IMA Journal of Applied Mathematics - Advance Access

The discrete diffraction transform

Sedelnikov, I., Averbuch, A., Shkolnisky, Y. — 2008-04-29

In this paper, we define a discrete analogue of the continuous diffracted projection. We define the discrete diffraction transform (DDT) as a collection of the discrete diffracted projections (DDPs) taken at specific set of angles along specific set of lines. The ‘DDP’ is defined to be a discrete transform that is similar in its properties to the continuous diffracted projection. We prove that when the DDT is applied to a set of samples of a continuous object, it approximates a set of continuous vertical diffracted projections of a horizontally sheared object and a set of continuous horizontal diffracted projections of a vertically sheared object. A similar statement, where diffracted projections are replaced by the X-ray projections, that holds for the 2D discrete Radon transform (DRT), is also proved. We prove that the DDT is rapidly computable and invertible.

Dynamics of electrorheological clutch and a problem for non-linear parabolic equation with non-local boundary conditions

Litvinov, W. G. — 2008-03-28

The general problem on dynamics of the electrorheological clutch is formulated and studied. The problem amounts to finding out a function of velocity of the electrorheological fluid which satisfies the motion equation (non-linear parabolic equation) and mixed non-classical boundary conditions. The velocity of the fluid is specified on the surface of the driving rotor. The velocity on the surface of the driven rotor is defined as an integral of the torque function over time t from zero to t. The torque function is computed upon integrating the shear stresses (non-linear functions of derivatives of the velocity) over the surface of the driven rotor. Approximate problem is formulated in a form of a problem with a delay. It is proved the existence and the uniqueness of the solution of the initial and approximate problems and the convergence of the solutions of the approximate problem to the solution of the initial problem as the parameter of delay tends to zero.

Wavefronts for a non-local reaction-diffusion population model with general distributive maturity

Weng, P., Wu, J. — 2008-03-28

We consider the Al-Omari and Gourley non-local reaction–diffusion population model with distributed maturity. Existence of monotone wavefronts for some particular probability distribution of maturity that permits the linear chain trick was previously obtained; here, we consider the most general form of such a distribution using some comparison arguments for abstract functional differential equations with infinite delay and a fixed point approach combined with the upper and lower solutions technique.

Trapped modes in 3D topographically varying plates

Postnova, J., Craster, R. V. — 2008-03-18

Trapped modes in 3D elastic plates are considered as a model of waves that are guided along, and localized to the vicinity of, welds. These waves propagate unattenuated along the weld and exponentially decay with distance transverse to it. In the direction of propagation (y), there is no change in geometry and we assume that waves have the form exp(iβy). An asymptotic long-wave theory provides numerical values of the trapped mode frequencies and gives conditions at which trapping can occur; these depend on the components of the wave number in different directions and variations of the plate thickness. The results of this long-wave theory are compared with a numerical solution of the full governing equations.

Homogenization of the 1D Vlasov-Maxwell equations

Bostan, M. — 2008-03-03

In this paper, we investigate the homogenization of the 1D Vlasov–Maxwell system. We indicate the rate of convergence towards the limit solution. In the non-relativistic case, we compute explicitly the limit solution. The theoretical results are illustrated by some numerical simulations.

An improved multimodal approach for non-uniform acoustic waveguides

Hazard, C., Luneville, E. — 2008-02-29

This paper explores from the point of view of numerical analysis a method for solving the acoustic time-harmonic wave equation in a locally non-uniform 2D waveguide. The multimodal method is based on a spectral description of the acoustic field in each transverse section of the guide, using Fourier-like series. In the case of sound-hard boundaries, the weak point of the method lies in the poor convergence of such series due to a poor approximation of the field near the boundaries. A remedy was proposed by Athanassoulis & Belibassakis (1999, J. Fluid Mech., 389, 275–301), where the convergence is improved thanks to the use of enhanced series. The present paper proposes a theoretical analysis of their idea and studies further improvements. For a general model which includes different kinds of non-uniformity (varying cross section, bend and inhomogeneous medium), a hybrid spectral/variational formulation is introduced. Error estimates are provided for a semi-discretized problem which concerns the effect of spectral truncation. These estimates are confirmed by numerical results.

Spreading speeds and travelling waves in a delayed population model with stage structure on a 2D spatial lattice

Cheng, C.-P., Li, W.-T., Wang, Z.-C. — 2008-02-27

In this paper, we derive a lattice model for a single species in a 2D patchy environment with infinite number of patches connected locally by diffusion and global interaction by delay. The important feature of the model is the reflection of the joint effect of the diffusion dynamics, the non-local delayed effect and the direction of propagation. We study the well-posedness of the initial-value problem and establish the existence of monotone travelling waves for wave speed c ≥ c_*() > 0, where is any fixed direction of propagation. In particular, we show that the minimal wave speed c_*() coincides with the asymptotic speed of spread for any fixed direction . Moreover, we find that the asymptotic speed of spread depends on not only the maturation period and the diffusion rate of mature population monotonically but also the direction of propagation, which is different from the case when the spatial variable is continuous.

A potential multipole theory for the hydrodynamics of bubble clouds

Wilson, M., Blake, J. R., Haese, P. M. — 2008-01-11

The occurrence of bubbles in clouds, or assemblies, is observed in a wide range of fluid flows. In an otherwise quiescent fluid, their free rise due to buoyancy may lead to transient planar clusters and ultimately to non-spherical bubble behaviour. This paper concentrates on the application of multipole methods to study the fluid dynamics of such clouds. Under the assumptions of an irrotational flow, the boundary-value problem for the fluid velocity potential is determined from the requirement that the fluid kinetic energy, in suitable integral form, is stationary with respect to a small variation in the potential. Two alternative formulations for the bubble dynamics are then presented. The first relates the trajectories of each spherical bubble to their driving forces through the Kelvin impulse. The coupled two-phase flow equations are posed in a convenient matrix form, and solved numerically to predict the transient behaviour of the assembly. To model the more general free-surface flow, a second novel approach uses a weighted residual method. Simulations on the rise of small spherical, buoyant, bubbles indicate that compact, transient, clusters are formed in a short length of time. As non-uniformities in the bubble radii strongly influence the interaction, the smaller bubbles tend to streamline around the larger ones, leading to dispersion.

Analysis of a 1D incompressible two-fluid model including artificial diffusion

Holmas, H., Sira, T., Nordsveen, M., Langtangen, H. P., Schulkes, R. — 2008-01-07

This article examines a 1D incompressible two-fluid model including artificial tensor diffusion. The aim is to obtain a formulation that provides convergent numerical solutions for all flow conditions within the stratified and the stratified wavy flow regime. With appropriate simplifications, the two-fluid model reduces to one momentum balance, one mass conservation and two algebraic equations. It has previously been established that a formulation that is well posed in possessing exclusively real characteristics can be obtained by including an axial diffusion term in the momentum balance. In this article, however, we demonstrate that this is not sufficient to obtain a system suitable for numerical simulations. Although the unbounded growth rates of the standard two-fluid model are eliminated, linear stability theory predicts that infinitesimal wavelengths still experience finite growth. This entails that grid refinement always will result in new unstable wavelengths being resolved. On the other hand, if artificial axial diffusion is added to both the mass and the momentum equations as suggested here, a cut-off wavelength is established below which all wavelengths are stable. Thus, a numerically converging model is formed, which retains the long-wavelength properties of the standard two-fluid model. The conclusions of the mathematical analysis are substantiated by numerical simulations of 1D gravity waves.

A procedure for the reconstruction of a stochastic stationary temperature field

Johansson, B. T. — 2007-12-14

An iterative procedure is proposed for the reconstruction of a temperature field from a linear stationary heat equation with stochastic coefficients, and stochastic Cauchy data given on a part of the boundary of a bounded domain. In each step, a series of mixed well-posed boundary-value problems are solved for the stochastic heat operator and its adjoint. Well-posedness of these problems is shown to hold and convergence in the mean of the procedure is proved. A discretized version of this procedure, based on a Monte Carlo Galerkin finite-element method, suitable for numerical implementation is discussed. It is demonstrated that the solution to the discretized problem converges to the continuous as the mesh size tends to zero

Experimental evidence of non-unique solutions to a steady non-linear coating flow

Marston, J. O., Decent, S. P., Simmons, M. J. H. — 2007-12-04

Experimental observations of a coating flow are described, including evidence of a new qualitative feature, namely, that the steady stable flow field is not always unique for certain fixed values of the physical parameters. Since the mathematical modelling of these processes is the subject of some controversy, this potentially provides a qualitative test of competing proposed models.

On the method of conformal mapping applied to the flow around thin cylindrically curved aerofoils

Visser, F. C. — 2007-12-03

This article discusses the method of conformal mapping applied to the flow around thin cylindrically curved aerofoils. The velocity distribution of the flow around this type of lifting vanes is solved in closed form and expressions for the circulation are presented. The results are used to compute the lift exerted on the profiles at zero and non-zero incidence angle. Furthermore, the pressure distribution and profile loading are discussed. The treatise given is a classical one, and the formulae presented provide an extension to some famous results from the theory of aerofoils.

Determination of unknown coefficient in a non-linear elliptic problem related to the elastoplastic torsion of a bar

Hasanov, A., Erdem, A. — 2007-11-17

The inverse problem of determining the unknown coefficient of the non-linear differential equation of torsional creep is studied. The unknown coefficient g = g(²) depends on the gradient : = |u| of the solution u(x), x Rⁿ, of the direct problem. It is proved that this gradient is bounded in C-norm. This permits one to choose the natural class of admissible coefficients for the considered inverse problem. The continuity in the norm of the Sobolev space H¹() of the solution u(x;g) of the direct problem with respect to the unknown coefficient g = g(²) is obtained in the following sense: ||u(x;g) – u(x;g_m)||₁ -> 0 when g_m() -> g() point-wise as m -> . Based on these results, the existence of a quasi-solution of the inverse problem in the considered class of admissible coefficients is obtained. Numerical examples related to determination of the unknown coefficient are presented.