Page 213 - Process Modelling and Simulation With Finite Element Methods

200 Process Modelling and Simulation with Finite Element Methods

Since the eigenvalues reported by this method are the decay rates if positive, we
can conclude that all the eigenmodes are decaying, although one is neutral. SO
the eigensystem stability analysis shows that the viscous fingering instability
simulated here is unconditionally stable, even at parameter values that the
analytic theory finds a long wave instability.
We can visualize the eigenmode solutions by tricking FEMLAB 's built-in
postplot facility to see them as solutions.
>>fem.sol.u=V(:,l)
corresponds to the eigenvector with the smallest magniture, h=0.0000. A
standard use of postplot to give concentration contours visualizes the
eigenfunction.
postplot(fem,. . .
'geomnum', 1 ,. . .
'context','local',. . .
'contdata', ( 'c','cont','internal'} ,. . .
'contlevels',20,.. .
'contstyle','color', ...
'contlabel','off ,...
'contmaxmin','off', ...
'contbar','on', ...
'contmap','cool', ...
'geom', 'on',. . .
'geomcol','bginv',. . .
'refine', 3, ...
'contorder',2,.. .
'phase', 0, ...
'title', 'Time=200 lambda=0.0000 Contour: concentration of c ', ...
'renderer','zbuffer',. . .
'solnum', 15, ...
'axisvisible','on')
Figures 5.12 and 5.13 demonstrate that the eigenvectors found this way represent
discernable patterns. That they all decay implies that there is no pattern
formation due to instability. Note that the eigenmodes shown satisfy the pde and
the appropriate homogeneous boundary conditions as well - no vertical flux
(flat) and uniform outflow.
We are left in this subsection with the apparent disagreement between linear
stability theory [ 131 and linearization of the full solution. Careful examination
of the theory and the simulations, however, suggests that the simulations in
FEMLAB are too restrictive in the imposition of uniform inlet and outlet
boundary conditions. Logically, if the inlet and outlet boundary conditions are