Showing papers on "Free boundary problem published in 1969"

Self-induced separation

Abstract: A rational theory is developed to explain the initial pressure rise and consequent separation of a laminar boundary layer when it interacts with a moderately strong shock. In this theory, which is firmly based on the linearized theory of Lighthill (1953), the region of interest is divided into three parts: the major part of the boundary layer, which is shown to change under largely inviscid forces, the supersonic main stream just adjacent to the boundary layer in which the pressure variation is small; and a region close to the wall, on boundary-layer scale, in which the relative variation of the velocity is large but is controlled by the incompressible boundary-layer equations, together with novel boundary conditions. We find that the first two parts can be handled in a straightforward way and the problem of self-induced separation reduces, in its essentials, to the solution of a single problem in the theory of incompressible boundary layers. It is found that this problem has three solutions, one of which corresponds to undisturbed flow and another describes a boundary layer which, spontaneously, generates an adverse pressure gradient and a decreasing skin friction which eventually vanishes and then downstream a reversed flow is set up. The third solution generates a favourable pressure gradient and is not relevant to the present study. Although there has hitherto been no valid numerical method of integrating a boundary layer with reversed flow, we find that an ad hoc method seems to lead to a stable solution which has a number of the properties to be expected of a separated boundary layer. Comparison with experiment gives qualitatively good agreement, but quantitatively errors of the order of 20% are found. It is believed that these errors arise because the Reynolds numbers at which the experiments were carried out are too small.

The resolvent of an elliptic boundary problem.

Abstract: This paper and its successor, [10], extend to boundary value problems the analysis of the powers of an elliptic operator given in [9]. (Similar methods are applied in [3], [5], and [6] for the case without boundary; [5a] announces boundary results.) Although [9] treated general elliptic pseudo-differential operators on compact manifolds, we restrict ourselves here to differential operators, except in the final section, where we correct some (fortunately inconsequential) errors in the proof given in [9]. We consider a q X q system A = > aaDa of Cdifferential operators

The optimization of methods of solving boundary value problems with a boundary layer

Abstract: THE solutions of a number of value problems for differential equations with small parameters having higher derivatives possess singularities of the boundary layer type [1, 2]. For the solution of such problems by finite-difference methods the integration step near the boundary must be substantially less than the thickness of the boundary layer which is a characteristic dimension of the problem. In the case of constant steps throughout the whole region of integration this circumstance leads to a considerable increase in the volume of calculations when the parameters are reduced with higher derivatives. An exception may be only the so-called “quasiclassical approximations” of [3], adapted specially for the solution of problems with small parameters having higher derivatives, but these can only be written for isolated classes of ordinary differential equations. The use of asymptotic methods of solution [2, 4] requires these parameters to be rather small, and their coefficients to be very smooth. The same often applies to the use of numerical methods. Below we construct, for two model boundary value problems (a set of ordinary differential equations and an elliptic equation), methods of solution on a network with variable steps and an evaluation of the error which is homogeneous in the small parameter, but with few constraints on the smoothness of the coefficients.

The Boundary Value Problems

Abstract: Now we ask if we can determine a solution of the Maxwell equations in such a way that their boundary values assume specified values on a closed regular surface. It has already been shown that a field E, H which satisfies the equations exterior to a regular region, the radiation condition, and on the boundary surface F, vanishes identically. This result means that a solution of our equations which in addition satisfies the radiation conditions is uniquely determined by the value n × E on the boundary surface. Now the question arises as to whether we can prescribe these boundary values.

A variational principle for boundary value problems in kinetic theory

Abstract: A variational principle which applies directly to the integrodifferential form of the linearized Boltzmann equation is introduced. Extremely general boundary conditions and collision terms are allowed. For a class of interesting problems, the value of the functional to be varied is shown to be closely related to quantities of great physical interest. The formalism is applied to the treatment of plane Couette flow for different forms of the collision term (BGK model, rigid spheres, Maxwell's molecules).

Diffracted Wave Fields Expressible by Plane-Wave Expansions Containing only Homogeneous Waves

Abstract: The angular-spectrum representation of wave fields is shown to yield scalar Helmholtz-equation solutions (in a half-space) that tend in the L2 limit of the mean to prescribed boundary values on an infinite plane boundary for all square-integrable boundary values. This result is employed to obtain several important properties of source-free wave fields, i.e., wave fields (satisfying square-integrable boundary values) that contain only homogeneous plane waves in their angular-spectrum representations: (1) the function describing a source-free wave field can be extended into the region behind the plane boundary to give a bounded continuous solution of the scalar Helmholtz equation in all space; (2) the function describing a wave field can be extended to the whole space of three complex variables as an entire function satisfying a certain inequality if, and only if, the wave field is source free; (3) the two-dimensional autocorrelation function of the wave field on planes parallel to the boundary plane is independent of the distance of the plane from the boundary if, and only if, the field is source free; and (4) a boundary value exists that produces a three-dimensional, pseudoscopic, real image of a wave field if, and only if, the wave field is source free. A series-mode expansion for source-free fields in terms of the boundary value is derived and is shown to be absolutely and uniformly convergent. The series is transformed into a two-dimensional Taylor series (with coefficients determined in terms of the boundary value) and another series that displays explicitly the contribution to the wave field due to each partial derivative of boundary value. The series are valid representations of wave fields in a broad class that is the natural extension of the source-free class of fields when the restriction that the boundary value be square integrable is removed. The series are used to derive two angular-spectrum representations for wave fields in the extended class and to discuss the properties of those wave fields.

Some aspects of the method of the hypercircle applied to elliptic variational problems.

Abstract: : The hypercircle method is studied from the point of view of the recently developing theory of Galerkin-type approximations in Sobolev spaces using spline functions. Given an elliptic boundary value problem it is shown how to obtain a conjugate problem - thereby interchanging the roles of 'forced' and 'natural boundary conditions. Given approximate solutions for both problems their errors can be estimated 'a posteriori'. The approximate solution of the primal problem being well-known, the authors consider the approximation of the solution of the conjugate problem, obtaining theorems of convergence and estimates of the rate of convergence. (Author)

A method of ascent for solving boundary value problems

Abstract: Stefan Bergman [ l ] and Ilya Vekua [4] have given representation formulas for solutions of the partial differential equation (1). We obtain an improvement of their results for the case of two independent variables (namely equation (2) with n set equal to 2). Furthermore, we are able to extend our result to higher dimensions (the ascent) by a remarkably simple variation of this two dimensional formula. Our representation (2) also contains Vekua's formulas [4, p. 59], for the Helmholtz equation in n è 2 variables.

On a Method to Subtract off a Singularity at a Corner for the Dirichlet or Neumann Problem

Abstract: Let D be a plane domain partly bounded by two line segments which meet at the origin and form there an interior angle 7ra > 0. Let U(x, y) be a solution in D of Poisson's equation such that either U or a U/an (the normal derivative) takes prescribed values on the boundary segments. Let U(x, y) be sufficiently smooth away from the corner and bounded at the corner. Then for each positive integer N there exists a function VN(X, y) which satisfies a related Poisson equation and which satisfies related boundary conditions such that U - VN is N-times con- tinuously differentiable at the corner. If 1/a is an integer VN may be found ex- plicitly in terms of the data of the problem for U. a In solving an elliptic partial differential equation by numerical methods the results proved about convergence of the numerical approximation to the actual solution frequently depend on differentiability properties of the (unknown) solu- tion. In the work of Gerschgorin (2) and other papers written since, it is assumed that the solution of the partial differential equation has derivatives of order four which are continuous up to the boundary. If the boundary and all the data are sufficiently smooth there is, of course, no problem. In many cases, however, the boundary pos- sesses a finite number of singularities, usually (in the two-dimensional case) in the. form of corners; occasionally too, the boundary data may have jumps. Laasonean (3) has proved that convergence of the discrete solution to the actual solution holds for the Dirichlet problem, but that the convergence is slow in a neighborhood of the corner. In this paper we will consider a method to subtract off the singularity. The method is quite old (see Fox (1)), but includes results on the asymptotic behavior of solutions near a corner. In this light see the works of Lewy (4), Lehman (5), Wasow (6), and the author (7). We consider a problem for which the solution is not known too be smooth. We then find, explicitly in terms of the boundary data, a solution to a related problem; then the difference between these two solutions is a solution to a. third problem, and is sufficiently well-behaved to insure convergence of difference schemes. Finally, the sought solution can be found by adding the explicitly given one to the numerically-solved one. Let D be a plane domain partly bounded by two open line segments ri and r2, which share the origin as a common endpoint and form there an interior angle 7ra > 0. We assume that ri is a subset of the positive x-axis and r2 makes an angle 7ra > 0 with the positive x-axis. Let F(x, y) be given in D and ib(x, y) (respectively

A numerical method for calculating unsteady two-dimensional laminar boundary layers

Abstract: A method for calculating unsteady two dimensional boundary layers in laminar incompressible flow has been developed and tested. No restrictive assumptions are made regarding the time-dependent terms in the boundary-layer equation. The differential equations are solved with an implicit difference scheme similar to that employed for steady two-dimensional boundary layers. At each step, here, known conditions at three stations are used to calculate the conditions at a new (fourth) station. The entire field is covered by a succession of these steps.

Optimization of a non-linear distributed parameter system using periodic boundary control†

Abstract: Conditions for loeal optimality are worked out using simple calculus of variations. To find the optimum control, a two-point boundary value problem in space and time has to be solved, which involves the solution of the adjoint differential equation together with the prooess equation. The method is applied to the optimization of a periodic process, consisting of a tubular reactor where a second-order homogeneous reaction takes place and a periodicity oondition of the state is satisfied everywhere along the reactor. Tho plug flow and the diffusion model are assumed. In the first case an exact solution is carried out. The improvements in yield compared with steady-state conditions are obtained and shown in graphs.

A free boundary problem and an extension of Muskat's model

Abstract: I t is the purpose of this paper to derive and solve a mathematical model for the following physical problem. Suppose that in a homogeneous compressible porous medium one incompressible fluid is displacing another. The problem is to describe the motion of the fluids, in particular, the motion of the interface between the fluids, if the initial velocity distribution, or equivalently, the initial pressure distribution, of the fluids is given, together with appropriate boundary data. We assume the flow to be in the horizontal x-direction, say, and neglect gravitational effects. We further assume the two fluids are immiscible so that for each time t there is a well defined interface between the fluids whose location is given by x =~(t). To the left of ~(t) we denote the velocity of the fluid by u(x, t) and its pressure by p(x , t), and to the right we denote velocity and pressure by v(x, t) and q(x, t) respectively. The pressures and velocities are related by Darcy's law: