Global subsonic Euler flows in an infinitely long axisymmetric nozzle

In this paper, we consider global subsonic compressible flows through an infinitely long axisymmetric nozzle. The flow is governed by the steady Euler equations and has boundary conditions on the nozzle walls. Existence and uniqueness of global subsonic solution are established for an infinitely long axisymmetric nozzle, when the variation of Bernoulli's function in the upstream is sufficiently small and the mass flux of the incoming flow is less than some critical value. The results give a strictly mathematical proof to the assertion in Bers (1958) [2]: there exists a critical value of the incoming mass flux such that a global subsonic flow exists uniquely in a nozzle, provided that the incoming mass flux is less than the critical value. The existence of subsonic flow is obtained by the precisely a priori estimates for the elliptic equation of two variables. With the assumptions on the nozzle in the far fields, the asymptotic behavior can be derived by a blow-up argument for the infinitely long nozzle. Finally, we obtain the uniqueness of uniformly subsonic flow by energy estimate and derive the existence of the critical value of incoming mass flux.     

Global subsonic and subsonic-sonic flows through infinitely long axially symmetric nozzles

In this paper, we establish existence of global subsonic and subsonic-sonic flows through infinitely long axially symmetric nozzles by combining variational method, various elliptic estimates and a compensated compactness method. More precisely, it is shown that there exist global subsonic flows in nozzles for incoming mass flux less than a critical value; moreover, uniformly subsonic flows always approach to uniform flows at far fields when nozzle boundaries tend to be flat at far fields, and flow angles for axially symmetric flows are uniformly bounded away from π/2; finally, when the incoming mass flux tends to the critical value, subsonic-sonic flows exist globally in nozzles in the weak sense by using angle estimate in conjunction with a compensated compactness framework.     

Global subsonic compressible flows through a two-dimensional periodic duct

We deal with two-dimensional compressible potential subsonic flows in an infinitely long duct with periodic walls. It is shown that there exists a critical value of mass flux: If the incoming mass flux is less than the critical value, then the flow is also periodic. Existence, uniqueness and regularity of the periodic solution are obtained by techniques of elliptic equations.     

Existence of steady subsonic Euler flows through infinitely long periodic nozzles

In this paper, we study the global existence of steady subsonic Euler flows through infinitely long nozzles which are periodic in $x_1$-direction with the period L. It is shown that when the variation of Bernoulli function at some given section is small and mass flux is in a suitable regime, there exists a unique global subsonic flow in the nozzle. Furthermore, the flow is also periodic in $x_1$-direction with the period L. If, in particular, the Bernoulli function is a constant, we also get the existence of subsonic-sonic flows when the mass flux takes the critical value.     

Uniqueness of symmetric steady subsonic flows in infinitely long divergent nozzles

We prove that the spherically symmetric subsonic flows in an infinitely long straight divergent nozzle with arbitrary smooth cross-section are unique for the three-dimensional steady potential flow equation. The proof depends on an extreme principle for elliptic equations in an unbounded conical domain, under the assumption that the gradient of the solution is of order \({O\left(\frac{1}{|x|}\right)}\) as \({|x|\rightarrow\infty}\) . Similar result holds for steady subsonic Euler flows in two-dimensional infinitely long straight divergent nozzles.     

Uniqueness of symmetric steady subsonic flows in infinitely long divergent nozzles

We prove that the spherically symmetric subsonic flows in an infinitely long straight divergent nozzle with arbitrary smooth cross-section are unique for the three-dimensional steady potential flow equation. The proof depends on an extreme principle for elliptic equations in an unbounded conical domain, under the assumption that the gradient of the solution is of order O(\frac1|x|){O\left(\frac{1}{|x|}\right)} as |x|?￥{|x|\rightarrow\infty} . Similar result holds for steady subsonic Euler flows in two-dimensional infinitely long straight divergent nozzles.     

Collision of incompressible inviscid fluids effluxing from two nozzles

This paper concerns the mathematical theory of the collision problem of two-dimensional incompressible inviscid fluids issuing from two given nozzles. The main result reads that for given two co-axis symmetric semi-infinitely long nozzles with arbitrary variable sections, imposing the incoming mass fluxes in two nozzles, there exists a smooth impinging outgoing jet, such that the two free boundaries of the impinging jet initiate smoothly at the endpoints of the nozzles and approach to some asymptotic direction in downstream, and the pressure on the free surface remains a constant. Furthermore, we show that there exists a unique smooth surface separating the two nonmiscible fluids and there exists a unique stagnation point in the fluid region and its closure. Moreover, some results on the uniqueness and the estimates of the location of the impinging outgoing jet are also established. Finally, the asymptotic behaviors, the precise estimate to the deflection angle and other properties to the impinging outgoing jet are also considered.     

Note on the uniqueness of subsonic Euler flows in an axisymmetric nozzle

In this paper, we consider the uniqueness of globally subsonic compressible flows through an infinitely long axisymmetric nozzle. The flow is governed by the steady Euler equations and satisfies no-flow boundary conditions on the nozzle walls. We will show that for given mass flux and Bernoulli’s function in the upstream, the subsonic flow is unique in the class of all axisymmetric solutions, which possess the asymptotic behaviors at the far fields. This result extends the uniqueness of solutions in the previous paper Du and Duan (2011) [1].     

Stabilization of a nonlinear flow-plate interaction via component-wise decomposition

Asymptotic-in-time interior feedback control of a panel interacting with an inviscid, subsonic flow is considered. The classical model [8] is given by a clamped nonlinear plate strongly coupled to a convected wave equation on the half space. In the absence of energy dissipation the plate dynamics converge to a compact and finite dimensional set [6, 7]. With a sufficiently large velocity feedback control on the structure we show that the full flow-plate system exhibits strong convergence to the set of stationary states in the natural energy topology. We show a decomposition of the dynamics into “smooth” component and global-in-timeHadamard continuous component, thus permitting approximation by smooth data. That the flows are subsonic is critical for our approach. Our result implies that flutter (a periodic or chaotic end behavior) is not present in subsonic flows with sufficient viscous damping in the structure.     

Three dimensional steady subsonic Euler flows in bounded nozzles

The existence and uniqueness of three dimensional steady subsonic Euler flows in rectangular nozzles were obtained when prescribing normal component of momentum at both the entrance and exit. If, in addition, the normal component of the voriticity and the variation of Bernoulli's function at the entrance are both zero, then there exists a unique subsonic potential flow when the magnitude of the normal component of the momentum is less than a critical number. As the magnitude of the normal component of the momentum approaches the critical number, the associated flows converge to a subsonic–sonic flow. Furthermore, when the normal component of vorticity and the variation of Bernoulli function are both small, the existence and uniqueness of subsonic Euler flows with non-zero vorticity are established. The proof of these results is based on a new formulation for the Euler system, a priori estimate for nonlinear elliptic equations with nonlinear boundary conditions, detailed study for a linear div–curl system, and delicate estimate for the transport equations.     

The optimal convergence rates of non-isentropic subsonic Euler flows through the infinitely long three-dimensional axisymmetric nozzles

This paper is concerned about the optimal convergence rates of non-isentropic subsonic flows at far fields in three-dimensional infinitely long axisymmetric nozzles. By using the stream function formulation for the compressible Euler equations, the subsonic Euler flows are equivalent to a quasilinear elliptic equation of the stream function. The key points to prove the convergence rates of subsonic flows at far fields are the choice of compared functions and the maximum principles.     

Compressible subsonic jet flows issuing from a nozzle of arbitrary cross-section

We are concerned with the well-posedness theory of two-dimensional compressible subsonic jet flow issuing from a semi-infinitely long nozzle of arbitrary cross-section. Given any atmospheric pressure $p_0$, we show that there exists a critical mass flux $m_{cr}$ depending on $p_0$ and Ω, such that if the incoming mass flux $m_0$ is less than the critical value, then there exists a unique smooth subsonic jet flow, issuing from the given nozzle. The jet boundary is a free streamline, which initiates from the end point of the nozzle smoothly and extends to the infinity. One of the key observations in this paper is that the restriction of the incoming mass flux guarantees completely the subsonicity of the compressible jet in the whole flow field, which coincides with the observation on the compressible subsonic flows in an infinitely long nozzle without free boundary in [8].     

Steady subsonic potential flows through infinite multi-dimensional largely-open nozzles

We proved existence, regularity and uniqueness of steady subsonic potential flows in n-dimensional (n ≥ 3) infinite nozzles with largely-open convergent and divergent parts when the total mass flux is less than a certain value. Such nozzles consist of two cones with arbitrary open angles and an arbitrary smooth bounded tubular part. The existence of a weak solution is proved by applying the direct method of calculus of variation to a carefully chosen functional defined on a Hilbert space based upon Hardy inequality. Hölder gradient regularity of weak solution is shown by using Moser iteration to quasilinear elliptic equations in divergence form. Also, the obtained solution is unique in the class of functions with finite kinetic energy by modulo a constant.     

Three dimensional non-isentropic subsonic Euler flows in rectangular nozzles

In this paper, we study the existence and uniqueness of three dimensional non-isentropic subsonic Euler flows in rectangular nozzles. This work is an extension of Chen and Xie’s work on isentropic subsonic Euler flows. If, besides small normal component of vorticity, Bernoulli’s function and entropy function with small variations are given on the entrance, the existence and uniqueness of non-isentropic subsonic Euler flows are established.     

Global Conic Shock Wave for the Steady Supersonic Flow Past a Large Curved Cone at Infinity

In this paper, we establish the global existence and stability of a steady symmetric shock wave for the constant supersonic flow past an infinitely long and large curved conic body. The flow is assumed to be polytropic, isentropic and described by a steady potential equation. Through looking for the suitable “dissipative” boundary conditions on the shock and the conic surface together with the special form of shock equation, we show that the conic shock attached at the vertex of the cone exists globally in the whole space when the speed of the supersonic incoming flow is appropriately large.     

Transonic nozzle flows and free boundary problems for the full Euler equations

We establish the existence and uniqueness of transonic flows with a transonic shock through a two-dimensional nozzle of slowly varying cross-sections. The transonic flow is governed by the steady, full Euler equations. Given an incoming smooth flow that is close to a constant supersonic state (i.e., smooth Cauchy data) at the entrance and the subsonic condition with nearly horizontal velocity at the exit of the nozzle, we prove that there exists a transonic flow whose downstream smooth subsonic region is separated by a smooth transonic shock from the upstream supersonic flow. This problem is approached by a one-phase free boundary problem in which the transonic shock is formulated as a free boundary. The full Euler equations are decomposed into an elliptic equation and a system of transport equations for the free boundary problem. An iteration scheme is developed and its fixed point is shown to exist, which is a solution of the free boundary problem, by combining some delicate estimates for the elliptic equation and the system of transport equations with the Schauder fixed point argument. The uniqueness of transonic nozzle flows is also established by employing the coordinate transformation of Euler-Lagrange type and detailed estimates of the solutions.     

Incompressible limit of solutions of multidimensional steady compressible Euler equations

A compactness framework is formulated for the incompressible limit of approximate solutions with weak uniform bounds with respect to the adiabatic exponent for the steady Euler equations for compressible fluids in any dimension. One of our main observations is that the compactness can be achieved by using only natural weak estimates for the mass conservation and the vorticity. Another observation is that the incompressibility of the limit for the homentropic Euler flow is directly from the continuity equation, while the incompressibility of the limit for the full Euler flow is from a combination of all the Euler equations. As direct applications of the compactness framework, we establish two incompressible limit theorems for multidimensional steady Euler flows through infinitely long nozzles, which lead to two new existence theorems for the corresponding problems for multidimensional steady incompressible Euler equations.     

Local Interaction Theory for Laminar Transonic Flows in Slender Channels

Transonic flows through channels so narrow that the classical boundary layer approach fails are considered. As a consequence the properties of the inviscid core and the viscosity dominated boundary layer region can no longer be determined in subsequent steps but have to be calculated simultaneously. The resulting viscous inviscid interaction problem for weakly three dimensional laminar flows is formulated for perfect gases under the requirement that the channel is sufficiently narrow so that the flow outside the viscous wall layers becomes planar in the leading order approximation. Representative solutions for subsonic as well as for supersonic flows disturbed by three dimensional surface mounted obstacles will be presented. (© 2009 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Existence of a solitary planar bubble in symmetric channels

Two-dimensional steady ideal fluids with gravity in finite height symmetric channels are considered. This model is well-known, but the existence results seem rare in the literature. For suitable incoming vertical velocity and the divergent nozzles, there exists a solitary bubble starting at the symmetric axis and extending downwards without limit. Finally, the inclination angle at the vertex of this bubble is investigated.     

On the Existence and Stability of a Global Subsonic Flow in a 3D Infinitely Long Cylindrical Nozzle

This paper is concerned with the problem on the global existence and stability of a subsonic flow in an infinitely long cylindrical nozzle for the 3D steady potential flow equation. Such a problem was indicated by Courant-Friedrichs in [8, p. 377]： A flow through a duct should be considered as a cal symmetry and should be determined steady, isentropic, irrotational flow with cylindriby solving the 3D potential flow equations with appropriate boundary conditions. By introducing some suitably weighted HSlder spaces and establishing a priori estimates, the authors prove the global existence and stability of a subsonic potential flow in a 3D nozzle when the state of subsonic flow at negative infinity is given.