Regular Perturbation Analysis Research Articles

Perturbation analysis of radiative effect on free convection flows in porous medium in the presence of pressure work and viscous dissipation

The effect of thermal radiation with a regular three-parameter perturbation analysis has been studied for the effects in some free convection flows of Newtonian fluid-saturated porous medium. The effects of the thermal radiation, permeability of the porous medium, pressure stress work and viscous dissipation on the flows and temperature fields have been included in the analysis. Four different vertical flows have been analyzed, those adjacent to an isothermal surface, uniform heat flux surface, a plane plume and flow generated from a horizontal line energy source, and, a vertical adiabatic surface. Rosseland approximation is used to describe the radiative heat flux in the energy equation. The numerical results of the perturbation analysis for four conditions are solved numerically by the fourth-order Runge–Kutta integration scheme. Numerical values of the main physical quantities are the skin friction and a heat transfer and total heat and mass convected downstream are presented in a tabular form with the parameters characterizing the radiation, permeability of the porous medium, pressure stress work and viscous dissipation. The obtained results are compared and a representative set is displayed graphically to illustrate the influences of the radiation, permeability of the porous medium, pressure stress work and viscous dissipation on the velocity and the temperature profiles.

A martingale control variate method for option pricing with stochastic volatility

A generic control variate method is proposed to price options under stochastic volatility models by Monte Carlo simulations. This method provides a constructive way to select control variates which are martingales in order to reduce the variance of unbiased option price estimators. We apply a singular and regular perturbation analysis to characterize the variance reduced by martingale control variates. This variance analysis is done in the regime where time scales of associated driving volatility processes are well separated. Numerical results for European, Barrier, and American options are presented to illustrate the effectiveness and robustness of this martingale control variate method in regimes where these time scales are not so well separated.

An analytical model for high velocity impacts on thin CFRPs woven laminated plates

An analytical model to study the impact process of a spherical projectile penetrating at high velocity into a carbon/epoxy plain woven laminate is developed in this work. The model is based on an energy balance, where the kinetic energy of the projectile is absorbed by the laminate by three different mechanisms: laminate crushing, linear momentum transfer and tensile fiber failure. A non-homogeneous differential equation is obtained. A subsequent simplification using regular perturbation analysis gives a closed-form solution that allows the approximative calculation of the residual velocity and hence the ballistic limit. The model is validated with the results of experimental tests in which the residual velocity is measured by means of high speed cameras.

Nonlinear analysis of arterial blood flow—steady streaming effect

Recent interest in the nonlinear analysis of oscillatory flow in an annular region has been motivated by the desire to apply the results of modelling to clinical medicine—especially to estimate the catheter-induced errors in blood pressure measurement. Insertion of catheters in the blood vessels brings in additional impedance or frictional resistance to flow and will alter the pressure distribution. The nonlinear analysis of Sarkar and Jayaraman (J. Biomech. 31 (1998) 781–791; Phys. Fluids 13(10) (2001) 2901–2912) provided a correction to the mean pressure drop predicted earlier by linear theory. A serious pathological condition is encountered when some blood constituents deposited on the artery wall get detached from the wall, join the blood stream again and form a clot. This can lead to partial or even complete blockage of the blood vessels. The seriousness can be very acute in the case of a stenotic artery. In this paper, the model experiment proposed by Doffin and Chagneau (J. Biomech. 14(3) (1981) 143–148) to understand such a serious situation and its repercussions in terms of vortices and separation in the flow of blood in the artery is analyzed in detail taking into account the nonlinear characteristic of the model. The mathematical treatment of the problem includes regular perturbation analysis when frequency parameter is small and matched asymptotic expansions magnifying the boundary layer characteristics near the walls when the frequency parameter is large. Important fluid dynamic parameters like wall shear stress, pressure drop and streamline patterns are discussed.

DNA separation by EFFF in a microchannel

This paper theoretically explores the application of electric field flow fractionation (EFFF) for the size-based separation of DNA strands in a microchannel. An axial electric field cannot separate DNA strands in solution because the electrical mobility of the strands is independent of the length. However, lateral electric fields coupled with an axial Poiseuille flow can separate the DNA strands of different sizes. By using regular perturbation analysis, we obtain the effective diffusivity and the mean velocity of the DNA molecules that are undergoing a pressure driven Poiseuille flow in a 2D channel in presence of a lateral electric field. The mean velocities and the dispersion coefficients are then utilized to determine the scaling for length of the channel and the time required for separation of DNA molecules in different parameter regimes. The results show that EFFF can separate DNA strands in the range of 10 kbp that differ in size by about 2.5 kbp in about half an hour in a 1 cm long channel. While DNA strands can be separated by EFFF, the performance of devices based on EFFF seems to be at best comparable to other techniques such as entropic trapping.

Variance reduction for Monte Carlo methods to evaluate option prices under multi-factor stochastic volatility models

We present variance reduction methods for Monte Carlo simulations to evaluate European and Asian options in the context of multiscale stochastic volatility models. European option price approximations, obtained from singular and regular perturbation analysis [Fouque J P, Papanicolaou G, Sircar R and Solna K 2003 Multiscale stochastic volatility asymptotics, SIAM J. Multiscale Modeling and Simulation 2], are used in importance sampling techniques, and their efficiencies are compared. Then we investigate the problem of pricing arithmetic average Asian options (AAOs) by Monte Carlo simulations. A two-step strategy is proposed to reduce the variance where geometric average Asian options (GAOs) are used as control variates. Due to the lack of analytical formulas for GAOs under stochastic volatility models, it is then necessary to consider efficient Monte Carlo methods to estimate the unbiased means of GAOs. The second step consists in deriving formulas for approximate prices based on perturbation techniques, and in computing GAOs by using importance sampling. Numerical results illustrate the efficiency of our method.

On the occurrence of even harmonics in the shear stress response of viscoelastic fluids in large amplitude oscillatory shear

The non-linear response of polymeric liquids observed experimentally in large amplitude oscillatory shear (LAOS) is generally characterized by the presence of odd harmonics of the excitation frequency in the Fourier spectrum for the shear stress. Even harmonics of relatively smaller amplitude have also been observed, whose appearance is usually attributed to wall slip phenomena. In the present work, we show that wall slip is not a necessary condition for the occurrence of even harmonics. To this end, we perform a non-linear study of planar LAOS flow between two infinite parallel plates using either a monotone or non-monotone viscoelastic constitutive equation (i.e., respectively, the Giesekus and Johnson–Segalman models). The analysis allows for spatially non-homogeneous velocity and stress fields. We assume no-slip boundary conditions, and investigate the combined effects of inertia, elasticity, and shear thinning by means of spectral methods. A regular perturbation analysis is also conducted in the inertialess monotone case. Results for the Giesekus model show that combination of elasticity and shear thinning yields transient even harmonics in shear stress whose life span and intensity are considerably increased by inertia. Furthermore, the one-dimensional flow is unstable to finite two-dimensional perturbations under inertia and at high elasticity. This results in the development of secondary flows and saturation of even harmonics into small but finite values. Simulations for the non-monotone Johnson–Segalman model predict even harmonics of relatively larger amplitude that settle in dynamic equilibrium. Furthermore, the fluid’s response is quasi-periodic with the appearance of incommensurate frequencies.

A Numerical Method for the Viscoelastic Melt-Spinning Model with Radial Resolutions of Temperature and Stress Fields

A numerical method to compute the viscoelastic melt-spinning model with radial resolutions of temperature and stress fields is formulated and applied to the low speed range. The starting framework is the reduction of the complete continuous model into both the perturbed two-dimensional model and the perturbed average model obtained from a first-order regular perturbation analysis available in the literature. The polymer rheology is described with the nonisothermal Phan-Thien and Tanner and Giesekus constitutive equations. By using the implicit tridiagonal scheme of finite differences coupled to the fourth-order Runge−Kutta method, an iterative numerical algorithm is proposed for the computation of the coupled balance equations. The temperature and stress fields in the filament as functions of axial and radial positions are obtained for a well-refined mesh and with high numerical precision. The numerical algorithm considers the appropriate interplay between axial and radial varying temperature and stress fields and the rigorous averaged balances of momentum and energy and averaged constitutive equation. The development of a skin-core structure is predicted with the two rheological models.

A poroelastic continuum model of the cupula partition and the response dynamics of the vestibular semicircular canal.

Using mixture theory, an axisymmetric continuum model is presented describing the response dynamics of the vestibular semicircular canals to canal-centered head rotation in which the cupula partition is modeled as a poroelastic mixture of interpenetrating solid and fluid constituents. The solid matrix of the cupula is assumed to behave as a linear elastic material, whereas the fluid constituent is assumed to be Newtonian. A regular perturbation analysis of the fluid dynamics in the canal provides a dynamic boundary condition, which acts across the cupula partition. Numerical solution of the coupled system of momentum equations provides the spatio-temporal displacement fields for both the fluid and solid constituents of the cupula. Results indicate that at frequencies above 1 Hz, the fluid constituent is dynamically entrained by the solid matrix such that their motions are bound as if to exist as a single component. The resulting high-frequency response is consistent with the macromechanical response predicted by single-component viscoelastic models of the cupula. Below 1 Hz, the dynamic coupling between the fluid and solid constituents weakens and the transcupular differential pressure is sufficient to force fluid through the mixture with little deformation of the solid matrix. Results are sensitive to the precise value of the cupular permeability. One of the most important distinctions between the present analysis and previous impermeable models of the cupula arises at the micromechanical level in terms of the local fluid flow that is predicted to occur within the cupula and around the ciliary bundles and sensory hair cells. Another important result reveals that the permeation dynamics predicted below 1 Hz gives rise to the same low-frequency macromechanical response as would occur with an impermeable viscoelastic structure having a much greater stiffness. Current estimates of the mechanical stiffness of the cupula, based solely on afferent nerve data, may therefore overestimate the true value intrinsic to the solid matrix by as much as an order of magnitude.

Radiative effects in natural convection flows

A regular perturbation analysis is presented for the radiative effects on the following laminar natural convection flows with temperature dependent viscosity: a freely-rising plane plume, the flow above a horizontal line source on an adiabatic surface (a plane wall plume) and the flow adjacent to a vertical uniform flux surface. While these flows have well-known power-law similarity solutions when the fluid viscosity is taken to be constant, they are non-similar when the viscosity is considered to be a function of the temperature. A single similar flow, that adjacent to a vertical isothermal surface is also analyzed for comparison in order to estimate the accuracy. Numerical results with various parameters are tabulated.

Analysis of the transpiration cooling of a thin porous plate in a hot laminar convective flow

This paper deals with the asymptotic and numerical analysis for the steady-state transpiration cooling of a thin porous flat plate in a laminar hot convective flow, taking into account the streamwise heat conduction through the plate. For high conductivity plates, a regular perturbation analysis has been carried out, yielding a three-term asymptotic solution for the distribution of plate temperature. In the limit of a very poorly conducting plate, a singular perturbation technique, based on matched asymptotic expansions, is employed to solve the governing equations. We also solved the equations numerically using a quasilinearization technique. The numerical results are in good agreement with the asymptotic solution close to the asymptotic limits studied.

General Trifurcation Analysis in Elastic Stability*

ABSTRACT A general theory of trifurcation analysis is presented, in the context of the elastic stability of discrete systems. Under certain conditions, an elastic structure can display critical states for which there is coincidence of the critical load and the critical direction. Such states are called trifurcation, because three equilibrium paths intersect at the critical state. As in the general theory of bifurcation, a regular perturbation analysis is employed for each one of the emerging path. It is shown that methods of solution of perturbation equations in trifurcation analysis are different from those employed in bifurcation. The present development is based in the V-formulation; i.e., the total potential energy is written in terms of the original generalized coordinates of the system.

Natural Convection from a Vertical Plate in a Porous Medium Using Brinkman's Model

A regular perturbation analysis is presented for the following laminar natural convection flows of Newtonian fluids with temperature-dependent effective viscosity: a freely-rising plane plume, the flow above a horizontal line source on an adiabatic surface (a plane wall plume) and the flow adjacent to a vertical uniform flux surface for porous medium. The temperature-dependent effective viscosity introduces nonsimilarity into the governing equations. Numerical results are presented for the flow and heat transfer characteristics.

Effects of a transverse flows with variable viscosity in micropolar fluids

A regular perturbation analysis is presented for three laminar natural convection flows in micropolar fluids in liquids with temperature dependent viscosity: a freely-rising plane plume, the flow above a horizontal line source on an adiabatic surface (a plane wall plume) and the flow adjacent to a vertical uniform flux surface. While these flows have well-known power-low similarity solutions when the fluid viscosity is taken to be constant, they are non-similar when the viscosity is considered to a function of temperature. A single similar flow, that adjacent to a vertical isothermal surface, is also analysed for comparison in order to estimate the extent of validity of perturbation analysis. The formulation used here provides a unified treatment of variable viscosity effects on those four flows. Computed first-order perturbation quantities are presented for all four flows. Numerical results for velocity, angular velocity and thermal functions has been shown graphically or tabulated for different values of micropolar parameters.

Radiative Effect on Natural Convection Flows in Porous Media

A regular two-parameter perturbation analysis based upon the boundary layer approximation is presented here to study the radiative effects of both first- and second-order resistances due to a solid matrix on the natural convection flows in porous media. Four different flows have been studied, those adjacent to an isothermal surface, a uniform heat flux surface, a plane plume and the flow generated from a horizontal line energy source on a vertical adiabatic surface. The first-order perturbation quantities are presented for all these flows. Numerical results for the four conditions with various radiation parameters are tabulated.

Reservoir Description by Well-Test Analysis by Use of Cyclic Flow-Rate Variation

Summary This work investigates the possibility of characterizing the permeability distribution in areally heterogeneous reservoirs through the interpretation of pressure response to cyclic flow-rate variations. We formulated a mathematical model for the pressure behavior caused by sinusoidal flow rates in reservoirs with continuously and smoothly varying radial-permeability distributions. The solution is obtained by the application of a regular perturbation analysis, and a correlation between the radius of cyclic influence and the frequency of a sinusoidal flow rate is developed. The analytical perturbation solution allows a qualitative assessment of the effects of areal reservoir heterogeneities on the pressure response either at the active well or at an observation well. Because the perturbation method results in only an approximate solution, we propose a quantitative method to describe the permeability distribution based on the fully correct multicomposite radial reservoir model. The permeability in each zone of the equivalent reservoir is estimated by using a nonlinear regression procedure. The frequency correlation may be used to design pulse-test flowrate schedules so that an optimum result can be achieved in terms of parameter estimation. We investigated several factors affecting the quality and the uncertainty in the estimated permeability distribution, along with examples including a variety of geometric and permeability distribution configurations. These factors include the type of test (constant-rate drawdown or pulse test) and reservoir configuration. In particular, we found the frequency of flow-rate changes and the point in the system at which to measure and interpret the pressure data to be important.

Nonlinear theory of the interaction of sound waves with an inhomogeneous magnetized plasma in the resonant slow wave layer

The nonlinear theory of driven magnetohydrodynamic (MHD) waves in the resonant slow wave dissipative layer developed by Ruderman, Hollweg, and Goossens [Phys. Plasmas 4, 75 (1997)] is used to study the interaction of sound waves with a static one-dimensional planar magnetic plasma configuration. This configuration is a nonhomogeneous magnetic slab (region II) sandwiched by a homogeneous magnetic field free plasma (regions I) and a homogeneous magnetic plasma (region III). The equilibrium magnetic field is unidirectional. An incoming sound wave is launched from region I, and the equilibrium quantities and the parameters of the incoming sound wave are chosen in such a way that the wave is evanescent in region III and resonates with a slow local MHD wave in region II. The analysis is restricted to incoming sound waves with wave vectors in the plane determined by the equilibrium magnetic field and the direction of inhomogeneity, so that there is no resonance with a local Alfvén wave. Partial reflection of the incoming sound wave from region II generates an outgoing sound wave in region I. The nonlinearity parameter introduced by Ruderman et al. is assumed to be small, and a regular perturbation analysis is used to determine the wave solution. The present analysis shows that nonlinearity in the resonant slow wave dissipative layer causes the following new effects in comparison with results obtained on the basis of linear theory: (i) higher harmonic contributions are generated in the wave solution in the dissipative layer and also in the outgoing sound wave in addition to the fundamental harmonic. Outside the dissipative layer the amplitude of each harmonic contribution is of a higher order with respect to the small nonlinearity parameter than inside the dissipative layer. This property reflects the fact that nonlinearity is only taken into account in the dissipative layer, so that the higher harmonic contributions to the outgoing wave are only due to the interaction of the external plasma motions with the plasma motions in the dissipative layer; (ii) the coefficient of wave energy absorption and the absolute value of the jump in the normal velocity are decreased.

Asymptotic methods for the analysis of wave propagation in nonlinear, lossless transmission lines and equivalent circuits

AbstractPerturbation methods are employed to determine the small‐signal behavior of an LC‐ladder nonlinear, lossless transmission line. It is shown that a straightforward, regular perturbation analysis results in nonuniformly valid expansions, which can be rendered uniformly valid by stretching the time variable. Perturbation methods are also used to show that, when dispersion effects are accounted for, solitons may propagate along the nonlinear, lossless transmission line. The dispersion characteristics of some equivalent circuits derived from a lossless transmission line are also analyzed. © 1995 John Wiley & Sons. Inc.

Hybrid approach to solution of optimal control problems

This paper proposes a solution approach for nonlinear optimization problems that seeks to combine the desirable features of analytic methods that are based on use of simplified models and numerical methods that use piecewise elementary interpolation functions to represent the solution. The approach is illustrated by a combination of regular perturbation analysis and the method of collocation, although other combinations of model simplification and numerical solution methods can also be envisioned. A simple fourth-order nonlinear system is used to illustrate the conceptual approach for several possible levels of approximation.

Concentration fields in dusty gas flows

It is well-known that particle inertia in dusty gas flows may cause a certain complex behavior of the concentration fields, such as the enrichment of particle concentration along particle trajectories. Traditional analyses of these and similar problems were made by treating the gas and the particles as two interpenetrating continua and using the two-phase flow equations for the description of the two phases. For cases with relatively low Stokes number, the concentration field may be obtained by a regular perturbation analysis of the equations of change using the Stokes number as a parameter. In the present study, we adopted the Lagrangian view of particle motion and obtained particle concentrations by applying mass balance to neighboring trajectories. In contrast to the earlier works, this approach can be used for all Stokes numbers and its application (at least in the cases considered) is straightforward. To demonstrate its utility, we present the results of our analyses of two specific problems and compare our results with those obtained by previous investigators.