Transcript of On the steady compressible flows in a nozzle Zhouping Xin The Institute of Mathematical Sciences,...
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- On the steady compressible flows in a nozzle Zhouping Xin The
Institute of Mathematical Sciences, The Chinese University of Hong
Kong 2008, Xiangtan
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- Contents 1 Introduction Compressible Euler system and transonic
flows Global subsonic flows Subsonic-Sonic flows Transonic flows
with shocks ** A Problem due to Bers ** A problem due to
Courant-Friedrich on transonic- shocks in a nozzle
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- 2Global Subsonic and Subsonic-Sonic Potential Flows in Infinite
Long Axially Symmetric Nozzles Main Results Ideas of Analysis
3Global Subsonic Flows in a 2-D Infinite Long Nozzles Main
Results
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- 4 Transonic Shocks In A Finite Nozzle Uniqueness Non-Existence
Well-posedness for a class of nozzles
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- 1 Introduction The ideal steady compressible fluids are
governed by the following Euler system: where
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- Key Features: nonlinearities ( shocks in general) mixed-type
system for many interesting wave patterns (change of types,
degeneracies, etc.) It seems difficult to develop a general theory
for such a system. However, there have been huge literatures
studying some of important physical wave patterns, such as Flows
past a solid body; Flows in a nozzle; Wave reflections, etc.
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- Even for such special flow patterns, there are still great
difficulties due to the change type of the system, free boundaries,
internal and corner singularities etc.. Some simplified models:
Potential Flows: Assume that In terms of velocity potential, Then
(0.1) can be replaced by the following Potential Flow
Equation.
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- with and the Bernoullis law which can be solved to yield hereis
the enthalpy given by.
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- Remark 1: The potential equation (0.4) is a 2nd order
quasilinear PDE which is Remark 2: (0.4) also appears in geometric
analysis such as mean curvature flows. 2-D Isentropic Euler Flows
Assume that S = constant. Then the 2-D compressible flow equations
are
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- The characteristic polynomial of (0.7) has three roots given as
Thus, (0.7) is hyperbolic for supersonic flows (0.7) is coupled
elliptic-hyperbolic for subsonic flow (0.7) is degenerate for sonic
flow CHALLENGE: Transonic Flow patterns
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- Huge literatures on the studies of the potential equation
(0.4). In particular for subsonic flows. The most significant work
is due to L. Bers (CPAM, Vol. 7, 1954, 441-504): MM
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- Fact: For 2-D flow past a profile, if the Mach number of the
freestream is small enough, then the flow field is subsonic outside
the profile. Furthermore, as the freestream Mach number increases,
the maximum of the speed will tend to the sound speed. (See also
Finn-Gilbarg CPAM (1957) Vol. 10, 23-63). These results were later
generalized to 3-D by Gilbarg and then G. Dong, they obtained
similar theory. And recently, a weak subsonic-sonic around a 2-D
body has been established by Chen-Dafermos-Slemrod-Wang.
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- A lot of the rich wave phenomena in M-D compressible fluids
appear in steady flows in a nozzle, which are important in fluid
dynamics and aeronautic. In his famous survey (1958), Bers proposed
the following problem: For a given infinite long 2-D or 3-D axially
symmetric solid nozzle, show that there is a global subsonic flow
through the nozzle for an appropriately given incoming mass flux
One would expect a similar theory as for the airfoil would hold for
the nozzle problem. Question: How the flow changes by varying m 0
?
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- However, this problem has not been solved dispite many studies
on subsonic flows is a finite nozzle. One of Keys: To understand
sonic state s
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- Our main strategy to studying compressible flows in a nozzle
is: Step 1 Existence of subsonic flow in a nozzle for suitably
small incoming mass flux It is expected that if the incoming mass
flux is small, then global uniform subsonic flow in a nozzle
exists. Some of the difficulties are: Global problem with different
far fields, so the compactification through Kelvin-type transmation
becomes impossible;
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- Possibility of appearance of sonic points For rotational flows,
it is unclear how to formulate a global subsonic problem. Step 2
Transition to subsonic-sonic flow We study the dependence of the
maximum flow speed on the incoming flux and to investigation
whether there exists a critical incoming fluxsuch that if the
incoming mass flux m increases to, then the corresponding maximum
flow speedapproaches the sound speed.
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- Step 3 Obtain a subsonic-sonic flow in a nozzle as a limit of a
sequence of subsonic flows. Assume that Step 1 and Step 2 have been
done. Let Let be the corresponding subsonic flow velocity field in
the nozzle. Questions: 1. 2. Can solve (0.4)?
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- If both questions can be answered positively, then will yield a
subsonic-sonic flow in a general nozzle!! Remark: Due to the strong
degeneracy at sonic state, it is a long standing open problem how
to obtain smooth flows containing sonic states, exceptions:
accelerating transonic flows (Kutsumin, M. Feistauer) (for special
nozzles and special B.C.) subsonic flow which becomes sonic at the
exit of a straight expanding nozzle (Wang-Xin, 2007)
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- Finally, we deal with transonic flows with shocks. When, in
general, transonic flows must appear. However, it can be shown that
smooth transonic flows must be unstable (C. Morawetz). shock
wave
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- Thus, SHOCK WAVES must appear in general, and the flows
patterns can become extremely complicated. Then the analysis of
such flow patterns becomes a challenge for the field due to:
complicated wave reflections, degeneracies, free boundaries, change
type of equations, mixed-type equations, etc.
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- Thus, Morawetz proposed to study the general weak solution by
the framework of Compensated-Compactness for the 2-D potential
flows. Yet, this approach has not been successful so far. The
quasi-1D model has been successfully analyzed by many people,
Embid-Majda-Goodam, Gamba, Liu, etc. Some special steady
multi-dimensional transonic wave patterns with shock have been
investigated recently by Chan- Feldman, Xin-Yin, S. Chen, Fang
etc.
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- Motivated by engineering studies, Courant-Friedrichs proposed
the following problem on transonic shock phenomena in a de Laval
nozzle: pepe 0, (q 0, 0, 0)
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- Consider an uniform supersonic flow entering a de Laval nozzle.
Given an appropriately large receiver pressure p e at the exit of
the nozzle, if the supersonic flow extends passing through the
throat of the nozzle, then at the certain place of the divergent
part of the nozzle, a shock wave must intervene and the flow is
compressed and slowed down to a subsonic speed, and the location
and strength of the shock are adjusted automatically so that the
pressure at the exit becomes the given pressure p e.
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- Experimentally and physically, it seems to be a very reasonable
conjecture. Indeed, there are cases, such as quasi- one-dimensional
model, the conjecture is definitely true. As we will show later, it
also holds for symmetric flows. Unfortunately, this seems to be a
very tricky question in general as we will show later. Some
surprising facts appear!!! general uniqueness results non-existence
well-posedness for a class of nozzles
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- 2Global Subsonic and Subsonic-Sonic Potential Flows in Infinite
Long Axially-Symmetric Nozzles We first give a complete positive
answer to the problem of Bers on global subsonic flows a general
infinite nozzle. Furthermore, we will obtain a subsonic-sonic flow
in the nozzle also as mentioned in the introduction. 2.1
Formulation of the problem Consider 3-D potential equation (0.4)
with
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- Set and assume that Bernoullis law, (0.5), becomes with being
the maximal speed.
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- Normalize the flow by the critical speed Then (2.2) can be
rewritten as For example, for polytrophic gases,, (2.4) is
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- Some facts: 1. Subsonic 2. is a two-valued function ofand
subsonic branch corresponds to 1
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- 3. Let H be the specific volume, i.e., Then
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- Now G = G (q 2 ) such that then Then the potential equation can
be rewritten as
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- Assume that the nozzle is axi-symmetric and given by where is
assumed to be smooth such that for some Assume also that the nozzle
wall is impermeable, so that the boundary condition is
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- Note that for any smooth solution to (2.9) satisfying the
boundary condition (2.12), the mass flux through any section of the
nozzle transversal to the x-axis is constant, the nozzle problem
can be formulated as: Find a solution to (2.9) and (2.12) such that
where s is a section of the nozzle transversal to x-axis, and is
the normal of s forming an accurate angle with x-axis.
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- 2.2 The Main Results Then the following existence results on
the global uniform subsonic flow in the nozzle hold: Theorem 2.1
(Xie-Xin) Assume that nozzle is given by (2.10) satisfying (2.11).
Then a positive constant, which depends only on f, such that if,
the boundary value problem (2.9), (2.12) and (2.13) has a smooth
solution, such that and the flow is axi-symmetric in the sense that
where, (U, V) (x, r) are smooth, and V (x, 0) = 0.
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- To study some important properties of the subsonic flows in a
nozzle, in particular, the dependence of the flows on the incoming
mass flux m 0, we assume that the wall of the nozzle tends to be
flat at far fields, say (rescaling if necessary) Then we have
following sharper results. Theorem 2.2 (Xie-Xin) Let the nozzle
satisfy (2.11) and (2.16). Then a positive constant with the
following properties:
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- (1) axially-symmetric uniformly subsonic solution to the
problem (2.19), (2.12), and (2.13) with the properties and
uniformly in r, where G is given in (2.8).
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- (2) is critical in the sense that ranges over [0,1) as m 0
varies in [0, ). (3) For, the axial velocity is always positive in,
i.e., (4) (Flow angle estimates): For, the flow angle satisfies
where
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- (5) (Flow speed estimates) For any, In particular, (No
stagnation uniformly). Finally, we show the asymptotic behavior of
these subsonic solutions when the incoming mass flux m 0 approaches
the critical value. Based on Theorem 2.2 and a framework of
compensated- compactness, we can obtain the existence of a global
subsonic-sonic weak solution to (2.9), (2.12) and (2.13).
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- Theorem 2.3 (Xie-Xin) Assume that (i)The nozzle given by (2.10)
satisfies (2.11) and (2.16). (ii)The fluids satisfy (iii)Let m n be
any sequence such that Denote by the global uniformly subsonic flow
corresponding to m n. Then subsequence of m n, still labeled as m
n, such that
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- with almost every where convergence. Moreover, the limit yields
a 3-D flow with density and velocity satisfying in the sense of
distribution, and for any.
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- Remark 1 (2.26) implies that the boundary condition (2.12) is
satisfied by the limiting velocity field as the normal trace of the
divergence free field on the boundary. Remark 2 Similar theory
holds for the 2-D flows (of Xie-Xin).
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- Remark 3Compared with 3-D airfoil problem, the main difficulty
is how to obtain the uniform ellipticity of (2.9). Remark 4 Key
ideas of analysis: - Cut-off and desigularization; - Hodograph
transformation part-hodograph transformation; - Rescaling and
blow-up estimates for uniformly elliptic equations of two
variables; - Compensated-compactness.
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- 3 Global Isentropic Subsonic Euler Flow in a Nozzle In this
section, we present some results on the existence of global
subsonic isentropic flows through a general 2-D infinite long
nozzle. Formulation of the problem Note that the steady, isentropic
compressible flow is governed by (0.7), which is a coupled
elliptic-hyperbolic system.
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- Let the 2-D nozzle be with boundaries: Assumptions on s i
:
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- Impermeable Solid Wall Condition: Incoming Mass Flux: Let l be
any smooth curve transversal to the x 1 -direction, and is the
normal of l in the positive x 1 -axis direction, l
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- Set which is a constant independent of l. Due to the hyperbolic
mode, one needs to impose one boundary condition at infinity. Set
whereis the anthalpy normalized so that h(0) = 0.
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- Then we propose the following boundary condition on B where B(x
2 ) is smooth given function defined on [0,1]. Problem (*): Find a
global subsonic solution to (0.7) on satisfying (3.3), (3.4), and
(3.6).
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- Main Results Theorem 3.1 (Xie-Xin) Assume that 1. (3.2) holds,
2. Then such that if thenwith the property that for all, the
problem (*) has a solution such that the following properties hold
true:
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- uniquely determined by m, B(x 2 ), and b a such that
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- uniformly on any sets k 1 cc (0, 1), and k 2 cc (a, b). 4. The
solution to the problem (*) is unique under the additional
assumptions (3.10)- (3.11). Furthermore, is the upper critical mass
flux for the existence of subsonic flow in the following sense, it
holds that either
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- or such that for all the problem (*) has a solution with the
properties (3.9)-(3.11) and Remark 1: Similar results hold for the
full non-isentropic Euler system if, in addition, the entropy is
specified at the upstream.
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- Remark 2: One of main steps in the proof of the main results is
to reduce a non-local boundary value problem for a coupled-
hyperbolic-elliptic system (0.7) to a standard boundary value
problem for a 2nd quasilinear equations on a unbounded domain. The
key to this is a stream-function formulation of the problem. Assume
u > 0. Then (0.7) (3.14) (3.15)
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- in terms of stream function
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- where and J (M, S) can be derived from the equation of
state.
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- Remark 3 Open problems: Uniqueness of Subsonic flows Regularity
of the Subsonic-Sonic flows and Geometry of the degeneracies
General 3-D Nozzle Existence of smooth subsonic-sonic flows
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- 4 Transonic Shock in a nozzle In this section, we will present
some recent progress on transonic flows with shock in a nozzle due
to Courant- Friedrichs. For simplicity in presentation, we will
concentrate on 2D, steady, isentropic Euler equations. 4.1
Formulation of The Problem Consider a uniform supersonic flow (q
0,0) with constant density 0 > 0 which enters a nozzle with
slowly-varying sections
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- .. x2x2 x1x1 p e = p( e ) x 2 = f 2 (x 1 ) x 2 = f 1 (x 1 ) x 1
= (x 2 )
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- Letbe the shock surface we are looking for, which is assumed to
go through a fixed point on the wall sometimes, i.e.,
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- The across the shock surface, we require that
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- The boundary conditions can be described as: where the given
large density at the exit satisfies
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- with the constant statesatisfying Thus the problem is to find a
piecewise smooth solution solving (0.7) with conditions (4.3), and
(4.5)-(4.9). Then we have the following uniqueness results.
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- Theorem 4.1 (Xin-Yan-Yin) a positive constant such that if and
(4.2) and (4.10) hold, then the transonic shock problem(0.7),
(4.3), and (4.5)-(4.9) has no more than one solution such
thatsatisfy the following estimates with :
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- Remark 4.1 It should be emphasized that although one of the key
issues to solve some mixed boundary value problem with corners, and
thus may be a reasonable class for well-posedness, yet the
regularity assumptions in Theorem 3.1 are plausible. Indeed,
implies that R-H condition (4.5) is compatible with solid-wall B.C.
(4.8), while yields the compatibility of (4.8) and (4.9) at the
fixed corners. Then regularity of a special class of 2nd order
elliptic equations can be improved.
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- Remark 4.2 Similar uniqueness holds if the end pressure p e is
prescribed on a c 3 -smooth curve which is a small perturbation of
x 1 =1. Remark 4.3 For general nozzle, the condition (4.3) is
required for uniqueness, due to the example of flat nozzles. Remark
4.4 The condition (4.2) is necessary for the transonic shock wave
patterns conjectured by Courant-Friedrichs in general. Since,
otherwise, there might be supersonic shocks in the supersonic
region and supersonic bulbs in the subsonic flows. Remark 4.5
Similar results holds for non-isentropic flows & in 3-D.
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- 4.2 Non-Existence Although the formulation of the transonic
shock problem, (0.7), (4.3), and (4.5)(4.9) looks reasonable
physically, this problem HAS NO SOLUTION in general, indeed, we can
show that for a class of nozzles, there exists no such transonic
solutions for general given supersonic incoming flow and end
pressure. Our first example is 2-D nozzles with flat walls.
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- Theorem 4.2 (Xin-Yan-Yin) Assume that the nozzle is flat, i.e.
Then for the constant supersonic incoming flow with, and the end
pressure, the Euler equations (0.7), with boundary condition
(4.5)-(4.9) has no transonic solutions so that satisfies the
following requirements with some:
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- where is a suitable small constant which depends only on the
Remark 4.6 It should be emphasized that Theorem 4.2 does not
require that the transonic shock wave goes through a fixed point,
i.e. we do not assume (4.3). Remark 4.7 For flat nozzles, similar
non-existence results hold true for the non-isentropic Euler system
with a similar analysis.
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- 4.3 Well-posedness We now solve the conjecture of
Courant-Friedrich for a class of nozzle. We consider a class of
non-flat nozzles which c 5 -regular, whose wall consist of two
parts on [-1,1].
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- - - x2x2 (1,0) x1x1
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- Set Let satisfy
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- so that for a symmetric shock for an angular section nozzle.
Furthermore, assume the incoming supersonic flow is symmetric on in
the sense that and is a small perturbation of. Indeed of (4.3), the
shock is assumed to go through (0,0). and instead of (4.9), one
imposes the B.C. at the exit as
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- Theorem 4.3 (Xin-Yan-Yin) Let the nozzle be given as above and
be suitably small. Then the transonic shock problem (0.7) with
boundary condition (4.3), (4.5)-(4.8), and (4.9) is ill-posed for
large ||. More precisely, supersonic incoming flows, which are
small perturbations of, such that the problem (0.7), (4.3),
(4.5)-(4.8), and (4.9) has no transonic shock solution with
satisfying the following properties for some.
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- where are the intersection points of the shock wave curve with
the solid wall respectively. Remark 4.8 Similar results hold for
non-isentropic flow and 3D fluids. Despite the non-existence
results in above, it is possible to have the transonic shock wave
pattern conjecture by
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- Courant-Friedrichs for some interesting class of nozzle and
special exit boundary condition. Instead, consider the nozzle give
in (4.14). If one gives up the requirement (4.3), that is, the
shock positive is completely free, then it is possible to have a
solution. Indeed, one has Theorem 4.4 (Xin-Yan-Yin) Let the nozzle
be given in (4.14) and the incoming supersonic flow be described as
in Theorem 4.3. Then (1) positive constants p 1 and p 2, p 1 < p
2, which are determined by the incoming flow and the shape of the
nozzle,
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- such that for a given constant pressure, symmetric transonic
shock solution to the problem (0.7), (4.5)- (4.8), (4.9) with the
shock location at which depends on p e monotonically. Furthermore,
in the subsonic region, the solution is denoted by
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- (2) Let. Then the above symmetric transonic shock are unique in
the class for suitably small. (3) Such a transonic-shock is
dynamically stable! Finally, we consider the general case that the
exit and pressure is a variable withsuitably small,
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- Then we have the following general results: Theorem 4.5
(Li-Xin-Yin) Let the nozzle be given as in (4.14) and the incoming
supersonic flow be described as in Theorem 4.3 such that Then
constant such that for all the transonic shock problem (0.7), (4.5)
(4.8), (4.17) (here (4.7) becomes )
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- has a unique solutionwith the following properties: (i) (ii)
with being the subsonic region
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- Remark 4.9 Same results hold for non-isentropic flow. Remark
4.10 In fact, the shock position depends on the exit and pressure
monotonically, this is the key for the proof of the existence in
Theorem 4.5. The proof of this depends crucially on the properties
of incoming supersonic flow. Remark 4.11 Similar results have been
obtained by Li-Xin- Yin for 3-D case.
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- Thank You!