Nonbound dislocations in hexagonal patterns: pentagon lines
in surface-tension-driven Be
´
nard convection
Kerstin Eckert
1
and Andre
´
Thess
2
1
Institute for Aerospace Engineering, Dresden University of Technology, 01062 Dresden, Germany
2
Department of Mechanical Engineering, Ilmenau University of Technology, P.O. Box 100565, 98684 Ilmenau, Germany
Received 5 December 1997; revised manuscript received 10 June 1999
We report on a novel class of defects in a hexagonal pattern which we call pentalines. They are built up of
two nonbound dislocations and are orientated parallel to the roll axis of the mode free of a dislocation. A
pentaline has its origin in a transformation of the penta-hepta defect PHD, taking place at higher supercriti-
cality. The underlying mechanism consists in a combination of glide and climb motion of the original dislo-
cations bound to the PHD. We demonstrate that the pentalines play an important role within the transition from
hexagonal towards square convection cells, observed in surface-tension-driven Be ´nard convection.
S1063-651X9906310-2
PACS numbers: 47.54.+r, 47.20.Dr
I. INTRODUCTION
Defects strongly influence the dynamics of hydrodynamic
systems. Impressive examples are the change of the charac-
teristic macroscopic length scale due to the defect motion
e.g., 1,2 and the mediating of the transition to a weakly
turbulent state 3,4, or to a pattern with different symmetry
5–9. The defect type which has been studied most exten-
sively is the dislocation in a roll pattern for an overview see,
e.g., 10. The dislocation represents the generic point de-
fect of the roll pattern and can in principle either move par-
allel climb or perpendicular glide to the roll axis. The
present paper is concerned with formation and evolution of
defects in a hexagonal pattern. Hexagonal patterns are the
result of a superposition of three roll systems whereby the
angle between them amounts to 120°. We find that changes
in the defect structure in such a composite pattern can be
traced back to the two elementary types of dislocation mo-
tion in the particular modes. Therefore it makes sense to
summarize what the results of previous experimental and
theoretical studies on pure roll patterns are with respect to
the dislocation motion. We first consider the Rayleigh-
Be
´
nard convection RBC. The majority of studies is focused
to the climb motion of dislocations. In an early experiment
1 it was shown by means of a row of dislocations that
climbing provides a size-adjustment mechanism. The climb
velocity was found to be inversely proportional to the Prandtl
number of the fluid. The Prandtl number Pr= / is the
ratio between the kinematic viscosity and the thermal dif-
fusivity . This experiment motivated later theoretical stud-
ies 11,12. Both works demonstrate that the climb velocity
is proportional to k
3/2
provided the dynamics can be derived
from a potential. Here, k =k
opt
-k where k
opt
stands for the
wave number at which the dislocation is stationary. For non-
potential systems the climb velocity obeys a linear depen-
dency on k 12. The uniform climb velocity, predicted by
11, was experimentally verified in 2. Furthermore, it was
shown in 2 that the climb velocity behaves approximately
proportional to k
3/2
. At a given k , the climb velocity in-
creases with the control parameter.
In contrast to the climb motion, the glide motion occurs
only in nonpotential systems 12 and involves nonsymmet-
ric pinching off and reforming of rolls. From the theoretical
side, it is therefore much harder to treat than climbing 12.
Presently, an elaborated theory does not exist. Some qualita-
tive features of gliding are given in 13. Gliding occurs with
small and nonuniform velocity and is stopped at high Ray-
leigh numbers. It is favored if k k
opt
, or if the layer rotates
14. In the latter work was found that the gliding motion and
the subsequent annihilation of defects causes a reorientation
of the whole structure.
We briefly turn to anisotropic flows, for which electrohy-
drodynamic convection EHC is an important example. The
preparation of well separated defects can be achieved more
simply than in RBC which supported a better understanding
of the dynamics and interaction of defects in EHC. For an
overview we refer to 15. According to theory 16–18,a
dislocation can display both climbing and gliding. Climbing
occurs if the wavenumber deviates from the optimal one
while gliding is preferred if the rolls are tilted against the
normal direction. The motion of a pair of dislocations with
opposite charge has been investigated experimentally in 18–
21. The study at vanishing external stresses 21, i.e., at zero
wave number mismatch, revealed a universal length scale for
both climb and glide motion below which the mutual inter-
action of defects becomes important. Studies with imposed
external stresses show that climbing occurs in two stages
18,20. At large distances between both defects the velocity
is constant. Below a crossover distance the motion acceler-
ates. In contrast to the smooth climbing motion the gliding
shows a steplike behavior 18,20. In accordance with 14
for RBC, the annihilation of dislocations by gliding leads to
a reduction of the original tilt angle 19. This observation
supported the idea that the glide motion is a selection process
for the pattern orientation.
We now leave the pure roll pattern to turn to defects in the
composite hexagonal pattern which recently have attracted
attention 22–29. Typically, hexagonal patterns occur in
presence of a vertical asymmetry, caused, e.g., by
temperature-dependent material parameters or by a differ-
ence between top and bottom boundary conditions. Promi-
PHYSICAL REVIEW E OCTOBER 1999 VOLUME 60, NUMBER 4
PRE 60 1063-651X/99/604/41178/$15.00 4117 © 1999 The American Physical Society