Showing papers on "Method of matched asymptotic expansions published in 2019"

First analytical calculation of black hole shadow in McVittie metric

Abstract: Cosmic expansion influences the angular size of black hole shadow. The most general way to describe a black hole embedded into an expanding universe is to use the McVittie metric. So far, the exact analytical solution for the shadow size in the McVittie metric, valid for arbitrary law of expansion and arbitrary position of the observer, has not been found. In this paper, we present the first analytical solution for angular size of black hole shadow in McVittie metric as seen by observer comoving with the cosmic expansion. We use a method of matched asymptotic expansions to find approximate solution valid within the entire range of possible positions of observer. As two particular examples, we consider black hole in de Sitter and matter dominated universe.

Viscoplastic fluids in 2D plane squeeze flow: A matched asymptotics analysis

Abstract: A matched asymptotic expansions approach is used to determine the flow behaviour of Casson and Herschel–Bulkley fluids between two parallel plates that are approaching each other with a constant velocity. The present study is based on the earlier work of Muravleva (2015), who has analyzed the squeeze flow of a Bingham fluid using the method of matched asymptotic expansions. A naive application of classical lubrication theory leads to a kinematic inconsistency in the predicted plug region - the well known “squeeze flow paradox” for a viscoplastic fluid. The objective of this work is to determine a consistent solution for the aforementioned constitutive equations. Based on the technique of matched asymptotic expansions, the solution is formulated in terms of separate expansions in the regions adjacent to the two plates where the shear stress is dominant, and a central pseudo-plug (plastic) region where the normal stresses become comparable to the shear stress; the two regions being separated by a pseudo-yield surface. In this manner, a complete asymptotic solution is developed for the squeeze flow of both Casson and Herschel–Bulkley fluid models. Using this solution, we derive expressions for the velocity, pressure and stress fields, and the squeeze force acting to retard the plates. The effect of the yield threshold on the pseudo-yield surface that separates the sheared and plastic zones, pressure distribution and squeeze force is investigated.

Topological Derivatives of Shape Functionals. Part I: Theory in Singularly Perturbed Geometrical Domains

Abstract: Mathematical analysis and numerical solutions of problems with unknown shapes or geometrical domains is a challenging and rich research field in the modern theory of the calculus of variations, partial differential equations, differential geometry as well as in numerical analysis. In this series of three review papers, we describe some aspects of numerical solution for problems with unknown shapes, which use tools of asymptotic analysis with respect to small defects or imperfections to obtain sensitivity of shape functionals. In classical numerical shape optimization, the boundary variation technique is used with a view to applying the gradient or Newton-type algorithms. Shape sensitivity analysis is performed by using the velocity method. In general, the continuous shape gradient and the symmetric part of the shape Hessian are discretized. Such an approach leads to local solutions, which satisfy the necessary optimality conditions in a class of domains defined in fact by the initial guess. A more general framework of shape sensitivity analysis is required when solving topology optimization problems. A possible approach is asymptotic analysis in singularly perturbed geometrical domains. In such a framework, approximations of solutions to boundary value problems (BVPs) in domains with small defects or imperfections are constructed, for instance by the method of matched asymptotic expansions. The approximate solutions are employed to evaluate shape functionals, and as a result topological derivatives of functionals are obtained. In particular, the topological derivative is defined as the first term (correction) of the asymptotic expansion of a given shape functional with respect to a small parameter that measures the size of singular domain perturbations, such as holes, cavities, inclusions, defects, source terms and cracks. This new concept of variation has applications in many related fields, such as shape and topology optimization, inverse problems, image processing, multiscale material design and mechanical modeling involving damage and fracture evolution phenomena. In the first part of this review, the topological derivative concept is presented in detail within the framework of the domain decomposition technique. Such an approach is constructive, for example, for coupled models in multiphysics as well as for contact problems in elasticity. In the second and third parts, we describe the first- and second-order numerical methods of shape and topology optimization for elliptic BVPs, together with a portfolio of applications and numerical examples in all the above-mentioned areas.

Singularly Perturbed Vector Field Method (SPVF) Applied to Combustion of Monodisperse Fuel Spray

Abstract: In this paper we present the concept of singularly perturbed vector field (SPVF) method, and its application to thermal explosion of diesel spray combustion. Given a system of governing equations, which consist of hidden Multi-scale variables, the SPVF method transfer and decompose such system to fast and slow singularly perturbed subsystems. The resulting subsystem enables us to understand better the complex system, and to simplify the calculations. Later powerful analytical, numerical and asymptotic methods [e.g method of integral (invariant) manifold, the homotopy analysis method etc.] can be applied to each subsystem. In this paper, we compare the results obtained by the methods of integral invariant manifold and SPVF as applied to the spray droplets combustion model.

Droplet dynamics on heterogeneous surfaces

Abstract: The motion of a liquid drop over solid surfaces is easy to visualise, yet, from a scientific standpoint is inherently challenging to study. This arises from the multi-scale nature of the governing physics, including gravity and capillarity in the macro-scale, and slip close to the contact line. This thesis studies droplets through a combined numerical and analytical approach to extract physical insights in complex scenarios. Using the lubrication approximation, the Stokes equations are combined with the appropriate boundary conditions to derive a non-linear partial differential equation for the fluid thickness. To determine how the droplet evolves in time, we develop solution methods to the full equation using a pseudospectral collocation approach in both two-, and three-dimensional settings. Using the boundary integral formulation we also develop a hybrid method which is combined with the analysis to offer an attractive compromise between the low-order models and full-scale computing. Analytical progress is made in the slow spreading and negligible gravity regime by utilising the method of matched asymptotic expansions which has been successful in related works to derive low-order approximate models that predict the solutions of the full equations. Specifically, we consider droplets spreading over flat and horizontal substrates with mass transfer that may occur at free surface, or by evaporation which is maximised close to the contact line. Extensions are also made by considering topographically varying substrates with sufficiently small amplitudes. The outcomes of the analysis are contrasted to simulations of the governing equation for a number of cases. We present convincing numerical evidence that suggest that the reduced models can replace the full model within their domain of validity, and thus mitigate considerably the associated high computational costs required for such simulations, at the same time, uncover experimentally observed phenomena, such as pinning, stick-slip, and hysteresis-type effects induced through surface features.

Multi-point Monin–Obukhov similarity in the convective atmospheric surface layer using matched asymptotic expansions

Abstract: The multi-point Monin–Obukhov similarity (MMO) was recently proposed (Tong & Nguyen, J. Atmos. Sci., vol. 72, 2015, pp. 4337–4348) to address the issue of incomplete similarity in the framework of the original Monin–Obukhov similarity theory (MOST). MMO hypothesizes the following: (1) The surface-layer turbulence, defined to consist of eddies that are entirely inside the surface layer, has complete similarity, which however can only be represented by multi-point statistics, requiring a horizontal characteristic length scale (absent in MOST). (2) The Obukhov length is also the characteristic horizontal length scale; therefore, all surface-layer multi-point statistics, non-dimensionalized using the surface-layer parameters, depend only on the height and separations between the points, non-dimensionalized using . However, similar to MOST, MMO was also proposed as a hypothesis based on phenomenology. In this work we derive MMO analytically for the case of the horizontal Fourier transforms of the velocity and potential temperature fluctuations, which are equivalent to the two-point horizontal differences of these variables, using the spectral forms of the Navier–Stokes and the potential temperature equations. We show that, for the large-scale motions (wavenumber ) in a convective surface layer, the solution is uniformly valid with respect to (i.e. as decreases from to ), where is the height from the surface. However, for the solution is not uniformly valid with respective to as it increases from to , resulting in a singular perturbation problem, which we analyse using the method of matched asymptotic expansions. We show that (1) is the characteristic horizontal length scale, and (2) the Fourier transforms satisfy MMO with the non-dimensional wavenumber as the independent similarity variable. Two scaling ranges, the convective range and the dynamic range, discovered for in Tong & Nguyen (2015) are obtained. We derive the leading-order spectral scaling exponents for the two scaling ranges and the corrections to the scaling ranges for finite ratios of the length scales. The analysis also reveals the dominant dynamics in each scaling range. The analytical derivations of the characteristic horizontal length scale ( ) and the validity of MMO for the case of two-point horizontal separations provide strong support to MMO for general multi-point velocity and temperature differences.

Diffusiophoresis of a Colloidal Cylinder at Small Finite Péclet Numbers

Abstract: The diffusiophoretic migration of a circular cylindrical particle in a nonelectrolyte solution with a solute concentration gradient normal to its axis is analytically studied for a small but finite Peclet number P e . The interfacial layer of interaction between the solute molecules and the particle is taken to be thin, but the polarization of its mobile molecules is allowed. Using a method of matched asymptotic expansions, we solve the governing equations of conservation of the system and obtain an explicit formula for the diffusiophoretic velocity of the cylinder correct to the order P e 2 . It is found that the perturbed solute concentration and fluid velocity distributions have the order P e , but the leading correction to the particle velocity has the higher order P e 2 ln P e . The correction to the particle velocity to the order P e 2 can be either positive or negative depending on the polarization parameter of the thin interfacial layer, establishing that the solute convection effect is complicated and can enhance or retard the diffusiophoretic motion. The particle velocity at P e = 0.6 can be about 17% smaller or 0.2% greater than that at P e = 0 . Under practical conditions, the solute convection effect on the diffusiophoretic velocity is much greater for a cylindrical particle than for a spherical particle, whose leading correction has the order P e 2 .

Trapped modes in thin and infinite ladder like domains. Part 2: Asymptotic analysis and numerical application

Abstract: We are interested in a 2D propagation medium obtained from a localized perturbation of a reference homogeneous periodic medium. This reference medium is a " thick graph " , namely a thin structure (the thinness being characterized by a small parameter e > 0) whose limit (when e tends to 0) is a periodic graph. The perturbation consists in changing only the geometry of the reference medium by modifying the thickness of one of the lines of the reference medium. In the first part of this work, we proved that such a geometrical perturbation is able to produce localized eigenmodes (the propagation model under consideration is the scalar Helmholtz equation with Neumann boundary conditions). This amounts to solving an eigenvalue problem for the Laplace operator in an unbounded domain. We used a standard approach of analysis that consists in (1) find a formal limit of the eigenvalue problem when the small parameter tends to 0, here the formal limit is an eigenvalue problem for a second order differential operator along a graph; (2) proceed to an explicit calculation of the spectrum of the limit operator; (3) deduce the existence of eigenvalues as soon as the thickness of the ladder is small enough. The objective of the present work is to complement the previous one by constructing and justifying a high order asymptotic expansion of these eigenvalues (with respect to the small parameter e) using the method of matched asymptotic expansions. In particular, the obtained expansion can be used to compute a numerical approximation of the eigenvalues and of their associated eigenvectors. An algorithm to compute each term of the asymptotic expansion is proposed. Numerical experiments validate the theoretical results.

Modes transformation for a Schroedinger type equation: avoided and unavoidable level crossings.

Abstract: An asymptotic approach for a Schroedinger type equation with a non selfadjoint slowly varying Hamiltonian of a special type is developed. The Hamiltonian is assumed to be the result of a small perturbation of an operator with a twofold degeneracy (turning) point, which can be diagonalized at this point. The non-adiabatic transformation of modes is studied in the case where two small parameters are dependent: the parameter characterizing an order of the perturbation is a square root of the adiabatic parameter. The perturbation of the Hamiltonian produces a close pair of simple degeneracy points. Two regimes of mode transformation for the Schroedinder type equation are identified: avoided crossing of eigenvalues, corresponding to complex degeneracy points, and an explicit unavoidable crossing (with real degeneracy points). Both cases are treated by a method of matched asymptotic expansions in the context of a unifying approach. An asymptotic expansion of the solution near a crossing point containing the parabolic cylinder functions is constructed, and the transition matrix connecting the coefficients of adiabatic modes to the left and to the right of the degeneracy point is derived. Results are illustrated by an example: fermion scattering governed by the Dirac equation.

On liners viscoacoustic impedance boundary conditions for an array of Helmholtz resonators in 3D.

Abstract: The present work deals with the resolution of the Linearized Navier-Stokes problem in a domain made of an array that consists into a repetition of elongated resonators connected to an half-space. We provide and justify a limit equivalent model which takes into account the presence of resonators array as an equivalent boundary condition. Our approach combines the method of matched asymptotic expansions and the method of periodic surface homogenization adapted to more than two scales, and a complete justification is included in the paper.

A dam break driven by a moving source: a simple model for a powder snow avalanche

Abstract: We study the two-dimensional, irrotational flow of an inviscid, incompressible fluid injected from a line source moving at constant speed along a horizontal boundary, into a second, immiscible, inviscid fluid of lower density. A semi-infinite, horizontal layer sustained by the moving source has previously been studied as a simple model for a powder snow avalanche, an example of an eruption current, Carroll et al. (Phys. Fluids, vol. 24, 2012, 066603). We show that with fluids of unequal densities, in a frame of reference moving with the source, no steady solution exists, and formulate an initial/boundary value problem that allows us to study the evolution of the flow. After considering the limit of small density difference, we study the fully nonlinear initial/boundary value problem and find that the flow at the head of the layer is effectively a dam break for the initial conditions that we have used. We study the dynamics of this in detail for small times using the method of matched asymptotic expansions. Finally, we solve the fully nonlinear free boundary problem numerically using an adaptive vortex blob method, after regularising the flow by modifying the initial interface to include a thin layer of the denser fluid that extends to infinity ahead of the source. We find that the disturbance of the interface in the linear theory develops into a dispersive shock in the fully nonlinear initial/boundary value problem, which then overturns. For sufficiently large Richardson number, overturning can also occur at the head of the layer.