Regular Perturbation Expansion Research Articles

Dynamics of a viscous vesicle in linear flows

An analytical theory is developed to describe the dynamics of a closed lipid bilayer membrane (vesicle) freely suspended in a general linear flow. Considering a nearly spherical shape, the solution to the creeping-flow equations is obtained as a regular perturbation expansion in the excess area. The analysis takes into account the membrane fluidity, incompressibility, and resistance to bending. The constraint for a fixed total area leads to a nonlinear shape evolution equation at leading order. As a result two regimes of vesicle behavior, tank treading and tumbling, are predicted depending on the viscosity contrast between interior and exterior fluid. Below a critical viscosity contrast, which depends on the excess area, the vesicle deforms into a tank-treading ellipsoid, whose orientation angle with respect to the flow direction is independent of the membrane bending rigidity. In the tumbling regime, the vesicle exhibits periodic shape deformations with a frequency that increases with the viscosity contrast. Non-Newtonian rheology such as normal stresses is predicted for a dilute suspension of vesicles. The theory is in good agreement with published experimental data for vesicle behavior in simple shear flow.

Models for acoustic scattering in high contrast media

In this work a numerical method for evaluating backscatter from a three-dimensional medium having high acoustic contrast is presented. The solution is sought in terms of a perturbation expansion in the contrast amplitude. It is shown that limitations of the regular perturbation expansion can be overcome by recasting the perturbation sequence as a rational fraction using Padé approximants. The resulting solution allows for an accurate representation of the pressure and allows for the poles in the frequency response to be modeled. The determination of the pulse-echo response for a high-contrast medium is discussed and presented.

Multiscale Intensity Models for Single Name Credit Derivatives

We study the pricing of defaultable derivatives, such as bonds, bond options, and credit default swaps in the reduced form framework of intensity-based models. We use regular and singular perturbation expansions on the intensity of default from which we derive approximations for the pricing functions of these derivatives. In particular, we assume an Ornstein-Uhlenbeck process for the interest rate, and a two-factor diffusion model for the intensity of default. The approximation allows for computational efficiency in calibrating the model. Finally, empirical evidence on the existence of multiple scales is presented by the calibration of the model on corporate yield curves.

Evolving eddy structures in oscillatory Stokes flows in domains with sharp corners

Stokes flow of a viscous fluid in a cylindrical container driven by time-periodic forcing, either at the boundary or through oscillation of the cylinder about an axis parallel to its generators, is considered. The behaviour is governed by a dimensionless frequency parameter η and by the geometry of the cylinder cross-section. Various cross-sections (square, rhombus, and sector of a circle) are first treated by either finite-difference or analytic techniques and typical transitions of streamline topology during flow reversals are identified. Attention is then focused on the asymptotic behaviour near any sharp corner on the boundary. For small η, a regular perturbation expansion reveals the manner in which local flow reversal proceeds during each half-cycle of the flow. The behaviour depends on the corner angle, and different regimes are identified for both types of forcing. For example, for an oscillating square domain, eddies grow symmetrically from each corner and participate in the subsequent flow reversal in the interior. For large η, the corner eddies merge into Stokes-type boundary layers which drive the interior flow-reversal process. In general, the local corner analysis provides a key to an understanding of the global flow evolution.

On the interplay between inertial and viscoelastic effects for the flow in weakly modulated channels

AbstractThe flow inside a spatially modulated channel is examined for viscoelastic fluids of the Oldroyd‐B type. The lower wall is flat and the upper wall is sinusoidally modulated. The modulation amplitude is assumed to be small. Thus, a regular perturbation expansion of the flow field coupled to a variable‐step finite‐difference scheme is used to solve the problem. Convergence and accuracy assessment against earlier experimental results indicate that there is a significant range of validity of the perturbation approach. The influences of wall geometry, inertia and viscoelasticity on the flow kinematics and stresses are investigated systematically. In particular, the interplay between the flow and fluid parameters effects on the conditions for the onset of backflow, number of vortices, their size and location is revealed. The distance between the flow separation and reattachment locations identifies the vortex size. Non‐monotonic dependence of the vortex size on elasticity is reported. The critical conditions for the onset of negative elasticity effects on vortex size are identified. The critical Reynolds number for the onset of backflow initially decreases then levels off or even increases as elasticity increases. For highly elastic fluid and large enough Reynolds number, more than one vortex appear near the lower wall. Copyright © 2005 John Wiley & Sons, Ltd.

Pattern formation in weakly forced Taylor–Couette flow of viscoelastic fluids

The influence of elasticity, inertia and modulation on the Taylor–Couette flow with spatially modulated cylinders is examined for an Oldroyd-B fluid. Only the inner cylinder rotates, but one or two cylinders can be modulated. The modulation of the cylinders is periodic and the modulation amplitude is assumed to be small. A regular perturbation expansion is used to determine the flow field at small to moderately large Taylor and elasticity numbers. The flow is assumed to be axisymmetric and only the steady state is studied. For a Newtonian fluid, the increase in inertia leads to an intensification of the secondary (Taylor-vortex) flow, but the vortex structure and number remain unchanged as Ta increases, regardless of the type of modulation. Fluid elasticity, for both weakly and strongly elastic fluids, is found to alter significantly the vortex structure. When the outer cylinder is modulated, fluid elasticity is found to intensify the flow to a maximum level at an elasticity number E = E max, at which point the vortices are “squeezed” toward the inner cylinder, resulting in the weakening of the secondary flow as E is further increased beyond E max. When only the inner cylinder is modulated, two additional vortices appear, originating at the outer cylinder. In this case, the flow intensity exhibits a minimum as E is varied. The combined effect of inner- and outer-cylinder modulation has also been examined. Finally, there is an optimal modulation wavenumber for which the secondary vortex structure is best visible.

Steady flows in pipes with finite curvature

Motivated by blood flow in a curved artery, we consider fluid flow through a uniformly curved pipe, driven by a steady pressure gradient. Previous studies have typically considered pipes with asymptotically small curvature in which only the centrifugal effect is retained in the governing equations. Here we consider flows in pipes of finite curvature, and determine the effects of both the centrifugal and Coriolis forces on the flow. The flow behavior is governed by two dimensionless parameters: the curvature, δ, and the Dean number, D. Asymptotic solutions are developed for D⪡1 (using regular perturbation expansions) and D⪢1 (using matched asymptotic expansions and the Pohlhausen method). For intermediate values of D we use a pseudospectral code to obtain solutions. For values of D greater than a critical value, Dc, multiple steady solutions to the governing equations exist; we determine how Dc and the form of the solutions depends on δ. The results indicate that while many of the qualitative features of steady curved-pipe flows are captured by the asymptotic results, the effects of finite curvature can lead to substantial quantitative differences, e.g., in the wall shear stress distribution. The physiological implications of the results for blood flow in arteries are discussed.

Shear-thinning flow in weakly modulated channels

The steady flow inside a spatially modulated channel is examined for shear-thinning and shear-thickening fluids. The flow is induced by the translation of the lower plate. The modulation amplitude is assumed to be small. A regular perturbation expansion of the flow field is used, coupled to a variable-step finite-difference scheme, to solve the problem. Convergence and accuracy assessment against finite-volume calculations indicates that there is a significant range of validity of the perturbation approach. The influence of the wall geometry, inertia and non-Newtonian effects are investigated systematically. In particular, the influence of the flow and fluid parameters is examined on the conditions for the onset of separation, vortex size and location. Copyright © 2005 John Wiley & Sons, Ltd.

Torque measurements on ferrofluid cylinders in rotating magnetic fields

We study the response of magnetic nanoparticle suspensions (ferrofluids) to uniform rotating magnetic fields generated by a two-pole three-phase magnetic induction motor stator winding. Measurements of the torque required to rotate a polycarbonate spindle submerged in ferrofluid subjected to co-rotating and counter-rotating fields yield experimental observations of negative magnetoviscosity in a cylindrical Couette geometry, conceptually similar to the observations of Bacri et al. (Phys. Rev. Lett. 75 (1995) 2128) in a Poiseuille flow under an oscillating magnetic field. Further measurements are presented for the torque required to restrain a spindle when it is (i) entirely filled with ferrofluid, (ii) entirely surrounded with ferrofluid, and (iii) both entirely filled and surrounded with ferrofluid. Some of the results for the spindle either entirely filled or entirely surrounded with ferrofluid are compared to theoretical expressions obtained from the ferrohydrodynamic equations using a rigorous regular perturbation expansion in the small parameter Ω τ , where Ω is the applied field frequency and τ is the effective magnetic relaxation time of the suspension.

Prandtl number dependence of unsteady natural convection along a vertical plate in a stably stratified fluid

The Prandtl number dependence of unsteady laminar natural convection along an infinite vertical plate in a thermally stratified fluid is investigated. Flows are induced by an impulsive (step) change in plate temperature and by a suddenly imposed plate heat flux. Analytical solutions of the viscous equations of motion and thermodynamic energy are obtained for Prandtl numbers near unity by the method of Laplace transforms and a regular perturbation expansion. The zeroth-, first- and second-order terms in the expansion are obtained for an impulsive change in plate temperature, while the zeroth- and first-order terms are obtained for a sudden application of a plate heat flux. The developing boundary layers are thicker, more vigorous, and more sensitive to the Prandtl number at smaller Prandtl numbers (<1) than at larger Prandtl numbers (>1). The analytical results are confirmed and extended with results from numerical simulations for Prandtl numbers strongly deviating from unity.

Pattern formation in weakly forced Taylor-Couette flow.

Low-inertia vortex formation and pattern selection are examined for axisymmetric Taylor-Couette flow with spatially modulated cylinders. The forcing is arbitrary but remains periodic. The modulation amplitude is assumed to be small, and a regular perturbation expansion is used to determine the flow field at small to moderately large Taylor numbers (below the critical threshold). It is found that the presence of a weak modulation leads unambiguously to the emergence of steady Taylor-vortex flow even at vanishingly small Taylor number. This situation is closely reminiscent of the effect of end plates, and the consequent onset of imperfect bifurcation. The vortex structure is found to have the same periodicity as the forcing when only one of the cylinders is modulated, or when the modulations are commensurate. For incommensurate modulations, the vortex pattern is quasiperiodic, with regions of almost purely azimuthal flow. When the counter-rotation speed of the outer cylinder increases, the original vortices are gradually replaced by new ones that end up spanning the entire gap width, and in turn break up into two vortices resulting in two rows of vortices commensurate with each cylinder modulation. It is also shown that, for any modulation amplitude, the forcing wave number that generates the most intense vortex flow for a given Taylor number varies monotonically with Ta, but always reaches the critical value predicted by linear stability analysis for straight cylinders, regardless of which cylinder is modulated.

Turbulence with pressure: anomalous scaling of a passive vector field.

The field theoretic renormalization group (RG) and the operator-product expansion are applied to the model of a transverse (divergence-free) vector quantity, passively advected by the "synthetic" turbulent flow with a finite (and not small) correlation time. The vector field is described by the stochastic advection-diffusion equation with the most general form of the inertial nonlinearity; it contains as special cases the kinematic dynamo model, linearized Navier-Stokes (NS) equation, the special model without the stretching term that possesses additional symmetries and has a close formal resemblance with the stochastic NS equation. The statistics of the advecting velocity field is Gaussian, with the energy spectrum E(k) proportional to k(1-epsilon) and the dispersion law omega proportional to k(-2+eta), k being the momentum (wave number). The inertial-range behavior of the model is described by seven regimes (or universality classes) that correspond to nontrivial fixed points of the RG equations and exhibit anomalous scaling. The corresponding anomalous exponents are associated with the critical dimensions of tensor composite operators built solely of the passive vector field, which allows one to construct a regular perturbation expansion in epsilon and eta; the actual calculation is performed to the first order (one-loop approximation), including the anisotropic sectors. Universality of the exponents, their (in)dependence on the forcing, effects of the large-scale anisotropy, compressibility, and pressure are discussed. In particular, for all the scaling regimes the exponents obey a hierarchy related to the degree of anisotropy: the more anisotropic is the contribution of a composite operator to a correlation function, the faster it decays in the inertial range. The relevance of these results for the real developed turbulence described by the stochastic NS equation is discussed.

Investigation of harmonic generation in thermoacoustic engines

One factor limiting the efficiency of thermoacoustic engines and refrigerators is the generation of harmonics. Not only do thermoviscous losses increase in relation to the generation of higher frequencies, but the stack is not designed to optimize thermoacoustic processes at these frequencies. We describe here a semi-analytical investigation of nonlinear effects associated with harmonic generation. The approach is based on a regular perturbation expansion of the acoustical quantities, combined with the assumptions and methodology underlying the linear theory developed by Rott. At first order, Rott’s equations for the pressure and particle velocity are obtained and solved numerically. The second-order system consists of Rott’s equations for the second harmonic driven by a forcing function containing products of the first-order solutions. The goal is to solve the third-order system, given by Rott’s equations for the fundamental driven by products of the first- and second-order solutions, to determine the nonlinear correction to the fundamental component. The investigation was not completed at the time of writing, and we therefore present here the intermediate results for second-harmonic generation. These results provide partial explanations of how harmonic generation influences thermoacoustics. [Work supported by ONR and ARL:UT IR&amp;D.]

Transient Chaos in Coupled Oscillators with Shape Deformable Potential

The dynamics of a perturbed system consisting of a particle embedded in a strongly nonlinear potential V(ϕ; r, η) whose shape can be varied continuously as a function of r in the range -1 &lt; r &lt; 1, and coupled to an harmonic oscillator is analyzed. The perturbations are made of the coupling and damping forces. When they are removed, the subsystem formed by the anharmonic oscillator contains saddle points connected by homoclinic loops. Thus, the whole unperturbed system has saddle-centre points whose stable and unstable manifolds coincide in three-dimensional manifold. We concentrate our analysis of the perturbed system in the regions near these saddle-centre points. First, the Melnikov theory is used to investigate the presence of horseshoes chaos in the dynamics. Due to some discrepancies of the Melnikov theory, we next rederive the boundary between the regions of regular and irregular motions in the parameters space by using regular perturbation expansion. It is found here that for small values of the natural pulsation ω of the linear oscillator, complicated behavior intensifies in the system as the absolute value of the shape parameter r approaches zero. On the contrary, chaotic motion intensifies according as r decreases from values close to 1 to values close to -1, for large ω. The numerical analysis, which includes computation of maximal Lyapunov exponent, bifurcation diagrams and Poincaré sections, shows that the system exhibits transient stochastic behavior, but ultimately settles down on a simple set which is either a fixed point or a limit cycle. The dependence of this transient behavior on the system parameters agrees qualitatively well with the analytical predictions.

Nonlinear stratified spin-up

Both a weakly nonlinear analytic theory and direct numerical simulation are used to document processes involved during the spin-up of a rotating stratified fluid driven by wind-stress forcing for time periods less than a homogeneous spin-up time. The strength of the wind forcing, characterized by the Rossby number ε, is small enough (i.e. ε[Lt ]1) that a regular perturbation expansion in ε can be performed yet large enough (more specifically, ε∝E1/2, where E is the Ekman number) that higher-order effects of vertical diffusion and horizontal advection of momentum/density are comparable in magnitude. Cases of strong stratification, where the Burger number S is equal to one, with zero heat flux at the upper boundary are considered. The Ekman transport calculated to O(ε) decreases with increasing absolute vorticity. In contrast to nonlinear barotropic spin-up, vortex stretching in the interior is predominantly linear, as vertical advection negates stretching of interior relative vorticity, yet is driven by Ekman pumping modified by nonlinearity. As vertical vorticity is generated during the spin-up of the fluid, the vertical vorticity feeds back on the Ekman pumping/suction, enhancing pumping and vortex squashing while reducing suction and vortex stretching. This feedback mechanism causes anticyclonic vorticity to grow more rapidly than cyclonic vorticity. Strict application of the zero-heat-flux boundary condition leads to the growth of a diffusive thermal boundary layer E−1/4 times thicker than the Ekman layer embedded within it. In the Ekman layer, vertical diffusion of heat balances horizontal advection of temperature by extracting heat from the thermal boundary layer beneath. The flux of heat extracted from the top of the thermal boundary layer by this mechanism is proportional to the product of the Ekman transport and the horizontal gradient of the temperature at the surface. The cooling caused by this heat flux generates density inversions and intensifies lateral density gradients where the wind-stress curl is negative. These thermal gradients make the potential vorticity strongly negative, conditioning the fluid for ensuing symmetric instability which greatly modifies the spin-up process.

On the validity of the perturbation approach for the flow inside weakly modulated channels

AbstractThe equations governing the flow of a viscous fluid in a two‐dimensional channel with weakly modulated walls have been solved using a perturbation approach, coupled to a variable‐step finite‐difference scheme. The solution is assumed to be a superposition of a mean and perturbed field. The perturbation results were compared to similar results from a classical finite‐volume approach to quantify the error. The influence of the wall geometry and flow Reynolds number have extensively been investigated. It was found that an explicit relation exists between the critical Reynolds number, at which the wall flow separates, and the dimensionless amplitude and wavelength of the wall modulation. Comparison of the flow shows that the perturbation method requires much less computational effort without sacrificing accuracy. The differences in predicted flow is kept well around the order of the square of the dimensionless amplitude, the order to which the regular perturbation expansion of the flow variables is carried out. Copyright © 2002 John Wiley &amp; Sons, Ltd.

Dispersion by Pressure-Driven Flow in Serpentine Microfluidic Channels

Microfluidic hydrodynamic chromatography performed in serpentine microchannels etched on chips is analyzed in the limiting case of chips containing a large number of periodically arrayed turns. Comparison is made between these results and those for a straight channel of the same length as the curvilinear channel, all other things being equal. Explicitly, generalized Taylor−Aris dispersion (macrotransport) theory for spatially periodic systems is adapted to compute the chip-scale solute velocity Ū* and dispersivity D* for effectively point-size, physicochemically inert Brownian particles entrained in a low Reynolds number, pressure-driven solvent flow occurring within the curvilinear interstices of such serpentine devices. Attention is focused upon relatively thin channels of uniform cross section, enabling the various transport fields pertinent to the problem to be expressed as regular perturbation expansions with respect to a small dimensionless parameter e, representing the ratio of channel half-width ...

Crack-tip field for fast fracture of an elastic–plastic–viscoplastic material coupled with quasi-brittle damage. Part 2. Small damage regime

A parameter perturbation technique is used to obtain asymptotic solutions that apply to a fast moving crack-tip, where small damage condition prevails. The material can be described by an elastic–plastic–viscoplastic constitutive relation including quasi-brittle damage. A dimensionless coefficient, which shows the characteristic damage within this regime, is taken as a perturbation parameter. A set of asymptotic equations is derived in terms of a regular perturbation expansion procedure. Asymptotic solutions are obtained for radial and angular variations of stresses and velocities with first- and second-order accuracy.

Singular perturbations for sensitivity analysis in symmetric bifurcation buckling

AbstractA direct procedure for the evaluation of imperfection‐sensitivity in bifurcation problems is presented. The problems arise in the context of the general theory of elastic stability (Koiter's theory) for discrete structural systems, in which the total potential energy is employed together with a stability criterion based on energy derivatives. The imperfection sensitivity of critical states, such as bifurcations and trifurcations, is usually represented as a plot of the critical load versus the amplitude ε of the imperfection considered. However, such plots have a singularity at the point with ε=0, so that a regular perturbation expansion of the solution is not possible. In this work, we describe a direct procedure to obtain the sensitivity of the critical load (eigenvalue of the bifurcation problem) and the sensitivity of the critical direction (eigenvector of the bifurcation problem) using singular perturbation analysis. The perturbation expansions are constructed as a power series in terms of the imperfection amplitude, in which the exponents and the coefficients are the unknowns of the problem. The solution of the exponents is obtained by means of trial and error using a least degenerate criterion, or by geometrical considerations. To compute the coefficients a detailed formulation is presented, which employs the conditions of equilibrium and stability at the critical state and their contracted forms. The formulation is applied to symmetric bifurcations, and the coefficients are solved up to third‐order terms in the expansion. The algorithmis illustrated by means of a simple example (a beam on an elastic foundation under axial load) for which the coefficients are computed and the imperfection‐sensitivity is plotted. Copyright © 2001 John Wiley &amp; Sons, Ltd.

A new approach to evaluate imperfection sensitivity in asymmetric bifurcation buckling analysis

A direct procedure for the evaluation of imperfection sensitivity in bifurcation problems is presented. The problems arise in the context of the general theory of elastic stability for discrete structural systems, in which the energy criterion of stability of structures and the total potential energy formulation are employed. In cases of bifurcation buckling the sensitivity of the critical load with respect to an imperfection parameter e is singular at the state given by epsilon =0, so that, a regular perturbation expansion of the solution is not possible. In this work we describe a direct procedure to obtain the relations between the critical loads, the generalized coordinates at the critical state, the eigenvector, and the amplitude of the imperfection, using singular perturbation analysis. The expansions are assumed in terms of arbitrary powers of the imperfection parameter, so that both exponents and coefficients of the expansion are unknown. The solution of the series exponents is obtained by searching the least degenerate solution. The formulation is here applied to asymmetric bifurcations, for which explicit expressions of the coefficients are obtained. The use of the method is illustrated by a simple example, which allows consideration of the main features of the formulation.