Boundary Layer Phenomena Research Articles

Propagation of singularities and structure of layers in shells: Hyperbolic case

We consider the boundary layer phenomena which appear in thin shell theory as the relative thickness ε tends to zero. We consider here the hyperbolic case. We focus our attention on the internal layers associated with propagation of singularities along the characteristics. The special structure of the limit problem often implies solutions which exhibit distributional singularities along the characteristics. The corresponding layers for small ε have a very large intensity. Layers along the characteristics have a special structure implying local locking. Numerical experiments using adaptive meshes exhibit the behavior foreseen by using asymptotic methods.

A macroscopic model of the diffusion and heat transfer processes in a periodically micro-stratified solid layer

A macroscopic mathematical model for the analysis of the diffusion and heat transfer processes in a bounded periodically micro-stratified thick solid layer will be derived. Contrary to the known asymptotic homogenization approach the proposed modelling method retains the microstructure length in a macroscopic description of the process. It will be shown that this effect makes it possible to formulate correctly boundary conditions on the boundaries which intersect the stratification and is responsible for certain transient and boundary layer phenomena.

Boundary layers in thin elastic shells with developable middle surface

We consider the boundary layer phenomena which appear in thin shell theory as the relative thickness e tends to zero. We deal with a developable middle surface. Boundary layers along and across the generators (which are the characteristics of the underlying system) have very different structure. It also appear internal layers associated with propagation of singularities along the characteristics. The special structure of the limit problem often implies solutions which exhibit distributional singularities along the characteristics. The corresponding layers for small e have a very large intensity. Layers along the characteristics have a special structure involving subspaces, the corresponding Lagrange multipliers are exhibited. Numerical experiments show the advantage of adaptive anisotropic meshes in these problems.

Boundary Layers and Boundary Hinge-Support Conditions for Thin Plates

The boundary-layer phenomenon is investigated for a thin three-dimensional plate. An arbitrary anisotropy of elastic properties and nonhomogeneity in longitudinal and transversal directions are assumed. Several initial asymptotic terms are constructed and a way of continuing the asymptotic procedure is described. Precise estimates of the difference between the exact and approximate solutions are obtained for the energy norm. An amalgamated problem is formed that gives the two-term asymptotics of the fields of displacements and stresses at a distance from the lateral side of the plate, whereas the edge effects are simulated by conditions of elastic fastening containing integral characteristics of the boundary-layer problem in the semistrip. Hinge-support conditions due to a sufficiently narrow clamped zone at the edge of the plate are derived and justified as well. Bibliography: 48 titles.

The University of Glasgow unmanned aerial vehicle system

States that at present wind tunnels are the most commonly used methods for obtaining data on aerofoils and investigating boundary layer phenomena in order to improve laminar flows. Looks at an alternative to wind tunnels – an unmanned aerial vehicle developed by the University of Glasgow – the GUAV‐1. Discusses the work undertaken by the GUAV‐1 as well as its intended uses and aspirations.

On a boundary layer phenomenon in Mindlin-Reissner plate theory for laminated circular sector plates

In this article, the edge-zone equation of Mindlin-Reissner plate theory, for composite plates laminated of transversely isotropic layers is studied. Analytical solutions are obtained for both circular sector and completely circular plates with various boundary conditions. The boundary-layer function and its effect on the stresses are numerically studied. Effects of plate thickness and boundary conditions are investigated. The results for circular and completely circular plates are exactly the same as those of rectangular plates in our previous work. Therefore, this boundary layer phenomenon seems to be geometry independent.

Boundary Layer Solutions in Elastic Solids

Circumferential shear deformation in an annular domain is studied for a large class of incompressible isotropic elastic materials. It is demonstrated that large strains are confined in a region adjacent to a boundary, in analogy to the boundary layer phenomenon in fluid mechanics. The size of this region is quantified. An approximate solution technique for the deformation of nonlinear elastic solids, proposed by Rajagopal [7], is further studied. In this solution, akin to the boundary layer approximation in classical fluid mechanics, the full nonlinear problem is solved in a relatively small region of large strain, while the linearized problem is solved in the remaining region. Error estimates for the approximate solution are obtained.

The NetLander atmospheric instrument system (ATMIS): description and performance assessment

The pointwise meteorological observations of the Viking Lander and Mars Pathfinder as well as the orbital mapping and sounding performed by, e.g., Mariner 9, Viking Orbiters and the Mars Global Surveyor have given a good understanding of the basic behaviour of the Martian atmosphere. However, the more detailed characterisation of the Martian circulation patterns, boundary layer phenomena and climatological cycles requires deployment of meteorological surface networks. The European NetLander concept comprising four well-instrumented landers is being studied for launch in 2005 and operations spanning at least a Martian year in 2006–2008. The landers are to be deployed to areas in both Martian hemispheres from equatorial regions to low mid-latitudes. The NetLander atmospheric instrument system (ATMIS) on board each of the landers is designed to measure atmospheric vertical profiles of density, pressure and temperature during the descent onto the surface, as well as pressure, atmospheric and ground temperatures, wind, atmospheric optical thickness and humidity through a full Martian year, possibly beyond. The main operational objective of this meteorological experiment is to provide a regular time series of the meteorological parameters as well as accelerated measurement campaigns. Such a data set would substantially improve our understanding of the atmospheric structure, dynamics, climatological cycles, and the atmosphere–surface interactions. The ATMIS sensor systems and measurement approaches described here are based on solutions and technologies tested for similar observations on Mars-96, Mars Pathfinder, Huygens, and Mars Polar Lander. Although the number of observation sites only permits characterisation of some components of the general circulation, the NetLander ATMIS will more than double the number of in situ vertical profiles (only three profiles — two from Viking Landers and one from Mars Pathfinder — are currently available and as envisioned at the time of writing, none of the 2001 and 2003 landers’ payloads include entry phase measurements of pressure or temperature), perform the first in situ meteorological observations in the southern low- and mid-latitudes and provide the first simultaneous in situ multi-site observations of the local and general circulation patterns, in a variety of locations and terrains. As such, NetLander ATMIS will be the precursor of more comprehensive meteorological surface networks for future Mars exploration.

Flow of fluids with substantially different mobilities through a porous medium in the presence of phase transitions: Boundary layer phenomena, spatial phase structures, and instability of the flow

A model of flow through a porous medium with phase transitions which permits an efficient qualitative investigation is proposed for two fluids with sharply different (high-contrast) mobilities. It is shown that the model problem of flow toward a unit sink is singularly perturbed and can be solved using analytic asymptotic matching methods. The nature of the singularity is associated with violation of the condition of the flow contrast in certain zones. The solution can be unstable depending on the direction of interphase mass transfer and the zone in which the process takes place.

Outer solution for elastic torsion by the method of boundary layer residual states

By the method of boundary layer residual state (BLRS), it is possible to specify the unknown parameters in the general form of the outer asymptotic solution of the governing differential equations for linear boundary value problems (BVP) without any reference to the inner asymptotic solutions of the same problem and the matching procedure. The method accomplishes this task by rationally assigning a portion of the prescribed boundary data to the outer solution. Specifically, the method requires certain weighted averages of the outer solution to be equal to the same averages of the data over the (localized) boundary where the data is prescribed. These weighted averages are consequences of a reciprocity relation inherent in the BVP and the stipulation that the difference between the outer solution and the exact solution (called the residual solution) of the BVP be a boundary layer phenomenon.¶The weighted average requirements are only necessary conditions for the residual state to be a boundary layer. Unfortunately, there are generally countably infinite number of (2) states, many more than the available degrees of freedom in the outer solution to satisfy them. We must show that there is no over-determination or non-uniqueness of the outer asymptotic solution, the abundance of necessary conditions notwithstanding. The present note describes an approach to assuring a well-specified outer solution (up to the expected accuracy) by way of the problem of Saint-Venant torsion. The same approach also also applies to other linear BVP, deducing the appropriate outer solution whenever the determination of the relevant inner solutions is not practical.

A model problem for boundary layers of thin elastic shells

We consider a model problem (with constant coefficients and simplified geometry) for the boundary layer phenomena which appear in thin shell theory as the relative thickness e of the shell tends to zero. For e = 0 our problem is parabolic, then it is a model of developpable surfaces. Boundary layers along and across the characteristic have very different structure. It also appears internal layers associated with propagations of singularities along the characteristics. The special structure of the limit problem often implies solutions which exhibit distributional singularities along the characteristics. The corresponding layers for small e have a very large intensity. Layers along the characteristics have a special structure involving subspaces; the corresponding Lagrange multipliers are exhibited. Numerical experiments show the advantage of adaptive meshes in these problems.

A model of the elastodynamic interaction of a thin-walled inclusion with a matrix under antiplanar shear

Using the example of antiplanar shear we give a generalized scheme for constructing models of elastodynamic interaction of an elastic matrix with a thin-walled elastic curvilinear inclusion lying in the matrix under conditions of ideal contact subject to steady vibrations, by introducing suitable conditions on the middle line of the inhomogeneity. The procedure for the study is based on the application of the methods of the theory of singular perturbations. We describe the behavior of the stress-strain state of the body near the edges of the inhomogeneity, where boundary-layer phenomena appear.

Mathematical Analysis of a Bonded Joint with a Soft Thin Adhesive

This paper considers the problem of two adherents joined by a soft thin adhesive along their common surface. Using the asymptotic expansion method, the authors obtain a simplified model in which the adhesive is treated as a material surface and is replaced by returning springs. The authors show weak and strong convergence of the exact solution toward the solution of the limit problem. The singularities of the limit problem are analyzed, and it is shown that typically they are logarithmic. Furthermore, the authors investigate the phenomenon of boundary layer by studying the correctors, the extra terms, which must be added to the classical asymptotic expansion to verify the boundary conditions. The correctors show that, contrary to the adherents, in the adhesive there are power-type singularities, which are at the base of the failure of the assemblage.

Model problem of singular perturbation without limit in the space of finite energy and its computation

We consider elliptic problems of singular perturbation which depend on a small parameter e, the right hand side f belonging to the dual of the energy space for ɛ > 0. For ɛ = 0, / belongs neither to the energy space nor to the admissible spaces of Lions and Magenes, taking account of the behaviour at the boundary. We exhibit boundary layer phenomena with large intensity. These boundary layers determine u 0( x, ɛ) (the leading term) which explodes at an order of ɛ depending on f. Numerical computations are faithful only with a very thin mesh in the boundary layers. We also give some results concerning the case when f can be treated within the framework of Lions and Magenes theory for the limit problem.

Acoustic boundary layer phenomena near a corrugated surface

Acoustic resonator techniques for measuring thermophysical properties require a quantitative understanding of the viscous and thermal boundary layers. Near smooth solid boundaries, well-known analytic models of boundary layers have been confirmed by careful measurements. Little is known either experimentally or theoretically about the effects of surface roughness on the boundary layers. Roughness effects have been investigated through numerical calculations of the coupled acoustic, thermal, and shear fields near a corrugated interface, i.e., a surface with a profile y=A cos qx. The calculations were carried out in the limit where the acoustic wavelength is much larger than the scale of surface roughness (A,2π/q) or the penetration lengths δt and δv. Boundary layer phenomena cause phase shifts and energy absorption in the reflection of a plane acoustic wave by a smooth surface. These effects are further modified at a corrugated reflecting surface. The calculated reflection coefficients have been applied to the model relating the speed of sound to the frequencies of spherical and cylindrical acoustic cavity resonances, and to the theoretical model of the Greenspan acoustic viscometer. [Research supported in part by the Office of Naval Research.]

Asymptotic solution for the perturbed Stokes problem in a bounded domain in two and three dimensions

We consider the Stokes problem with a small viscosity. When the viscosity goes to zero, the boundary-layer phenomenon can appear. In this case, the solution of the given perturbed Stokes equation cannot be properly approximated by the solution of its limiting equation ‘near’ the boundary Γ of the domain of study, say Ω To overcome this problem, we need to construct a corrector term in the neighbourhood of Γ Lions has studied this problem and has constructed a corrector for the case where Ω is a half space in ℝ2. The case where Ω is an open and bounded domain of ℝ2 or ℝ3, which remained unsolved, is the concern of this paper. The construction of the corrector to the perturbed Stokes equation depends heavily on the geometry of Ω In two dimensions, we construct the corrector in the form of a stream function, while in ℝ3 we construct it in the form of a potential vector. The corrector acts effectively in a neighbourhood of Γ that is the boundary layer. Using similar methods to those of Baranger and Tartar, we define the thickness of the boundary layer in a natural way. In addition, in this paper we study the behaviour of the corrected solution in some Hölder spaces.

Asymptotic behaviour of solutions of boundary-value problems for equations with rapidly oscillating coefficients in a domain with a small cavity

Asymptotic representations of the solutions of boundary-value problems for a second-order equation with rapidly oscillating coefficients in a domain with a small cavity (of diameter comparable with the period of oscillation) are found and substantiated. Dirichlet or Neumann conditions are set at the boundary of the domain. In addition to an asymptotic series of structure standard for homogenization theory there occur terms describing the boundary layer phenomenon near the opening, while the solutions of the homogenized problem and their rapidly oscillating correctors acquire singularities at the contraction point of the openings. The dimension of the domain and some other factors influence even the leading term of the asymptotic formula. Some generalizations, including ones to the system of elasticity theory, are discussed.

Meteorological observations on Martian surface: met-packages of Mars-96 Small Stations and Penetrators

The scientific objectives of a meterological experiment on the Martian surface are defined, and the meteorological equipment of the landing elements of the Mars-96 mission are described with emphasis on the applicability for re-use in forthcoming Mars missions. The general strategy for atmospheric surface observations is discussed. Meteorological surface observations are of utmost value in studying the Martian atmosphere. The climatological cycles and atmospheric circulations, as well as the boundary layer phenomena can be understood thoroughly only, if the contribution of in situ surface measurements are amalgamated with the remote observations. The Mars-96 mission had an ambitious goal of deploying four versatile payloads at four Northern hemispheric sites. The observations of pressure, temperature, wind, atmospheric optical thickness and humidity, as well as pressure and temperature measurements during the atmospheric descent were included in the meteorology experiment. Even though the Mars-96 mission was unsuccessful, the objectives and implementation of the meteorology experiment are applicable to any forthcoming landing mission to Mars. This applies both to a mission having a number of observation sites spread all over the surface of Mars, and to a single lander or rover. The main operational objective of this meteorological experiment is to provide a regular time series of the meteorological parameters with accelerated measurement campaigns during dawn and dusk. Such a data set would substantially improve our understanding of the atmospheric structure, dynamics, climatological cycles, and the atmosphere-surface interactions. The implementation of the meteorology instrument features advanced sensor technology and flexible system design. The application on the Mars-96 landing elements was, however, severely constrained by the limited power supply. The usefulness of the system can be substantially enhanced by modest additional resources and with few or no design modifications.

Vertical profiles of the scattering coefficient of dry atmospheric particles over Europe normalized to air at standard temperature and pressure

Simultaneous vertical profiles of the scattering coefficient of the air at one wavelength between 0.475 and 0.75 μm, temperature, pressure, and relative humidity have been measured from 1973 to 1978, mostly in Central and Western Europe. From these results, vertical profiles of the normalized (to air at standard temperature and pressure) scattering coefficients of dry particles were derived. These standardized scattering coefficients are independent of the thermodynamic state of the atmosphere and thus strictly comparable. Mean vertical profiles of the normalized scattering coefficients of dry particles from 400 up to 5000 m above mean sea level are obtained as results. These profiles are different in continental and maritime air as well as during the summer and the winter half-years. The results allow the formulation of simple profile models for the light scattering by mid–latitude particles in the lower troposphere over Central and Western Europe. These models are in accordance with our knowledge about the atmospheric boundary layer phenomena and indicate that in the mean the major particle sources should be at or near the ground. Integrals over the mean profiles agree with results from sun photometry.

The hp finite element method for problems in mechanics with boundary layers

We consider the numerical approximation of boundary layer phenomena occurring in many singularly perturbed problems in mechanics, such as plate and shell problems. We present guidelines for the effective resolution of such layers in the context of existing, commercial p and hp finite element (FE) version codes. We show that if high order, ‘spectral’ elements are available, then just two elements are sufficient to approximate these layers at a near-exponential rate, independently of the problem parameters thickness or Reynolds number. We present hp mesh design principles for situations where both corner singularities and boundary layers are present.