spatio-temporal dynamics in a phase-field model with phase-dependent heat absorption

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


Problem formulation


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


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


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


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[


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


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


a) Sinusoidal initial profile.

b) Initial profile in the form of tanh


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


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


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


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


Evans function results


Transverse stabilityConsider now the stability of travelling waves in the direction

orthogonal to the basic direction of propagation. To model

this, we replace .2yyxx


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


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


Thanks for your attention

spatio-temporal dynamics in a phase-field model with phase-dependent heat absorption

Documents