The Laplace transform method for Burgers' equation

The Laplace transform method (LTM) is introduced to solve Burgers' equation. Because of the nonlinear term in Burgers' equation, one cannot directly apply the LTM. Increment linearization technique is introduced to deal with the situation. This is a key idea in this paper. The increment linearization technique is the following: In time level t, we divide the solution u(x, t) into two parts: u(x, t^k) and w(x, t), t^k⩽t⩽t^k+1, and obtain a time‐dependent linear partial differential equation (PDE) for w(x, t). For this PDE, the LTM is applied to eliminate time dependency. The subsequent boundary value problem is solved by rational collocation method on transformed Chebyshev points. To face the well‐known computational challenge represented by the numerical inversion of the Laplace transform, Talbot's method is applied, consisting of numerically integrating the Bromwich integral on a special contour by means of trapezoidal or midpoint rules. Numerical experiments illustrate that the present method is effective and competitive. Copyright © 2009 John Wiley & Sons, Ltd.     

Aspect ratio effects on three-dimensional incompressible flow in a two-sided non-facing lid-driven parallelepiped cavity

A numerical study of the three-dimensional fluid flow has been carried out to determine the effects of the transverse aspect ratio, Ay, on the flow structure in two-sided non-facing lid-driven cavities. The flow is complex, unstable and can undergo bifurcation. The numerical method is based on the finite volume method and multigrid acceleration. Computations have been investigated for several Reynolds numbers and various aspect ratio values. At a fixed Reynolds number, Re=500, the three-dimensional flow characteristics are analyzed considering four transverse aspect ratios, Ay=1,0.75,0.5 and 0.25. It is observed that the transition to the unsteady regime follows the classical scheme of a Hopf bifurcation. An analysis of the flow evolution shows that, at Ay=0.75, the flow bifurcates to a periodic regime at (Re=600) with a frequency f=0.093 less than the predicted value in the cubical cavity. A correlation is established when Ay=0.5 and gives the critical Reynolds number value. At Ay=0.25, the periodic regime occurs at high Re value beyond 3500, after which the flow becomes chaotic. It is shown that, when increasing Ay over the unit, the flow in the cavity exhibits a complex behavior. The kinetic energy transmission from the driven walls to the cavity center is reduced at low Ay values.     

The effects of boundary to the ECRH and the energy absorption of ICRH

Electron-cyclotron resonant heating (ECRH) of Tokamak plasma is examined. When plasma is heated by waves, we must consider the distribution of incident wave energy toO andX modes as the wave is incident from vacuum to the surface of plasma as well as the absorption efficiency ofO mode andX mode. Numerical calculation shows that for small incident angle, the incident energy transfers principally intoO mode when the electric fieldE ⁱ of incident wave is parallel to the incident plane, therefore it is efficient to heat the plasma byO mode. WhenE ⁱ is perpendicular to the incident plane, the energy transfers principally intoX mode and heating the plasma byX mode is efficient. Ion-cyclotron resonant heating (ICRH) is also considered, the formula of the energy of ion-cyclotron wave absorbed by plasma is obtained.     

Stochastic resonance in a bias monostable system driven by a periodic rectangular signal and uncorrelated noises

The stochastic resonance in a bias monostable system driven by a periodic rectangular signal and uncorrelated noises is investigated by using the theory of signal-to-noise (SNR) in the adiabatic limit. The analytic expression of the SNR is obtained for arbitrary signal amplitude without being restricted to small amplitudes. The SNR is a nonmonotonic function of intensities of multiplicative and additive noises and the noise intensity ratio R=D/Q, so stochastic resonance exhibits in the bias monostable system. We investigate the effect of any system parameter (such as D,Q,R,r) on the SNR. It is shown that the SNR is a nonmonotonic function of the static asymmetry r, also; the SNR is decreased when |r| is increased. Moreover, the SNR is increased when the noise intensity ratio R=D/Q is increased.     

Discontinuous Solutions of the Navier-Stokes Equations for Multidimensional Flows of Heat-Conducting Fluids

We prove the global existence of weak solutions of the Navier-Stokes equations for compressible, heat-conducting fluids in two and three space dimensions when the initial density is close to a constant in L ²∩L ^∞, the initial temperature is close to a constant in L ², and the initial velocity is small in H ^s ∩L ⁴, where s=0 when n=2 and when n=3. (The L ^p norms must be weighted slightly when n=2.) In particular, the initial data may be discontinuous across a hypersurface of ⁿ . A great deal of qualitative information about the solution is obtained. For example, we show that the velocity, vorticity, and temperature are relatively smooth in positive time, as is the “effective viscous flux”F, which is the divergence of the velocity minus a certain multiple of the pressure. We find that F plays a central role in the entire analysis, particularly in closing the required energy estimates and in understanding rates of regularization near the initial layer. Moreover, F is precisely the quantity through which the hyperbolicity of the corresponding equations for inviscid fluids shows itself, an effect which is crucial for obtaining time-independent pointwise bounds for the density. (Accepted June 13, 1996)     

Streamwise evolution of a high-Schmidt-number passive scalar in a turbulent plane wake

The concentration fluctuation c of diluted fluorescein dye, a high-Schmidt-number passive scalar (Sc=ν/D ≈ 2000, ν and D are the fluid momentum and dye diffusivities, respectively), is measured in the wake of a circular cylinder using a single-point laser-induced fluorescence (SPLIF) technique. The streamwise decay rate of the mean and rms values of c is slow in comparison to that of θ, the temperature fluctuation for which the molecular Prandtl number Pr=ν/κ is about 0.7 (κ is the thermal diffusivity). The comparison between mean and rms distributions of c and θ highlights the combined role the Reynolds and Schmidt numbers play in terms of dispersing the scalar. The streamwise evolution of the probability density functions (pdfs) of c and θ suggest that while p(θ) is approximately Gaussian in the intermediate wake (x/d ≈ 80), p(c) is strongly non-Gaussian, and depends on both x/d and Re. The skewness of c is larger than that of θ along the wake centreline. Arguably, the asymmetry of p(c) reflects the relatively strong organisation of the large-scale motion in the far-wake. Received: 27 July 2000/Accepted: 22 December 2000     

A Summary of New Predictive High Frequency Thermo-Vibrational Models in Porous Media

In this paper, we consider the effect of mechanical vibration on the onset of convection in porous media. The porous medium is saturated either by a pure fluid or by a binary mixture. The importance of a transport model on stability diagrams is presented and discussed. The stability threshold for the Darcy–Brinkman case in the Ra _Tc -R and k _c -R diagrams is presented (where Ra _Tc , k _c and R are the critical Rayleigh number, the critical wave number and the vibration parameters, respectively). It is shown that there is a significant deviation from the Darcy model. In the thermo-solutal case with the Soret effect, the influence of vibration on the reduction of multi-cellular convection is emphasized. A new analytical relation for obtaining the threshold of mono-cellular convection is derived. This relation shows how the separation factor Ψ is related to the controlling parameters of the problem, Ψ = f (R, ε*, Le), when the wave number k → 0. The importance of vibrational parameter definition is highlighted and it is shown how, by using a proper definition for vibrational parameter, we may obtain compact relationship. It is also shown how this result may be used to increase component separation.     

Non-Newtonian flow over a wedge with suction

A pseudo-similarity solution is obtained for the flow of an incompressible fluid of second grade past a wedge with suction at the surface. The non-linear differential equation is solved using quasi-linearization and orthonormalization. The numerical method developed for this purpose enables computation of the flow characteristics for any values of the parameters K, a and b, where K is the dimensionless normal stress modulus of the fluid, a is related to the wedge angle and b is the suction parameter. A significant effect of suction on the wall shear stress is observed. The present results match exactly those from an earlier perturbation analysis for Kx^2a ? 0·01 but differ significantly as Kx^2a increases.     

Grain boundary oxidation and fatigue crack growth at elevated temperatures

Fatigue crack growth rate at elevated temperatures can be accelerated by grain boundary oxidation. Grain boundary oxidation kinetics and statistical distribution of grain boundary oxide penetration depth were studied.At a constant ΔK-level and at a constant test temperature, fatigue crack growth rate, da/dN, is a function of cyclic frequency, ν. A fatigue crack growth model of intermittent micro-ruptures of grain boundary oxide is constructed. The model is consustent with the experimental observations that, in the low frequency region, da/dN is inversely proportional to ν, and fatigue crack growth is intergranular.     

Hyperbolic heat conduction in the semi-infinite body with a time-dependent laser heat source

 The Cattaneo hyperbolic and classical parabolic models of heat conduction in the laser irradiated materials are compared. Laser heating is modelled as an internal heat source, whose capacity is given by g(x,t)= I(t)(1−R)μexp(−μx). Analytical solution for the one-dimensional, semi-infinite body with the insulated boundary is obtained using Laplace transforms and the discussion of solutions for different time characteristics of the heat source capacity (constant, instantaneous, exponential, pulsed and periodic) is presented. Received on 18 May 1999     

Hydromagnetic free convection flow with induced magnetic field effects

An exact solution is presented for the hydromagnetic natural convection boundary layer flow past an infinite vertical flat plate under the influence of a transverse magnetic field with magnetic induction effects included. The transformed ordinary differential equations are solved exactly, under physically appropriate boundary conditions. Closed-form expressions are obtained for the non-dimensional velocity (u), non-dimensional induced magnetic field component (B _x ) and wall frictional shearing stress i.e. skin friction function (τ _x ) as functions of dimensionless transverse coordinate (η), Grashof free convection number (G _r ) and the Hartmann number (M). The bulk temperature in the boundary layer (Θ) is also evaluated and shown to be purely a function of M. The Rayleigh flow distribution (R) is derived and found to be a function of both Hartmann number (M) and the buoyant diffusivity parameter (ϑ ^*). The influence of Grashof number on velocity, induced magnetic field and wall shear stress profiles is computed. The response of Rayleigh flow distribution to Grashof numbers ranging from 2 to 200 is also discussed as is the influence of Hartmann number on the bulk temperature. Rayleigh flow is demonstrated to become stable with respect to the width of the boundary layer region and intensifies with greater magnetic field i.e. larger Hartman number M, for constant buoyant diffusivity parameter ϑ ^*. The induced magnetic field (B _x ), is elevated in the vicinity of the plate surface with a rise in free convection (buoyancy) parameter G _r , but is reduced over the central zone of the boundary layer regime. Applications of the study include laminar magneto-aerodynamics, materials processing and MHD propulsion thermo-fluid dynamics.     

Equivariant bifurcation theory and symmetry breaking

A study is made of the failure of the Maximal Isotropy Subgroup Conjecture for the Weyl group seriesW(D) _k . As part of the investigation, a general genericity and stability theorem is proved for bifurcation diagrams in equivariant bifurcation theory. As well, a concept of determinacy for equivariant bifurcation theory is introduced and it is shown that, for all compact Lie groupsG and absolutely irreducibleG-representationsV, G-equivariant bifurcation problems onV are finitely determined.     

New numerical method for partial differential equations. 1: Application to the diffusion equation

In this paper a new, highly accurate method called PH is presented for the numerical integration of partial differential equations. The method is applied for the solution of the one-dimensional diffusion equation. Upon integrating the equation within a subdomain of space and time using the prismoidal approximation, a three-point implicit scheme is obtained with a truncation error of order O(k⁴, h⁶), where k and h represent the time and space steps respectively. The method is stable under the condition s = αk/h² ? S(δ), where the function S(δ) increases as the parameter δ decreases from 1/12 to negative values. In practice the method behaves as unconditionally stable upon choosing an appropriate value for δ. A new formula is also adopted for the implementation of a Neumann boundary condition, introducing a truncation error of order O(h⁴). Numerical solutions are obtained incorporating Dirichlet and Neumann boundary conditions. The results prove that our method is far more accurate than any other-implicit or explicit method.     

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.     

Bifurcations of Heteroclinic Orbits

The search for traveling wave solutions of a semilinear diffusion partial differential equation can be reduced to the search for heteroclinic solutions of the ordinary differential equation ü − cu̇ + f(u) = 0, where c is a positive constant and f is a nonlinear function. A heteroclinic orbit is a solution u(t) such that u(t) → γ ₁ as t → −∞ and u(t) → γ ₂ as t → ∞ where γ ₁, γ ₂ are zeros of f. We study the existence of heteroclinic orbits under various assumptions on the nonlinear function f and their bifurcations as c is varied. Our arguments are geometric in nature and so we make only minimal smoothness assumptions. We only assume that f is continuous and that the equation has a unique solution to the initial value problem. Under these weaker smoothness conditions we reprove the classical result that for large c there is a unique positive heteroclinic orbit from 0 to 1 when f(0) = f(1) = 0 and f(u) > 0 for 0 < u < 1. When there are more zeros of f, there is the possibility of bifurcations of the heteroclinic orbit as c varies. We give a detailed analysis of the bifurcation of the heteroclinic orbits when f is zero at the five points −1 < −θ < 0 < θ < 1 and f is odd. The heteroclinic orbit that tends to 1 as t → ∞ starts at one of the three zeros, −θ, 0, θ as t → −∞. It hops back and forth among these three zeros an infinite number of times in a predictable sequence as c is varied.     

Energy release or absorption due to simultaneous rotation of two nano voids in plane elastic materials as influenced by both surface effect and interacting effect

In this paper, the L-integral analysis for two nano-sized voids in plane elasticity under uniaxial loading is present. Three surface parameters are considered including the surface tension and two surface Lamé constants. Attention is focused on the mutual influence on the L-integral from both the surface effect at voids’ rims and the interacting effect between voids. A close-form expression of L-integral for multiple nano voids is obtained. Comparing with those in macro fracture mechanics, the L-integral shows some different features when the surface effect is taken into account. It is concluded that under tensile loading and due to the mutual influence, the L-integral might be either positive or negative, depending on the loading level. The numerical results show that the surface tension is the dominant one in surface parameters on impacting the L-integral. It is also concluded that the surface effect shields the energy release (positive L-integral value) while enhances the energy absorption (negative L-integral value). The two-state L-integral analysis is performed to clarify the way that the surface effect impacts the L-integral. It is concluded that the contribution to L-integral from the voids’ configuration could either be negative or positive, while that from the surface effect is always negative. Besides, the size dependence in the present problem is studied in detail.     

N-space collision model for rotation-translation energy exchange in DSMC collisions

A new discrete simulation Monte Carlo (DSMC) collision model for molecules possessing an integer number of classical degrees of freedom for molecular structure energy is proposed. The total molecular energy (translation plus molecular structure) is represented by a velocity in five-dimensional space. Each collision is an elastic N-sphere collision in N-space, where N= 3, 4, or 5. For N=5, there is a maximum chance of exchange of energy between the two components of velocity, which represent the rotation energy and the three components that represent the translational velocity. For N=3, there is no change in the rotation energy of each molecule, and for N=4, there is an intermediate chance that rotation and translation energy will be exchanged. The exchange probability ϕ can be set to give the desired rotational relaxation rate. To achieve any realistic viscosity μ(T), the N-space model must be coupled with a modified collision procedure known as ν-DSMC. The new model is shown to match the results of molecular dynamics calculations for the internal structure of a Mach 7 shock, with a saving of about 20% in CPU time compared to standard DSMC using the standard Borgnakke-Larsen exchange model.     

Short- and long-time behavior of eddy-viscosity models

The short-time behavior of the turbulent viscosity is inferred from the immediate response of the Reynolds stress deduced by Crow [1] for the problem of isotropic turbulence subjected to a mean strain at time t=0. The turbulent viscosity v is defined for t 0 by the relation T _ij =–2vS _ij , where T _ij is the Reynolds stress and S _ij is the mean rate of strain. It follows that the viscosity is v=O(t) for t 0. Matching the short- and long-time behaviors, we propose an analytic expression for the effective viscosity valid for all time. Using the proposed viscosity, the K – E model for homogeneously sheared turbulence is reformulate to be valid in both the short- and long-time limits. Previously, the K – E model has been used with the long-time form of the effective viscosity for all time. Comparison of theoretical predictions with the results of physical and numerical experiments is presented. Implications of the short-time response for large-eddy simulations and spectral-space closure theories are discussed.Support for this work was provided by contract AFOSR-90-0124.     

Extended Lagrangian formalism and the corresponding energy relations

In this paper an extended Lagrangian formalism for the rheonomic systems with the nonstationary constraints is formulated, with the aim to examine more completely the energy relations for such systems in any generalized coordinates, which in this case always refer to some moving frame of reference. Introducing new quantities, which change according to the law τ_a=φ_a(t), it is demonstrated that these quantities determine the position of this moving reference frame with respect to an immobile one. In the transition to the generalized coordinates qⁱ they are taken as the additional generalized coordinates q^a=τ_a, whose dependence on time is given a priori. In this way the position of the considered mechanical system relative to this immobile frame of reference is determined completely.Based on this and using the corresponding d'Alembert–Lagrange's principle, an extended system of the Lagrangian equations is obtained. It is demonstrated that they give the same equations of motion qⁱ=qⁱ(t) as in the usual Lagrangian formulation, but substantially different energy relations. Namely, in this formulation two different types of the energy change law dE/dt and the corresponding conservation laws are obtained, which are more general than in the usual formulation. So, under certain conditions the energy conservation law has the form E=T+U+P=const, where the last term, so-called rheonomic potential expresses the influence of the nonstationary constraints.Afterwards, a detailed analysis of the obtained results and their connection with the usual formulation of mechanics are given. It is demonstrated that so formulated energy relations are in full accordance with the corresponding ones in the usual vector formulation, when they are expressed in terms of the rheonomic potential. Finally, the obtained results are illustrated by several simple, but characteristic examples.     

A photoelastic study of friction at multipoint contacts

A new method of measuring the normal and sliding loads associated with multiple-point contact is introduced. A multiple-point contact is modeled with a steel die with a profile that simulates a rough surface. A very large scale factor is used in modeling this surface. The steel die is placed in contact with a photoelastic model of a half plane and is subjected to a normal load. This normal load is partitioned over the multiple points of contact producing an isochromatic fringe pattern that describes the stress distribution in the local neighborhood of the contact points. A sliding load is then imposed on the model which destroys the symmetry of this fringe pattern. The fringe data in this pattern are sufficient to determine the local loadsP _i andQ _i and the local coefficient of frictionf _i =Q _i /P _i at each contact point. An overdeterministic method is introduced which gives the solution forP _i ,Q _i andf _i using many data points taken from the isochromatic pattern in the local neighborhood of the contacts.Paper was presented at the 1991 SEM Spring Conference on Experimental Mechanics held in Milwaukee, WJ on June 9–13.