Topic
Method of matched asymptotic expansions
About: Method of matched asymptotic expansions is a research topic. Over the lifetime, 4233 publications have been published within this topic receiving 73311 citations.
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TL;DR: In this article, the asymptotic estimate of the solution y ( t ) of linear difference equations with almost constant coefficients and the condition of equivalence between y( t ) and y 0 ( t ), where y 0 is the corresponding solution of linear equations with constant coefficients are given.
29 citations
TL;DR: In this paper, the viscous compressible flow in the vicinity of a right-angle corner, formed by the intersection of two perpendicular flat plates and aligned with the free stream, is investigated.
Abstract: The viscous compressible flow in the vicinity of a right-angle corner, formed by the intersection of two perpendicular flat plates and aligned with the free stream, is investigated. In the absence of viscous-inviscid interactions and imbedded shock waves, a theory is developed that is valid throughout the subsonic and supersonic Mach number range. Within this limitation and the additional assumptions of unit Prandtl number and a linear viscosity-temperature law, a consistent set of governing equations and boundary conditions is derived. The method of matched asymptotic expansions is applied in order to distinguish the relevant regions in the flow field.In the corner region the Crocco integral is shown to apply, even for a three-dimensional flow field. The equations governing the flow in the corner layer consist of four coupled nonlinear elliptic partial differential equations of the Poisson variety. Since they do not lend themselves to analytic solution, numerical methods are employed. Two such methods used here are the Gauss-Seidel explicit technique and the alternating direction implicit method. The merits of both techniques are discussed with regard to convergence rate, accuracy and stability. The calculations show that in cases where the Gauss-Seidel method fails to give converged solutions, owing to instability, the alternating direction implicit method does provide converged solutions. However, in cases where both methods are convergent, there is no appreciable difference in convergence rates. The numerical calculations were done on a CDC 6600 computer.Results of calculations are presented for representative compressible-flow conditions. The extent of the corner disturbance is controlled by the Mach number and wall temperature ratio in a manner analogous to the two-dimensional boundary layer. A swirling motion is noted in the corner layer which is influenced to a great extent by the asymptotic cross-flow profiles. The skin-friction coefficient is shown to increase monotonically from zero at the corner point to its asymptotic two-dimensional value. For cold wall cases, this value is approached more rapidly. The asymptotic analysis indicates that for even colder wall cases, not considered here, an overshoot is possible.
29 citations
TL;DR: In this article, the sharp interface limit for diffusive interface models with the generalized Navier boundary condition was derived for the moving contact line problem, and the leading order dynamic contact angle is the same as the static one satisfying the Young's equation.
Abstract: Using method of matched asymptotic expansions, we derive the sharp interface limit for the diffusive interface model with the generalized Navier boundary condition recently proposed by Qian, Wang and Sheng in (9, 11) for the moving contact line problem. We show that the leading order problem satisfies a boundary value problem for a coupled Hale-Shaw and Navier-Stokes equations with the interface being a free boundary, and the leading order dynamic contact angle is the same as the static one satisfying the Young's equation.
29 citations
TL;DR: In this article, a differentially heated rectangular cavity of small aspect ratio A was examined and it was shown that when the capillary number is C=O(A3), the interface undergoes an O(1) deformation from its flat position and the flow inside the cavity becomes nonparallel everywhere.
Abstract: Steady thermocapillary convection is examined in a differentially heated rectangular cavity of small aspect ratio A. It is shown that when the capillary number is C=O(A3), the interface undergoes an O(1) deformation from its flat position and the flow inside the cavity becomes nonparallel everywhere. The velocity and temperature profiles and the shape of the deformed interface are derived using the method of matched asymptotic expansions.
29 citations
TL;DR: A mathematical model of a steady-state diffusion process through a periodic membrane for a wide class of periodic membranes is dealt with and formulas used in physical, chemical, and biological investigations to describe effective membrane properties are analyzed.
Abstract: The paper deals with a mathematical model of a steady-state diffusion process through a periodic membrane. For a wide class of periodic membranes, we define the effective permeability and obtain upper and lower estimates of the effective permeability. For periodic membranes made from two materials with different absorbing properties, we study the asymptotic behavior of the effective permeability when the fraction of one material tends to zero (low concentration asymptotics). When the low fraction material forms homothetically vanishing disperse periodic inclusions in the host material, low concentration approximations are built by the method of matched asymptotic expansions. We also show that our results are consistent with those which can be obtained by a boundary homogenization. Finally, we analyze formulas used in physical, chemical, and biological investigations to describe effective membrane properties.
29 citations