Linear Perturbation Method Research Articles

Stability of a semi-infinite strip inthermoelastic contact with a rigid wall

When heat is conducted across an interface between two dissimilar materials, thermoelasticdistortion affects the contact pressure distribution. The existence of a pressure-sensitive thermal contact resistance at the interface can cause such systems to be unstable in the steady state. The stability of a semi-infinite strip in contact with a rigid wall is investigated using linear perturbation methods. The strip is assumed to be in uniform contact with the wall in the steady-state, with one-dimensional heat conduction along the strip and across the interface, where there is a pressure-dependent thermal contact resistance. Possible perturbations are expressed in the form of an eigenfunction series, using the Papkovich-Fadle eigenfunctions for the strip and related eigenfunctions for the thermoelastic particular solution. Selection of perturbations that can grow exponentially in time lead to an eigenvalue problem for the coefficients of the series. Results show that stability is governed by a symmetric perturbation with approximately sinusoidal form across the width of the strip. The stability boundary is quite well approximated by a simplified analysis assuming an exactly sinusoidal perturbation, particularly when the dimensionless thermal contact resistance in the steady state is small.

Finite Element Analysis of Thermoelastic Contact Stability

When heat is conducted across an interface between two dissimilar materials, theimoelastic distortion affects the contact pressure distribution. The existence of a pressure-sensitive thermal contact resistance at the interface can cause such systems to be unstable in the steady-state. Stability analysis for thermoelastic contact has been conducted by linear perturbation methods for one-dimensional and simple two-dimensional geometries, but analytical solutions become very complicated for finite geometries. A method is therefore proposed in which the finite element method is used to reduce the stability problem to an eigenvalue problem. The linearity of the underlying perturbation problem enables us to conclude that solutions can be obtained in separated-variable form with exponential variation in time. This factor can therefore be removed from the governing equations and the finite element method is used to obtain a time-independent set of homogeneous equations in which the exponential growth rate appears as a linear parameter. We therefore obtain a linear eigenvalue problem and stability of the system requires that all the resulting eigenvalues should have negative real part. The method is discussed in application to the simple one-dimensional system of two contacting rods. The results show good agreement with previous analytical investigations and give additional information about the migration of eigenvalues in the complex plane as the steady-state heat flux is varied.

Gravitational instabilities in a proto-planetary disk including the effects of magnetic fields

We investigate the gravitational instability of a thin, Keplerian protoplanetary disk including the effects of a largely azimuthal magnetic field. The model follows that of our previous work (Noh, Vishniac, & Cochran 1991) except for the inclusion of a magnetic field. The disk is assumed to consist of neutral and ionized gas and neutral dust which are coupled by gravity and friction. The growth rates and eigenfunctions are calculated numerically using nonaxisymmetric linear perturbation methods. The results show that the growth rate has a maximum at some intermediate azimuthal number m, but for each value of m it is reduced relative to the unmagnetized case. The effects of the magnetic field appear more strongly on small scales. As the strength of the equilibrium magnetic field increases the growth rates decrease, and the maximum instability occurs at a lower value on m due to the increasing magnetic pressure. The response of each component to the magnetic field is discussed using the behavior of the eigenfunctions in the radial direction. With the inclusion of the magnetic field, the effects of the ionization fraction and friction on the growth rates also appear to be important for high m modes. Increasing the ionization fraction or the friction suppresses instability, but only slightly changes the maximally unstable azimuthal scales. The enhanced growth rates due to a dust component for which thermal pressure is negligible are somewhat reduced by the inclusion of a magnetic field. The effects of different boundary conditions (reflecting and transmitting) on the growth rates are also shown.

Hydrodynamic Instability of Solar Thermosyphon Water Heaters

The flow instability of a solar thermosyphon water heater is studied analytically. A system dynamics model is derived by means of a one-dimensional approach and a linear perturbation method. The characteristic equation is obtained and the Nyquist criterion is used to examine the flow stability. The parameter M is a dimensionless parameter of system stability. The stability maps are plotted in terms of 14 parameters. The occurrence of hydrodynamic instability is determined by comparing the stability curves and the designed values of M. Flow instability is shown not to occur in most of solar water heaters commercially available, because the loop friction is relatively high in the design and because solar irradiation in field operation is still not high enough to cause flow instability.

Frequency modification using Newton's method and inverse iteration eigenvector updating

An iterative procedure is proposed for determining structural changes necessary for modifying natural frequencies of a structure in a prescribed manner. The procedure uses a Newton's method approach to determine a structural change that satisfies the equation of motion of the modified structure. The Newton equation is the same as that derived from sensitivity analysis and from linear perturbation methods. Iterative application of this equation can be used to predict accurately the structural change for either large or small frequency modifications. Inverse iteration is used to update the structural eigenvector between Newton iterations. The application of this procedure to a finite element analysis is described. Natural frequency modification of a finite element structural model is shown to converge rapidly to accurate results.

Nonlinear eigenmodes of a three-core fibre coupler

For a nonlinear three-core fibre coupler an important subclass of solutions has been investigated analytically. These are the stationary solutions or nonlinear eigenmodes. Their stability is checked using an exact method as well as the linear perturbation method, and numerical tests.

Gravitational instabilities in a proto-planetary disk

Gravitational instabilities were studied in a thin, Keplerian proto-planetary disk with a uniform entropy. The disk is a two-fluid system, composed of dust and gas, coupled by gravity and friction. The growth rates and the eigenfunctions are computed by using the nonaxisymmetric linear perturbation method in two-dimensional cylindrical coordinates. While recent work by Adams et al. (1989, ApJ, 347) has been done on the dipole mode, this study includes modes with high values of m. It is found that for the usual values of input parameters (mass of disk) the disk is unstable to nonaxisymmetric perturbations for some range of m in addition to m = 1.

Linear operator perturbation method in minimax control of linear dynamic systems

A perturbation method is proposed for solving some minimax control problems. The method is based on perturbation of eigenvalues of linear operators in a Hilbert space and Raleigh formulas. An explicit solution of the control problem under uncertainty is obtained.

Thermoelastic instability in planar solidification

A linear perturbation method is used to examine the stability of a unidirectional solidification problem in which a liquid, initially at the melting temperature, becomes solidified by heat transfer across a pressure-dependent thermal contact resistance to a plane mold. The contact pressure will be influenced by thermal distortion in response to the instantaneous temperature field in the solidified shell. The heat transfer and thermal stress problems are therefore coupled through the boundary conditions. The temperature and stress fields are assumed to consist of a unidirectional component and a small spatially-sinusoidal perturbation which can vary with time. Analysis of the thermoelastic problem for the solidified shell leads to an ordinary differential equation relating the perturbation in heat flux at the mold/casting interface and the corresponding perturbation in contact pressure. A second equation relating the same two variables is obtained by linear perturbation of the relation for heat conduction across the thermal contact resistance. These are then reduced to a single equation which is solved numerically. The results show that a small initial perturbation will grow substantially during the solidification process if the thermal contact resistance is very sensitive to pressure.

Spontaneous pattern on the corroded surface of metal: Phenomenon and theory

A spontaneous pattern was found on the corroded surface of 1Cr18Ni9Ti stainless steel which was subjected to a crude oil containing naphthenic acid at temperature about 330 °C. The high regularity, the periodicity, and SO2 symmetry of the pattern imply that a spontaneous dissipative structure arose in this corrosion system. It was also found that the distribution of chemical elements on the corroded surface was in accordance with the morphologic pattern. A new theory, which is based on the linear perturbation method and in which the shift of boundary is considered, is proposed to characterize the formation of the large-scale spontaneous pattern, reveal the mechanism, and explain this phenomenon in the corrosion system.

Sinusoidal perturbation solutions for planar solidification

A linear perturbation method is used to solve the two-dimensional heat conduction problem in which a liquid, initially at the melting temperature, becomes solidified by heat transfer to a plane mold the temperature of which is approximately uniform, but contains a small sinusoidal perturbation in one space dimension. Results are obtained for the consequent sinusoidal perturbation in the melt/solid boundary as a function of time and for the temperature distribution throughout the solid shell. These results are expressed in terms of a series of confluent hypergeometric functions which shows good convergence for practical values of material properties and temperatures. The inverse problem, in which the melt/solid boundary is prescribed and the mold temperature is to be determined, is also briefly discussed.

An asymptotic stability condition for inhomogeneous simple shear

Analytic steady solutions of inhomogeneous simple shear with isothermal and stress boundary conditions are found. The material is assumed to be thermoviscous and inertia and heat conduction effects are included. The basic inhomogeneous solution is spatially dependent, but time independent. Bifurcation of this solution, as the parameters vary, is analyzed. It is shown that there is a critical value of the parameter, corresponding to thermal softening, below which two steady state solutions exist for specified values of other parameters. A linear perturbation method, which gives rise to a linear set of partial differential equations (with spatially dependent coefficients), is used to distinguish the stable branch of the bifurcation diagram. After separation of variables, the existence of eigenvalues and eigenfunctions of the resulting fourth-order system is demonstrated. An asymptotic solution to the eigenvalue problem, for the case when an appropriate parameter is set equal to zero, is obtained explicitly. An integral method is then used in the general case to obtain a sufficient condition for stability.

Analytical network model on the flow and thermal characteristics of cyclic flow cryogenic regenerators

The flow and thermal characteristics of rapid cyclic flow cryogenic regenerators and their interrelationship have been analysed using the linear network theory and perturbation method. The effect of flow resistance, gas storage in void volume, temperature distribution, interaction of pressure wave and cyclic mass flow, and the factor of real gas were considered. A computer simulation program, CFCRX, was developed and the numerical results obtained are presented. This theory provides a better understanding of the working mechanism of cryogenic regenerators.

The influence of a flow field on the stability of the force-free magnetic field

In order to analyse the convective instability of the force-free magnetic field, an exact solution of the MHD equation for the magnetic field (1) together with the flow field (2) of constant speed V 0 making an angle θ with the magnetic field, was chosen as the unperturbed state. The stability of the fields between two parallel conducting walls of seperation d was studied by a linear perturbation method, which led to the eigenvalue problem (12), X being given by (13). It was shown by an approximate variational method that instability will set in by the flow field if V 0 is greater than 1/ 3 times Alfven velocity V A. For β= V 2 o V 2 A < 1 3 , the stability of the force-free field (1) is not influenced by the flow field, which may still be significant in other respects. Perturbations transverse to the magnetic field were found to be the most unstable modes.

Rationale for a Linear Perturbation Method for the Flow Field Induced by Fluid-Structure Interactions

A formal justification is developed for a method in which hydrodynamic data for a transient in a rigid-wall system (derived, for example, from a small-scale experimental simulation) is used as input in a linear computation for the perturbation flow field due to actual wall flexibility. The method is useful in problems where the basic flow transient is so complex that it can be quantified only empirically, and where the fluid-structure interaction is too complex for the fluid side to be represented by a priori defined equivalent mass.

Thermal instability in spherical liquid shells induced by surface tension

A linear perturbation method is employed to determine the condition for neutral stability in spherical liquid shells induced by surface tension mechanism. Three possible boundary conditions are considered: at least one boundary free or both. The critical Marangoni numbers for the onset of cellular convections are found for two types of steady radial temperature distributions in the spherical shells. Results are compared with those induced by buoyancy mechnism. It is concluded that surface tension forces are much more effective than buoyancy forces in producing thermal instability and a parabolic steady temperature distribution is more susceptible than a linear one to thermal disturbances due to surface tension forces. Heat transfer between a free surface and the ambient promotes thermal stability in liquid shells.

The propagation of waves in shallow water flow with lateral inflow / La propagation des ondes dans l'écoulement des eaux peu profondes avec afflux latéral

Abstract The propagation characteristics of small amplitude waves in shallow water flow with lateral inflow are determined using a linear perturbation method. Expressions relating the celerity, attenuation and wave number of the waves are given and used to establish two critical values of the Froude number.

Evaluation of Multi-Frequency-Microwave-Radiometer-System performance for oceanography

A technique is presented for constructing a mathematical model of a Multi-Frequency Microwave Radiometer System. The technique combines the different responses of microwave radiometers with models of the sea surface, the effects of the Earth's atmosphere and of the sky emission. A linear perturbation method and a more accurate non-linear method are outlined for processing of data simultaneously collected by the radiometers. The mathematical model of the Radiometer System combined with the linear data-processing method is useful for predicting in-flight sensor performance. Based on a chosen performance function, an evaluation of a spaceborne Multi-Frequency Microwave Radiometer System is given.

The computation of optimal control programmes using a modified successive sweep method†

A second-order method for numerically solving control optimization problems has been developed. The method, referred to as the modified sweep method (MSM), differs from the successive sweep method (SSM) proposed by McReynolds and Bryson (1965) in that the conditions for local control optimality are used to determine the control as an explicit function of the state variables and time. The control is eliminated from the problem and the solution to the resulting two-point boundary value problem can be obtained by linear perturbation methods. The modified sweep method proposed here uncouples the perturbation equations for the state variables and the Lagrange multipliers by using a generalized matrix-Riccati transformation of variables. The resulting algorithm for the numerical iteration process is concerned with determining the initial values of a set of Lagrange multipliers rather than correcting a numerical control programme over the entire time interval of interest.

On the Stability of Precipitate–Matrix Interfaces in Dilute Ternary Systems

Using linear perturbation methods, a precipitate–matrix interface stability criterion is obtained for dilute ternary systems. The calculation takes account of capillarity, transport in the precipitate, and ternary diffusional interaction. The general stability criterion contains a term reflecting the destabilizing influence of the solute gradients (i.e., the point effect of diffusion) and a term representing the stabilizing influence of capillarity. Two limiting cases are treated in detail—insignificant transport in the precipitate and negligible ternary diffusional interaction. It is demonstrated that in these two cases the two solutes behave in a cumulative fashion.