Singularity structure and chaotic behavior of the homopolar disk dynamo

We study in great detail a system of three first-order ordinary differential equations describing a homopolar disk dynamo (HDD). This system displays a large variety of behaviors, both regular and chaotic. Existence of periodic solutions is proved for certain ranges of parameters. Stability criteria for periodic solutions are given. The nonintegrability aspects of the HDD system are studied by investigating analytically the singularity structure of the system in the complex domain. Coexisting attractors (including period-doubling sequence) and coexisting strange attractors appear in some parametric regimes. The gluing of strange attractors and the ungluing of a strange attractor are also shown to occur. A period of bifurcation leading to chaos, not observed for other chaotic systems, is shown to characterize the chaotic behavior in some parametric ranges. The limiting case of the Lorenz system is also studied and is related to HDD.

Unsteady laminar compressible swirling flow with massive blowing

THE study of swirling boundary layers is of considerable importance in many rotodynamic machines such as rockets, jet engines, swirl generators, swirl atomizers, arc heaters, etc. For example, the introduction of swirl in a flow acceleration device such as a nozzle in a rocket engine promises efficient mass flow control. In nuclear rockets, swirl is used to retain the uranium atoms in the rocket chamber. With these applications in mind, Back1 and Muthanna and Nath2 have obtained the similarity solutions for a low-speed three-dimensional steady laminar compressible boundary layer with swirl inside an axisymmetric surface of variable cross section. The aim of the present analysis is to study the effect of massive blowing rates on the unsteady laminar swirling compressible boundary-layer flow of an axisymmetric body of arbitrary cross section when the freestream velocity and blowing rate vary with time. The type of swirl considered here is that of a free vortex superimposed on the longitudinal flow of a compressible fluid with variable properties. The analysis is applicable to external flow over a body as well as internal flow along a surface. For the case of external flow, strong blowing can have significant use in cooling the surface of hypervelocity vehicles, particularly when ablation occurs under large aerodynamic or radiative heating, but there may not be such an important application of strong blowing in the case of internal flow. The governing partial differential equations have been solved numerically using an implicit finite difference scheme with a quasilinearization technique.3 High temperature gas effects, such as radiation, dissociation, and ionization, etc., are not investigated. The nomenclature is usually that of Ref. 4 and is listed in the full paper.

Critical buckling temperature of single-walled carbon nanotubes embedded in a one-parameter elastic medium based on nonlocal continuum mechanics

In this paper, the critical budding temperature of single-walled carbon nanotubes (SWCNTs), which are embedded in one-parameter elastic medium (Winkler foundation) is estimated under the umbrella of continuum mechanics theory. Nonlocal continuum theory is incorporated into Timoshenko beam model and the governing differential equations of motion are derived. An explicit expression for the non-dimensional critical buckling temperature is also derived in this work. The effect of the nonlocal small scale coefficient, the Winkler foundation parameter and the ratio of the length to the diameter on the critical buckling temperature is investigated in detail. It can be observed that the effects of nonlocal small scale parameter and the Winkler foundation parameter are significant and should be considered for thermal analysis of SWCNTs. The results presented in this paper can provide useful guidance for the study and design of the next generation of nanodevices that make use of the thermal buckling properties of embedded single-walled carbon nanotubes. (C) 2011 Elsevier B.V. All rights reserved.

Unsteady flow and heat transfer on a semi-infinite flat plate with an aligned magnetic field

The unsteady laminar boundary layer flow of an electrically conducting fluid past a semi-infinite flat plate with an aligned magnetic field has been studied when at time t > 0 the plate is impulsively moved with a constant velocity which is in the same or opposite direction to that of free stream velocity. The effect of the induced magnetic field has been included in the analysis. The non-linear partial differential equations have been solved numerically using an implicit finite-difference method. The effect of the impulsive motion of the surface is found to be more pronounced on the skin friction but its effect on the x-component of the induced magnetic field and heat transfer is small. Velocity defect occurs near the surface when the plate is impulsively moved in the same direction as that of the free stream velocity. The surface shear stress, x-component of the induced magnetic field on the surface and the surface heat transfer decrease with an increasing magnetic field, but they increase with the reciprocal of the magnetic Prandtl number. However, the effect of the reciprocal of the magnetic Prandtl number is more pronounced on the x-component of the induced magnetic field. (C) 1999 Elsevier Science Ltd. All rights reserved.

Velocity distribution for a dilute vibrated granular material

The velocity distribution for a vibrated granular material is determined in the dilute limit where the frequency of particle collisions with the vibrating surface is large compared to the frequency of binary collisions. The particle motion is driven by the source of energy due to particle collisions with the vibrating surface, and two dissipation mechanisms-inelastic collisions and air drag-are considered. In the latter case, a general form for the drag force is assumed. First, the distribution function for the vertical velocity for a single particle colliding with a vibrating surface is determined in the limit where the dissipation during a collision due to inelasticity or between successive collisions due to drag is small compared to the energy of a particle. In addition, two types of amplitude functions for the velocity of the surface, symmetric and asymmetric about zero velocity, are considered. In all cases, differential equations for the distribution of velocities at the vibrating surface are obtained using a flux balance condition in velocity space, and these are solved to determine the distribution function. It is found that the distribution function is a Gaussian distribution when the dissipation is due to inelastic collisions and the amplitude function is symmetric, and the mean square velocity scales as [[U-2](s)/(1 - e(2))], where [U-2](s) is the mean square velocity of the vibrating surface and e is the coefficient of restitution. The distribution function is very different from a Gaussian when the dissipation is due to air drag and the amplitude function is symmetric, and the mean square velocity scales as ([U-2](s)g/mu(m))(1/(m+2)) when the acceleration due to the fluid drag is -mu(m)u(y)\u(y)\(m-1), where g is the acceleration due to gravity. For an asymmetric amplitude function, the distribution function at the vibrating surface is found to be sharply peaked around [+/-2[U](s)/(1-e)] when the dissipation is due to inelastic collisions, and around +/-[(m +2)[U](s)g/mu(m)](1/(m+1)) when the dissipation is due to fluid drag, where [U](s) is the mean velocity of the surface. The distribution functions are compared with numerical simulations of a particle colliding with a vibrating surface, and excellent agreement is found with no adjustable parameters. The distribution function for a two-dimensional vibrated granular material that includes the first effect of binary collisions is determined for the system with dissipation due to inelastic collisions and the amplitude function for the velocity of the vibrating surface is symmetric in the limit delta(I)=(2nr)/(1 - e)much less than 1. Here, n is the number of particles per unit width and r is the particle radius. In this Limit, an asymptotic analysis is used about the Limit where there are no binary collisions. It is found that the distribution function has a power-law divergence proportional to \u(x)\((c delta l-1)) in the limit u(x)-->0, where u(x) is the horizontal velocity. The constant c and the moments of the distribution function are evaluated from the conservation equation in velocity space. It is found that the mean square velocity in the horizontal direction scales as O(delta(I)T), and the nontrivial third moments of the velocity distribution scale as O(delta(I)epsilon(I)T(3/2)) where epsilon(I) = (1 - e)(1/2). Here, T = [2[U2](s)/(1 - e)] is the mean square velocity of the particles.

A study on determining the theoretical dispersion curve for Rayleigh wave propagation

A method has been presented to establish the theoretical dispersion curve for performing the inverse analysis for the Rayleigh wave propagation. The proposed formulation is similar to the one available in literature, and is based on the finite difference formulation of the governing partial differential equations of motion. The method is framed in such a way that it ultimately leads to an Eigen value problem for which the solution can be obtained quite easily with respect to unknown frequency. The maximum absolute value of the vertical displacement at the ground surface is formed as the basis for deciding the governing mode of propagation. With the proposed technique, the numerical solutions were generated for a variety of problems, comprising of a number of different layers, associated with both ground and pavements. The results are found to be generally satisfactory. (C) 2011 Elsevier Ltd. All rights reserved.

MHD flow over a moving plate in a rotating fluid with magnetic field, Hall currents and free stream velocity

The non-similar boundary layer flow of a viscous incompressible electrically conducting fluid over a moving surface in a rotating fluid, in the presence of a magnetic field, Hall currents and the free stream velocity has been studied. The parabolic partial differential equations governing the flow are solved numerically using an implicit finite-difference scheme. The Coriolis force induces overshoot in the velocity profile of the primary flow and the magnetic field reduces/removes the velocity overshoot. The local skin friction coefficient for the primary flow increases with the magnetic field, but the skin friction coefficient for the secondary flow reduces it. Also the local skin friction coefficients for the primary and secondary flows are reduced due to the Hall currents. The effects of the magnetic field, Hall currents and the wall velocity, on the skin friction coefficients for the primary and secondary flows increase with the Coriolis force. The wall velocity strongly affects the flow field. When the wall velocity is equal to the free stream velocity, the skin friction coefficients for the primary and secondary flows vanish, but this does not imply separation. (C) 2002 Published by Elsevier Science Ltd.

A new beam finite element for the analysis of functionally graded materials

A new beam element is developed to study the thermoelastic behavior of functionally graded beam structures. The element is based on the first-order shear deformation theory and it accounts for varying elastic and thermal properties along its thickness. The exact solution of static part of the governing differential equations is used to construct interpolating polynomials for the element formulation. Consequently, the stiffness matrix has super-convergent property and the element is free of shear locking. Both exponential and power-law variations of material property distribution are used to examine different stress variations. Static, free vibration and wave propagation problems are considered to highlight the behavioral difference of functionally graded material beam with pure metal or pure ceramic beams. (C) 2003 Elsevier Science Ltd. All rights reserved.

Effect of longitudinal surface curvature on boundary layers

We consider here the higher order effect of moderate longitudinal surface curvature on steady, two-dimensional, incompressible laminar boundary layers. The basic partial differential equations for the problem, derived by the method of matched asymptotic expansions, are found to possess similarity solutions for a family of surface curvatures and pressure gradients. The similarity equations obtained by this anaylsis have been solved numerically on a computer, and show a definite decrease in skin friction when the surface has convex curvature in all cases including zero pressure gradient. Typical velocity profiles and some relevant boundary-layer characteristics are tabulated, and a critical comparison with previous work is given.

Unsteady free convection flow over an infinite vertical porous plate due to the combined effects of thermal and mass diffusion, magnetic field and Hall currents

The unsteady free convection flow over an infinite vertical porous plate, which moves with time-dependent velocity in an ambient fluid, has been studied. The effects of the magnetic field and Hall current are included in the analysis. The buoyancy forces arise due to both the thermal and mass diffusion. The partial differential equations governing the flow have been solved numerically using both the implicit finite difference scheme and the difference-differential method. For the steady case, analytical solutions have also been obtained. The effect of time variation on the skin friction, heat transfer and mass transfer is very significant. Suction increases the skin friction coefficient in the primary flow, and also the Nusselt and Sherwood numbers, but the skin friction coefficient in the secondary flow is reduced. The effect of injection is opposite to that of suction. The buoyancy force, injection and the Hall parameter induce an overshoot in the velocity profiles in the primary flow which changes the velocity gradient from a negative to a positive value, but the magnetic field and suction reduce this velocity overshoot.

Right-Definite Multiparameter Sturm–Liouville Problems With Eigenparameter-Dependent Boundary Conditions

We study a system of ordinary differential equations linked by parameters and subject to boundary conditions depending on parameters. We assume certain definiteness conditions on the coefficient functions and on the boundary conditions that yield, in the corresponding abstract setting, a right-definite case. We give results on location of the eigenvalues and oscillation of the eigenfunctions.

Doubly spectral stochastic finite element method (DSSFEM) for random field problems

Uncertainties in complex dynamic systems play an important role in the prediction of a dynamic response in the mid- and high-frequency ranges. For distributed parameter systems, parametric uncertainties can be represented by random fields leading to stochastic partial differential equations. Over the past two decades, the spectral stochastic finite-element method has been developed to discretize the random fields and solve such problems. On the other hand, for deterministic distributed parameter linear dynamic systems, the spectral finite-element method has been developed to efficiently solve the problem in the frequency domain. In spite of the fact that both approaches use spectral decomposition (one for the random fields and the other for the dynamic displacement fields), very little overlap between them has been reported in literature. In this paper, these two spectral techniques are unified with the aim that the unified approach would outperform any of the spectral methods considered on their own. An exponential autocorrelation function for the random fields, a frequency-dependent stochastic element stiffness, and mass matrices are derived for the axial and bending vibration of rods. Closed-form exact expressions are derived by using the Karhunen-Loève expansion. Numerical examples are given to illustrate the unified spectral approach.

Tracking analysis of an LMS decision feedback equalizer for a wireless channel

We consider a time varying wireless fading channel, equalized by an LMS Decision Feedback equalizer (DFE). We study how well this equalizer tracks the optimal MMSEDFE (Wiener) equalizer. We model the channel by an Autoregressive (AR) process. Then the LMS equalizer and the AR process are jointly approximated by the solution of a system of ODEs (ordinary differential equations). Using these ODEs, we show via some examples that the LMS equalizer moves close to the instantaneous Wiener filter after initial transience. We also compare the LMS equalizer with the instantaneous optimal DFE (the commonly used Wiener filter) designed assuming perfect previous decisions and computed using perfect channel estimate (we will call it as IDFE). We show that the LMS equalizer outperforms the IDFE almost all the time after initial transience.

Filtered density function for large eddy simulation of turbulent reacting flows

A methodology termed the “filtered density function” (FDF) is developed and implemented for large eddy simulation (LES) of chemically reacting turbulent flows. In this methodology, the effects of the unresolved scalar fluctuations are taken into account by considering the probability density function (PDF) of subgrid scale (SGS) scalar quantities. A transport equation is derived for the FDF in which the effect of chemical reactions appears in a closed form. The influences of scalar mixing and convection within the subgrid are modeled. The FDF transport equation is solved numerically via a Lagrangian Monte Carlo scheme in which the solutions of the equivalent stochastic differential equations (SDEs) are obtained. These solutions preserve the Itô-Gikhman nature of the SDEs. The consistency of the FDF approach, the convergence of its Monte Carlo solution and the performance of the closures employed in the FDF transport equation are assessed by comparisons with results obtained by direct numerical simulation (DNS) and by conventional LES procedures in which the first two SGS scalar moments are obtained by a finite difference method (LES-FD). These comparative assessments are conducted by implementations of all three schemes (FDF, DNS and LES-FD) in a temporally developing mixing layer and a spatially developing planar jet under both non-reacting and reacting conditions. In non-reacting flows, the Monte Carlo solution of the FDF yields results similar to those via LES-FD. The advantage of the FDF is demonstrated by its use in reacting flows. In the absence of a closure for the SGS scalar fluctuations, the LES-FD results are significantly different from those based on DNS. The FDF results show a much closer agreement with filtered DNS results. © 1998 American Institute of Physics.

Tracking performance of an LMS-Linear Equalizer for fading channels

We consider a time varying wireless fading channel, equalized by an LMS linear equalizer. We study how well this equalizer tracks the optimal Wiener equalizer. We model the channel by an Auto-regressive (AR) process. Then the LMS equalizer and the AR process are jointly approximated by the solution of a system of ODEs (ordinary differential equations). Using these ODEs, the error between the LMS equalizer and the instantaneous Wiener filter is shown to decay exponentially/polynomially to zero unless the channel is marginally stable in which case the convergence may not hold.Using the same ODEs, we also show that the corresponding Mean Square Error (MSE) converges towards minimum MSE(MMSE) at the same rate for a stable channel. We further show that the difference between the MSE and the MMSE does not explode with time even when the channel is unstable. Finally we obtain an optimum step size for the linear equalizer in terms of the AR parameters, whenever the error decay is exponential.