Nonlinear Free Boundary Value Problems Research Articles

Structural stability of transonic shocks for the full compressible Euler system in a two-dimensional general nozzle with an external force

In this paper, we investigate the transonic shock solutions to the full compressible Euler system in a general two-dimensional nozzle with an external force. In their book, Courant and Friedrichs [Supersonic Flow and Shock Waves (Interscience Publishers, Inc., New York, 1948)] described the transonic shock phenomena in a de Laval nozzle: given the appropriate receiver pressure, if the upcoming flow is still supersonic after passing through the throat of the nozzle, then a shock front intervenes at some place in the diverging part of the nozzle, and the gas is compressed and slowed down to subsonic speed. We first establish the existence and uniqueness of one-dimensional transonic shock solutions to the steady full Euler system with an external force by prescribing suitable pressure at the exit of the nozzle when the upstream flow is a uniform supersonic flow. With the help of external forces, both existence and uniqueness are established by solving a nonlinear free boundary value problem in a weighted Hölder space.

Existence of transonic shocks in a cylindrical nozzle with given exit normal momentums

In this paper, we study the existence of transonic shocks in a cylindrical nozzle. More precisely, given the compatible outer normal momentum at the exit of a cylindrical nozzle, there exists a transonic shock solution such that the incoming supersonic flow is compressed to some subsonic flow. The shock position is uniquely determined by the exit normal momentum when we fix one point of the shock. This problem can be reduced to solve a nonlinear free boundary value problem for a hyperbolic–elliptic coupled system and the key ingredient is to solve a second‐order quasilinear elliptic equation with a nonlocal term by the method of separation of variables.

Transonic Shocks for the Full Compressible Euler System in a General Two-Dimensional De Laval Nozzle

In this paper, we study the transonic shock problem for the full compressible Euler system in a general two-dimensional de Laval nozzle as proposed in Courant and Friedrichs (Supersonic flow and shock waves, Interscience, New York, 1948): given the appropriately large exit pressure p e(x), if the upstream flow is still supersonic behind the throat of the nozzle, then at a certain place in the diverging part of the nozzle, a shock front intervenes and the gas is compressed and slowed down to subsonic speed so that the position and the strength of the shock front are automatically adjusted such that the end pressure at the exit becomes p e(x). We solve this problem completely for a general class of de Laval nozzles whose divergent parts are small and arbitrary perturbations of divergent angular domains for the full steady compressible Euler system. The problem can be reduced to solve a nonlinear free boundary value problem for a mixed hyperbolic–elliptic system. One of the key ingredients in the analysis is to solve a nonlinear free boundary value problem in a weighted Holder space with low regularities for a second order quasilinear elliptic equation with a free parameter (the position of the shock curve at one wall of the nozzle) and non-local terms involving the trace on the shock of the first order derivatives of the unknown function.

Circular map for supercavitating flow in a multiply connected domain

A nonlinear free boundary-value problem of supercavitating flow past n + 1 hydrofoils is analyzed. To describe the cavities' closure mechanism, the Tulin-Terent'ev single-spiral-vortex model is employed. The flow domain is considered as the image of an (n + 1)-connected circular domain. The conformal map is constructed in terms of the solutions to two Riemann-Hilbert problems of the theory of symmetric automorphic functions. One of the problems is homogeneous and its coefficients are continuous functions while the second problem is inhomogeneous and has discontinuous coefficients. The exact solutions to the problems are found by using quasiautomorphic and quasimultiplicative analogs of the Cauchy kernel. The case of a single plate is considered in detail and the numerical results are reported.

HIGHER-ORDER METHOD FOR SOLVING FREE BOUNDARY-VALUE PROBLEMS

We present an extended one-step method of order 5 for the solution of the third-order, nonlinear free boundary-value problem. The Falkner-Skan equation as well as the Blasius equation are special cases of this problem. We prove that this method is A- and L-stable. The method uses a coordinate transformation to map a semi-infinite physical domain to the open unit interval (0, 1). Comparisons are made between the numerical solutions obtained by the proposed method and those obtained by previous authors.

Oscillatory Third-Order Perturbation Solutions for Sums of Interacting Long-Crested Stokes Waves on Deep Water

Oscillatory third-order perturbation solutions for sums of interacting long-crested Stokes waves on deep water are obtained. A third-order perturbation expansion of the nonlinear free boundary value problem, defined by the coupled Bernoulli equation and kinematic boundary condition evaluated at the free surface, is solved by replacing the exponential term in the potential function by its series expansion and substituting the equation for the free surface into it. There are second-order changes in the frequencies of the first-order terms at third order. The waves have a Stokes-like form when they are high. The phase speeds are a function of the amplitudes and wave numbers of all of the first-order terms. The solutions are illustrated. A preliminary experiment at the United States Naval Academy is described. Some applications to sea keeping are bow submergence and slamming, capsizing in following seas and bending moments.

On permanent capillar-heavy waves in infinite channels

This work is based on the constructive existence proof of solutions of a comprehensive class of non-linear free boundary-value problems of plane hydrodynamics by E. Zeidler (1971) and a general computational method for constructing the solutions numerically given by the author (1991). The case of permanent capillar-heavy waves in infinite channels will be considered.

Obstacle problem for von Kármán equations

The free boundary value problem in obstacle problem for von Kármán equations is studied. By using the method of complementarity analysis, Rockafellar's theory of duality is generalized to the nonlinear variational problems and a complementarity theory of obstacle problem for von Kármán plates is established. We prove that the uniqueness and existence of solution directly depend on a complementary gap function. Moreover, a generalized dual extreme principle is established. We prove that the nonlinear primal variational inequality problem is eventually equivalent to a semi-quadratic dual optimization problem defined on a statically admissible space. This equivalence can be used to develop an effective numerical method for solving nonlinear free boundary value problems.

Vertical slender jets with surface tension

The shape of a vertical slender jet of fluid falling steadily under the force of gravity is studied. The problem is formulated as a nonlinear free boundary-value problem for the potential. Surface-tension effects are included and studied. The use of perturbation expansions results in a system of equations that can be solved by an efficient numerical procedure. Computations were made for jets issuing from three different orifice shapes, which were an ellipse, a square, and an equilateral triangle. Computational results are presented illustrating the effects of different values for the surfacetension coefficient on the shape of the jet and the periodic nature of the cross-sectional shapes.

A Numerical Method for Computing the Shape of a Vertical Slender Jet

A numerical method is presented for computing the shape of a vertical slender jet of fluid falling steadily under the force of gravity. The problem to be solved is fomulated as a nonlinear free boundary value problem for the cross-sectional shape of the jet. The numerical method of solution treats the boundary conditions of the problem as a pair of nonlinear hyperbolic pseudo-differential equations to be integrated in the stream-wise direction. The original differential equation appears as an auxiliary condition. This formulation is shown to be well-posed. The numerical method is found to be stable and second-order accurate. Computations are presented for jets issuing from several different orifice shapes. The numerical method of solution appears to be new and may be applicable to other nonlinear free boundary value problems.

Vertical slender jets

The shape of a vertical slender jet of fluid falling steadily under the force of gravity is studied. The problem is formulated as a nonlinear free boundary-value problem for the potential. Surface tension effects are neglected. The use of perturbation expansions results in a system of equations that can be solved by an efficient numerical procedure. Computations were made for jets issuing from orifices in various shapes including an ellipse, a rectangle, and an equilateral triangle. Computational results are presented illustrating the propagation of discontinuities and the formation of thin sheets of fluid.

Existence theorem for a nonlinear free boundary-value problem

Existence theorem for a nonlinear free boundary-value problem