Showing papers on "Free boundary problem published in 1975"
TL;DR: In this paper, the amplitude ratio of constant-frequency disturbances as a function of Reynolds number for insulated and cooled-wall flat-plate boundary layers between Mach numbers 1.3 and 5.8 is calculated.
Abstract: Compressible linear stability theory is first reviewed and then used to calculate the amplitude ratio of constant-frequency disturbances as a function of Reynolds number for insulated and cooled-wall flat-plate boundary layers between Mach numbers 1.3 and 5.8. These results are used to examine the consequences of using a fixed disturbance amplitude of the most unstable frequency as a transition criterion. The effect of the freestream Mach number M1 on the transition of insulated-wall boundary layers is calculated using two different assumptions concerning the initial boundary-layer disturbance amplitude A0. It is found that the shape of the transition Reynolds number Ret vs MI curve observed in wind tunnels can be closely duplicated. As a second example, the effect of wall cooling at MI = 3.0 is calculated. A much faster increase of Re, with cooling is obtained than is observed experimentally. However, when A0 is determined from the forced response of the boundary layer to irradiated sound and from the measured freestream power spectrum, a rise in Re, similar to what is observed is obtained for a certain amplitude criterion.
515 citations
TL;DR: The convergence rate for difference approximations to mixed initial boundary value problems has been shown to be linear in the convergence rate of the difference approximation as mentioned in this paper, which is the best known convergence rate.
Abstract: The convergence rate for difference approximations to mixed initial boundary value problems
386 citations
168 citations
TL;DR: In this article, a free boundary value problem arising in plasma physics is reduced to a non-linear eigenvalue problem of a nonclassical type, and the existence of solutions of such a problem is established.
Abstract: A free boundary value problem arising in plasma physics is reduced to a non-linear eigenvalue problem of a non-classical type. We establish the existence of solutions of the non-linear eigenvalue problem; these solutions are critical points of appropriate functionals.
155 citations
TL;DR: In this article, the shape and smoothness of the free boundary for the variational inequality (1) was considered, where the parabolic operator has variable coefficients and the space dimension is any number n > 1.
147 citations
TL;DR: The main result shows, essentially, that, if the nonlinear problem has an isolated solution and the approximating family has stable Lipschitz continuous linearizations, then the approximation problem has a stable solution which is close to the exact solution.
Abstract: General nonlinear problems in the abstract form F(x) = 0 and corresponding families of approximating problems in the form Ffi(xn) = 0 are considered (in an appropriate Banach space setting) The relation between "isolation" and "stability" of solutions is briefly studied The main result shows, essentially, that, if the nonlinear problem has an isolated solution and the approximating family has stable Lipschitz continuous linearizations, then the approximating problem has a stable solution which is close to the exact solution Error estimates are obtained and Newton's method is shown to converge quadratic-ally These results are then used to justify a broad class of difference schemes (resembling linear multistep methods) for general nonlinear two-point boundary value problems
106 citations
TL;DR: In this article, a discussion of resistance laws for barotropic planetary boundary layers is presented, and concrete expressions are obtained for universal functions A(μ0), B(μ 0), C (μ 0) of Rossby number similarity theory for the Ekman boundary layer as well as of analogous functions for non-steady boundary layers.
Abstract: A further discussion of the resistance laws for barotropic planetary boundary layers is presented. Concrete expressions are obtained for universal functions A(μ0), B(μ0) C(μ0)of Rossby number similarity theory for the Ekman boundary layer as well as of analogous functions for non-steady boundary layers. Prediction equations are proposed, which describe variations of the boundary layer depth for unstable and stable stratifications.
104 citations
100 citations
TL;DR: In this paper, the local character of the elastostatic field in plane strain near a point that separates a free from an adjoining fixed segment of a rectilinear boundary component is investigated.
78 citations
TL;DR: In this paper, a fixed domain numerical method is presented which is motivated by a theoretical formulation developed by Rogers, and the frontal generation appears by way of a simple algebraic comparison operation involving truncation of the computed approximation.
Abstract: We consider the numerical solution of an implicit moving free boundary problem which arises in the study of diffusion and consumption of oxygen in tissue. A fixed domain numerical method is presented which is motivated by a theoretical formulation developed by Rogers. Our numerical method uses any convenient finite difference or finite element scheme which converges to the underlying partial differential equation. The frontal generation appears by way of a simple algebraic comparison operation involving truncation of the computed approximation. Higher space dimensions are treated with equal ease. Results of numerical experiments are presented. A convergence proof for the truncation method is given.
71 citations
TL;DR: In this paper, the stability of a general toroidal MHD equilibrium with a continuous pressure profile is investigated for the case where the fluid is surrounded by vacuum and it is found that a disturbance which is localized near the free boundary grows exponentially in time unless a certain necessary criterion is satisfied.
Abstract: The stability of a general toroidal MHD equilibrium with a continuous pressure profile is investigated for the case where the fluid is surrounded by vacuum. It is found that a disturbance which is localized near the free boundary grows exponentially in time unless a certain necessary criterion is satisfied. Because of the weaker boundary condition, this criterion imposes a more stringent restriction on the configuration than does Mercier's criterion near the boundary. For the cylindrically symmetric case the criterion requires a decrease of the rotational transform and yields a critical relation between the shear and the pressure gradient.
TL;DR: In this paper, a numerical method to compute the solution of a system of conservation laws subject to an initial condition and certain boundary conditions which involve a free boundary is described, which is specially designed to solve the Eulerian equations of compressible flow.
TL;DR: In this article, a non-linear boundary value problem is treated using the principle of T. Wazewski, where one can describe the sets of initial data which satisfy the boundary conditions at + ∞ and at -∞ and to show how they intersect.
Abstract: A non-linear boundary value problem is treated using the principle of T. Wazewski. The equation is d
2/d, x
2
p (x)+s (x) p (1−p)=0 where s (x) is non zero near ±∞. The boundary condition on p at ±∞ is 0 and 1 according as sgn s (±∞) is −1 or +1. Two essentially different cases are treated, namely sgn s (+ ∞)= ±sgn s (-∞). A radially symmetric problem with x e R
2 is also discussed. The Wazewski principle allows one to describe the sets of initial data which satisfy the boundary conditions at + ∞ and at -∞ and to show how they intersect. The problem arises in the study of clines in population genetics theory.
TL;DR: In this paper, a variational formulation for hydrodynamic lubrication is presented and the free boundary problem associated with the Reynolds boundary condition is shown to arise naturally from this formulation.
TL;DR: In this paper, a new method is presented to obtain a state feedback form solution to an optimal control problem with nonlinear dynamics and a quadratic performance index, based on solving an integral equation equivalent to the two-point boundary-value problem related to the optimization problem by applying an inverse theorem concerning analytic nonlinear operators.
Abstract: A new method is presented to obtain a state feedback form solution to an optimal control problem with nonlinear dynamics and a quadratic performance index. The method is based on solving an integral equation equivalent to the two-point boundary-value problem related to the optimization problem by applying an inverse theorem concerning analytic nonlinear operators. Compared with the previous methods, this one is straightforward, more generally applicable, and gives important additional knowledge about the solution. An example is presented to illustrate the use of the method.
01 Jan 1975
TL;DR: In this paper, the first twenty coefficients for a boundary value problem are approximated and a conformal transformation is given to give a useful conformal transform for the boundary value problems.
Abstract: : The first twenty coefficients for a boundary value problem are approximated This gives a useful conformal transformation
TL;DR: A survey of algorithms for solving the eigenproblem for a class of matrices of nearly tridiagonal form that arise from eigenvalue problems for differentia1 equations where the solution is subject to periodic boundary conditions.
Abstract: A survey of algorithms for solving the eigenproblem for a class of matrices of nearly tridiagonal form is given. These matrices arise from eigenvalue problems for differentia1 equations where the solution is subject to periodic boundary conditions. Algorithms both for computing selected eigenvalues and eigenvectors and for solving the complete eigenvalue problem are discussed.
TL;DR: In this paper, the question of boundary conditions for the given differential equations of a continuum theory, is treated by the method of nonequilibrium thermodynamics, and the underlying applications now are rarefied polyatomic gases within walls, perhaps in an external field.
Abstract: The question which boundary conditions are appropriate for the given differential equations of a continuum theory, is treated by the method of nonequilibrium thermodynamics. In the first part the previous investigation1) which had been confined to ordinary hydro(aero-)dynamics is generalized by taking into account higher derivatives in the continuum constitutive laws. This kind of generalization becomes important when a rarefied gas with boundary is treated phenomenologically (wall influence, slip-flow regime). In the second part an even more general scheme is discussed. The underlying applications now are rarefied polyatomic gases within walls, perhaps in an external field. So, the starting point is a general set of transport-relaxation equations with more variables than the hydrodynamical ones. By considering the corresponding entropy production, especially its part due to the boundary, it is again possible to set up constitutive laws, i.e., matching or boundary conditions, at an interface or a surface.
TL;DR: In this paper, the authors considered the problem of bifurcation problems of the form LyAXf(y), By =0, where y is a scalar function, X is a real scalar, L is a linear differential operator and By = 0 represents some linear homogeneous two-point boundary conditions.
Abstract: Numerical methods for bifurcation problems of the form (*) LyAXf(y), By =0, where f(0) = 0 and f'(0) * 0, are considered. Here y is a scalar function, X is a real scalar, L is a linear differential operator and By = 0 represents some linear homogeneous two-point boundary conditions. Under certain assumptions, it is shown that if (*) is replaced by an appropriate difference scheme, then there exists a unique branch of nontrivial solutions of the discrete problem in a neighborhood of a branch of nontrivial solutions of (*) bifurcating from the trivial solution and that the discrete branch converges to the continuous one. Error estimates are derived and an illustrative numerical example is included.
TL;DR: In this article, an implicit non-steady free boundary problem is transformed into a variational inequality, which is solved by means of a semi-discretization technique, and the problem is then solved by a semidiscrete solution.
Abstract: An implicit non-steady free boundary problem is transformed into a variational inequality, which is solved by means of a semi-discretization technique.
TL;DR: In this article, the authors considered the nonlinear differential equation (1.1) where x ∊l = [a, ∞] and made the following assumptions (A) f is continuous on [a ∞) × Rn, (B) solutions of initial value problems (I.V.P.'s) are unique and extend throughout [a and ∞).
Abstract: We consider here the nonlinear differential equation (1.1) where x ∊l= [a, ∞). We will make the following assumptions (A) f is continuous on [a, ∞) × Rn, (B) solutions of initial value problems (I.V.P.'s) are unique and extend throughout [a, ∞). (C)
TL;DR: In this article, a finite difference scheme is given for the numerical approximation of the real solution of the second order linear differential equation, lacking the first derivative, with mixed boundary conditions.
Abstract: A finite difference scheme is given for the numerical approximation of the real solution of the second order linear differential equation, lacking the first derivative, with mixed boundary conditions. The matrix associated with the resulting system of linear equations is tridiagonal and the overall discretization error isO (h4). The derived error bound is at most four times larger than the observed maximum error in absolute value for the numerical problem considered.