A mathematical analysis of nonlinear waves in a fluid filled visco-elastic tube

Our investigations in this paper are centred around the mathematical analysis of a ldquomodal waverdquo problem. We have considered the axisymmetric flow of an inviscid liquid in a thinwalled viscoelastic tube under certain simplifying assumptions. We have first derived the propagation space equations in the long wave limit and also given a general procedure to derive these equations for arbitrary wave length, when the flow is irrotational. We have used the method of operators of multiple scales to derive the nonlinear Schrödinger equation governing the modulation of periodic waves and we have elaborated on the ldquolong modulated wavesrdquo and the ldquomodulated long wavesrdquo. We have also examined the existence and stability of Stokes waves in this system. This is followed by a discussion of the progressive wave solutions of the long wave equations. One of the most important results of our paper is that the propagation space equations are no longer partial differential equations but they are in terms of pseudo-differential operators.Die vorliegenden Untersuchungen beziehen sich auf die mathematische Behandlung des ldquorModalwellenrdquo-Problems. Die achsensymmetrische Strömung einer nichtviskosen Flüssigkeit in einem dünnwandigen viskoelastischen Rohr, unter bestimmten vereinfachenden Annahmen, wird betrachtet. Zuerst werden die Gleichungen des Ausbreitungsraumes im Langwellenbereich abgeleitet und eine allgemeine Methode zur Herleitung dieser Gleichungen für beliebige Wellenlängen bei nichtrotierender Strömung angegeben. Eine Operatorenmethode mit multiplem Maßstab wird verwendet zur Herleitung der nichtlinearen Schrödinger-Gleichung für die Modulation der periodischen Wellen, und die ldquorlangmodulierten Wellenrdquo sowie die ldquormodulierten Langwellenrdquo werden aufgezeigt. Weiters wird die Existenz und die Stabilität der Stokes-Wellen im System untersucht. Anschließend werden die progressiven Wellenlösungen der Langwellengleichungen diskutiert. Eines der wichtigsten Ergebnisse dieser Arbeit ist, daß die Gleichungen des Ausbreitungsraumes keine partiellen Differentialgleichungen mehr sind, sondern Ausdrücke von Pseudo-Differentialoperatoren.

Performance comparison of batch and continuous flow surface aeration systems

The oxygen transfer rate and the corresponding power requirement to operate the rotor are vital for design and scale-up of surface aerators. The aeration process can be analyzed in two ways such as batch and continuous systems. The process behaviors of batch and continuous flow systems are different from each other. The experimental and numerical results obtained through the batch systems cannot be relied on and applied for the designing of the continuous aeration tank. Based on the experimentation on batch and continuous type systems, the present work compares the performance of both the batch and continuous surface aeration systems in terms of their oxygen transfer capacity and power consumption. A simulation equation developed through experimentation has shown that continuous flow surface aeration systems are taking more energy than the batch systems. It has been found that batch systems are economical and better for the field application but not feasible where large quantity of wastewater is produced.

Spectral element approach to wave propagation in uncertain beam structures

This paper presents a study on the uncertainty in material parameters of wave propagation responses in metallic beam structures. Special effort is made to quantify the effect of uncertainty in the wave propagation responses at high frequencies. Both the modulus of elasticity and the density are considered uncertain. The analysis is performed using a Monte Carlo simulation (MCS) under the spectral finite element method (SEM). The randomness in the material properties is characterized by three different distributions, the normal, Weibull and extreme value distributions. Their effect on wave propagation in beams is investigated. The numerical study shows that the CPU time taken for MCS under SEM is about 48 times less than for MCS under a conventional one-dimensional finite element environment for 50 kHz loading. The numerical results presented investigate effects of material uncertainties on high frequency modes. A study is performed on the usage of different beam theories and their uncertain responses due to dynamic impulse load. These studies show that even for a small coefficient of variation, significant changes in the above parameters are noticed. A number of interesting results are presented, showing the true effects of uncertainty response due to dynamic impulse load.

Alfvén waves in inhomogeneous magnetic fields: An integrodifferential equation formulation

An integrodifferential formulation for the equation governing the Alfvén waves in inhomogeneous magnetic fields is shown to be similar to the polyvibrating equation of Mangeron. Exploiting this similarity, a time‐dependent solution for smooth initial conditions is constructed. The important feature of this solution is that it separates the parts giving the Alfvén wave oscillations of each layer of plasma and the interaction of these oscillations representing the phase mixing.

Erosion and cavity characteristics in rotating components

An investigation of the initiation and growth of erosion and of the effect of velocity and pressure on erosion in a rotating disk is presented. Also, the role of an intervening noncavitating period on erosion is studied. The results indicate that at high intensities the peak rate of erosion decreases with increases in pressure. The erosion rate/time curves obtained for metallic materials are explained by the eroded particle distribution and the cavity size. The average size of the eroded particles decreased when pressure and tensile strength of the material were increased. The erosion rate peaked after an intervening noncavitating period. The use of the rate of erosion, defined as an average over the entire test duration, in the equation governing the theory of erosion resulted in reasonably good correlations. The correlations reveal that it is possible to predict the length, width, and area of a cavity when the cavitation parameter σ is known. The normalized width of a cavity may be estimated if its normalized length is known.

Fatigue crack propagation model for plain concrete - An analogy with population growth

In this work, an analytical model is proposed for fatigue crack propagation in plain concrete based on population growth exponential law and in conjunction with principles of dimensional analysis and self-similarity. This model takes into account parameters such as loading history, fracture toughness, crack length, loading ratio and structural size. The predicted results are compared with experimental crack growth data for constant and variable amplitude loading and are found to capture the size effect apart from showing a good agreement. Using this model, a sensitivity analysis is carried out to study the effect of various parameters that influence fatigue failure. (C) 2010 Elsevier Ltd. All rights reserved.

A new approach to the kinetics of size reduction in ball milling

The rate of breakage of feed in ball milling is usually represented in the form of a first-order rate equation. The equation was developed by treating a simple batch test mill as a well mixed reactor. Several case of deviation from the rule have been reported in the literature. This is attributed to the fact that accumulated fines interfere with the feed material and breaking events are masked by these fines. In the present paper, a new rate equation is proposed which takes into account the retarding effect of fines during milling. For this purpose the analogy of diffusion of ions through permeable membranes is adopted, with suitable modifications. The validity of the model is cross checked with the data obtained in batch grinding of ?850/+600 ?m size quartz. The proposed equation enables calculation of the rate of breakage of the feed at any instant of time.

Dynamic growth of tensile cracks by ductile and brittle fracture mechanisms in a viscoplastic material

In this paper, a finite element analysis of steady-state dynamic crack growth under Mode I, plane strain, small-scale yielding conditions is performed in a rate dependent plastic material characterized by the over-stress model. The main objective of the paper is to obtain theoretically the dependence of dynamic fracture toughness on crack speed. Crack propagation due to a ductile (micro-void) mechanism or a brittle (cleavage) mechanism, as well as transition from one mode to another are considered. The conversion from ductile to brittle has been observed experimentally but has received very little attention using analytical methods. Local fracture criteria based on strains and stresses are used to describe ductile and brittle fracture mechanisms. The results obtained in this paper are in general agreement with micro-structural observations of mode conversion during fracture initiation. Finally, the particular roles played by material rate sensitivity and inertia are examined in some detail.

Relaxation system based sub-grid scale modelling for large eddy simulation of Burgers' equation

An implicit sub-grid scale model for large eddy simulation is presented by utilising the concept of a relaxation system for one dimensional Burgers' equation in a novel way. The Burgers' equation is solved for three different unsteady flow situations by varying the ratio of relaxation parameter (epsilon) to time step. The coarse mesh results obtained with a relaxation scheme are compared with the filtered DNS solution of the same problem on a fine mesh using a fourth-order CWENO discretisation in space and third-order TVD Runge-Kutta discretisation in time. The numerical solutions obtained through the relaxation system have the same order of accuracy in space and time and they closely match with the filtered DNS solutions.

A thermodynamic protocol for explaining subcriticality and other subtle features in self-deflagrating solids

A novel universal approach to understand the self-deflagration in solids has been attempted by using basic thermodynamic equation of partial differentiation, where burning mte depends on the initial temperature and pressure of the system. Self-deflagrating solids are rare and are reported only in few compounds like ammonium perchlorate (AP), polystyrene peroxide and tetrazole. This approach has led us to understand the unique characteristics of AP, viz. the existence of low pressure deflagration limit (LPL 20 atm), hitherto not understood sufficiently. This analysis infers that the overall surface activation energy comprises of two components governed by the condensed phase and gas phase processes. The most attractive feature of the model is the identification of a new subcritical regime I' below LPL where AP does not burn. The model is aptly supported by the thermochemical computations and temperature-profile analyses of the combustion train. The thermodynamic model is further corroborated from the kinetic analysis of the high pressure (1-30 atm) DTA thermograms which affords distinct empirical decomposition rate laws in regimes I' and 1 (20-60 atm). Using Fourier-Kirchoff one dimensional heat transfer differential equation, the phase transition thickness and the melt-layer thickness have been computed which conform to the experimental data.

Continuity of plastic strain rate in stress relaxation tests

Stress relaxation testing is often utilised for determining whether athermal straining contributes to plastic flow; if plastic strain rate is continuous across the transition from tension to relaxation then plastic strain is fully thermally activated. This method was applied to an aged type 316 stainless steel tested in the temperature range 973–1123 K and to a high purity Al in the recrystallised annealed condition tested in the temperature range 274–417 K. The results indicated that plastic strain is thermally activated in these materials at these corresponding test temperatures. For Al, because of its high strain rate sensitivity, it was necessary to adopt a back extrapolation procedure to correct for the finite period that the crosshead requires to decelerate from the constant speed during tension to a dead stop for stress relaxation.

Effect of tolerance on the propagation characteristic of a superconducting thin film microstrip delay line

The propagation constant of a superconducting microstrip transmission delay line is evaluated using the spectral domain immitance approach, modelling the superconductor as a surface current having an equivalent surface impedance found through the complex resistive boundary condition. The sensitivity approach is used to study the beta variations with substrate parameters and film characteristics. Results show that the surface impedance does not have much influence on beta sensitivities with respect to epsilon r, W and h. However, it can be observed that the surface impedance plays a crucial role in determining the optimum design.

Exact analysis of Burgers equation on semiline with flux condition at the origin

The initial boundary value problem for the Burgers equation in the domain x greater-or-equal, slanted 0, t > 0 with flux boundary condition at x = 0 has been solved exactly. The behaviour of the solution as t tends to infinity is studied and the “asymptotic profile at infinity” is obtained. In addition, the uniqueness of the solution of the initial boundary value problem is proved and its inviscid limit as var epsilon → 0 is obtained.