Sixth-order System Research Articles

A sixth-order parabolic system in semiconductors

The authors investigate the global existence and semiclassical limit of weak solutions to a sixth-order parabolic system, which is a quantum-corrected macroscopic model derived recently to simulate the quantum effects in miniaturized semiconductor devices.

Elastic interactions in phase-field crystal models: numerics and postprocessing

Abstract In this paper, numerical and implementation aspects of the phase-field crystal model with elastic interactions are addressed. This model leads to a time-dependent, sixth-order system of differential equations which yields a continuous density field, from which individual atom positions can be extracted and tracked in time. We solve the system using high-order finite elements, and reconstruct the elastic properties of the crystal from the atomic displacements with a meshless interpolation scheme. Numerical simulations are performed of benchmark experiments, which include deformation and dislocation dynamics.

Determination of stability regions for controlled mechanical systems without friction and with position feedback

A method for determining the possible ranges of control coefficients for a sixth-order mechanical system is proposed. The equations describing this system have only even derivatives. The method is illustrated by an example.

Evaluation of eigenfunctions from compound matrix variables in non-linear elasticity – II. Sixth order systems

We show how the compound matrix method can be extended to give eigenfunctions as well as eigenvalues to bifurcation problems in non-linear elasticity. The non-trivial boundary conditions create some difficulties and we find that sixth order systems for elasticity problems will require a shooting method for two functions of two unknown parameters over and above the calculations required for comparable problems in fluids.

Non-equilibrium effects on fast reactions in a heterogeneous batch reactor

Recent studies by the authors of the heterogeneous catalysis of fast binary reactions have taken a dynamical systems approach, assuming that fast enough reactions are confined to a manifold upon which surface equilibrium holds. This approximation makes substantial simplification possible, for instance in the case of a batch reactor, it allows a naturally sixth order system to be approximated by a two dimensional manifold for the dynamics of two modified Thiele moduli. Nevertheless, a proper assessment of how much faster must the velocity of surface reaction be than the velocity of mass transfer to the catalytic surface before the quasi-equilibrium on the surface holds should be made. In this paper, a theory for the systematic correction to infinitely fast reactions is made for large but finite velocity reactions. It is compared to full numerical solutions to the model equations. Recommendations about the regime of applicability of the quasi-equilibrium approximation are made. In general, the predictions of the quasi-equilibrium theory hold for ratios of mass transfer coefficients to reaction velocity ξ of less than 1 / 1000 , with qualitative agreement in regimes of less than 1 / 100 . The general trend, however, is that the stronger the kinetic asymmetry between the mass transfer coefficients of the reactants, the slower the reaction rate can be and still have the quasi-equilibrium theory hold. A perturbation analysis demonstrates that the quasi-equilibrium theory is a regular limit of the fast non-equilibrium theory. In the irreversible case, a matched asymptotic analysis gives the same prediction for the switch time from effective surface depletion of one reagent to the other as the quasi-equilibrium theory. Furthermore, it gives an estimate of the smoothing out of the transition zone with a temporal width of ξ 1 / 2 . It should be noted that the continual drive for improved catalyst activities inevitably leads to mass transfer limited reactions, and thus this regime is not uncommon.

Solitary solutions of some nonlinear evolution equations

In this paper, the traveling wave reduction combined with the homogeneous balance method (HBM) is used to find new exact solutions by using less-studied nonlinear partial differential equations (nPDG) of higher order. The usual starting point is a special transformation converting the nPDG in its two variables x and t into an nonlinear ordinary differential equation (nODE) in the single variable ξ. Using the hyperbolic tangent method, new exact solutions for the three nPDGs are studied. The new feature of this paper is of course the fact that we are dealing with nPDGs which cannot be found by studying the relevant literature, therefore we believe it is time for publishing some special solutions of this interesting kind of equations. Clearly we do not use a new formalism, but with the aid of the “tanh-method” we are able to calculate at least exact traveling wave solutions (solitary waves) which can physically interpreted as an action of wave propagation. Simultaneous we point out the necessity of such sophisticated methods since a general theory of nPDGs does not exist. The existence of nontrivial, noncomplex valued solutions of the Ramani equation enables us to generate families of real valued solutions of a nonlinear sixth order system. Special differential transformations are leading to connections between traveling wave solutions of associated nonlinear evolution equations.

On non-linear stability in unconstrained non-linear elasticity

A method of obtaining a full three-dimensional non-linear Hadamard stability analysis of inhomogeneous deformations of arbitrary, unconstrained, hyperelastic materials is presented. The analysis is an extension of that given by Chen and Haughton (Proc. Roy. Soc. London A 459 (2003) 137) for two-dimensional incompressible problems. The process that we present replaces the second variation condition expressed as an integral involving a quadratic in three arbitrary perturbations, with an equivalent sixth-order system of ordinary differential equations. The positive definiteness condition is thereby reduced to the simple numerical evaluation of zeros of a well-behaved function. The general theory is illustrated by applying it to the problem of the inflation of a thick-walled spherical shell. The present analysis provides a simpler alternative approach to bifurcation problems approached by using the incremental equations of non-linear elasticity.

Solitary waves and the structures of discontinuities in non-dissipative models with complex dispersion

Stationary solutions of reversible evolutionary equations of mechanics with higher derivatives are analysed. A two-dimensional graphical method for investigating the solutions of systems of ordinary differential equations is described, which enables one to find special types of solutions: periodic waves, solitary waves and the structures of discontinuities. At the same time, solitary waves can be obtained by taking the limit of sequences of periodic waves and the structures of discontinuities obtained by taking the limit of sequences of solitary waves. This general approach has enabled the existence of all earlier predicted structures to be verified has enabled new types of structures (three-wave structures) to be revealed and has enabled all the necessary conditions at the discontinuities to be found. All the previously known types of solitary waves are found and new types of solitary waves are revealed (generalized ordinary and 1:1 multisolitons). Methods of finding generalized solitary waves, including those with a finite amplitude of the periodic component, are determined. Examples of the solution of the following problems are given for a fourth-order system: generalized solitary waves as the limiting solutions of two-wave resonance solutions, generalized solitary waves and the structure of a discontinuity with three waves, a 1:1 soliton and the structure of a discontinuity with a single radiated wave, a solitary wave with fixed propagation velocity, and the structure of a discontinuity in the form of a kink with radiation. A generalized 1:1 soliton and the structure of a discontinuity with two radiated waves is considered in the case of sixth-order systems. The discussion is mainly based on the example of travelling waves described by the generalized Korteweg-de Vries equations. Other models with complex dispersion (a plasma and a stratified fluid) are also considered.

AXISYMMETRIC VIBRATIONS OF CONCENTRIC DISSIMILAR ORTHOTROPIC COMPOSITE ANNULAR PLATES

The axisymmetric vibrational behaviour of two concentric dissimilar orthotropic composite annuli is investigated. Sixth order systems of equations of motion for each annulus are presented together with the matching conditions at their interface. The common frequency of vibration of the structure is determined numerically using a finite difference method. Bi-layered, triple-layered and bi-+triple-layered material compositions were considered. Computed results for a wide range of possible orthotropic material combinations indicate the complex role the material orthotropies play in determining the annuli's vibrational response. It is shown that the frequency increase factor (defined as the ratio between the highest and lowest frequencies obtained for all inner and outer material combinations considered) is very sensitive to the geometric, heterogeneity and orthotropy parameters of the composite annular plates. Unfortunately, no simple rule of thumb formula for estimating the effect of the material composition on the frequency appears to be deducible. However, in keeping with previous results for concentric dissimilar isotropic annuli, the geometry (expressed via the radii of the inner and outer annuli) and the different material compositions were found to exert a considerable influence on the natural frequency of vibration. In the absence of appropriate analytical expressions, optimization and, thereby, control of the natural frequency of composite plate-like structures, through geometry and material composition, must make use of parametric studies similar to the current one.

Cantilever Boundary Condition, Deflections, and Stresses of Sandwich Beams

An extended set of Timoshenko beam equations is presented, which results in a sixth-order system and allows the three boundary conditions of zero deflection, zero slope, and zero rotation angle to be satisfied at a cantilever end. The analysis assumes the core material of the sandwich provides shear stiffness, but negligible extensional stiffness to the beam, and the two skins remain equidistant from one another. The equations are found to depend on essentially two nondimensional parameters, a shear parameter g, and a skin-core thickness ratio h/h c . Closed-form solutions are given for the deflections and stresses of an end-loaded, cantilever sandwich beam. Trends over a wide range of shear parameter g indicate that the sandwich behavior varies smoothly from that of a conventional Ruler beam at large g to two separate Euler beams of the skins at low g. In the region of the cantilever end, an edge zone develops, which transitions the beam from the conventional Timoshenko behavior to a behavior that satisfies all three boundary conditions there. Simple estimates of the extent of the edge zone are given. The edge zone also causes a redistribution of the normal and shear stresses occurring there. Extension to vibrations of these beams is indicated.

Modelling of viscoelastic plume-lithosphere interaction using the adaptive multilevel wavelet collocation method

SUMMARY Modelling of mantle flows with sharp viscosity contrasts in a viscoelastic medium is a challenging computational problem in geodynamics because of its multiple-scale nature in space and time. We have employed a recently developed adaptive multilevel wavelet collocation algorithm to study the dynamics of a small rising diapir interacting with a stiff lithosphere in a Maxwell viscoelastic mantle. In this kinematic model we have prescribed the upward velocity of the diapir and then we need to integrate in time only the momentum equation governing the temporal evolution of the pressure, stress and velocity components, which together constitute a sixth-order system in time. The total number of collocation points did not exceed 10 4 , compared to more than 10 6 gridpoints using conventional evenly spaced grid methods. The viscosity of the diapir is 10 x4 times lower than that of the surrounding mantle, while the viscosity of the thin lithosphere, about 5‐10 per cent of the entire layer depth, is 10 4 ‐10 8 times stiffer than the ambient mantle. Our results demonstrate the efficacy of wavelets to capture the sharp gradients of the stress and pressure fields developed in the diapiric impingement process. The interaction of the viscoelastic lithosphere with the rising viscoelastic diapir results in the localization of stress within the lithosphere. The magnitude of the stress fields can reach around 100‐300 MPa. Our simple kinematic model shows clearly that viscoelasticity can potentially play an important role in the dynamics of the lithosphere, especially concerning the potential severage of the lithosphere by mantle upwellings.

Axisymmetric vibrations of concentric dissimilar isotropic composite plates

The axisymmetric vibrational behaviour of two concentric dissimilar isotropic composite plates is analysed. A sixth order system of equations for each plate is solved in closed form. The common frequency of vibration of the structure is determined from a transcendental equation derived from application of boundary conditions and matching conditions at the interface between the inner and outer plate. Numerical examples for both annular+full plate and annular+annular plate combinations reveal that the major factors influencing the natural frequency of vibration of these structures are (a) the different material compositions of the two plates, and (b) the plates radius. It was found that for certain combinations of material lay-ups higher frequencies of vibration could be achieved than for an equivalent single plate structure. In addition, the higher frequencies could be optimised by altering the radius-to-thickness ratio of the inner plate. The results point to the usefulness of constructing composite plate-like structures in this fashion, thus enabling control of their dynamical behaviour by efficient use of the materials heterogeneity and their geometry.

Modulation and symmetry changes in stellar dynamos

Stellar dynamos are governed by non-linear partial differential equations (PDEs) which admit solutions with dipole, quadrupole or mixed symmetry (i.e. with different parities). These PDEs possess periodic solutions that describe magnetic cycles, and numerical studies reveal two different types of modulation. For modulations of Type 1 there are parity changes without significant changes of amplitude, while for Type 2 there are amplitude changes without significant changes in parity. In stars like the Sun, cyclic magnetic activity is interrupted by grand minima that correspond to Type 2 modulation. Although the Sun’s magnetic field has maintained dipole symmetry for almost 300 yr, there was a significant parity change at the end of the Maunder Minimum. We infer that the solar field may have flipped from dipole to quadrupole polarity (and back) after deep minima in the past and may do so again in the future. Other stars, with different masses or rotation rates, may exhibit cyclic activity with dipole, quadrupole or mixed parity. The origins of such behaviour can be understood by relating the PDE results to solutions of appropriate low-order systems of ordinary differential equations (ODEs). Type 1 modulation is reproduced in a fourth-order system while Type 2 modulation occurs in a third-order system. Here we construct a new sixth-order system that describes both types of modulation and clarifies the interactions between symmetry-breaking and modulation of activity. Solutions of these non-linear ODEs reproduce the qualitative behaviour found for the PDEs, including flipping of polarity after a prolonged grand minimum. Thus we can be confident that these patterns of behaviour are robust, and will apply to stars that are similar to the Sun.

The stability of rotating-disc boundary-layer flow over a compliant wall. Part 2. Absolute instability

A numerical study has been undertaken of the influence of a compliant boundary on absolute instability. In a certain parameter range absolute instability occurs in the boundary layer on a rotating disc, thereby instigating rapid transition to turbulence. The conventional use of wall compliance as a laminar-flow control technique has been to lower growth rates of convective instabilities. This has the effect of reducing amplification of disturbances as they propagate downstream. For absolute instability, however, only the suppression of its onset would be a significant gain. This paper addresses the question of whether passive wall compliance can be advantageous when absolute instability exists in a boundary layer.A theoretical model of a single-layer viscoelastic compliant wall was used in conjunction with the sixth-order system of differential equations which govern the stability of the boundary-layer flow over a rotating disc. The absolute/convective nature of the flow was ascertained by using a spatio-temporal analysis. Pinch-point singularities of the dispersion relation and a point of zero group velocity identify the presence of absolute instability. It was found that only a low level of wall compliance was enough to delay the appearance of absolute instability to higher Reynolds numbers. Beyond a critical level of wall compliance results suggest that complete suppression of absolute instability is possible. This would then remove a major route to transition in the rotating-disc boundary layer.

The stability of rotating-disc boundary-layer flow over a compliant wall. Part 1. Type I and II instabilities

A theoretical study into the effects of wall compliance on the stability of the rotating-disc boundary layer is described. A single-layer viscoelastic wall model is coupled to a sixth-order system of fluid stability equations which take into account the effects of viscosity, Coriolis acceleration, and streamline curvature. The coupled system of equations is integrated numerically by a spectral Chebyshev-tau technique.Travelling and stationary modes are studied and wall compliance is found to greatly increase the complexity of the eigenmode spectrum. It is effective in stabilizing the inviscid Type I (or cross-flow) instability. The effect on the viscous (Type II) eigenmode is more complex and can be strongly destabilizing. An analysis of the energy flux indicates that this destabilization arises as a result of a large degree of energy production by viscous stresses at the wall/flow interface.The Type I and II instabilities are shown to be negative and positive energy waves respectively. The co-existence of eigenmodes of opposite energy type indicates the possibility of modal interaction and coalescence. It is found that, compared with the rigid disc, wall compliance promotes the interaction and coalescence of the Type I and II eigenmodes. There is an associated strong instability which appears to be characterized by marked horizontal motion of the compliant surface. Modal coalescence is interpreted physically as producing local algebraic growth which could advance the onset of nonlinear effects.

Elliptical vortices in shear: Hamiltonian moment formulation and Melnikov analysis

The equations of motion for interacting elliptical vortices in a background shear flow are derived from a Hamiltonian moment formulation. The equations reduce to the sixth order system of Melander et al. [J. Fluid Mech. 167, 95 (1986)] when a pair of vortices is considered and shear is neglected. The equations for a pair of identical vortices are analyzed using a number of methods, with particular emphasis on the implications for vortex merger. The splitting distance between the stable and unstable manifolds connecting the hyperbolic fixed points of the intercentroidal motion—the separatrix splitting—is estimated with a Melnikov analysis. This analysis differs from the standard time-periodic Melnikov analysis on two counts: (a) the ‘‘periodic’’ perturbation arises from a second degree of freedom in the system which is not wholly independent of the first degree of freedom, the intercentroidal motion; (b) this perturbation has a faster time scale than the intercentroidal motion. The resulting Melnikov integral appears to be exponentially small in the perturbation as the latter goes to zero. Numerical simulations, notably Poincaré sections, provide a global view of the dynamics and indicate that, as observed in previous studies, there are essentially two modes of merger. The effect of the shear on chaotic motion is also discussed.

On the stability of attachment-line boundary layers. Part 1. The incompressible swept Hiemenz flow

The stability of the incompressible attachment-line boundary layer is studied by solving a partial-differential eigenvalue problem. The basic flow near the leading edge is taken to be the swept Hiemenz flow which represents an exact solution of the Navier-Stokes (N-S) equations. Previous theoretical investigations considered a special class of two-dimensional disturbances in which the chordwise variation of disturbance velocities mimics the basic flow and renders a system of ordinary-differential equations of the Orr-Sommerfeld type. The solution of this sixth-order system by Hall, Malik & Poll (1984) showed that the two-dimensional disturbance is stable provided that the Reynolds number . Both symmetric and antisymmetric two- and three-dimensional eigenmodes can be amplified. These unstable modes with the same spanwise wavenumber travel with almost identical phase speeds, but the eigenfunctions show very distinct features. Nevertheless, the symmetric two-dimensional mode always has the highest growth rate and dictates the instability. As far as the special two-dimensional mode is concerned, the present results are in complete agreement with previous investigations. One of the major advantages of the present approach is that it can be extended to study the stability of compressible attachment-line flows where no satisfactory simplified approaches are known to exist.

Double averaging approach to the study of spinup dynamics of flexible satellites

We study the dynamics of a class of flexible dual-spin satellites in which the rotor is spun up by a small constant torque T applied by the platform. We use a previously published zero torque solution that was obtained using the Krylov-Bogoliubov-Mitropolski averaging method. A second application of averaging developed herein leads to a reduction of the equations of motion from a sixth-order system to a single first-order equation describing the slow evolution of energy as a function of the axial angular momentum of the rotor. The geometrical interpretation of this reduction involves projecting solutions onto a certain bifurcation diagram. Numerical solutions of the averaged equation agree with numerical solutions to the full sixth-order system and show that the flexible spacecraft behaves essentially the same as its rigid counterpart during the spinup maneuver.

Incremental stresses in loaded orthotropic circular membrane tubes—II. Numerical solution

In Part I of this article (1995, Int. J. Solids Structures32, 1907–1925) a simplified theory for the incremental state of stress in an orthotropic circular membrane tube was presented. In this second part, a numerical method is devised to solve the resulting elliptic sixth-order system of equations, for tubes which contain no slits or holes. In this method, Fourier decomposition is used in the circumferential direction and finite element discretization is used in the longitudinal direction. Special finite elements with six degrees-of-freedom are employed. Using this numerical method, the solutions of several specific problems of membrane tubes are obtained, presented and discussed. These include two problems concerning the decay of boundary disturbances. The theoretical treatment of such decay is also discussed.

Three-dimensional stress analysis of orthotropic and cross-ply laminated hollow cylinders and cylindrical panels

This paper deals with a detailed, three-dimensional stress and displacement analysis of transversely loaded, laminated complete hollow cylinders and open cylindrical panels having a symmetric or an antisymmetric cross-ply lay-up. The analysis is based on a refined, static version of a successive approximation approach which was first proposed by Soldatos and Hadjigeorgiou (J. Sound Vibration 137 (1990) 369–384) in connection with corresponding dynamic problems. However, the present refined formulation makes use of a recursive rather than a successive approximation formula. As a result, independently of the number of layers employed, the solution of the bending problem considered always ends with the solution of a sixth order system of simultaneous algebraic equations. The effectivenesss of the approach employed is checked by comparing numerical results with corresponding three-dimensional elasticity results published elsewhere in the literature. Then, the proposed formulation is further used towards a detailed stress and displacement analysis of the afore-mentioned structural elements.