Post on 19-Jan-2016
27 October 200527 October 2005 Konstantin Blyuss, University of Exeter Konstantin Blyuss, University of Exeter 11
Spatio-temporal dynamics in a phase-field model with phase-dependent heat
absorption
Konstantin Blyuss
(joint work with Peter Ashwin, David Wright and Andrew Bassom)
University of Exeter, UK
27 October 2005
27 October 200527 October 2005 Konstantin Blyuss, University of Exeter Konstantin Blyuss, University of Exeter 22
Problem formulation
27 October 200527 October 2005 Konstantin Blyuss, University of Exeter Konstantin Blyuss, University of Exeter 33
Phase-field model
The dynamics is characterized by two fields: the temperature
T(x,t) and the phase x,t). The convention is is melt
and is solid.
Phase-field equation has the form
where
Here, p is the interface thickness, is the strength of coupling
between the phase field and the temperature field.
,)(ˆ)( 22 pguft
.53
2)( ,
42)(
5342 gf
27 October 200527 October 2005 Konstantin Blyuss, University of Exeter Konstantin Blyuss, University of Exeter 44
Temperature evolution is determined by
with being the radiative absorption coefficients, b is the
thermal emission coefficient, Ta is the ambient temperature
and d is a thermal diffusivity.
Basic features:
• energy throughput of the system is much larger than the
latent heat
• two phases have different rates of heat absorption
,)(2
][ 21111 TdTTb
IaaaaT at
1a
27 October 200527 October 2005 Konstantin Blyuss, University of Exeter Konstantin Blyuss, University of Exeter 55
Steady statesUniform equilibria of the model solve the followingsystem of equations
Two of the roots are
where
Provided
we have This also necessarily requires a-1>a1.
.0][
2
1
,01~
1
1111
22M
2
aTTbIaaaa
TT
T,
),,1(, and ),1(, 11 TTTT
.11 b
IaTT a
,1M
1
b
IaTT
b
Iaa
.1M1 TTT
27 October 200527 October 2005 Konstantin Blyuss, University of Exeter Konstantin Blyuss, University of Exeter 66
Assuming one has a cubic equation for the equilibrium
phase
where
Depending on the relation between A, B and there are one or
more roots with These roots correspond to the mushy
layers.
For our system the steady states with intermediate values of the
phase and temperature can be interpreted as the states of mushy
layer, where a transition from melt to solid takes place.
,1
,01)(~ 2 BA
.2
and ,2
11M
11
TTBT
TTA
].1,1[
27 October 200527 October 2005 Konstantin Blyuss, University of Exeter Konstantin Blyuss, University of Exeter 77
Dispersion relation for stability of the steady states is
From this relation it follows that the linearly stable steady
states are further stabilised by diffusion.
The possibility of linearly unstable steady states is not
excluded, but the Turing instability cannot occur since we
have an inhibitor-inhibitor system.
.01)ˆ(ˆ
~4ˆ31)()(1
~
2
1
]1)ˆ(ˆ~
4ˆ31)([
2M
2222211
22
2M
2222
TTdkbdkbkpIaa
TTkpdb
27 October 200527 October 2005 Konstantin Blyuss, University of Exeter Konstantin Blyuss, University of Exeter 88
Travelling wavesThese solutions have the form
Substituting this into the system gives
Two equilibria are and a travelling wave is a
heteroclinic connection between them. Spectrum of the
linearization near the steady states has two positive and two
negative real eigenvalues, and so heteroclinic connections
can exist only for isolated values of velocity c.
. ),( ),( ctxzzSTz
.0)(
2])([
,011)(~
1111
22M
2
STbI
aaaaScSd
TScp
a
),,1(),( 1 TS
27 October 200527 October 2005 Konstantin Blyuss, University of Exeter Konstantin Blyuss, University of Exeter 99
a) Sinusoidal initial profile.
b) Initial profile in the form of tanh
27 October 200527 October 2005 Konstantin Blyuss, University of Exeter Konstantin Blyuss, University of Exeter 1010
Stability of travelling wavesLinearizing the system near the travelling wave solution
and looking for solutions of the form
one arrives at the following eigenvalue problem for the growth
rate :
with the auxiliary function This
system can be recast as a first-order system
)),(),(( zSz
,)(
2
1)(
,0)(1~
)()](31[
11
2222
IaaTbTcTd
zzPzcp
)].(1)[)()((~
4)( 2 zTzSzzP M
,))(ˆ),(ˆ)(exp(Re))(),(()),(),,(( zTztzSztxTtx
.),,,(,),( Tzzz TTz v vAv
27 October 200527 October 2005 Konstantin Blyuss, University of Exeter Konstantin Blyuss, University of Exeter 1111
Stability of travelling wavesIn the limit the matrix A reduces to a constant matrix
with the eigenvalues
Let U+(z,) be the two-dimensional subspace of solutions that
decay exponentially as and U–(z,) be a two-
dimensional subspace decaying exponentially as The
non-trivial intersection of these two subspaces indicates the
presence of unstable eigenmodes.
z),,(lim)( z
zAA
d
bd
pspec
)(,
2)(
A
z.z
27 October 200527 October 2005 Konstantin Blyuss, University of Exeter Konstantin Blyuss, University of Exeter 1212
One can define the Evans function as
Zeros of this function correspond to the eigenvalues of the
linearized stability problem.
For actual evaluation it is convenient to integrated the induced
systems which describe the dynamics of the corresponding
subspaces. Also, in this way one can preserve analyticity in
spectral parameter.
).,(),()( zUzUE
27 October 200527 October 2005 Konstantin Blyuss, University of Exeter Konstantin Blyuss, University of Exeter 1313
satisfies the following induced system
Here acts on a decomposable two-form
as
The limiting matrix has an eigenvalue
with the corresponding eigenvector
Similarly, for the adjoint system
The Evans function can now be written as
),( zU
).(),(lim,),()2(
zzdz
dU UAU
)()(: 6262)2( CC A
2xxx 1 .2121)2( AxxxAxxA
),(lim)( )2()2( zz
AA
)( ).(
).(),(lim,)],([ )2(
zzdz
dz
T V VA-V
.),0(),,0()(6
UVE
27 October 200527 October 2005 Konstantin Blyuss, University of Exeter Konstantin Blyuss, University of Exeter 1414
Evans function results
27 October 200527 October 2005 Konstantin Blyuss, University of Exeter Konstantin Blyuss, University of Exeter 1515
Transverse stabilityConsider now the stability of travelling waves in the direction
orthogonal to the basic direction of propagation. To model
this, we replace .2yyxx
27 October 200527 October 2005 Konstantin Blyuss, University of Exeter Konstantin Blyuss, University of Exeter 1616
Looking for solutions of the linearized problem in the form
one arrives at the eigenvalue problem with transverse
wavenumber as a parameter:
Evans function can be defined in the same way as before.
,))(ˆ),(ˆ)(exp(Re))(),(()),,(),,,(( zTztikyzSztyxTtyx
.)(
2
1)(
,0)(1~
)(])(31[
112
222222
IaaTdkbTcTd
zzPpkzcp
27 October 200527 October 2005 Konstantin Blyuss, University of Exeter Konstantin Blyuss, University of Exeter 1717
Conclusions
• Phase-dependent absorption is sufficient to provide
a bi-stability in the system
• Travelling fronts are stable with respect to both
longitudinal and transversal perturbations
• The model can be extended to study explosive
crystallisation
27 October 200527 October 2005 Konstantin Blyuss, University of Exeter Konstantin Blyuss, University of Exeter 1818
Thanks for your attention