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.

Unsteady mixed convection flow from a rotating vertical cone with a magnetic field

An analysis is developed to study the unsteady mixed convection flow over a vertical cone rotating in an ambient fluid with a time-dependent angular velocity in the presence of a magnetic field. The coupled nonlinear partial differential equations governing the flow have been solved numerically using an implicit finite-difference scheme. The local skin friction coefficients in the tangential and azimuthal directions and the local Nusselt number increase with the time when the angular velocity of the-cone increases, but the reverse trend is observed for decreasing angular velocity. However, these are not mirror reflection of each other. The magnetic field reduces the skin friction coefficient in the tangential direction and also the Nusselt number, but it increases the skin friction coefficient in the azimuthal direction. The skin friction coefficients and the Nusselt number increase with the buoyancy force.

Unsteady MUD rotating flow over a rotating sphere near the equator

The unsteady rotating flow of a laminar incompressible viscous electrically conducting fluid over a rotating sphere in the vicinity of the equator has been studied. The fluid and the body rotate either in the same direction or in opposite directions. The effects of surface suction and magnetic field have been included in the analysis. There is an initial steady state that is perturbed by a sudden change in the rotational velocity of the sphere, and this causes unsteadiness in the flow field. The nonlinear coupled parabolic partial differential equations governing the boundary-layer flow have been solved numerically by using an implicit finite-difference scheme. For large suction or magnetic field, analytical solutions have also been obtained. The magnitude of the radial, meridional and rotational velocity components is found to be higher when the fluid and the body rotate in opposite directions than when they rotate in the same direction. The surface shear stresses in the meridional and rotational directions change sign when the ratio of the angular velocities of the sphere and the fluid lambda greater than or equal to lambda(0). The final (new) steady state is reached rather quickly which implies that the spin-up time is small. The magnetic field and surface suction reduce the meridional shear stress, but increase the surface shear stress in the rotational direction.

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.

Two-Stage Extended Kalman Filters with Derivative-Free Local Linearizations

This paper proposes a derivative-free two-stage extended Kalman filter (2-EKF) especially suited for state and parameter identification of mechanical oscillators under Gaussian white noise. Two sources of modeling uncertainties are considered: (1) errors in linearization, and (2) an inadequate system model. The state vector is presently composed of the original dynamical/parameter states plus the so-called bias states accounting for the unmodeled dynamics. An extended Kalman estimation concept is applied within a framework predicated on explicit and derivative-free local linearizations (DLL) of nonlinear drift terms in the governing stochastic differential equations (SDEs). The original and bias states are estimated by two separate filters; the bias filter improves the estimates of the original states. Measurements are artificially generated by corrupting the numerical solutions of the SDEs with noise through an implicit form of a higher-order linearization. Numerical illustrations are provided for a few single- and multidegree-of-freedom nonlinear oscillators, demonstrating the remarkable promise that 2-EKF holds over its more conventional EKF-based counterparts. DOI: 10.1061/(ASCE)EM.1943-7889.0000255. (C) 2011 American Society of Civil Engineers.

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.

Simulation of RNA silencing pathway for time-dependent transgene transcription rate

The synthesis of dsRNA is analyzed using a pathway model with amplifications caused by the aberrant RNAs. The transgene influx rate is assumed time-decaying considering the fact that the number of transgenes can not be infinite. The dynamics of the transgene induced RNA silencing is investigated using a system of coupled nonautonomous ordinary nonlinear differential equations which describe the model phenomenologically. The silencing phenomena are detected after a period of transcription. Important contributions of certain parameters are discussed with several numerical examples.

Exact controllability of generalized Hammerstein type equation

In this article, we study the exact controllability of an abstract model described by the controlled generalized Hammerstein type integral equation $$ x(t) = int_0^t h(t,s)u(s)ds+ int_0^t k(t,s,x)f(s,x(s))ds, quad 0 leq t leq T less than infty, $$ where, the state $x(t)$ lies in a Hilbert space $H$ and control $u(t)$ lies another Hilbert space $V$ for each time $t in I=[0,T]$, $T$ greater than 0. We establish the controllability result under suitable assumptions on $h, k$ and $f$ using the monotone operator theory.

Noise reduction with perforated three-duct muffler components

This paper describes the authors’ distributed parameter approach for derivation of closed-form expressions for the four-pole parameters of the perforated three-duct muffler components. In this method, three simultaneous second-order partial differential equations are first reduced to a set of six first-order ordinary differential equations. These equations are then uncoupled by means of a modal matrix. The resulting 6 × 6 matrix is reduced to the 2 × 2 transfer matrix using the relevant boundary conditions. This is combined with transfer matrices of other elements (upstream and downstream of this perforated element) to predict muffler performance like noise reduction, which is also measured. The correlation between experimental and theoretical values of noise reduction is shown to be satisfactory.