Roll patterns

The fourth basic convection pattern, vermiculated rolls, was associated with the presence of a nonvolatile material of large molecular size in the surface region. For liquids containing such a material, a vermiculated roll pattern was found to prevail at intermediate depths (3-6 mm). At depths... 

A similar analysis can be carried out in the general case. One can show that solutions Vn = i /V )dnm n = 1,..., A for any m, m = 1,..., A, which correspond to rolls of different orientahon, are stable, while all other stahonary solutions are unstable. Generally, the same result on the stability of roll patterns and instability of any other patterns is obtained for the more general system. 

Figure 9. Phase portrait of the system (25) for N=2 corresponding to the selection of (a) roll patterns (b) square patterns.

Because of the competitive nonlinear interaction, only one species, i.e. a particular roll pattern, survives at large T. 

So, in the interval 0 < F < Fi neither quiescent state nor roll patterns are stable. One has to investigate other critical points of the Lyapunov function... 

Because F2 < 0, both the quiescent state and the upper branch of hexagons are stable, i.e. provide a local miifimum of the Lyapunov function U, in the interval F2 < F <0. Thus, the transition between the quiescent state and the hexagonal pattern in the presence of a cubic term in the Lyapunov function is similar to a first order phase transition which takes place in the presence of a cubic term in the free energy Ginzburg-Landau functional [44], in contradistinction to the transition between the quiescent state and the roll pattern in the... 

Recall that the roll pattern becomes stable for T > Ti = 4/3. Hence, in the interval Ti < T < Ts the Lyapunov function has 4 local minima, three of them correspond to three types of roll patterns, and one of them corresponds to hexagons. The basins of attractions between them are separated by stable manifolds of some additional saddle-point stationary solutions, corresponding to squares (e.g. R = R2 0, Rs = 0) and skewed hexagons" (e.g. R = R2 7 R3 7 0). Finding the latter solutions is suggested to the readers as an exercise. 

We come to the conclusion that only the roll patterns inside the stability interval 0 < K < 1/ /3 are stable. The stability interval is also called the Busse balloon, for it was first discovered by Busse et al. in the context of the Rayleigh-Benard convection patterns [40]. See the diagram in Fig. 12. 

Derivation of the nonlinear phase diffusion equation. The longwave nature of the two basic instabilities of roll patterns described above shows that longwave distortions of rolls are of major interest. Let us consider longwave solutions of the NWS equation (73)... 

We shall apply this equation for the consideration of defects in roll patterns. 

Dislocation. According to relation (84), the phase 9 is defined modulo 27r at points where A / 0, and undefined at points where A = 0. The roll pattern can contain a point defect of the following structure. The amplitude A = 0 at a certain point, say the point X = Y = 0. Except for this point, the phase is smooth, but going around the point X = Y = 0 along a closed circle y2 leads to a phase increment 2mt, where n = 1. Such a... 

Nonlinear theory of the zigzag instability. In the previous subsection, we found that a roll pattern with K < 0 is subject to a transverse (zigzag) instability with Kx = 0, Ky 7 0. In order to investigate the temporal evolution of a zigzag disturbance on the background of a roll pattern, substitute 9 = -f (y, r) into the nonlinear phase equation (87). We obtain ... 

The ansatz (99) resembles (21), but there is an essential difference now the functions A depend on the slow coordinate ri. Therefore we can consider different roll patterns localized in different regions rather than uififormly superposed. 

Stationary domain walls. For the sake of simplicity, let us consider a plane domain wall perpendicular to the axis X, which separates two semi-infinite roll systems with wave vectors ki and lc2 [51], as shown in Fig. 15a. Because the Lyapunov functional densities of both roll patterns are equal, there is no reason for a motion of the domain wall, hence it is motionless [52]. The problem is governed by the following system of ordinary differential equations ... 

As we know, the wavevector k of the roll pattern is not unique. Therefore, we can imagine a situation when the local wavevector of a roll pattern is a slow function of the coordinate r, and it can slowly change in time ... 

Let us find the solution of the Swift-Hohenberg equation (120), corresponding to a slowly distorted roll pattern with the wavevector field (122),... 

The solution 0o coincides with the 27t-periodic function f 9) described above, which corresponds to an undistorted roll pattern with the wavenumber k. 

Equation (127) is called Cross-Newell equation. This equation is universal and can be derived for any rotationally isotropic system which produces roll patterns due to a short-wave monotonic instability. Each particular problem is characterized by specific T k) and B k). 

Disclinations. The Cross-Newell equation can be used for studying special type of defects in roll patterns, disclinations [3], [54]. When going around the... 

The stability analysis of roll patterns performed in the framework of the system (148), (149), reveals a skewed-varicose instability. Numerical simulations predict the development of labyrinthine, spiral and target patterns. 

Electrically driven convection in nematic liquid crystals [6,7,16] represents an alternative system with particular features listed in the Introduction. At onset, EC represents typically a regular array of convection rolls associated with a spatially periodic modulation of the director and the space charge distribution. Depending on the experimental conditions, the nature of the roll patterns changes, which is particularly reflected in the wide range of possible wavelengths A found. In many cases A scales with the thickness d of the nematic layer, and therefore, it is convenient to introduce a dimensionless wavenumber as q = that will be used throughout the paper. Most of the patterns can be understood in terms of the Carr-Helfrich (CH) mechanism [17, 18] to be discussed below, from which the standard model (SM) has been derived... 

Figure 3. Cross section of a roll pattern at the direct onset to EC in the planar geometry (indicated by the small dashes at the confining plates). Double arrows denote the director modulations, which are maximal at the midplane. The lines follow the stream lines. The symbols -f and — denote the sign of the induced charges, shown at a phase of the apphed ac voltage where the electric field points downward.

Another example is related to the motion of defects (dislocations in the roll pattern) which constitutes the basic mechanism of wavevector selection. In the normal-roll regime, the stationary structure is characterized by the condition q II H. However, when changing the field direction one can easily induce a temporary wavevector mismatch Aq = q ew — qoid which relaxes via a ghde (v II q) motion of defects. Experiments have confirmed the validity of detailed theoretical predictions, both with respect to the direction (v J. Aq) and the magnitude (consistent with logarithmic divergence at Aq —> 0) of the defect velocity v [42]. 

Figure 10.4 Schematic representation of flow patterns near a moving contact line during immersion of a solid substrate into a pool of liquid, (a) Split-injection streamline in phase B and rolling pattern in phase A. (b) Transition flow pattern with motionless interface and rolling motion in phases A and B. (c) Rolling motion in phase B and split-ejection streamline in phase A...

Before discussing these experiments we will present new measurements using the nematic Phase 5, which show a similar crossover at ft < 0.1 Hz. Thus for f < ft there are flexodomains as a first instability while for f > ft there are the usual EC roll patterns with conductive symmetry. Above the Lifshitz frequency //, 40 Hz there are normal rolls, which are replaced by... 

Figure 8.20 Typical rolling pattern after the formation of the longitudinal joint.

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