Showing papers on "Method of matched asymptotic expansions published in 2011"

Thermodynamically Consistent, Frame Indifferent Diffuse Interface Models for Incompressible Two-Phase Flows with Different Densities

Abstract: A new diffuse interface model for a two-phase flow of two incompressible fluids with different densities is introduced using methods from rational continuum mechanics. The model fulfills local and global dissipation inequalities and is frame indifferent. Moreover, it is generalized to situations with a soluble species. Using the method of matched asymptotic expansions we derive various sharp interface models in the limit when the interfacial thickness tends to zero. Depending on the scaling of the mobility in the diffusion equation we either derive classical sharp interface models or models where bulk or surface diffusion is possible in the limit. In the latter case a new term resulting from surface diffusion appears in the momentum balance at the interface. Finally, we show that all sharp interface models fulfill natural energy inequalities.

The Stability and Dynamics of Localized Spot Patterns in the Two-Dimensional Gray–Scott Model

Abstract: The dynamics and stability of multispot patterns to the Gray–Scott (GS) reaction-diffusion model in a two-dimensional domain is studied in the singularly perturbed limit of small diffusivity $\varepsilon$ of one of the two solution components. A hybrid asymptotic-numerical approach based on combining the method of matched asymptotic expansions with the detailed numerical study of certain eigenvalue problems is used to predict the dynamical behavior and instability mechanisms of multispot quasi-equilibrium patterns for the GS model in the limit $\varepsilon\to 0$. For $\varepsilon\to 0$, a quasi-equilibrium k-spot pattern is constructed by representing each localized spot as a logarithmic singularity of unknown strength $S_j$ for $j=1,\ldots,k$ at an unknown spot location ${\mathbf x}_j\in \Omega$ for $j=1,\ldots,k$. A formal asymptotic analysis is then used to derive a differential algebraic ODE system for the collective coordinates $S_j$ and ${\mathbf x}_j$ for $j=1,\ldots,k$, which characterizes the slo...

Asymptotic Behavior of Generalized Functions

Abstract: Asymptotic Behavior of Generalized Functions: S-Asymptotics F'g Quasi-Asymptotics in F' Applications of the Asymptotic Behavior of Generalized Functions: Asymptotic Behavior of Solutions to Partial Differential Equations Asymptotics and Integral Transforms Summability of Fourier Series and Integrals.

Analysis of Wetting and Contact Angle Hysteresis on Chemically Patterned Surfaces

Abstract: Wetting and contact angle hysteresis on chemically patterned surfaces in two dimensions are analyzed from a stationary phase-field model for immiscible two phase fluids. We first study the sharp-interface limit of the model by the method of matched asymptotic expansions. We then justify the results rigorously by the $\Gamma$-convergence theory for the related variational problem and study the properties of the limiting minimizers. The results also provide a clear geometric picture of the equilibrium configuration of the interface. This enables us to explicitly calculate the total surface energy for the two phase systems on chemically patterned surfaces with simple geometries, namely the two phase flow in a channel and the drop spreading. By considering the quasi-static motion of the interface described by the change of volume (or volume fraction), we can follow the change-of-energy landscape which also reveals the mechanism for the stick-slip motion of the interface and contact angle hysteresis on the che...

Using reproducing kernel for solving a class of singularly perturbed problems

Abstract: This paper is concerned with a new algorithm for giving the analytical and approximate solutions of a class of partial differential equations with a singularly perturbed term in a reproducing kernel space. Two numerical examples are studied to demonstrate the accuracy of the present method. Results obtained by the method indicate the method is simple and effective.

Asymptotic behavior of positive solutions of sublinear differential equations of Emden-Fowler type

Abstract: This paper is concerned with asymptotic analysis of positive solutions of the second-order nonlinear differential equation (A)x^'^'(t)+q(t)@f(x(t))=0, where q:[a,~)->(0,~) is a continuous function which is regularly varying and @f:(0,~)->(0,~) is a continuous increasing function which is regularly varying of index @c@?(0,1). An application of the theory of regular variation gives the possibility of determining precise information about the asymptotic behavior at infinity of intermediate solutions of Eq. (A).

The asymptotic behavior of solutions to three-dimensional Navier–Stokes equations with nonlinear damping

Abstract: This paper concerns the asymptotic behavior of solutions to three-dimensional Navier–Stokes equations with nonlinear damping | u | β − 1 u . We first study the L 2 decay of weak solutions with β ≥ 10 / 3 by developing the classic Fourier splitting method. Second, for 7 / 2 ≤ β 5 , we prove the optimal upper bounds of the higher-order derivative of the strong solution by employing a new analysis technique. Finally, we investigate the asymptotic stability of the large solution to the system with β ≥ 7 / 2 under large initial perturbation.

On periodic solutions to singularly perturbed parabolic problems in the case of multiple roots of the degenerate equation

Abstract: For singularly perturbed parabolic problems, asymptotic expansions of time-periodic solutions with boundary layers in a neighborhood of interval’s endpoints are constructed and justified in the case where the degenerate equation has a double or a triple root.

Step-like contrast structure of singularly perturbed optimal control problem

Abstract: In this paper, the existence of step-like contrast structure for a class of singularly perturbed optimal control problem is shown by the contrast structure theory. By means of direct scheme of boundary function method, we construct the uniformly valid asymptotic solution for the singularly perturbed optimal control problem. Finally, an example is presented to show the result. 2000 Mathematics Subject Classification. 34B15; 34E15.

Stationary and slowly moving localised pulses in a singularly perturbed Brusselator model

Abstract: Recent attention has focused on deriving localised pulse solutions to various systems of reaction–diffusion equations. In this paper, we consider the evolution of localised pulses in the Brusselator activator–inhibitor model, long considered a paradigm for the study of non-linear equations, in a finite one-dimensional domain with feed of the inhibitor through the boundary and global feed of the activator. We employ the method of matched asymptotic expansions in the limit of small activator diffusivity and small activator and inhibitor feeds. The disparity of diffusion lengths between the activator and inhibitor leads to pulse-type solutions in which the activator is localised while the inhibitor varies on an O(1) length scale. In the asymptotic limit considered, the pulses become spikes described by Dirac delta functions and evolve slowly in time until equilibrium is reached. Such quasi-equilibrium solutions with N activator pulses are constructed and a differential-algebraic system of equations (DAE) is derived, characterising the slow evolution of the locations and the amplitudes of the pulses. We find excellent agreement for the pulse evolution between the asymptotic theory and the results of numerical computations. An algebraic system for the equilibrium pulse amplitudes and locations is derived from the equilibrium points of the DAE system. Both symmetric equilibria, corresponding to a common pulse amplitude, and asymmetric pulse equilibria, for which the pulse amplitudes are different, are constructed. We find that for a positive boundary feed rate, pulse spacing of symmetric equilibria is no longer uniform, and that for sufficiently large boundary flux, pulses at the edges of the pattern may collide with and remain fixed at the boundary. Lastly, stability of the equilibrium solutions is analysed through linearisation of the DAE, which, in contrast to previous approaches, provides a quick way to calculate the small eigenvalues governing weak translation-type instabilities of equilibrium pulse patterns.

Parameter uniform numerical method for singularly perturbed differential–difference equations with interior layers

Abstract: The present study is devoted to the numerical study of boundary value problems for singularly perturbed linear second-order differential–difference equations with a turning point. The points of the domain where the coefficient of the convection term in the singularly perturbed differential equation vanishes are known as the turning points. The solution of such type of differential equations exhibits boundary layer(s) or interior layer(s) behaviour depending upon the nature of the coefficient of convection term and the reaction term. In particular, this paper focuses on problems whose solution exhibits interior layers. In the development of numerical schemes for singularly perturbed differential–difference equations with a turning point, we use El-Mistikawy–Werle exponential finite difference scheme with some modifications. Some priori estimates have been established and parameter uniform convergence analysis of the proposed scheme is also discussed. Several examples are considered to demonstrate the perfo...

Dynamics of a Rational System of Difference Equations in the Plane

Abstract: We consider a rational system of first-order difference equations in the plane with four parameters such that all fractions have a common denominator. We study, for the different values of the parameters, the global and local properties of the system. In particular, we discuss the boundedness and the asymptotic behavior of the solutions, the existence of periodic solutions, and the stability of equilibria.

Local expansions of solutions to the fifth Painlevé equation

Abstract: In this work, the methods of power geometry are used to find asymptotic expansions of solutions to the fifth Painleve equation as x → 0 for all values of its four complex parameters. We obtain 30 families of expan� sions, of which 22 are obtained from published expan� sions of solutions to the sixth Painleve equation. Among the other eight families, one was previously known and two can be obtained from the expansions of solutions to the third Painleve equation. Three fami�

On singularly perturbed q-difference-differential equations with irregular singularity

Abstract: We study a q-analog of a singularly perturbed Cauchy problem with irregular singularity in the complex domain. We construct solutions of this problem that are holomorphic on open half-q-spirals. Using a version of a q-analog of the Malgrange---Sibuya theorem obtained by J.-P. Ramis, J. Sauloy, and C. Zhang, we show the existence of a formal power-series solution in the perturbation parameter which is the q-asymptotic expansion of these holomorphic solutions.

Asymptotic estimates of hydrodynamic loads in the early stage of water entry of a circular disk

Abstract: The hydrodynamic loads generated during the entry of a circular disk into deep water are evaluated with the help of the method of matched asymptotic expansions. It is assumed that the liquid is initially at rest and the disk is floating on the still liquid surface. Then the disk suddenly starts its downward motion. The study is carried out under the assumption of an ideal and incompressible liquid. Attention is focused on the initial stage of the entry process. The solution is sought in the form of an asymptotic expansion of the velocity potential with the non-dimensional displacement of the disk being a small parameter of the problem. Gravity and surface-tension effects are shown to be of minor significance. Owing to the flow singularity at the edge of the disk, an inner problem is formulated and its solution is matched with the second-order outer velocity potential to achieve a uniformly valid solution. It is shown that the initial asymptotics of the hydrodynamic loads involves terms with \({h^{-\frac{1}{3}}}\) and log h where h(t) is the non-dimensional displacement of the disk. Both terms are unbounded in the limit of small penetration depth of the disk. The theoretical estimates are validated versus fully nonlinear numerical simulations of the problem during the later stage of the process. It is shown that the derived asymptotic estimates remain accurate, even for moderate displacements of the disk. The relative difference between the theoretical estimate of the hydrodynamic force and its numerical prediction is less than 5% when the penetration depth is smaller than \({\frac{1}{20}}\) of the disk radius. A way to use the theoretical estimates for practical applications is proposed and comparisons with experimental data available in the literature are also presented.

Hierarchy of the models of classical mechanics of inhomogeneous fluids

Abstract: The methods of perturbation theory and integral representations are used to analyze the general properties of a system of equations of the mechanics of inhomogeneous fluids including the equations of momentum, mass, and temperature transfer. We also consider various submodels of this system, including the reduced systems in which some kinetic coefficients are equal to zero and degenerate systems in which the variations of density or some other variables are neglected. We analyze both regularly perturbed and singularly perturbed solutions of the system. In the case of reduction or degeneration of solutions, the order of the system decreases. In this case, regularly perturbed solutions are preserved (with certain modifications) but the number of singularly perturbed components participating in the formation of the boundary layers on contact surfaces and their analogs in the bulk of the fluid, i.e., the elongated high-gradient interlayers, decreases. The interaction between all components of the currents is nonlinear, despite the fact that their characteristic scales are different.

On unsteady boundary-layer separation in supersonic flow. Part 1. Upstream moving separation point

Abstract: This study is concerned with the boundary-layer separation from a rigid body surface in unsteady two-dimensional laminar supersonic flow. The separation is assumed to be provoked by a shock wave impinging upon the boundary layer at a point that moves with speed V sh along the body surface. The strength of the shock and its speed V sh are allowed to vary with time t, but not too fast, namely, we assume that the characteristic time scale t « Re -1/2 / V 2 w . Here Re denotes the Reynolds number, and V w =-V sh is wall velocity referred to the gas velocity V ∞ in the free stream. We show that under this assumption the flow in the region of interaction between the shock and boundary layer may be treated as quasi-steady if it is considered in the coordinate frame moving with the shock. We start with the flow regime when V w = O(Re -1/8 ). In this case, the interaction between the shock and boundary layer is described by classical triple-deck theory. The main modification to the usual triple-deck formulation is that in the moving frame the body surface is no longer stationary; it moves with the speed V w =-V sh . The corresponding solutions of the triple-deck equations have been constructed numerically. For this purpose, we use a numerical technique based on finite differencing along the streamwise direction and Chebyshev collocation in the direction normal to the body surface. In the second part of the paper, we assume that 1 » V w » O(Re -1/8 ), and concentrate our attention on the self-induced separation of the boundary layer. Assuming, as before, that the Reynolds number, Re, is large, the method of matched asymptotic expansions is used to construct the corresponding solutions of the Navier-Stokes equations in a vicinity of the separation point.

Numerical Analysis of a System of Singularly Perturbed Convection-Diffusion Equations Related to Optimal Control

Abstract: We consider an optimal control problem with an 1D singularly perturbed differential state equation. For solving such problems one uses the enhanced system of the state equation and its adjoint form. Thus, we obtain a system of two convection- diffusion equations. Using linear finite elements on adapted grids we treat the effects of two layers arising at different boundaries of the domain. We proof uniform error estimates for this method on meshes of Shishkin type. We present numerical results supporting our analysis.

Asymptotic expansions in enzyme reactions with high enzyme concentrations

Abstract: In this paper, we find a new asymptotic expansion valid in enzymatic reactions, where the total amount of enzyme exceeds greatly the total amount of substrate. In such a case, it is well known that the Michaelis–Menten approximation is no longer valid; therefore our asymptotic expansion, which improves known results, is a new tool to approximate in a closed form the concentrations of the reactants in the presence of an enzyme excess. Copyright © 2011 John Wiley & Sons, Ltd.