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The Flip-Over Effect in Self-Similar Hydrodynamics

Permanent Link: http://ufdc.ufl.edu/UFE0021145/00001

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Title: The Flip-Over Effect in Self-Similar Hydrodynamics
Physical Description: 1 online resource (23 p.)
Language: english
Creator: Baxter, Nathan P
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2007

Subjects

Subjects / Keywords: Mathematics -- Dissertations, Academic -- UF
Genre: Mathematics thesis, M.S.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: A simplified model for the evolution of an expanding plasma is the so-called self-similar model. In this model it is assumed that the velocity vector field is proportional to the coordinate system with the center of the plasma plume at its origin. This model predicts that all quantities characterizing the plasma state have level surfaces that are spheres. In experiments, it has been shown that the shape of the plasma has a significant effect on the emitted radiation. Since plasmas generally do not expand as perfect spheres, we consider a recently proposed self-similar model, in which the level surfaces are ellipsoidal. In this model the plasma's evolution is fully determined by the principle axis as functions of time. The latter are determined by solving a Lagrangian dynamical system. In specific cases where the adiabatic constant of the gas used is 5/3, we are able to apply the Noether Theorem, which leads to the Noether integrals of motion. Our main focus was a phenomenon known as the flip-over effect as it occurs in such special cases. When the plasma is created it will have its longest axis in some direction. As it evolves with time, it will remain ellipsoidal, but the longest axis will be in the orthogonal direction. Our results show that the flip-over effect occurs exactly once.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Nathan P Baxter.
Thesis: Thesis (M.S.)--University of Florida, 2007.
Local: Adviser: Shabanov, Sergei.

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2007
System ID: UFE0021145:00001

Permanent Link: http://ufdc.ufl.edu/UFE0021145/00001

Material Information

Title: The Flip-Over Effect in Self-Similar Hydrodynamics
Physical Description: 1 online resource (23 p.)
Language: english
Creator: Baxter, Nathan P
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2007

Subjects

Subjects / Keywords: Mathematics -- Dissertations, Academic -- UF
Genre: Mathematics thesis, M.S.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: A simplified model for the evolution of an expanding plasma is the so-called self-similar model. In this model it is assumed that the velocity vector field is proportional to the coordinate system with the center of the plasma plume at its origin. This model predicts that all quantities characterizing the plasma state have level surfaces that are spheres. In experiments, it has been shown that the shape of the plasma has a significant effect on the emitted radiation. Since plasmas generally do not expand as perfect spheres, we consider a recently proposed self-similar model, in which the level surfaces are ellipsoidal. In this model the plasma's evolution is fully determined by the principle axis as functions of time. The latter are determined by solving a Lagrangian dynamical system. In specific cases where the adiabatic constant of the gas used is 5/3, we are able to apply the Noether Theorem, which leads to the Noether integrals of motion. Our main focus was a phenomenon known as the flip-over effect as it occurs in such special cases. When the plasma is created it will have its longest axis in some direction. As it evolves with time, it will remain ellipsoidal, but the longest axis will be in the orthogonal direction. Our results show that the flip-over effect occurs exactly once.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Nathan P Baxter.
Thesis: Thesis (M.S.)--University of Florida, 2007.
Local: Adviser: Shabanov, Sergei.

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2007
System ID: UFE0021145:00001


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THE FLIP-OVER EFFECT IN SELF-SIMILAR HYDRODYNAMICS


By
NATHAN P. BAXTER



















A THESIS PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
MASTER OF SCIENCE

UNIVERSITY OF FLORIDA

2007




































02007 Nathan P. Baxter









TABLE OF CONTENTS


pagfe

4


ABSTRACT .....

CHAPTER

1 INTRODUCTION

2 SELF-SIMILAR HYDRODYNAMICS

3 INTEGRALS OF MOTION ....

4 THE CASE OF AXIAL SYMMETRY

5 FLIP OVER EFFECT .....

REFERENCES .........









Abstract of Thesis Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Master of Science

THE FLIP-OVER EFFECT IN SELF-SIMILAR HYDRODYNAMICS

By

Nathan P. Baxter

August 2007

Cl.! ny~: Sergfei Shabanov
Major: Mathematics

A simplified model for the evolution of an expanding plasma is the so-called

self-sintilar model. In this model it is assumed that the velocity vector field is proportional

to the coordinate system with the center of the plasma plunle at its origin. This model

predicts that all quantities characterizing the plasma state have level surfaces that

are spheres. In experiments, it has been shown that the shape of the plasma has a

significant effect on the emitted radiation. Since plasmas generally do not expand as

perfect spheres, we consider a recently proposed self-sintilar model, in which the level

surfaces are ellipsoidal.

In this model the plasnia's evolution is fully determined by the principle axis as

functions of time. The latter are determined by solving a Lagrangian dynamical system.

In specific cases where the adiahatic constant of the gas used is 5/3, we are able to apply

the Noether Theorem, which leads to the Noether integrals of motion. Our main focus was

a phenomenon known as the flip-over effect as it occurs in such special cases. When the

plasma is created it will have its longest axis in some direction. As it evolves with time, it

will remain ellipsoidal, but the longest axis will be in the orthogonal direction. Our results

show that the flip-over effect occurs exactly once.









CHAPTER 1
INTRODUCTION

One of the techniques of remote chemical analysis is based on the analysis of radiation

spectra of laser induced plasmas. A laser pulse evaporates a small piece of material

turning it into a hot plasma. The plasma plume expands, cools and emits light. The

radiated light is collected by a spectrometer that produces the radiation intensity as a

function of light wavelength (frequency). This function is then used to reconstruct the

plasma initial state characterized by temperature, pressure, and concentrations of chemical

elements. Solving the inverse initial value problem requires knowledge of the plasma

evolution during which the spectral analysis is carried out [1-5].

General equations describing the time evolution of expanding plasmas are nonlinear

PDE's (based on the Navier-Stokes equations) whose solutions are difficult to obtain

even numerically (due to high computational costs). So, in practical applications various

simplified plasma evolution models are used. In particular, the so-called self-similar models

turn out to be accurate for practical purposes of remote chemical analysis. In the most

widely used self-similar model, it is assumed the velocity vector field is proportional to

the position vector in the coordinate system whose origin is the plasma plume center

[5-9]. The corresponding (self-similar) solutions of the general plasma evolution equations

have the characteristic feature that all quantities characterizing the current plasma state

(mass density, temperature, pressure) have level surfaces being spheres, while the sphere

radii are functions of time satisfying some differential equation of the second order (which

is often solved numerically). This solution clearly describes expansion of spherically

symmetric plasma plumes. It has been found that deviations of the plasma plume shape

from the spherical one can have a substantial effect on the emitted radiation. Hence, the

use of the spherical self-similar expansion model to reconstruct the initial mass densities

(concentrations of chemical elements) of the evaporated material sample might lead to

large errors or even incorrect results.










To take into account the shape effects on the emitted radiation, a new self-similar

model has been proposed [10]. Its characteristic feature is that all the quantities

characterizing the current plasma state as functions of spatial variables have level surfaces

being ellipsoids. The plasma time evolution is fully determined by the evolutions of the

principal axes of the ellipsoids which, in turn, is described by a Lagrangian dynamical

system. This system appears integrable due to symmetries leading to the Noether integrals

of motion in a practically important case of the axial symmetric initial plasma plumes and

of plasmas whose physical properties are close to those of ideal one atomic gases (which

means the adiabatic constant is 5/3 in the equation of state).

A remarkable phenomenon predicted by the .I-i-mmetric self-similar model is the

so-called flip over effect: If initially the level surfaces of the mass density were ellipsoids

with largest axes in a given direction, then the plasma evolves so that the level surfaces

become ellipsoids with the largest axis in the orthogonal direction. In other words, a

plasma plume which has initially a "p11 .In .:' shape turns into a plume with a "
shape and vice versa.

This effect was first found in numerical simulations and also proved for the case

when the .I-i-mmetry is small (the ratios of ellipsoids principal axis are close to one). The

latter study was based on perturbation theory. The flip-over effect has also been observed

experimentally. In the present study it is proved that the flip-over effect occurs exactly

once, i.e. no oscillations between "p ll., .l:- and "
expansion.









CHAPTER 2
SELF-SIMILAR HYDRODYNAMICS

In this section we give a derivation of an .I-i-mmetric self similar solution [10] of the

Navier-Stokes equations for hydrodynamics [5-9]. All vectors will be denoted by bold

characters.

Sep + V (pv) = 0 (2-1)

8~ty + ~71:vy~v = -1VP (2-2)

8t(Pe + pV2/2) = -V [v(pe + pV2/2) + PV] pH (2-3)

We let 8< denotes the partial derivative with respect to time t, p(t, x) is the mass

density, v(t, x) is the velocity vector field, e(t, x) is the internal energy density which

is proportional to the temperature, P(t, x) is the pressure, and H(t, x) is the energy

density loss due to radiation per unit time and unit mass. In our case the energy loss

due to radiation is small and can be neglected, so we set H = 0. We must also add the

equation of state that relates mass, energy density and pressure.


P = (y 1) pe (2-4)

Here y is the adiabatic constant of a particular gas. Beginning with the initial conditions

p(0, x) = po(x), e(0, x) = co(x), and v(0, x) = vo(x) we look for a solution to the initial

value problem (2-1) (2-3).

The .I-i-mmetric self-similar plasma expansion model is based on the assumption that the

velocity vector field has a specific dependence on x as follows,




where the functions (y(t) are to be determined. The velocity vector field cannot depend on

(4(0) because v is scale invariant, (4(t) cjA,(t). The constants cj are arbitrary and have









no effect on v. Thus (4 is dimensionless and we set


i(o(0 = 1, (i(0) = vy


(2-6)


so that vj(0, x) = vjxj. It is worth noting that Eq. (2-1) (2-3) are covariant under

general translations and rotations in space. So, Eq. (2-5) describes, in fact, a family of

solutions which can be obtained by translating and rotating the position vector x. Next

we set


p(t, x) = g,(t) f(t, x), g,(t) = [l(t)(t)l(23 -1,

and plug Eq. (2-5) and (2-7) into Eq. (2-1), which yields


(2-7)


8
Eq. (2-2) to obtain






V4Pj, j=1, 3, 3




h yP =l x, /( n e


atp


o(pv>


therefore ,


(2-8)













(2-9)


Next we substitute Eq. (2-5) into








which gives



We define the new variable y witl


.
q,(t) = 6, j = 1, 2, 3


(2-10)









It follows that Eq. (2-9) can be written in the following way


VjP = (1/O)8d,P

= 8, P, j = 1, 2, 3.

which leads to

8, P = --q~yyp, j = 1, 2, 3 (2-11)

The: inte~gr~ability condition: n ofr Eq. (2-11) te:lls us thlat 3 = ,1 whichl cairn o~nlyi be
true if f(t, x) = F(t, q) where q = 9 Ilyj .N~ow fromI Eq. (2-8) we get


iief = iitF + 8F x -





and finally

~(Z~2 (212)

The left hand side of Eq. (2-12) only depends on t and q, therefore the right hand side

should only depend on t and q. This is possible only if


rUj(t) = pjr(t), j = 1, 2, 3 (2-13)

where pj are just constants. Next we set




A solution to Eq. (2-12) is F(t, q) = G,(s), where G,(s) is a differentiable function. Thus,

a general solution under the assumption (2-5) should have the form

p(t, x) = g,(t)G,(s) (2-15)









where G, is determined by the initial mass density. Indeed, since g,(0) = 1, po(x) = G,(so)

and, so = s(0, x) = Ci Pixf. Therefore, the level surfaces of the mass density are ellipsoids

with principal semi axes being j. We will show that (j(t) > 0, t > 0. The parameters pj

determine the initial shape of the plasma. Next we put


P (t, x) = gp (t)Gp (S),


(2-16)


substitute it into Eq. (2-11), and infer that


dGp(s) 8(1/2)pjyj2
ds iiyj
dGp (s)


,j =1, 2, 3.
gy (t)


Hence ,


dG p
ds


-Gp, gP(t) = rl(t)gp(t),


(2-17)


From Eq. (2-4) it is clear that


e(t, x) = g,(t)G,(s)


(2-18)


where


Gp 1
G, =t g (2-19)
G, 1
Thus, the pressure and energy density (temperature) also have level surfaces being

ellipsoids with principal semi axis 6 .

It is left to determine the evolution equation for the function 6j, to do so we carry out

the differentiation in Eq. (2-3) and use Eqs. (2-1) and (2-2) to simplify. We get


8te~ = -v Ve V v
p


(2-20)









where the following relations have been used to carry out the differentiation in Eq. (2-3),


8< (pe) = pdte eV (pv),

8t (pV2/2) = V~ (pv) + pv 8ty,
2 2
v v
-V [py(e + ) + Pv] =-eV (pv) pv V e -V (pv)
2 2



Next we substitute (2-18) and (2-19) into (2-20) to obtain



n = 7 )/",(2-21)

which we get from


8te = geG, + g,[8sGets],







Integration of Eq. (2-21) yields

rl(t) = rl0(9p(t))y- (2-22)

where rlo = r1(0). The constant rlo can be related to the initial plasma state. For example,

by taking Eq. (2-4) for t = 0 and setting x = 0 (the center of the plasma plume), we see

that the constant rlo is related to the temperature at the plasma plume center [5-9]. From

Eq. (2-10) we get

= y-n- j 1, 3(2-23)

into which we substitute the explicit form of rl given in (2-22) and introduce new scaled

va~riables 1hy = (4/4 = lygy. As a result we obta~in a simple dyn~amlical m~odel thant









governs the time evolution of the plasma plume.


= ( ) -7, = r70(Plr2 3 (1 -7)/2, j = 1, 2,3.

Equations (2-24) can be seen as the Euler-Lagrange equations

d iiL 8iL
,td~ i~j=1,23


(2-24)


(2-25)


where the Lagrangian is given by


L( ,) ) = I, V,


V = ( 1 2 3 1-7.


(2-26)


It is important to note that if yj (t) is a solution of (2-24), then


a) '(t) = y' (t + a)


(2-27)



(2-28)


and


~,X(t)= Xa2/(137) y (At),


where a~ and A > 0 are constants, are also solutions of (2-24). The time translation

symmetry (2-27) and the scaling symmetry (2-28) lead to integrals of motion, which allow
us to investigate the behavior of solutions of (2-24) in some important special cases.









CHAPTER :3
INTEGRALS OF 1\OTION

In this section we describe the standard approach to finding integrals of motion using

the Noether Theorem. Let q : [t,, tb] i ,2 be a tra ectory of a dynamical system and

L : R's x R's x [to, tb] R hW e a Lagrangian of the syvstemn, where L = L(t', q, t) and t' = ~.

The action functional S is defined as a nmap of the space of all trajectories, 7T, into RW,


S= =l L(*, q, t)dt = Siq, t,, b] (3

The time evolution of a Lagrangian dynamical system is determined by the Hamilton least

action principle:
6S 'i~b /L dL _
-~ --6q +n S di = 0 (:32)

where the end points of the trajectory are fixed, 6q(t,) = 6q(tb) = 0. Front Eq. (:32) we
see that


/i'b S =t 0i4 iL iii4, (:33)
and therefore we obtain the Euler-Lagrange equations

d iiL iiL
= 0 (:34)
dt 8qk4 iik4

Consider a one-paranletric nmap fo : FT F that has the property that fo, a foo = fos

where Q3 = 0:4(01, 82) and fo = 1 (the identity nmap). We use the notation


fo (4(t)) = q o) (t) = Q (t, c0) (:35)

This nmap describes "rigid" transformations of a trajectory (e.g. rotation as a whole)

on the space in which the trajectory is embedded. Consider a one parametric nmap

To : [to, tbl [ta, tb], and define a new trajectory q as a mapping [t, tlb i I,2 obtained by

the composition q(t', c) = q(To(t')).










D definition A one-parametric differentiable I,, opp~.:: y q(t) Q(T o(t'), a) = Q(t ', a)

is called a one-parametric symmetry of the Lagr ,l:rit.o,: ]..;,,:rl,...;1lsystem if its action is
invariant under the tr i ..1..< ;;, notc y .. t': S[q, ta, b] = S[Q, t's) t]

Theorem Let q(t) be a solution of the Euler-Lagr rIety equation of motion. Suppose

the action is invariant under one-parametric symmetry transformations, Siq, to,,b]
S[Q, t's, tg]. Then I~/v~t), q/t)) = 0 where


I = viLI Q +I a L (3-6

that is, I is an :,:Ih lal of motion.
Sketch of the Proof

We shall use the notation (- )o for (- )a=o. By the hypothesis


(8L" ~~S[Q, t's, it] O(7

where

S [ Q, t 's t i ] = L Q t ( 3 -8 )


Q(t, >aQ) =q) a+ (t2)I (39)

Up to terms of O(Ct2) We have


(t ) = ( t)a / + + (3 -1 0 )



= (t) + B(t)a~


Where Q(t, a~) is defined by 3-5. Similarly we deduce

t' = To l(f) = t d -a=t t (-1


/tb-tba
S[Q", as[ t's, ti] = LJn~ v ,q+Btdi+Ot)(-2









Hence ,


a-L +T L +B dt1 (3-13)


Mlakin~g use of th~e Euler-L~agranlge equations (3-4) for ql, to express in? (3-13) weV infer
(BS tb d = b d dL
dc~~di + B(J di ,,= 0 (3-14)

Therefore I(t,) = I(tb), Where

ii BiL 8Qk ), T 8
I = + < L (3-15)
k=1 S~k 090 99

Since t, and tb are arbitrary, we conclude that I is an integral of motion.

End of proof.

Now we can apply the Noether theorem to the symmetries (2-27) and (2-28). In the

case of (2-27), it is easy to convince one self that the action is alr-li- invariant, provided

the Lagrangian does not depend on time explicitly, that is, L( ~), which is the case. The

corresponding integral of motion is known as the energy of the dynamical system. Put


fol = 1, T,(t) = t + a~ (3-16)

Hence

E = ()(3-17)

is an integral of motion. In the case (2-28), even though the equations of motion are

invariant, the action is not for all y, that is, the hypothesis of the Noether theorem is not

fulfilled by all y. Indeed, we have


f ( ) = X2/(1-3y)~() 38


T (t) = At, (3-19)

Lx = L(r ii i`(t), I, '"(t)) = A6(y-1)/(3y-1) L( (At), (At)) (3-20)









Sr = t/L dt A (3y-5)/(3y-1) Ldt = A(3y-5)/(3y-1)S (3-21)

From (3-21) it follows that only for y = 5/3 the action is invariant under the symmetry
transformation (2-28). In this case


(df ( )d x 1 (322


dT )=, t (3 23)

which leads to the corresponding Noether integral of motion


= +st- tL (3-24)





where we have used the fact that (3-17) is an integral of motion. The left hand side of

(3-25) is the total time derivative, hence we can integrate (3-25) and find that

W = ~) :I'?- Et2 p~t (3-26)


is an integral of motion. It is not hard to see that the constant p = 0 when vj = 0, that is,

when the plasma is at rest at the initial moment of time (2-6). The value of W is found

by setting t = 0.
W = (0 3-7









CHAPTER 4
THE CASE OF AXIAL SYMMETRY

We are interested in the case of the axial symmetry so that




We define 4,,(0) = le, I' (0) = lz, I = ,11~ cos #o = 1,/1, and sin no = lz/1. NUext we

change to polar coordinates where


~r = cos cp, #z = sin cp (4-2)

which ensures ~(0) = 1 and ~(0) = 0o. Making use of y = 5/3 and p = 0 and Eq. (3-26)
we obtain



From Eq. (3-17) we see that


~2~ + 4~2 ~ 0(cOS2 0 Sin 0~ 2/3 oo=np'' (4

We then switch to an auxiliary time defined as


7 = tan t) t> (4-5)

This change of variables allows us to rewrite Eq. (4-4) as the differential equation

ckp~ +V(cp, co) =1, V(cp, co) = o2co i ~ (4-6)
d-r cOS2 (p Sill p

can be viewed as a one dimensional mechanical system moving in the potential V(cp, co).

Equation (4-6) states that the total energy of this mechanical system is one. The potential

has a single minimum at cp, = tan- (1/z/) and is shown in Figure 1. The motion is

therefore oscillatory between the turning points co and p'o, where p'o can be found by

sin apo 3
sinI p'/o = 1 in














































III


We now turn to its analysis.


I
cn


Figure 4-1.


Potential energy V(cp, co). The total energy is fixed and equals one. At the
turning points 0: and V~(;o, ro) V(p'o, 90) 1. The motion ;(r) is
oscillatory between the turning points.


b. 6155









CHAPTER 5
FLIP OVER EFFECT

It has been shown that the expansion of a plasma plume with axial symmetry is

described by the initial value problem (4-6). The range of the auxiliary time r E [0, ~) is

in a one-to-one correspondence with the range of the physical time t E [0, 00). At every

mromrenlt of time thle plasmra plumre is: bounded by an ellipsoid +4i = 1 such~ that the

ratio of the axes is z/ tan cp = ~. The shape evolution of the plasma plume is described

by the initial value problem Eq. (4-6), where cp(0) = 0p. The function V(cp, co) has a

minimum at cp = cp, = an'1 Hence if j0 = p then q(T) = p. The ..-i-allll. I1.y

ratio tan 9, = compares to the spherical case, a = .

Definition The ftlip ov~er effect occurs if there exists 0 < r < ( such that (p(r) > 9,

if a~ < (Ps, or cp(-r) < cp, if a~ > (P.

In other words, if the plume is elongated along the z-axis (it has a cigar-like shape) at

t = 0, then at a later time it becomes more extended in the x-y plane (a pancake like

shape-like shape). The following theorem establishes the most important feature of the

shape evolution.

Theorem For the It;,,; ,.:;;/..l system (46) with cp0 E [0, cp,), the flip? over effect

occurs t er. At once.

Proof The solution of Eq. (4-6) with co E (0, cp) is


7 = (5-1)


First, we show that the flip over occurs at most once. Put

T( po) =(52
1 (po


T( po) is the time needed for the system to reach the other turning point p'o of the

potential. Hence co < c(-r) < (po. Therefore, if T( po) < ,, then at some value of r e (0, ()









the system must pass through cp = cp, < p'o. It is easy to see that V3/2 < V, and therefore


I~sino) = T( po)


By making the change of variable x = sin cp, we infer


(5-3)


Io)=


(x o(xo )


(5-4)


X")>


zo 1 xo > X"


ZO 1-x < 0 since xo E (0,). We then


where x'o


normalize the integration range to a unit interval,


1
I(xo) =
o t(1 t)(t + + 1)


(5-5)


sin2 w, t0 Obtain


where a~ = (x'n


xo). Finally we change the integration variable again, t


I(xo) =/ iif 2
sin


(5-6)


= .If d1")> 0, then x
where


where i3 = N~otic~e thlat I(0)

[0: ). Wle haRve alwo) dl(zo) do
dzo dp dzo '


I(0) < I(xo) < T( po), xo E


dp

dx~ro 4 (2 1 -


16 6xo

xo~ 02


(5-7)


is positive for anly xo e (0, ~). Therefore it is sufficient to show th~at > 0. By
differentiating the integrand we get


s \ fin'v w +: 1 1sn23/2


dl(xo)


(5-8)









We separate the integral into two parts, the first from 0 to and the second from to "

and shift the variable of the second integral by a so that

dl x/4COS2 W Sin2 w COS2 W Sin2
du i 2sn o1 sin P + Sin23/2 \C.OS2 0: ~ ( COS2 u 13/2
(5-9)

Since cos w > sin w when w E [0, x] we conclude that the integrand is positive and, hence,

li > 0. This inequality means that the other turning: point p'o can be reache~td at mnost

once during the entire evolution time. Therefore the plasma can flip over at most one

time.

It is left to show that the flip over effect ahr-l-w occurs. This can be done by proving

that the time T* needed to reach the point cp, is less than ( for all initial values co. To

do this, we replace V with a linear function that goes through the points ( po, 1) and

(P*, V(cp,)). We have


T* = < =i 2~ -1 + Ay, (5-10)
1 -V~, o fl+ A ~p A
co cp

Where
A=V(p,, co) 1 B oV(cp,, co) -P* (-1
0~o- 0P co- (P

The idea is to prove that the right hand side of this inequality is less than x, which is

equivalent to the following inequality

16 ( o (p) 2
2- 1 + V(cp, co) < 0, (5-12)

where V(cp, co) is the minimum value of the potential V. Put

16( po (p)2
f ( o) = 1 + V(cp, co). (5-13)

Note that f(cp,) = f/(cp,) = 0, because V(cp,, p,) = 1, V(cp, co) = 1/V( po, cp), and c = cp,

is a critical point of V. Hence f( po) < 0 if f///(co) > 0 and f/(cp,) a -2.7577, for all









co E [0, cp*). Indeed, we write the identity


f ( o) = f"(ri(d6dd. (5-14)

This implies that f( po) < 0 when co < (P if f"( po) I 0. The second derivative f"( po)
can be written as

f"(po) = /"( e,) + f(u ,(5-15)

Since f"(cp,) a -2.7577 I 0, the condition f'"(t) > 0 is sufficient for f( po) < 0. After
some algebraic transformations, we infer

27/3 t3 5t2 +6+ 1)
f// (t) = (5-16)
37/6t5/6( t37/6

where t = 3 cOS2 0p and t E [2, 3) So we need t3 5t2 + 6t + 1 > 0. By taking the derivative

of the left hand side of the inequality we see that it has a minimum in the range of t at
,+J ,71J? anld this mlinimrumr value is positivel showinailg thre ineqruality- holds for all

tE [2, 3).
Thus we have shown that the flip over effect occurs exactly once.









REFERENCES

[1] J. D. Winefordner, I. B. Gornushkin, T. Correll, E. Gibb, B. W. Smith, and N.
Onienetto, J. Anal. At. Spectront. 19, 1061 (2004).

[2] 31. N. Ashfold, F. C I i---- n!- G. 31. Fuge, and S. J. Henley, C'I. in! Soc. Rev. :$:, 2:3
(2004).

[:3] G. Conmpagnini, A. A. Scalisi, and O. Puglisi, J. Appl. Phys. 94, 7874 (200:3).

[4] I. B. Gornushkin, A. Ya. K~azakov, N. Onienetto, B. W. Smith, and J. D.
Winefordner, Spectochin1. Acta, Part B 59, 401 (2004).

[5] L. V. Ovsiannikov, Dokl. Akad. Nauk SSSR 111, 47 (1965).

[6] L. I. Sedov, Dokl. Akad. Nauk SSSR 40, 75:3 (195:3).

[7] S. I. Anisiniov and B. S. Luk'vanchuk, Usp. Fiz. Nauk 172, :301 (2002).

[8] L. Ferrario and D. Batani, Nuovo Cintento D 19D, 45 (1997).

[9] L. Ferrario, J. Plasma Phys. 64, 1 (2002).

[10] I. B. Gornushkin, S. V. Shabanov, N. Onienetto, and J. D. Winefordner, J. Appl.
Phys. 100, 07:3304 (2006).

[11] E. Noether. Invariante variationsprobleme. Goitt. Nachr., Pages 2:35-257, (1918).

[12] E. Noether. Invariant variation problems. Transport Theory Statist. Phys.,
1(:3):186-207, (1971).

[1:3] P.M. Morse, and H. Feshbach, Methods of Theoretical Physics, McGraw-Hill (195:3).





PAGE 1

THEFLIP-OVEREFFECTINSELF-SIMILARHY DRODYNAMICS By NATHANP.BAXTER ATHESISPRESENTEDTOTHEGRADUATESCHOOL OFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENT OFTHEREQUIREMENTSFORTHEDEGREEOF MASTEROFSCIENCE UNIVERSITYOFFLORIDA 2007 1

PAGE 2

c 2007 NathanP.Baxter 2

PAGE 3

page ABSTRACT ........................................ 4 CHAPTER 1INTRODUCTION .................................. 5 2SELF-SIMILARHYDRODYNAMICS ....................... 7 3INTEGRALSOFMOTION ............................. 13 4THECASEOFAXIALSYMMETRY ....................... 17 5FLIPOVEREFFECT ................................ 19 REFERENCES ....................................... 23 3

PAGE 4

AbstractofThesisPresentedtotheGraduateSchool oftheUniversityofFloridainPartialFulllmentofthe RequirementsfortheDegreeofMasterofScience THEFLIP-OVEREFFECTINSELF-SIMILARHY DRODYNAMICS By NathanP.Baxter August2007 Chair: SergeiShabanov Major:Mathematics Asimpliedmodelfortheevolutionofanexpandingplasmaistheso-called self-similarmodel.Inthismodelitisassumedthatthevelocityvectoreldisproportional tothecoordinatesystemwiththecenteroftheplasmaplumeatitsorigin.Thismodel predictsthatallquantitiescharacterizingtheplasmastatehavelevelsurfacesthat arespheres.Inexperiments,ithasbeenshownthattheshapeoftheplasmahasa signicanteectontheemittedradiation.Sinceplasmasgenerallydonotexpandas perfectspheres,weconsiderarecentlyproposedself-similarmodel,inwhichthelevel surfacesareellipsoidal. Inthismodeltheplasma'sevolutionisfullydeterminedbytheprincipleaxisas functionsoftime.ThelatteraredeterminedbysolvingaLagrangiandynamicalsystem. Inspeciccaseswheretheadiabaticconstantofthegasusedis5/3,weareabletoapply theNoetherTheorem,whichleadstotheNoetherintegralsofmotion.Ourmainfocuswas aphenomenonknownastheip-overeectasitoccursinsuchspecialcases.Whenthe plasmaiscreateditwillhaveitslongestaxisinsomedirection.Asitevolveswithtime,it willremainellipsoidal,butthelongestaxiswillbeintheorthogonaldirection.Ourresults showthattheip-overeectoccursexactlyonce.

PAGE 5

Oneofthetechniquesofremotechemicalanalysisisbasedontheanalysisofradiationspectraoflaserinducedplasmas.Alaserpulseevaporatesasmallpieceofmaterialturningitintoahotplasma.Theplasmaplumeexpands,coolsandemitslight.Theradiatedlightiscollectedbyaspectrometerthatproducestheradiationintensityasafunctionoflightwavelength(frequency).Thisfunctionisthenusedtoreconstructtheplasmainitialstatecharacterizedbytemperature,pressure,andconcentrationsofchemicalelements.Solvingtheinverseinitialvalueproblemrequiresknowledgeoftheplasmaevolutionduringwhichthespectralanalysisiscarriedout[ 1 { 5 ]. GeneralequationsdescribingthetimeevolutionofexpandingplasmasarenonlinearPDE's(basedontheNavier-Stokesequations)whosesolutionsarediculttoobtainevennumerically(duetohighcomputationalcosts).So,inpracticalapplicationsvarioussimpliedplasmaevolutionmodelsareused.Inparticular,theso-calledself-similarmodelsturnouttobeaccurateforpracticalpurposesofremotechemicalanalysis.Inthemostwidelyusedself-similarmodel,itisassumedthevelocityvectoreldisproportionaltothepositionvectorinthecoordinatesystemwhoseoriginistheplasmaplumecenter[ 5 { 9 ].Thecorresponding(self-similar)solutionsofthegeneralplasmaevolutionequationshavethecharacteristicfeaturethatallquantitiescharacterizingthecurrentplasmastate(massdensity,temperature,pressure)havelevelsurfacesbeingspheres,whilethesphereradiiarefunctionsoftimesatisfyingsomedierentialequationofthesecondorder(whichisoftensolvednumerically).Thissolutionclearlydescribesexpansionofsphericallysymmetricplasmaplumes.Ithasbeenfoundthatdeviationsoftheplasmaplumeshapefromthesphericalonecanhaveasubstantialeectontheemittedradiation.Hence,theuseofthesphericalself-similarexpansionmodeltoreconstructtheinitialmassdensities(concentrationsofchemicalelements)oftheevaporatedmaterialsamplemightleadtolargeerrorsorevenincorrectresults. 5

PAGE 6

10 ].Itscharacteristicfeatureisthatallthequantitiescharacterizingthecurrentplasmastateasfunctionsofspatialvariableshavelevelsurfacesbeingellipsoids.Theplasmatimeevolutionisfullydeterminedbytheevolutionsoftheprincipalaxesoftheellipsoidswhich,inturn,isdescribedbyaLagrangiandynamicalsystem.ThissystemappearsintegrableduetosymmetriesleadingtotheNoetherintegralsofmotioninapracticallyimportantcaseoftheaxialsymmetricinitialplasmaplumesandofplasmaswhosephysicalpropertiesareclosetothoseofidealoneatomicgases(whichmeanstheadiabaticconstantis5/3intheequationofstate). Aremarkablephenomenonpredictedbytheasymmetricself-similarmodelistheso-calledipovereect:Ifinitiallythelevelsurfacesofthemassdensitywereellipsoidswithlargestaxesinagivendirection,thentheplasmaevolvessothatthelevelsurfacesbecomeellipsoidswiththelargestaxisintheorthogonaldirection.Inotherwords,aplasmaplumewhichhasinitiallya"pancake"shapeturnsintoaplumewitha"cigar"shapeandviceversa. Thiseectwasrstfoundinnumericalsimulationsandalsoprovedforthecasewhentheasymmetryissmall(theratiosofellipsoidsprincipalaxisareclosetoone).Thelatterstudywasbasedonperturbationtheory.Theip-overeecthasalsobeenobservedexperimentally.Inthepresentstudyitisprovedthattheip-overeectoccursexactlyonce,i.e.nooscillationsbetween"pancake"and"cigar"shapesoccurduringtheplasmaexpansion. 6

PAGE 7

Inthissectionwegiveaderivationofanasymmetricselfsimilarsolution[ 10 ]oftheNavier-Stokesequationsforhydrodynamics[ 5 { 9 ].Allvectorswillbedenotedbyboldcharacters. Welet@tdenotesthepartialderivativewithrespecttotimet,(t;x)isthemassdensity,v(t;x)isthevelocityvectoreld,(t;x)istheinternalenergydensitywhichisproportionaltothetemperature,P(t;x)isthepressure,andH(t;x)istheenergydensitylossduetoradiationperunittimeandunitmass.Inourcasetheenergylossduetoradiationissmallandcanbeneglected,sowesetH=0.Wemustalsoaddtheequationofstatethatrelatesmass,energydensityandpressure. Hereistheadiabaticconstantofaparticulargas.Beginningwiththeinitialconditions(0;x)=0(x),(0;x)=0(x),andv(0;x)=v0(x)welookforasolutiontotheinitialvalueproblem( 2{1 )-( 2{3 ). Theasymmetricself-similarplasmaexpansionmodelisbasedontheassumptionthatthevelocityvectoreldhasaspecicdependenceonxasfollows, wherethefunctionsj(t)aretobedetermined.Thevelocityvectoreldcannotdependonj(0)becausevisscaleinvariant,j(t)!cjj(t).Theconstantscjarearbitraryandhave 7

PAGE 8

sothatvj(0;x)=jxj.ItisworthnotingthatEq.( 2{1 )-( 2{3 )arecovariantundergeneraltranslationsandrotationsinspace.So,Eq.( 2{5 )describes,infact,afamilyofsolutionswhichcanbeobtainedbytranslatingandrotatingthepositionvectorx.Nextweset andplugEq.( 2{5 )and( 2{7 )intoEq.( 2{1 ),whichyields NextwesubstituteEq.( 2{5 )intoEq.( 2{2 )toobtain j Wedenethenewvariableywithyj=xj=jandset 8

PAGE 9

2{9 )canbewritteninthefollowingway TheintegrabilityconditionofEq.( 2{11 )tellsusthat@2P @yj@yk=@2P @yk@yj,whichcanonlybetrueiff(t;x)=F(t;q)whereq=Pjjy2j.NowfromEq.( 2{8 )weget @qF=Xj_j ThelefthandsideofEq.( 2{12 )onlydependsontandq,thereforetherighthandsideshouldonlydependontandq.Thisispossibleonlyif wherejarejustconstants.Nextweset 2Xjjy2j:(2{14) AsolutiontoEq.( 2{12 )isF(t;q)=G(s),whereG(s)isadierentiablefunction.Thus,ageneralsolutionundertheassumption( 2{5 )shouldhavetheform 9

PAGE 10

substituteitintoEq.( 2{11 ),andinferthat FromEq.( 2{4 )itisclearthat where Thus,thepressureandenergydensity(temperature)alsohavelevelsurfacesbeingellipsoidswithprincipalsemiaxisj Itislefttodeterminetheevolutionequationforthefunctionj,todosowecarryoutthedierentiationinEq.( 2{3 )anduseEqs.( 2{1 )and( 2{2 )tosimplify.Weget rv(2{20) 10

PAGE 11

2{3 ), 2{18 )and( 2{19 )into( 2{20 )toobtain _=(1)_g whichwegetfrom@t=_gG+g[@sG@ts];@ts=Xj_j 2{21 )yields where0=(0).Theconstant0canberelatedtotheinitialplasmastate.Forexample,bytakingEq.( 2{4 )fort=0andsettingx=0(thecenteroftheplasmaplume),weseethattheconstant0isrelatedtothetemperatureattheplasmaplumecenter[ 5 { 9 ].FromEq.( 2{10 )weget j=j j;j=1;2;3(2{23) intowhichwesubstitutetheexplicitformofgivenin( 2{22 )andintroducenewscaledvariablesj=j=p 11

PAGE 12

jj=(123)1;=0(123)(1)=2;j=1;2;3:(2{24) Equations( 2{24 )canbeseenastheEuler-Lagrangeequations dt@L @_j=@L @j;j=1;2;3;(2{25) wheretheLagrangianisgivenby 2X_2jV;V= 1(123)1:(2{26) Itisimportanttonotethatifj(t)isasolutionof( 2{24 ),then and whereand>0areconstants,arealsosolutionsof( 2{24 ).Thetimetranslationsymmetry( 2{27 )andthescalingsymmetry( 2{28 )leadtointegralsofmotion,whichallowustoinvestigatethebehaviorofsolutionsof( 2{24 )insomeimportantspecialcases. 12

PAGE 13

InthissectionwedescribethestandardapproachtondingintegralsofmotionusingtheNoetherTheorem.Letq:[ta;tb]!RnbeatrajectoryofadynamicalsystemandL:RnRn[ta;tb]!RbeaLagrangianofthesystem,whereL=L(v;q;t)andv=dq dt.TheactionfunctionalSisdenedasamapofthespaceofalltrajectories,F,intoR, ThetimeevolutionofaLagrangiandynamicalsystemisdeterminedbytheHamiltonleastactionprinciple: q=Ztbta@L @_q_q+@L @qqdt=0(3{2) wheretheendpointsofthetrajectoryarexed,q(ta)=q(tb)=0.FromEq.( 3{2 )weseethat dt@L @_q+@L @qq=0(3{3) andthereforeweobtaintheEuler-Lagrangeequations dt@L @_qk@L @qk=0(3{4) Consideraone-parametricmapf:F!Fthathasthepropertythatf1f2=f3where3=3(1;2)andf0=1(theidentitymap).Weusethenotation Thismapdescribes"rigid"transformationsofatrajectory(e.g.rotationasawhole)onthespaceinwhichthetrajectoryisembedded.ConsideraoneparametricmapT:[t0a;t0b]![ta;tb],anddeneanewtrajectory~qasamapping[t0a;t0b]!Rnobtainedbythecomposition~q(t0;)=q(T(t0)). 13

PAGE 14

dtI(v(t);q(t))=0where @vk@Q @=0+vk@T @=0L@T @=0(3{6) where dt~Q;~Q;tdt(3{8) ~Q(t;)=q(t)+@~Q @!0+O(2)(3{9) UptotermsofO(2)wehave ~Q(t;)=q(t)+@Q @0+@Q @t0@T =q(t)+@Q @0+v(t)@T 3{5 .Similarlywededuce dt;q+B;tdt+O(2)(3{12) 14

PAGE 15

@0=L@T @0t=tb+L@T @0t=ta+Ztbta@L @vdB dt+@L @qBdt(3{13) MakinguseoftheEuler-Lagrangeequations( 3{4 )forq,toexpress@L @qin( 3{13 )weinfer @0=Ztbtad dtL@T @=0dt+Ztbtad dt@L @vBdt=0(3{14) ThereforeI(ta)=I(tb),where @vk@Qk @q0L@T @q0(3{15) Sincetaandtbarearbitrary,weconcludethatIisanintegralofmotion. Endofproof. NowwecanapplytheNoethertheoremtothesymmetries( 2{27 )and( 2{28 ).Inthecaseof( 2{27 ),itiseasytoconvinceoneselfthattheactionisalwaysinvariant,providedtheLagrangiandoesnotdependontimeexplicitly,thatis,L(_;),whichisthecase.Thecorrespondingintegralofmotionisknownastheenergyofthedynamicalsystem.Put Hence @_j_jL=1 2Xj_2+V()(3{17) isanintegralofmotion.Inthecase( 2{28 ),eventhoughtheequationsofmotionareinvariant,theactionisnotforall,thatis,thehypothesisoftheNoethertheoremisnotfullledbyall.Indeed,wehave 15

PAGE 16

From( 3{21 )itfollowsthatonlyfor=5=3theactionisinvariantunderthesymmetrytransformation( 2{28 ).Inthiscase 2(3{22) whichleadstothecorrespondingNoetherintegralofmotion 2j+_jttL(3{24) 2Xjj+_j+Et(3{25) wherewehaveusedthefactthat( 3{17 )isanintegralofmotion.Thelefthandsideof( 3{25 )isthetotaltimederivative,hencewecanintegrate( 3{25 )andndthat 2Xj2jEt2t(3{26) isanintegralofmotion.Itisnothardtoseethattheconstant=0whenj=0,thatis,whentheplasmaisatrestattheinitialmomentoftime( 2{6 ).ThevalueofWisfoundbysettingt=0. 2Xj2j(0)=1 2Xj1 16

PAGE 17

Weareinterestedinthecaseoftheaxialsymmetrysothat Wedener(0)=lr,z(0)=lz,l=p whichensures(0)=land(0)=0.Makinguseof=5=3and=0andEq.( 3{26 )weobtain FromEq.( 3{17 )weseethat Wethenswitchtoanauxillarytimedenedas ThischangeofvariablesallowsustorewriteEq.( 4{4 )asthedierentialequation d2+V(';'0)=1;V(';'0)=cos2'0sin'0 canbeviewedasaonedimensionalmechanicalsystemmovinginthepotentialV(';'0).Equation( 4{6 )statesthatthetotalenergyofthismechanicalsystemisone.Thepotentialhasasingleminimumat'=tan1(1=p sin'00=sin'0 4sin2'01=2(4{7) 17

PAGE 18

Figure4-1. PotentialenergyV(';'0).Thetotalenergyisxedandequalsone.Attheturningpointsd' dt=0,andV('0;'0)=V('00;'0)=1.Themotion'()isoscillatorybetweentheturningpoints. 18

PAGE 19

Ithasbeenshownthattheexpansionofaplasmaplumewithaxialsymmetryisdescribedbytheinitialvalueproblem( 4{6 ).Therangeoftheauxiliarytime20; c.TheshapeevolutionoftheplasmaplumeisdescribedbytheinitialvalueproblemEq.( 4{6 ),where'(0)='0.ThefunctionV(';'0)hasaminimumat'='=tan11 4{6 )with'02[0;'),theipovereectoccursexactlyonce. 4{6 )with'02(0;')is First,weshowthattheipoveroccursatmostonce.Put 19

PAGE 20

Bymakingthechangeofvariablex=sin',weinfer wherex00=x0 4x0>0,x000=x0 4x0<0sincex02(0;1 where=(x00x0).Finallywechangetheintegrationvariableagain,t=sin2w,toobtain where=x0 dx0,where dx0=166x0 4x03x02q 4x0(5{7) ispositiveforanyx02(0;1 d>0.Bydierentiatingtheintegrandweget 20

PAGE 21

d=2Z=40cos2wsin2w Sincecoswsinwwhenw2[0; Itislefttoshowthattheipovereectalwaysoccurs.ThiscanbedonebyprovingthatthetimeTneededtoreachthepoint'islessthan Where Theideaistoprovethattherighthandsideofthisinequalityislessthan 16('0')2 whereV(';'0)istheminimumvalueofthepotentialV.Put Notethatf(')=f0(')=0,becauseV(';')=1,V(';'0)=1=V('0;'),and'='isacriticalpointofV.Hencef('0)0iff000('0)>0andf00(')2:7577,forall 21

PAGE 22

Thisimpliesthatf('0)0when'0<',iff00('0)0.Thesecondderivativef00('0)canbewrittenas Sincef00(')2:75770,theconditionf000(t)0issucientforf('0)0.Aftersomealgebraictransformations,weinfer 37=6t5=6(1t=3)7=6;(5{16) wheret=3cos2'0andt2[2;3)Soweneedt35t2+6t+1>0.Bytakingthederivativeofthelefthandsideoftheinequalityweseethatithasaminimumintherangeoftat5+p 3;4714p 27,andthisminimumvalueispositive,showingtheinequalityholdsforallt2[2;3). Thuswehaveshownthattheipovereectoccursexactlyonce. 22

PAGE 23

[1] J.D.Winefordner,I.B.Gornushkin,T.Correll,E.Gibb,B.W.Smith,andN.Omenetto,J.Anal.At.Spectrom.19,1061(2004). [2] M.N.Ashfold,F.Claeyssens,G.M.Fuge,andS.J.Henley,Chem.Soc.Rev.33,23(2004). [3] G.Compagnini,A.A.Scalisi,andO.Puglisi,J.Appl.Phys.94,7874(2003). [4] I.B.Gornushkin,A.Ya.Kazakov,N.Omenetto,B.W.Smith,andJ.D.Winefordner,Spectochim.Acta,PartB59,401(2004). [5] L.V.Ovsiannikov,Dokl.Akad.NaukSSSR111,47(1965). [6] L.I.Sedov,Dokl.Akad.NaukSSSR40,753(1953). [7] S.I.AnisimovandB.S.Luk'yanchuk,Usp.Fiz.Nauk172,301(2002). [8] L.FerrarioandD.Batani,NuovoCimentoD19D,45(1997). [9] L.Ferrario,J.PlasmaPhys.64,1(2002). [10] I.B.Gornushkin,S.V.Shabanov,N.Omenetto,andJ.D.Winefordner,J.Appl.Phys.100,073304(2006). [11] E.Noether.Invariantevariationsprobleme.Gott.Nachr.,Pages235-257,(1918). [12] E.Noether.Invariantvariationproblems.TransportTheoryStatist.Phys.,1(3):186-207,(1971). [13] P.M.Morse,andH.Feshbach,MethodsofTheoreticalPhysics,McGraw-Hill(1953). 23


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