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1998

Volume 58, Issue 2, pp. 345-723


A Blunt-Nosed Thin Body in Hypersonic Flow

Julian D. Cole, Norman D. Malmuth, and Oleg S. Ryzhov

SIAM J. Appl. Math. 58, pp. 345-369 (25 pages)

Online Publication Date: July 26, 2006

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An inverse problem is posed to simulate the influence of the nose bluntness on the hypersonic inviscid steady flow field around a slender two-dimensional section. In this formulation a bow shock is given in advance whereas the body contour comes at the end of analysis being identified with a particular streamline. An equation for the shock shape involves two terms in the form of different powers of the distance measured along the direction of oncoming stream. The leading term derives from a classical similar solution for the strong viscous/inviscid interaction regime; a correction to it models bluntness effects at large distances downstream.
Matched asymptotic expansions are used to solve the problem within the hypersonic small-disturbance theory. In the outer region bounded by the shock, two sets of ordinary differential equations control the pressure, density, and velocity distributions. The second-order approximation admits of an explicit integral obtainable from a consideration of the finite drag exerted on the blunted nose of a section. The use of the momentum conservation law allows us to predict the power of the exponent of the correction term entering the shock equation. The asymptotic behavior of both first- and second-order approximations is established and employed for providing conditions for a solution in the inner region occupied by a high-entropy layer. Governing equations here are solved, explicitly determining a dependence of the body shape on the correction in the shock representation.
A thorough analysis of the Newtonian approach reveals certain limitations inherent in this simplified treatment of steady hypersonic flows.

An Analytic Solution for Low-Frequency Scattering by Two Soft Spheres

A. Charalambopoulos, G. Dassios, and M. Hadjinicolaou

SIAM J. Appl. Math. 58, pp. 370-386 (17 pages) | Cited 3 times

Online Publication Date: July 26, 2006

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A plane wave is scattered by two small spheres of not necessarily equal radii. Low-frequency theory reduces this scattering problem to a sequence of potential problems which can be solved iteratively. It is shown that there exists exactly one bispherical coordinate system that fits the given geometry. Then R-separation is utilized to solve analytically the potential problems governing the leading two low-frequency approximations. It is shown that the Rayleigh approximation is azimuthal independent, while the first-order approximation involves the azimuthal angle explicitly. The leading two nonvanishing approximations of the normalized scattering amplitude as well as the scattering cross-section are also provided. The Rayleigh approximations for the amplitude and for the cross-section involve only a monopole term, while their next order approximations are expressed in terms of a monopole as well as a dipole term. The dipole term disappears whenever the two spheres become equal, and this observation provides a way to determine whether the two spheres are equal or not, from far-field measurements. Finally, it is shown that for all practical purposes, first-order multiple scattering yields an excellent approximation of this scattering process.

Wetting Fronts in One-Dimensional Periodically Layered Soils

George Fennemore and Jack X. Xin

SIAM J. Appl. Math. 58, pp. 387-427 (41 pages) | Cited 3 times

Online Publication Date: July 26, 2006

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We study wetting front (traveling wave) solutions to the Richards equation that describe the vertical infiltration of water through one-dimensional periodically layered unsaturated soils. We prove the existence, uniqueness, and large time asymptotic stability of the traveling wave solutions under prescribed flux boundary conditions and certain constitutive conditions. The traveling waves are connections between two steady state solutions that form near the ground surface and towards the underground water table. We found a closed form expression of the wave speed. The speed of a traveling wave is equal to the ratio of the flux difference and the difference of the spatial averages of the two steady states. We give both analytical and numerical examples showing that the wave speeds in the periodic soils can be larger or smaller than those in the homogeneous soils which have the same mean diffusivity and conductivity. In our examples, if the phases of inhomogeneities in diffusivity and conductivity functions differ by half the period, then the periodic soils speed up the waves; if the phases are the same, then the periodic soils slow down the waves. We also present numerical solutions to the Richards equation using the finite difference method in regimes where our constitutive conditions are no longer valid, and we observe similar stable fronts.

Crack Tip Interpolation, Revisited

Glaucio H. Paulino and L. J. Gray

SIAM J. Appl. Math. 58, pp. 428-455 (28 pages) | Cited 18 times

Online Publication Date: July 26, 2006

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It is well known that the near tip displacement field on a crack surface can be represented in a power series in the variable $\sqrt r$, where r is the distance to the tip. It is shown herein that the coefficients of the linear terms on the two sides of the crack are equal. Equivalently, the linear term in the crack opening displacement vanishes. The proof is a completely general argument, valid for an arbitrary (e.g., multiple, nonplanar) crack configuration and applied boundary conditions. Moreover, the argument holds for other equations, such as Laplace. A limit procedure for calculating the surface stress in the form of a hypersingular boundary integral equation is employed to enforce the boundary conditions along the crack faces. Evaluation of the finite surface stress and examination of potentially singular terms lead to the result. Inclusion of this constraint in numerical calculations should result in a more accurate approximation of the displacement and stress fields in the tip region, and thus a more accurate evaluation of stress intensity factors.

The Dynamics of Thin Films I: General Theory

M. J. Miksis and M. P. Ida

SIAM J. Appl. Math. 58, pp. 456-473 (18 pages) | Cited 13 times

Online Publication Date: July 26, 2006

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The dynamics of a general three-dimensional thin film subject to van der Waals forces, surface tension, and surfactants is considered. Using an asymptotic analysis based upon the thinness of the film with respect to its lateral extent, evolution equations for the leading-order film thicknesses, tangential velocities, and surfactant concentrations are obtained.

The Dynamics of Thin Films II: Applications

M. J. Miksis and M. P. Ida

SIAM J. Appl. Math. 58, pp. 474-500 (27 pages) | Cited 8 times

Online Publication Date: July 26, 2006

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A system of equations for the dynamics of a general three-dimensional thin film subject to van der Waals forces, surface tension, and surfactants was derived in Part I of this work [M. P. Ida and M. J. Miksis, SIAM J. Appl. Math., 58 (1998), pp. 456--473]. Here we apply these equations to free films with various geometries. A linear and near-linear stability analysis and numerical simulations are done to the specialized equations.

Network Approximation for Transport Properties of High Contrast Materials

George C. Papanicolaou and Liliana Borcea

SIAM J. Appl. Math. 58, pp. 501-539 (39 pages) | Cited 19 times

Online Publication Date: July 26, 2006

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We show that the effective complex impedance of materials with conductivity and dielectric permittivity that have high contrast can be calculated approximately by solving a suitable resistor-capacitor network. We use variational principles extensively for the analysis and we assess the accuracy of the network approximation by numerical computations.

Large-Scale Instability of Generalized Oscillating Kolmogorov Flows

Alexander L. Frenkel and Xiaojing Zhang

SIAM J. Appl. Math. 58, pp. 540-564 (25 pages) | Cited 2 times

Online Publication Date: July 26, 2006

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The stability of an incompressible unidirectional flow that depends periodically, but otherwise arbitrarily, on a transverse coordinate and on time is considered. An iterative solution of an infinite-dimensional eigenvalue problem is constructed by a rigorous perturbation method. The critical Reynolds number Rc and the critical direction for which the large-scale "eddy viscosity" is minimum (and equal to zero) are determined by a system of two algebraic equations. For both time-independent and time-dependent cases, it turns out that the fastest-growing critical disturbances generally do not have the same transverse periodicity as that of basic flow. In the limit of large frequencies of oscillation, stability is essentially determined by the time-averaged flow. When the latter vanishes, the flow is absolutely stable for sufficiently large frequencies.

Computable Elastic Distances Between Shapes

Laurent Younes

SIAM J. Appl. Math. 58, pp. 565-586 (22 pages) | Cited 39 times

Online Publication Date: July 26, 2006

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We define distances between geometric curves by the square root of the minimal energy required to transform one curve into the other. The energy is formally defined from a left invariant Riemannian distance on an infinite dimensional group acting on the curves, which can be explicitly computed. The obtained distance boils down to a variational problem for which an optimal matching between the curves has to be computed. An analysis of the distance when the curves are polygonal leads to a numerical procedure for the solution of the variational problem, which can efficiently be implemented, as illustrated by experiments.

Motion and Homogenization of Vortices in Anisotropic Type II Superconductors

G. Richardson and S. J. Chapman

SIAM J. Appl. Math. 58, pp. 587-606 (20 pages) | Cited 2 times

Online Publication Date: July 26, 2006

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The motion of vortices in an anisotropic superconductor is considered. For a system of well-separated vortices, each vortex is found to obey a law of motion analogous to the local induction approximation, in which velocity of the vortex depends upon the local curvature and orientation. A system of closely packed vortices is then considered, and a mean field model is formulated in which the individual vortex lines are replaced by a vortex density.

Analysis of a Class of Models of Bursting Electrical Activity in Pancreatic $\beta$-Cells

Gerda de Vries and Robert M. Miura

SIAM J. Appl. Math. 58, pp. 607-635 (29 pages)

Online Publication Date: July 26, 2006

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Many models of bursting electrical activity (BEA) in pancreatic $\beta$-cells have been proposed. BEA is characterized by a periodic oscillation of the membrane potential consisting of a silent phase during which the membrane potential is varying slowly and an active phase during which the membrane potential is undergoing rapid oscillations. An important experimental observation of BEA is a correlation between the rate of insulin release from $\beta$-cells and the plateau fraction as a function of glucose concentration. The plateau fraction is the ratio of the duration of the active phase to the period of BEA. In [SIAM J. Appl. Math., 52 (1992), pp. 1627--1650], Pernarowski, Miura, and Kevorkian develop analytical techniques to determine the leading-order plateau fraction for one of the models, namely, the Sherman--Rinzel--Keizer (SRK) model [ Biophys. J., 54 (1988), pp. 411--425]. Applicability of these techniques depends critically on the fact that the fast subsystem of the SRK model is an integrable system to leading order. In this paper, we extend the techniques of Pernarowski, Miura, and Kevorkian to a class of models of BEA, namely, those first-generation models consisting of three first-order ordinary differential equations. We show that the fast subsystem of these models can be reformulated as an integrable system to leading order. The relative ease with which this reformulation can be done depends on a biological property of the models, namely, the value of the integer exponent of the activation variable in the description of the voltage-gated K+ current.

Riemann Problems for the Two-Dimensional Unsteady Transonic Small Disturbance Equation

Barbara Lee Keyfitz and Suncica Canic

SIAM J. Appl. Math. 58, pp. 636-665 (30 pages) | Cited 4 times

Online Publication Date: July 26, 2006

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We study a two-parameter family of Riemann problems for the unsteady transonic small disturbance (UTSD) equation, also called the two-dimensional Burgers equation, which is used to model the transition from regular to Mach reflection for weak shock waves. The related initial-value problem consists of oblique shock data in the upper half-plane, with two parameters a and b corresponding to the slopes of the initial shock waves. The study of quasi-steady solutions leads to a problem that changes type when written in self-similar coordinates. The problem is hyperbolic in the region where the flow is supersonic, and elliptic where the flow is subsonic.
In this paper we give a complete description of the flow in the hyperbolic region by resolving the hyperbolic wave interactions in the form of quasi-one-dimensional Riemann problems. In the region of physical space where the flow is subsonic, we pose the related free-boundary problems and discuss the behavior of the subsonic solution using results from our previous work. Based on this approach we establish the existence of regions of different qualitative behavior in parameter (a, b) space. Our results reveal that the UTSD equation seems to be particularly suitable for the study of the so-called von Neumann paradox in which linearly degenerate waves can be ignored. We establish the region in the parameter space where a prototype of von Neumann reflection takes place. In other regions of parameter space we find prototypes for Mach reflection, regular reflection, and transitional Mach reflection. The lack of linearly degenerate waves in this model is resolved by the presence of a small rarefaction wave emerging from the triple point.

On a Concept of Uniqueness in Inverse Scattering for a Finite Number of Incident Waves

Roland Potthast

SIAM J. Appl. Math. 58, pp. 666-682 (17 pages) | Cited 8 times

Online Publication Date: July 26, 2006

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We introduce a concept of uniqueness in inverse scattering for a finite number of incident plane waves. Our results hold for inverse acoustic scattering from a sound-soft obstacle, a sound-hard obstacle, or an inhomogeneous medium. It is shown that for $\epsilon>0$ we can find an $N\in \nat$ such that a knowledge of the far field for N incident waves determines the support of the scatterer up to an error of size $\epsilon$ in the Hausdorff distance. In the limit $N\rightarrow \infty$ we obtain well-known uniqueness theorems which hold if the far field for all incident plane waves is known. Our results contain a new proof for the uniqueness of the support of an inhomogeneous medium in this case.

Spectral Properties of Classical Waves in High-Contrast Periodic Media

A. Figotin and P. Kuchment

SIAM J. Appl. Math. 58, pp. 683-702 (20 pages) | Cited 25 times

Online Publication Date: July 26, 2006

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We introduce and investigate the band gap structure of the frequency spectrum for classical electromagnetic and acoustic waves in a high-contrast, two-component periodic medium. The asymptotics with respect to the high-contrast is considered. The limit medium is described in terms of appropriate self-adjoint operators and the convergence to the limit is proven. These limit operators give an idea of the spectral structure and suggest new numerical approaches as well. The results are obtained in arbitrary dimension and for rather general geometry of the medium. In particular, two-dimensional (2D) photonic band gap structures and their acoustic analogues are covered.

The Global Behavior of Elastoplastic and Viscoelastic Materials with Hysteresis-Type State Equations

Ivan G. Götz, Karl-Heinz Hoffmann, and Robert S. Anderssen

SIAM J. Appl. Math. 58, pp. 703-723 (21 pages) | Cited 1 time

Online Publication Date: July 26, 2006

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A one-dimensional model is derived in order to study how the elasticity (internal elastic energy) of viscoelastic and elastoplastic materials, such as biopolymers (muscles and grain flour dough) or metals, changes due to the action of external forces. For such materials, the model takes the form of an initial-boundary value problem, corresponding to Newton's second law, which is coupled to an auxiliary (stress-strain) state equation which characterizes the nature of the interaction between the material and the external forces. In the oscillatory loading of muscles and the mixing of grain flour, as well as of the fatiguing of metals, the state equation must model how the stress depends on the earlier history of the strain as well as describe how the material gains or loses elastic energy due to the action of the loading. One is thereby led to model the auxiliary stress-strain relationship as a constitutive relationship involving a Duhem--Madelung hysteresis operator.
As well as discussing the formulation of such models along with the properties of Duhem--Madelung hysteresis operators, this paper examines the existence and uniqueness for the solutions of such coupled systems. In addition, some global estimates are derived for these solutions, and their asymptotic behavior, as the time increases, is studied under the assumption that a part of the internal (elastic) energy dissipates during the interaction and, hence, the associated Duhem--Madelung hysteron has negative spin.
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