack and its front and the non-uniformity of the stresses along the latter induce the appearance of corrective terms in this expansion.     

Orbital Stability of Periodic Peakons to a Generalized μ-Camassa–Holm Equation

In this paper, we study the orbital stability of the periodic peaked solitons of the generalized μ-Camassa–Holm equation with nonlocal cubic and quadratic nonlinearities. The equation is a μ-version of a linear combination of the Camassa–Holm equation and the modified Camassa–Holm equation. It is also integrable with the Lax-pair and bi-Hamiltonian structure and admits the single peakons and multi-peakons. By constructing an inequality related to the maximum and minimum of solutions with the conservation laws, we prove that, even in the case that the Camassa–Holm energy counteracts in part the modified Camassa–Holm energy, the shapes of periodic peakons are still orbitally stable under small perturbations in the energy space.     

On the sensitivity of interconversion between relaxation and creep

The interconversion equation of linear viscoelasticity defines implicitly the interrelations between the relaxation and creep functions G(t) and J(t). It is widely utilised in rheology to estimate J(t) from measurements of G(t) and conversely. Because different molecular details can be recovered from G(t) and J(t), it is necessary to work with both. This leads naturally to the need to identify whether it is better to first measure G(t) and then determine J(t) or conversely. This requires an understanding of the stability (sensitivity) of the recovery of J(t) from G(t) compared with that of G(t) from J(t). Although algorithms are available that work adequately in both directions, numerical experimentation strongly suggests that the recovery of J(t) from G(t) measurements is the more stable. An elementary theoretical rationale has been given recently by Anderssen et al. (ANZIAM J 48:C346–C363, 2007) for single exponential models of G(t) and J(t). It explicitly exploits the simple algebra of such functions. In this paper, corresponding bounds are derived that hold for arbitrary sums of exponentials. The paper concludes with a discussion, from a practical rheological perspective, about the implications and implementations of the results.     

Interaction between an interface crack and subinterface microcracks in metal/piezoelectric bimaterials

The J-integral analysis is presented for the interaction problem between a semi-infinite interface crack and subinterface matrix microcracks in dissimilar anisotropic materials. After deriving the fundamental solutions for an interface crack subjected to different loads and the fundamental solutions for an edge dislocation beneath the interface, the interaction problem is deduced to a system of singular integral equations with the aid of a superimposing technique. The integral equations are then solved numerically and a conservation law among three values of the J-integral is presented, which are induced from the interface crack tip, the microcracks and the remote field, respectively. The conservation law not only provides a necessary condition to confirm the numerical results derived, but also reveals that the microcrack shielding effect in such materials could be considered as a redistribution of the remote J-integral. It is this redistribution that does lead to the phenomenological shielding effect.