213,242 research outputs found
The spatial stability of a class of similarity solutions
The spatial stability of a class of exact similarity solutions of the NavierâStokes equations whose longitudinal velocity is of the form xfâČ(y), where x is the streamwise coordinate and fâČ(y) is a function of the transverse, crossâstreamwise, coordinate y only, is determined. These similarity solutions correspond to the flow in an infinitely long channel or tube whose surface is either uniformly porous or moves with a velocity linear in x. Small perturbations to the streamwise velocity of the form x^λgâČ(y) are assumed, resulting in an eigenvalue problem for λ which is solved numerically. For the porous wall problem, it is shown that similarity solutions in which fâČ(y) is a monotonic function of y are spatially stable, while those that are not monotonic are spatially unstable. For the acceleratingâwall problem, the interpretation of the stability results is not unambiguous and two interpretations are offered. In one interpretation the conclusions are the same as for the porous problemâmonotonic solutions are stable; the second interpretation is more restrictive in that some of the monotonic as well as the nonmonotonic solutions are unstable
Free Boundary Formulation for BVPs on a Semi-Infinite Interval and Non-Iterative Transformation Methods
This paper is concerned with two examples on the application of the free
boundary formulation to BVPs on a semi-infinite interval. In both cases we are
able to provide the exact solution of both the BVP and its free boundary
formulation. Therefore, these problems can be used as benchmarks for the
numerical methods applied to BVPs on a semi-infinite interval and to free BVPs.
Moreover, we emphasize how for two classes of free BVPs, we can define
non-iterative initial value methods, whereas BVPs are usually solved
iteratively. These non-iterative methods can be deduced within Lie's group
invariance theory. Then, we show how to apply the non-iterative methods to the
two introduced free boundary formulations in order to obtain meaningful
numerical results. Finally, we indicate several problems from the literature
where our non-iterative transformation methods can be applied.Comment: 30 pages, 7 figures, 4 table
An orthogonality condition for a class of problems with high order boundary conditions: Applications in sound/structure interaction
This is a pre-copy-editing, author-produced PDF of an article accepted for publication in The Quarterly Journal of Mechanics and Applied Mathematics following peer review. The definitive publisher-authenticated version: Lawrie, J.B. & Abrahams, I.D. (1999) âAn orthogonality condition for a class of problems with high order boundary conditions; applications in sound/structure interaction.â Q. Jl. Mech. Appl. Math., 52(2), 161-181. is available online at: http://qjmam.oxfordjournals.org/cgi/content/abstract/52/2/161There are numerous interesting physical problems, in the fields of elasticity, acoustics and electromagnetism etc., involving the propagation of waves in ducts or pipes. Often the problems
consist of pipes or ducts with abrupt changes of material properties or geometry. For example, in car silencer design, where there is a sudden
change in cross-sectional area, or when the bounding wall is lagged. As the wavenumber spectrum in such problems is usually discrete, the wave-field is representable by a superposition of travelling or evanescent wave modes in each region of constant duct properties. The solution to the reflection or transmission of waves in ducts is therefore most frequently obtained by mode-matching across the interface
at the discontinuities in duct properties. This is easy to do if the eigenfunctions in each region form a complete orthogonal set of basis functions; therefore, orthogonality relations allow the eigenfunction coefficients to be
determined by solving a simple system of linear algebraic equations. The objective of this paper is to examine a class of problems in which the boundary conditions at the duct walls are not of
Dirichlet, Neumann or of impedance type, but involve second or higher derivatives of the dependent variable. Such wall conditions are found in models of fluid/structural
interaction, for example membrane or plate boundaries, and in electromagnetic wave propagation. In these models the eigenfunctions are not orthogonal, and also extra edge
conditions, imposed at the points of discontinuity, must be included when mode matching. This article presents a new orthogonality relation, involving eigenfunctions and their derivatives, for the general class of problems involving a scalar wave equation and
high-order boundary conditions. It also discusses the procedure for incorporating the necessary edge conditions. Via two specific examples from structural acoustics, both of which have exact solutions obtainable by other techniques, it is
shown that the orthogonality relation allows mode matching to follow through in the same manner as for simpler boundary conditions. That is, it yields coupled algebraic systems for the eigenfunction expansions which are easily solvable, and by which means more complicated cases, such as that illustrated in figure 1, are tractable
Lie symmetries of nonlinear boundary value problems
Nonlinear boundary value problems (BVPs) by means of the classical Lie
symmetry method are studied. A new definition of Lie invariance for BVPs is
proposed by the generalization of existing those on much wider class of BVPs. A
class of two-dimensional nonlinear boundary value problems, modeling the
process of melting and evaporation of metals, is studied in details. Using the
definition proposed, all possible Lie symmetries and the relevant reductions
(with physical meaning) to BVPs for ordinary differential equations are
constructed. An example how to construct exact solution of the problem with
correctly-specified coefficients is presented and compared with the results of
numerical simulations published earlier.Comment: Published versio
Some Results on the Boundary Control of Systems of Conservation Laws
This note is concerned with the study of the initial boundary value problem
for systems of conservation laws from the point of view of control theory,
where the initial data is fixed and the boundary data are regarded as control
functions. We first consider the problem of controllability at a fixed time for
genuinely nonlinear Temple class systems, and present a description of the set
of attainable configurations of the corresponding solutions in terms of
suitable Oleinik-type estimates. We next present a result concerning the
asymptotic stabilization near a constant state for general systems.
Finally we show with an example that in general one cannot achieve exact
controllability to a constant state in finite time.Comment: 10 pages, 4 figures, conferenc
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