Experiments on Thermally Driven Convection

Transcription

1 Experiments on Thermally Driven Convection Guenter Ahlers ABSTRACT Experimental results for thermally driven convection in a thin horizontal layer of a nematic liquid crystal heated from below or above and in a magnetic field are reviewed and, when possible, compared with theoretical calculations. For the case of planar alignment in a horizontal field, convection occurs only when heating is from below. The threshold is lowered dramatically by a heat-focusing mechanism. There is excellent agreement between experiment and theory for those properties which depend only on the linear terms in the equations of motion, namely for the primary bifurcation line and the wavevector as a function of magnetic field. Predictions for the location of and behavior near two Lifshitz points are quantitatively confirmed by the experiments. For the nonlinear properties, agreement with weakly nonlinear theory is not so satisfying. The predicted tricritical points have not been found in the measurements, and at modest fields the primary bifurcation appears to be supercritical and to lead to a state of spatio-temporal chaos instead of being subcritical. The experiments reveal an interesting parameter range where the primary bifurcation leads to a bimodal, or rhombic, structure which had not been predicted. For homeotropic alignment and a vertical field, convection can occur when heating is either from below or above. When heating from below, a subcritical Hopf bifurcation is predicted by theory and found in the experiments. Again there is quantitative agreement between the measured and the predicted bifurcation line as a function of magnetic field. The nonlinear state near the bifurcation is one of spatio-temporal chaos which seems to be the result of a zig-zag instability of the straight-roll state. When heating from above, convection occurs because of heat focusing. Experimental results for the bifurcation line exist only at small magnetic fields, but these agree quite well with the predictions. The bifurcation is supercritical and leads to a pattern consisting of squares. Recent measurements of two-phase convection are presented. Here the top of the fluid layer is in the nematic and the bottom in the isotropic phase. In a sufficiently large thermal gradient, the more dense phase can be stably stratified above the less dense one. Measurements of the bifurcation lines as a function of the vertical interface position are in good agreement with theory. A great diversity of patterns is observed in the nonlinear regime, including normal and parallel rolls, oblique rolls with defects, disordered rolls and circles, and cellular flow with upflow or downflow at the cell center.

2 Guenter Ahlers 1 Introduction This chapter presents a review of experimental results for thermally driven convection in a thin horizontal layer of a nematic liquid crystal (NLC) heated from below or above. Convection in an isotropic fluid heated from below is well known as Rayleigh-Bénard convection (RBC). [1, 2] However, this phenomenon is altered dramatically in the case of a NLC. When the director is orthogonal to the heat current Q, i.e. in the horizontal direction, a new mechanism involving heat-focusing [3 5] associated with the anisotropy of the thermal conductivity comes into play. On the other hand, when the director is parallel to Q, a competition between the time scales associated with director relaxation and thermal relaxation leads to a Hopf bifurcation, [6 8] i.e. to an instability at a finite frequency. In addition, when the fluid is a NLC, there are at least two other interesting cases. One of them is a thin horizontal layer heated from above. Again because of heat focusing [9] this can lead to convection when the director is parallel to Q. The other is two-phase convection. Here the fluid is heated from below, but the bottom (top) portion is at a temperature above (below) the nematic-isotropic transition temperature T NI, so that an interface between the isotropic and nematic phase exists in the sample. This situation leads to yet another instability mechanism associated with the interface which is quite different from the Rayleigh-Bénard case. [10] A further interesting aspect of thermal convection in NLCs is that an external magnetic field will couple to the fluid because the diamagnetic susceptibility is anisotropic. A field of modest strength can have a dramatic effect on the phenomena which are observed. This adds greatly to the richness of the physics accessible to the experimentalist. There have been several previous reviews of this topic. [5, 11, 12] The most recent one by Barratt [12] emphasizes theoretical developments. In the present chapter I will focus on the experimental contributions to the field, although these will be compared to the relevant theory. The earlier reviews [5, 11] were more broadly based, including other instabilities such as electroconvection [5] or shear-flow instabilities. [11] Although I will attempt to present a proper historical perspective, much of the information in this chapter is new and is the result of recent investigations in our laboratory. [13 17] The structure of this chapter is as follows. In Sects. 1.1 to 1.3 we will briefly discuss three aspects of thermal convection in NLCs, and consider various views from which they may be of broad interest. In the subsequent sections the experimental results will be reviewed in detail. Section 2 is devoted to convection in a nematic layer with planar alignment and in a horizontal magnetic field. Section 3 reviews the experimental results for pattern formation and bifurcation phenomena in a homeotropically aligned layer. Here Sect. 3.1 is devoted to the case where the heat is applied from below, and Sect. 3.2 discusses the case of

3 Experiments on Thermally Driven Convection heating from above. Section 4 is devoted to two-phase convection. In Appendix A, modern experimental methods for pattern-formation studies in NLCs are described. Appendix B gives a survey of the physical properties of 4 n pentyl 4 cyanobiphenyl (5CB, also known as K15 [18] or 5PP [19]) which may be useful for the interpretation of future experimental work. Lambda ( mw / cm deg C ) T ( deg C ) Fig Thermal conductivity λ of 5CB. The dash-dotted vertical line is at T NI. The circles were measured in a strong vertical magnetic field, and for T < T NI correspond to λ. The squares were obtained in a large horizontal field, and below T NI are λ. Adapted from Ref Instability Mechanisms There are several interesting aspects of convection in NLCs. From one point of view, it is of interest that new mechanisms for instability can come into play. One of these is due to heat focusing. [3 5, 9] The conductivity λ of a NLC is anisotropic, being largest when the temperature gradient is parallel to the director ˆn (λ ) and smallest when it is perpendicular to ˆn (λ ). This is illustrated by the measurements for 5CB which are shown in Fig [15] The anisotropy ratio λ /λ varies from over 1.7 at low temperatures to about 1.4 at T NI. This result is representative of other NLC s as well. [20 24] Let us consider first the case where the heat current Q is applied from below, and where ˆn is in the horizontal direction parallel to the confining plates (known as planar alignment). Figure 1.2a illustrates what happens when a spontaneous fluctuation of ˆn away from perfect horizontal alignment and with a wavelength approximately equal to twice the cell spacing occurs. In the figure, the double-headed nearly-horizontal arrows represent the local director orientation. The anisotropy of λ will lead to a heat current which has a periodic horizontal component. This yields a horizontal component of the temperature field in the sample interior, and to relatively hot (cold) regions near the plusses (minuses) in the figure. It yields a buoyancy force in the direction of the arrows below the plusses and minuses

4 Guenter Ahlers which will tend to induce fluid flow as indicated by the circles. The flow will be in the direction of the arrow heads on the circles. Since the velocity field is coupled to the director field, it will further enhance the perturbation of the director and thus of the laterally periodic component of the heat current, thereby enhancing the amplitude of the lateral temperature variation. The consequence is an enhanced buoyancy force. It is apparent that a small perturbation of the director will tend to be amplified by this heat focusing mechanism. [3 5, 25] Consequently, there is a dramatic reduction of the onset of convection below that of an isotropic fluid. This will be demonstrated in detail in Sect , and is shown below in Fig a Q b Q c Q Fig Schematic illustration of the heat-focusing mechanism. a: planar alignment heated from below. b: homeotropic alignment heated from below. c: homeotropic alignment heated from above. The double-headed arrows indicate the local director orientation. The plusses (minuses) show relatively hot (cold) regions. The flow which tends to be induced by the lateral temperature variation is shown by the circles with their arrow heads. Interestingly, just the opposite effect will come into play when ˆn is aligned vertically, i.e. perpendicular to the confining plates (known as homeotropic alignment). This case, when heated from below, is illustrated in Fig. 1.2b. It can be seen that under these conditions any flow induced by the lateral temperature variations suppresses the director fluctuations. In this case one would not expect a lowered threshold because of the anisotropy of λ, and (in the absence of other effects) convection should start roughly at the same Rayleigh number as in the isotropic case. However, another mechanism becomes important in a NLC as we shall see below. A particularly interesting aspect of heat focusing is that it can lead to an instability when heating is from above. [9, 26] It is easy to see from

5 Experiments on Thermally Driven Convection Fig. 1.2a that a sample with planar alignment and heated from above will not undergo convection. The overall density gradient is stable, and as in the homeotropic case heated from below, the director fluctuations are suppressed rather than enhanced by any flow which they induce. However, in the homeotropic case heated from above the fluctuations are enhanced, as is illustrated in Fig. 1.2c. In that case, convection will occur above a relatively low threshold even though the overall density gradient is stabilizing. The homeotropic case heated from below is interesting because a different mechanism occurs. [6, 7, 27] In this case, the usual Rayleigh-Bénard destabilization due to a thermally-induced density gradient is opposed by the stiffness of the director field which is coupled to and distorted by any flow. It turns out that relaxation times of the director field are much longer than thermal relaxation times. For that reason it is possible for director fluctuations and temperature/velocity fluctuations to be out of phase as they grow in amplitude. This situation typically leads to an oscillatory instability (also known as overstability), and the bifurcation at which these timeperiodic perturbations acquire a positive growth rate is known as a Hopf bifurcation. [8] This case is closely analogous to convection in binary-fluid mixtures with a negative separation ratio. [28, 29] In that case, concentration gradients oppose convection, and concentration diffusion has the slow and heat diffusion the fast time scale. It turns out that the Hopf bifurcation in the NLC case is subcritical, [27] and that the fully developed nonlinear state no longer is time periodic. Instead, the statistically stationary state above the bifurcation is one of spatio-temporal chaos with a typical time scale which is about two orders of magnitude slower than the inverse Hopf frequency. [17] However, it was possible to actually measure the Hopf frequency by looking at the growth or decay of small perturbations which were deliberately introduced when the system was close to the conduction state and near the bifurcation point. [27] Finally we mention that NLCs offer an experimentally convenient opportunity to study yet another instability mechanism which is important for convection in the presence of a first-order phase change. The nematicisotropic transition at T NI is of first order, with a latent heat and a discontinuity in the density. When the vertical temperature difference across a convection cell straddles T NI, an interface between the two phases exists at that vertical position where the local temperature is equal to T NI, with the low-temperature (more dense) nematic phase above it. From experiment it seems that the nematic nature of the upper phase has only a minor influence. For isotropic fluids, the problem was examined theoretically over two decades ago by Busse and Schubert, [10, 30] who identified the mechanism and carried out a linear stability analysis for certain special cases. Contrary to our more common experience with the Rayleigh-Bénard mechanism, it turns out that a vertical temperature gradient actually tends to stabilize the quiescent fluid. Qualitatively, this can be understood by first considering

6 Guenter Ahlers the instability mechanism in somewhat more detail. A small fluctuation with horizontal wavenumber k of the interface position will tend to be destabilizing for two reasons. First, it creates a fluid column of width π/k with a relatively large average density adjacent to a similar column of lesser average density, making it favorable for the heavy column to sink and the lighter one to rise. Second, a local upward displacement of the interface due to a fluctuation will release heat due to the associated conversion of the dense fluid to the less dense one. This leads to local heating in the neighborhood of the displacement, thereby adding to the already existing positive buoyancy force. Similarly, a downward displacement will absorb heat, producing cooling in its vicinity and thus enhancing the negative buoyancy which prevails there. It is seen that both of these factors further enhance the driving force for convection. In the absence of restraining influences, convection thus should be initiated by small fluctuations. This is indeed expected to be the case when the temperature gradient is vanishingly small, yet just large enough to straddle the transition. However, a large temperature gradient will tend to suppress the amplitudes of spontaneous interface fluctuations because the interface essentially is restrained to be located at that precisely defined vertical position where the local temperature equals the transition temperature. A larger temperature gradient thus provides a more severe constraint, and actually stabilizes the interface. This makes it possible for a more dense phase to be stably stratified above a less dense one. Of course, when the temperature gradient becomes large enough, the usual Rayleigh-Bénard mechanism leads to convection, and the instability is the result of a complicated interaction between the two phenomena. The theoretical work of Busse and Schubert was motivated by the relevance of the interface instability to geophysical and astrophysical problems. The stability of a dense phase above a less dense one plays a role in geothermal situations, where for instance water can be stably stratified above steam. [30, 31] Phase changes also are important for convection in the earth s mantle [32] and in stars. However, in those cases there are very large gravitational pressure gradients, and it is believed that the more dense phase is located below the less dense one. An instability can occur nonetheless because the latent heat which is released in an interface displacement can still destabilize the system. However, this situation is not readily achieved in the laboratory. Various types of interface instabilities, including ones with negative latent heat and when heating from above, have been enumerated by Busse. [30]. It turns out that two-phase convection also involves extremely interesting nonlinear problems, such as the exchange of stability between hexagons and rolls. There have been only qualitative experiments relevant to this interesting system [33 35] until very recently. [14, 16] A detailed account of our present state of experimental knowledge about the linear as well as the nonlinear aspects will be provided in Sect. 4.

7 Experiments on Thermally Driven Convection 1.2 Stability Analysis From another vantage point, NLC convection provides us with a system whose equations of motion are well known, [36, 37] but significantly more complicated than the Navier-Stokes equations for isotropic fluids. The usual viscosity and conductivity are replaced by five independent viscosities and two conductivities, and the equations for momentum and energy balance must be coupled to an equation for the director field which contains three elastic constants. In spite of these complexities, it has been possible to carry out quantitative stability analyses, and under some conditions predictions in the weakly nonlinear regime have been made. [25, 26, 38] Thus, one may argue that comparison of quantitative experiments on thermal convection in NLC s with corresponding detailed theoretical calculations provides an excellent testing ground for the applicability of methods of stability analysis and of weakly nonlinear theory to systems which are more complex than isotropic fluids. From this point of view, thermal convection in NLCs has an experimental advantage for instance over electro-convection [39, 40] (hydrodynamic flow in a thin layer of a NLC which is induced by the application of an electric field) in that adequately defined boundary conditions at the top and bottom plate are more easily established. For instance, when parallel alignment is desired, a slight tilt angle of the director adjacent to the boundaries becomes relatively unimportant in the relatively thick fluid layers which are needed, especially when a significant horizontal magnetic field is applied. On the other hand, the thick samples necessarily have a relatively small aspect ratio, having a radius which typically is no more than 10 times the layer thickness. For the latter reason, the experimentalist needs to be careful to distinguish between effects induced by the lateral sidewalls and phenomena associated with the laterally unbounded system usually envisioned in the theory. 1.3 Pattern Formation From a third point of view, thermal convection in NLCs provides a rich system for the study of general problems in pattern formation. During the last two decades interest in this nonlinear topic has seen a revival in the physics community, and a great deal has been learned from experiments about nonlinear pattern-forming dissipative systems. [41, 42] Of particular interest at the present are patterns which form in two spatial dimensions. Most of these have been studied in systems which are isotropic in the plane of the pattern. Notable examples are RBC in an isotropic fluid and the formation of Turing patterns in chemical reactions. [41] Thermal convection in NLCs with homeotropic alignment and in a vertical magnetic field also falls into this category. In those cases the patterns, e.g. the convection rolls, squares, or hexagons, can form with an arbitrary angular orientation which should be irreproducible in successive independent experiments

8 Guenter Ahlers unless the boundary conditions imposed by the apparatus break the rotational symmetry. NLCs have much to offer here. In particular, it was found in early experiments [9, 35] that heating from above will lead to squares, a case relatively rare in isotropic-fluid convection (squares do occur in the case of poorly conducting top and/or bottom boundaries, [43] and in the case of binary-mixture convection for certain parameter ranges [44, 45]). Another advantage of NLC thermal convection in the homeotropic case is that it should be relatively easy to break the (ideal) rotational symmetry by applying a small horizontal magnetic field in addition to a stronger vertical one. The opportunity thus exists to study the crossover from the isotropic to the uniaxial case. Finally we mention that the homeotropically aligned fluid when heated from below was predicted [6, 7] and found from experiment [27] to become unstable via a Hopf bifurcation. This bifurcation is subcritical, [26, 27] and leads immediately to a state of spatio-temporal chaos [17]. It is expected to terminate in a codimension-two point at high fields where the primary bifurcation should be to a stationary pattern. [26] For the most part, the phenomena described above have been explored by experiment only qualitatively, if at all. Thus there remains a large field with opportunities for numerous fruitful laboratory investigations. From the pattern-formation viewpoint it is particularly interesting to examine systems which have an intrinsic anisotropy which singles out a unique angular orientation ˆn. This anisotropy introduces new bifurcation phenomena which are absent in the isotropic case because the orientation of the pattern relative to ˆn can now change as a control parameter is varied. Examples of anisotropic systems are NLCs in which the director is oriented in a particular direction parallel to the flat plates which confine the fluid. [36] Such a sample may also be exposed to a horizontal magnetic field H parallel to ˆn. [4, 12, 46] This system was studied theoretically by several investigators, [3 5, 47] but most recently and in greatest detail by Feng et al. [25] Over an appropriate parameter range, RBC rolls are expected to form with an amplitude which grows continuously from zero as the temperature difference T is increased beyond its threshold value T c. The orientation of the roll axis is predicted to depend upon H. As H is increased beyond a critical value H L1, the roll-axis orientation at onset changes continuously from being normal to ˆn and H (normal rolls) to being oblique to ˆn and H (oblique rolls). When H reaches values beyond a second critical value H L2, the orientation is parallel to H and ˆn (parallel rolls). In analogy to equilibrium critical phenomena, [48] the two-parameter bifurcation-points at which the transitions from normal to oblique and oblique to parallel rolls occur have been called Lifshitz points. [49] Such a bifurcation phenomenon cannot exist in the isotropic case because the convection-roll orientation is arbitrary. Recent detailed experiments [13] are in quantitative agreement with the theoretical predictions. They will be reviewed in Sect. 2.

9 Experiments on Thermally Driven Convection 1.4 Materials The most common material for the study of both electro-convection and Rayleigh-Bénard convection has been p-methoxy benzylidene-p-nbutylaniline (MBBA). The reason for this apparently is historical; MBBA was the first material for which all relevant physical properties, which are necessary for comparison between experiment and theory, had been measured. However, a recent survey of the literature revealed that the properties of some of the cyano-biphenyls are known nearly as well. These materials are far more stable and less toxic than MBBA, and thus have advantages for precise experimental work. They are also relatively inexpensive, and this is an important factor for thermal convection because comparatively large amounts (typically perhaps 30 cm 3 ) are required. [50] The cyano-biphenyls are well suited for studies of thermal convection. However, they cannot be used for many types of investigations of electro-convection because in the case of parallel alignment their positive dielectric anisotropy leads to a Fréedericksz transition prior to the onset of electro-convection. In order to encourage its use in thermal-convection studies, a summary of the physical properties of 5CB is given in an appendix to this review. 2 Planar Alignment and a Horizontal Magnetic Field 2.1 Introductory Remarks Nematic liquid crystals with planar alignment offer an opportunity to examine pattern formation in systems which have an intrinsic anisotropy which singles out a particular angular orientation. [46] The anisotropy introduces new bifurcation phenomena which are absent in the isotropic case because the orientation of the pattern relative to ˆn can now change as a control parameter is varied. The seminal early theoretical work of Dubois-Violette et al. [3 5] identified the instability mechanism due to heat focusing (see Sect. 1.1), and made predictions about the threshold as a function of the strength of a magnetic field H parallel to the director. The same group of authors performed early experiments [4] which qualitatively confirmed the threshold prediction as a function of H, and which (for relatively small H) showed that the anisotropy led to convection rolls with their axes perpendicular to the director (normal rolls). Much more recently, quantitative measurements of both the linear and nonlinear properties of this system over a wide range of H have been performed, [13, 17] and in the present section we provide a review of the current state of our experimental knowledge. It turns out that the system is very rich. As H is varied, a number of new bifurcation phenomena as well as interesting nonlinear effects occur. Detailed predictions based on linear and weakly nonlinear theory have been made recently by Feng et al., [25], and the experimental results will be compared with these so far as possible.

10 Guenter Ahlers 2.2 Theoretical Predictions At very low fields, time-independent RBC rolls are expected to form via a supercritical bifurcation [8], that is with an amplitude which grows continuously from zero as the temperature difference T is increased beyond its threshold value T c. [25] For a range of modest field values between H t1 and H t2, the bifurcation is expected to become subcritical. Beyond the upper tricritical point at H t2, the onset of convection once again should be via a supercritical bifurcation, and is expected to lead to a time independent pattern. The orientation of the roll axes is predicted to depend upon H. As H is increased beyond a critical value H L1, the roll-axis orientation at onset changes continuously from being normal to ˆn and H (normal rolls) to being oblique to ˆn and H (oblique rolls). Depending on the fluid properties, H L1 may be larger or smaller than H t2. When H t2 > H L1, the predictions for the oblique-roll angle for H < H t2 pertains only to the infinitesimal perturbations of the linear state which first acquire a positive growth rate, and not necessarily to the fully developed nonlinear convecting state. When H reaches values beyond a second critical field H L2, the orientation is parallel to H and ˆn (parallel rolls). In analogy to equilibrium critical phenomena, [48] the two-parameter bifurcation points at which the transitions from normal to oblique and oblique to parallel rolls occur have been called Lifshitz points. [49] Quantitative measurements of the roll orientation as H is increased from below H L1 to above H L2 have been made. [13] There also exist results for T c (H) over a wide range of H. These experiments are in quantitative agreement with the calculations by Feng et al., [25] and will be reviewed in the next section. More limited predictions have been made for the nonlinear state which evolves above the bifurcation. [25] As will also be shown in the next section, some of these have been confirmed by experiment, but for other aspects of the nonlinear problem the theory seems to be incomplete at this time. Much work, both theoretical and experimental, remains to be done in this area. The quantitative aspects of the instabilities are determined by four dimensionless parameters which are formed from combinations of the fluid properties described in Appendix B. They are [25] the Prandtl number σ = (α 4/2) ρκ, (2.1) the ratio between the director relaxation time and the heat diffusion time F = γ 1κ k 11, (2.2) the Rayleigh number R = αgρd3 T (α 4 /2)κ, (2.3)

11 and the dimensionless magnetic field Experiments on Thermally Driven Convection h = H/H F, (2.4) with the Fréedericksz field H F, = π d k 11 ρχ a. (2.5) The time scale of transients and pattern dynamics is measured in terms of t v = d 2 /κ. (2.6) Both h and R are easily varied in an experiment, and may be regarded as two independent control parameters. The availability of h in addition to R makes it possible to explore the predicted two-parameter bifurcation phenomena associated with the tricritical and Lifshitz points. The parameters F, σ, and t v are essentially fixed once a particular NLC and temperature range have been chosen, and even between different NLCs there is not a great range at our disposal. For 5CB at 26 we have σ = 502 and F = 757. The critical value R c of R and the fluid parameters determine the critical temperature difference T c for a sample of a given thickness d. The realistic experimental requirement that T c a few C dictates that the sample thickness should be a few mm (except at very low fields where R c becomes small). For cell 2 (d = cm, see Appendix A) this leads to t v = 287 s. Since equilibration after control-parameter changes often requires time intervals of 10 to 100t v, careful experiments are extremely time consuming. In fact, it may be that some of the experimental observations of the convecting state cannot realistically correspond to stationary states if the time scale in establishing those states involves the director relaxation time F t v, which is 60 hours for cell 2. Typical values of H F, for samples of thickness a few mm are near 20 Gauss. Thus modest fields of less than a kgauss are adequate to explore the entire range of interest. 2.3 Experimental Results Linear Properties The critical temperature differences for the onset of convection were determined from heat-transport measurements. [13] These are usually expressed in terms of the Nusselt number N λ eff /λ. (2.7a) Here λ eff Qd/ T (2.7b)

12 R c /R c0 Guenter Ahlers is the effective conductivity and contains contributions from diffusive conduction and from hydrodynamic flow. It should be remembered that the diffusive contribution to λ eff is not the same as that of a sample with perfect planar alignment. The hydrodynamic flow couples to the director field and modifies the diffusive conductivity of the sample. However, the denominator λ in Eq. 2.7a is taken to be the diffusive conductivity in the absence of hydrodynamic flow with the director perpendicular to the heat current. The conductivity λ was measured below the convective onset [15] and is shown in Fig Figure 2.1 shows N at several fields for cell 2 (see Appendix A for details about the three cells which have been used). Nusselt Number N T ( deg C ) R c / R c H / H f F, Fig Nusselt numbers for cell 2 (d = 4.04 mm, H F, = 17.5 Gauss). From left to right, the data are for H = (h = ) 311 (17.8), 362 (20.7), 414 (23.7), 518 (29.6), 622 (35.5), and 1036 (59.2) Gauss. Fig Critical Rayleigh numbers as a function of the scaled magnetic field h = H/H F,. Squares: cell 1 (d = 4.98 mm, H F, = 14.2 Gauss). Circles: cell 2 (d = 4.04 mm, H F, = 17.5 Gauss). Crosses: cell 3 (d = 3.27 mm, H F, = 21.6 Gauss). Adapted from Ref. 13. In the high-field limit, the director and field direction are still expected to determine the roll orientation; but the threshold R c should be the same (R c = 1708) as that of an isotropic fluid. Figure 2.3a gives T c for cell 2 (d = cm) for a range of mean temperatures and for large fields (H > 1000 Gauss). The experimental values of T c vary by a factor of 2.8 in the nematic phase. The solid lines are derived from Eq. 2.3 with R = R c = 1708 and from the fluid properties given in Appendix B (for the isotropic phase, α 4 /2 in Eq. 2.3 is replaced by the shear viscosity η). The agreement with the data is seen to be excellent in both phases. The same results are shown with greater resolution in Fig. 2.3b as critical Rayleigh

13 Experiments on Thermally Driven Convection numbers, obtained from Eq. 2.3, the fluid properties, and the measured T c. The dashed horizontal line is the prediction R c = The vertical bar near T NI = 35 C corresponds to 1708 ± 85, that is its length corresponds to an error of ±5%. Most of the data for R c fall within this range. Exceptions are the points just below T NI where the fluid properties vary rapidly with temperature and are not known too well, and the data at the highest temperatures where the thermal expansion coefficient is rather uncertain. T c ( deg C ) (a) (b) T ( deg C ) R c T ( deg C ) Fig (a): Critical temperature differences and (b): critical Rayleigh numbers as a function of the mean temperature for cell 2. In (a), the solid line is the T c calculated from the fluid properties (Appendix B) and R c = In (b), the vertical bar corresponds to R c = 1708±5%. Less extensive measurements of T c for large H were made for the other two cells near 28 C. They gave R c = 1640 (cell 1, d = cm) and 1740 (cell 3, d = cm). Clearly, there is excellent agreement between experiment and theory for all three cells. In the isotropic phase, T c was found to be field independent, as expected. For instance, at a mean temperature of 37.9, it was found with cell 2 that T c = 5.40 and 5.38 for H = 40 and 1200 Gauss respectively. The dependence upon h (Eq. 2.4) of the measured T c in the nematic phase is compared with theory in Fig Shown as squares, circles, and crosses respectively are the ratios R c (h)/r c ( ) for cells 1, 2, and 3. Their good agreement with each other is consistent with the expected scaling of H with H F,. Note that H F, 1/d and that d varies from 3.3 mm (cell 3) to 5.0 mm (cell 1)! The solid line is the theoretical prediction. [25] Since the calculation contains no adjustable parameters, the agreement between

14 Guenter Ahlers theory and experiment is really very impressive. It should be noted that the theoretical bifurcation line is sensitive to the fluid parameters. The good agreement with experiment seen in Fig. 2.2 thus also reflects the fact that the properties of 5CB (see Appendix B) are known quite well. a b c d Fig Images of convection patterns in cell 1. The field and the director are in the horizontal direction. The examples are for a: 211 Gauss, ɛ = 0.036; b: 379 Gauss, ɛ = 0.035; c: 427 Gauss, ɛ = 0.069; and d: 442 Gauss, ɛ = The Fréedericksz field is 14.2 Gauss for this cell. Adapted from Ref. 13 Figure 2.4 shows images of the patterns at small ɛ T/ T c 1 at various H for cell 1. Here the entire cell is viewed from above (see Appendix A for the flow-visualization method), and the director and the field are in the horizontal direction in the figure. At relatively small H (4a), the roll axes are normal to the director and the field (normal rolls), and the wavenumber k decreases somewhat with increasing H. In this field range and in the rectangular cell, k tends to change discontinuously since structures commensurate with the cell length seem to be preferred. As H is further increased, we pass the predicted [25] Lifshitz point where there is a continuous transition to oblique rolls. The oblique rolls could have either of two orientations, corresponding to a positive or negative angle between their axes and ˆn. Over a field range with angles close to π/4, coexistence at the same spatial location of both orientations as a bimodal structure was observed (image 4c, see also Sect ). At the slightly higher field of 442 Gauss (image d) no bimodal structure was found near onset. At even higher fields than those represented in Fig. 2.4, no patterns were observable because the roll axes were parallel to the director and the field (parallel rolls) and thus the flow did not distort the director field. Compared to patterns in pure fluids, the rolls seem surprisingly little affected by the sidewalls, although there is a tendency for them to terminate at the sidewalls with their axes perpendicular to H as can be seen by close inspection of images b and d.

15 Experiments on Thermally Driven Convection Fig Top two rows: Images for cell 2. The field and the director are in the horizontal direction. The field magnitudes, in Gauss, are given in each figure. The Fréedericksz field for this cell is 17.5 Gauss. For the image at 1270 Gauss the temperature of the top plate was 31.5 and δt was large enough for the temperature of the bottom plate to be just above T NI. Bottom row: Grey-scale images of the structure factors S(k) of three of the images shown in the top two rows (again the field magnitudes, in Gauss, are given for each). The top two rows of Fig. 2.5 show patterns for cell 2. This cell had a circular cross section, and the entire pattern is shown. The field magnitude, in Gauss, is given in the upper left corner of each image. All except the one at 1270 Gauss are for ɛ Images for cell 3 at the same value of h look similar. As for cell 1, one sees the transition as a function of H from normal to oblique rolls. However, for cell 2 and near the lower Lifshitz field H L1, both roll orientations coexisted in the cell, each occupying a different portion with a boundary between them across the diameter parallel to H as seen for 441 Gauss. Since most phenomena depend upon H 2 and not on H, one would expect that there should also be patterns which correspond to a reflection through a plane perpendicular to the field direction. Even though experiments were carried out in which the final state was reached

16 Guenter Ahlers either by increasing T at constant H or by decreasing H at constant T, the search for these was unsuccessful. They also could not be produced when the direction of H was reversed, and all these experiments yielded only patterns like those shown for 441 Gauss. Presumably there was some slight asymmetry in the experimental cell which prevented access to the other patterns. For the field range where oblique rolls with their axes at an angle near π/4 to H are expected, bimodal structures were observed near onset also in this cell, as seen for 461 and 477 Gauss. Although this phenomenon involves the nonlinear interaction between the modes, we discuss its occurrence near onset already in this section rather that in Sect below. Two examples are shown in Fig. 2.5 so as to demonstrate that the bimodal structures actually exist over a field range, and not only at a unique field value. Near onset (ɛ 0.01), they formed over the range 461 H 487 Gauss, corresponding to 26.3 h The angle between the two modes varied from 104 at H = 461 Gauss to 94 at H = 477 Gauss. Thus, except possibly for a unique field value, these patterns are not squares but may be regarded as rhombi. [51] In this case, the rhombi arise as a small perturbation of squares. Rhombi which are closer to hexagons (with the angle between the roll axes close to 120 ) have been observed in chemical patterns by Ouyang et al. [52, 53] These authors, as well as Malomed et al., [54] discuss them in terms of coupled Ginzburg-Landau equations. At higher fields only one mode existed in a given pattern. In this parameter range either orientation occurred in separately prepared samples, although the one shown for H = 518 Gauss seemed to occur more frequently. Thus, at these fields any imperfection in the cell was not adequate to deny access to one of the ideally degenerate patterns, as apparently had been the case in the range near 440 Gauss. Above H L2, no patterns were visible because the flow did not distort the director field. However, in this range the flow became visible when the mean operating temperature was raised so that the temperature at the cell bottom exceeded T NI by a small amount when the bifurcation was reached; such experiments confirmed that the convection at high fields did indeed consist of parallel rolls. An example is shown in Fig. 2.5 for H = 1270 Gauss. The wavevector k has the two components q and p in the direction ˆn (and H) and perpendicular to ˆn (and H) respectively. In order to determine q and p quantitatively, images like those in Fig. 2.5 were processed further. The part of the image contained within 85 % of the cell radius was Fourier transformed (discarding the outer 15 % tended to reduce the influence of the sidewall). Grey-scale representations of the corresponding structure factors S(k) (the square of the modulus of the Fourier transform) are given in the bottom row of Fig The roll wavevectors could be determined from the locations of the peak pairs as obtained by computing the first moment of k over the nine pixels surrounding one of the peaks.

17 p or q or k Experiments on Thermally Driven Convection Over the oblique-roll range, the critical values q c and p c of q and p were determined as a function of H. Figure 2.6 shows qc, 2 p 2 c, and kc 2 = p 2 c +qc 2 and the corresponding theoretical [25] curves for cell 2 (circles and squares) and cell 3 (crosses). The lower Lifshitz point is located at the field value at which p 2 c begins to grow from zero, and the upper Lifshitz point corresponds to the field where qc 2 vanishes. There is quantitative agreement between the measurements for cells 2 and 3 and the theory. [25] From Fig. 2.6 it is apparent that p 2 c (q2 c ) varies linearly with H near H L1 (H L2 ). The implied square-root dependence of p c and q c on H near H Li is in agreement with general theoretical predictions [48] for Lifshitz points h Fig The squares p 2, q 2, and k 2 of the components p (squares) and q (circles) and the modulus k (triangles) of the wavevector k as a function of the reduced field h = H/H F,. These data are for cell 2. Corresponding results for two field values for cell 3 are given as exes. The continuous lines are the prediction based on Ref. 25. Adapted from Ref Nonlinear Properties General remarks. Two tools have been utilized for the study of the nonlinear states above the first bifurcation. One of them is measurements of the Nusselt number defined by Eq. (2.7), and the other is pattern visualization. In order to review the available information, we will proceed from high to low fields. We will first discuss the nature of the nonlinear state immediately above the primary bifurcation. Then we will consider secondary bifurcations, again starting at high and proceeding to low fields. The nature of the primary bifurcation. Representative results for N( T ) were given in Fig Additional results are given in Figs. 2.7 and 2.8. They (as well as many additional measurements very near the onset of convection) revealed no hysteresis and instead suggested that, within

18 Guenter Ahlers the resolution of the experiment, the primary bifurcation is supercritical over the entire range h > 17 which was investigated in cell 2. [55] The theory [25] had predicted that the bifurcation should be supercritical for h > h t2, where h t2 is an upper tricritical field. For a system with the physical properties of 5CB, h t2 was expected [56] to be very close to the lower Lifshitz field h L (see Fig. 2.6). The absence of hysteresis over the entire field range seems to be an important difference between the weakly nonlinear theory and the measurements. Of course we can not rule out that the hysteresis loop and the jump in N at T c are so small that they simply are not observable within the experimental resolution. In order to resolve this issue, more detailed, perhaps numerical, calculations would be very useful. 1.1 Nusselt Number N ε Fig The Nusselt number near onset for H = 466 Gauss(h = 26.6) and cell 2. The open circles were taken with increasing and the solid ones with decreasing heat current Nusselt Number N Nusselt Number N εε ε Fig Typical Nusselt numbers for cell 2 and three different fields as a function of ɛ. Diamonds: H = 1036 Gauss(h = 59). Circles: H = 404 Gauss(h = 23.1). Squares: H = 342 Gauss(h = 19.5). Open (solid) symbols: increasing (decreasing) T. When the bifurcation is supercritical, the Nusselt number near onset can be described by

19 Experiments on Thermally Driven Convection N = 1 + S 1 ɛ + S 2 ɛ (2.9) with ɛ T/ T c 1. A fit of Eq. (2.9) to the high-field data (h = 59.2) for cell 2 yielded S 1 = 1.37 and S 2 = The value of S 1 is close to the theoretical prediction for the isotropic [57] and the nematic case. Feng et al. [25] predicted S 1 = 1.45, in very satisfactory agreement with the experiment. Measurements of S 1 with pure fluids in circular cells of aspect ratios similar to that of cell 2 [58] usually gave values of S 1 somewhat below the theoretical prediction [57] for the laterally infinite system, presumably because of the influence of the sidewalls and of defects in the interior. The close agreement with the theory for the laterally infinite case again indicates that the sidewalls have only a minor effect on the flow in the planar NLC case. From measurements with isotropic fluids, S 2 is also negative and of about the same size as found here, but apparently no explicit prediction of S 2 has been made for either fluid. Slope S 1 of N h = H / H F Fig The initial slope S 1 of the Nusselt number for cell 2 as a function of the field h. The horizontal dashed line is the value measured at h = The horizontal solid line indicates the range of oblique rolls (see Fig. 2.6). For cell 2, S 1 was studied in detail as a function of h. Some of the results are given in Fig Above h L2 33, S 1 was nearly independent of h, in agreement with the calculations. As h decreased below h L2, S 1 increased and reached a maximum near h 29. Comparison with Fig. 2.6 shows that this is roughly in the middle of the oblique-roll range where p q, i.e. where the roll axes are at an angle of about 45 to the director. Perhaps the point of maximum S 1 corresponds to the theoretically predicted [25] tricritical field h t2. However, for 5CB h t2 had been expected [56] near h L1 24, which is somewhat smaller than the point of maximum S 1 in the experiment. Even below the field of maximum S 1 no measurable hysteresis occurred. This is shown by the data in Figs. 2.7 and 2.8. Figure 2.7 gives results for h = 26.6 which near onset are qualitatively representative of the range 23 < h < 29. Here the open circles (solid circles) correspond to data

20 Guenter Ahlers taken with increasing (decreasing) heat current. Near threshold, they agree with each other. In this range of h, actual measurements of S 1 became impossible because, as seen in Fig. 2.7, a hysteretic secondary bifurcation (see below) occurs at rather small ɛ. However, the initial rise of N(ɛ) is very rapid, corresponding to S 1 > 2 and perhaps to a vertical rise. This is not the behavior one might have expected over a range of fields below h t2. Although there are no explicit calculations of the size of the hysteresis loop, the absence of measurable hysteresis seems surprising. Additional Nusselt numbers in this range are shown in Fig. 2.8 for h = 23.1 and For comparison, some high-field data (h = 59) are shown there as well. The left figure gives a broad range of ɛ and N, whereas the right one displays details closer to the bifurcation. Again the open (solid) symbols correspond to points taken with increasing (decreasing) T. The steep initial rise of N(ɛ) for h = 23.1 is similar to what was seen at h = 26.6 in Fig The absence of measurable hysteresis at the primary bifurcation especially at h = 19.5 (which should be deep in the range below h t2 ) is noteworthy. For H < 380 Gauss(h < 22), S 1 once more is well defined by the data, as illustrated for h = 19.5 in Fig Here S 1 has values much smaller than typical of isotropic fluids, as shown in Fig. 2.9 over the range 300 to 380 Gauss (17 < h < 22). Figure 2.8 also illustrates the dramatic difference (by a factor of three) in the initial slope of N(ɛ) between the high- and the low-field data. Chaos at onset. So far we have discussed Nusselt numbers as though they were time independent. Actually, high-resolution measurements in cell 2 reveal that N was non-periodically time dependent immediately above onset over a wide parameter range. This is illustrated in Fig The top part (a) gives N(t) for ɛ 0.01, for the two fields h = 29.9 (lower curve) and 28.4 (upper curve) for which N was time independent. Any time dependence of the data reflects the instrumental noise and drift and initial transients which can be explained in terms of t v. Part (b) gives two examples at field values in the time-dependent range. There the upper curve is for h = 27.8, and the lower one for h = The data in (b) clearly reveal time dependent states. This phenomenon has not been investigated systematically, but it does persist down to the lowest values of h which have been studied. Here we should caution immediately that it is not possible to rule out that this time dependence is a transient which eventually would die out. If the establishment of a steady state involves the director relaxation time, then equilibration times may well be longer than any heretofore explored experimental time scales. As discussed in Sect. 2.2, the vertical director relaxation time is t d = F t v, which is several days. Processes involving phase adjustments in the lateral direction could conceivably require several times t d. Nonetheless, it is interesting that even with relatively short experimental time scales of order a few hours no time dependence was found near onset for h > 28. For now we will have to leave open the question of

21 Nusselt Number N Experiments on Thermally Driven Convection whether the chaos observed for h < 28 is a stationary process or whether it is a transient ( a ) ( b ) time ( t v ) Fig Nusselt numbers as a function of time for cell 2 at four different fields. In each case, the heat current was raised to about 2 % above critical at t = 0. a): Time-independent field range; the upper curve is for h = 28.4 (H = 497 Gauss), and the lower one for h = 29.9 (H = 523 Gauss). b): Non-periodically timedependent field range; the upper curve is for h = 27.8 (H = 487 Gauss), and the lower one for h = 25.5 (H = 445 Gauss). Time is scaled by the vertical thermal diffusion time t v = d 2 /κ = 242 s. The time dependence of the Nusselt number for h < 28 is associated with an amplitude instability of the pattern which, upon close inspection, is already noticeable in the image shown in Fig. 2.5 for H = 343 Gauss(h = 19.6). It turns out that the envelope of the rolls fluctuates in a seemingly random manner. This is illustrated in more detail by the images shown in Fig. 2.11, which are for H = 321 Gauss(h = 18.3) and ɛ = The top row shows three consecutive images at a spacing of one hour (15t v ) which were taken starting four hours after the heat current had been raised to above critical (i.e. at t = 60, 75, and 90; compare Fig. 2.10). The time dependence of the envelope is noticeable. However, it becomes more evident in the bottom row, which shows the envelopes itself as obtained by Fourier demodulation of the top images. Here the envelope is clearly seen to be fluctuating. Apparently we have an example of spatio-temporal chaos (or chaotic transients) immediately at onset. This was not anticipated by the weakly nonlinear theory, [25] although one might hope that the phenomenon could be treated by weakly nonlinear methods since the bifurcation is supercritical and the amplitude grows continuously from zero so far as we can tell from the experiment. For a more detailed study of chaos in a spatially extended system one would of course like to have a much larger sample. This is unlikely to become available for RBC in NLCs. However, it would be worth while to study the present sample (cell 2) in greater detail in order to see to what

22 Guenter Ahlers extent the methods of dynamical systems may be applicable to this case with limited spatial extent. a b c d e f Fig Top row: three temporally consecutive images, 15t v(1 h) apart, for H = 321 Gauss (h = 18.4), ɛ 0.01, and cell 2. Bottom row: The envelopes of the images in the top row, obtained by Fourier demodulation. Secondary bifurcations. We now turn to the issue of secondary bifurcations. There seem to be no predictions in the literature, although some calculations appear possible [56] within the general theoretical framework which has been developed. The experimental results are summarized in Fig. 2.12, which gives the various patterns which were found in the ɛ h plane. Each data point corresponds to an image which was acquired after equilibration for one to three hours. For H > 700 Gauss(h > 40) no secondary bifurcations were encountered over the range explored by the measurements (ɛ < 0.25). In this field range no pattern was visible, consistent with parallel alignment. The Nusselt numbers were well represented by Eq. (2.10). Close inspection of Fig. 2.1 already reveals that this is no longer the case for H = 622 Gauss(h = 35.5). This is demonstrated more clearly in a plot of (N 1)/ɛ vs. ɛ, as shown in Fig Here the intercept at ɛ = 0 corresponds to S 1, and the slope of a straight line through the data should have the value S 2. The high-field data are consistent with a single straight line, i.e. with Eq. (2.10). However, at the lower field the data reveal a bifurcation to a new state at ɛ L2 = Beyond ɛ L2, the flow patterns became visible, and they corresponded to oblique rolls. Thus we conclude that the observed secondary bifurcation is a point on a line of Lifshitz transitions emanating from ɛ = 0 and h L

23 Experiments on Thermally Driven Convection Schematically this is shown by the solid line in Fig Although the Nusselt-number data in Fig are not as detailed as might be desired, there seems to be no discontinuity in N at ɛ L εε h Fig Bifurcation diagram in the ɛ-field plane. All data points were obtained with increasing ɛ for cell 2. The open squares on the right and left correspond to parallel and normal rolls respectively. The circles are oblique-roll states. Here the solid circles correspond to bimodal patterns. The bimodal region is also indicated by shading. An estimate of the upper Lifshitz transition is given by the solid line. The diamonds are oblique-roll states which are reached via a hysteretic secondary bifurcation. The plusses are complicated intermediate states ( N - 1 ) / ε ε Fig Effective slope (N 1)/ɛ vs. ɛ for cell 2. Solid circles: H = 1036 Gauss(h = 59). Open circles: H = 622 Gauss(h = 35.5). Figure 2.14 shows the angle Θ of the roll wavevectors relative to the director and field direction for field values close to h L2. Here we measure Θ counterclockwise from the positive q-direction, so that parallel rolls have Θ = 90. The measurements at h = > h L2 are consistent with Θ growing continuously form 90 as ɛ increases beyond ɛ L The dashed line provides one possible qualitative estimate of the angle for h = h L2. Clearly it would be desirable to obtain much more detailed measurements near the Lifshitz line ɛ L2 (h). At the present, the shape and

24 Guenter Ahlers location of this line are known only approximately, and there are not enough data to decide definitively whether the bifurcation is supercritical or subcritical. The former seems likely since Θ seems to grow continuously from 90 when ɛ L2 is exceeded, and since the Nusselt numbers in Fig seem to be continuous at ɛ L2. Clearly, this is a problem on which additional work should be done. Ideally, one would like to find, both from experiment and from theory, a scaling function which gives the roll angle as a function of some suitably defined distance from the Lifshitz line. Theoretically, this problem is not easy because it involves pattern selection from a linearlystable range of patterns (roll angles). Theta ε Fig The angle of the wavevector relative to the director (in the counterclockwise direction) for several values of h = H/H F, for cell 2. The dotted line is one possible qualitative estimate of Θ(ɛ) for h = h L2. The field range 518 < H < 622 Gauss (29.6 < h < 35.5) was explored only for ɛ < 0.1 (see Fig. 2.12), and over that range no secondary bifurcations were found. The rolls were unobservable (parallel) for h > 33 and visible (oblique) below that field. At H = 518 Gauss(h = 29.1), a set of measurements was made up to ɛ = A secondary bifurcation was encountered at ɛ The transition was from single-mode oblique rolls with Θ 125 at small ɛ to a somewhat disordered zig-zag pattern (plusses in Fig. 2.12) in which the old roll orientation (zig) co-existed in different spatial regions with a new (zag), less-oblique one. In this range the somewhat disordered nature of the patterns suggests that perhaps longer equilibration times were needed to reach a truly steady state. As ɛ was increased further, the regions occupied by the new orientation grew, and finally near ɛ = 0.21 led to a single-mode oblique state (open diamonds in Fig. 2.12) with Θ 168. This less oblique state was reached also via a secondary bifurcation from the bimodal state at somewhat smaller fields, as shown by the diamonds in Fig Proceeding to smaller fields, we encounter the range in the ɛ h plane over which the bimodal patterns illustrated in Fig. 2.5 for H = 461 and ε

25 Experiments on Thermally Driven Convection 477 Gauss (h = 26.3 and 27.3) are stable. This range is shown in greater detail in Fig Here the left part contains data obtained with increasing, and the right one with decreasing ɛ. We see that, at finite ɛ, the bimodal region extends to fields above h 29. Comparison of the two parts of Fig shows that both bifurcations (single-mode oblique to bimodal and bimodal to the less-oblique large-ɛ state) are hysteretic. This is confirmed by detailed measurements of the Nusselt number, as illustrated in Fig. 2.7 for the particular case h = In the bimodal region at relatively large h, the angle between the rolls is pulled towards 90 as ɛ increases, well away from the angle of the single-mode state at smaller ɛ and at the same field. For instance, at h = 29, Θ = 127 in the single-mode state near the primary bifurcation. Over the bimodal ɛ-range from 0.05 to 0.12, Θ increases continuously with increasing ɛ from 127 to 138 [an angle of 90 between the two degenerate modes corresponds to Θ = 135 (or 45 )]. This leads to slightly disordered patterns, typically with singlemode oblique rolls of a lesser Θ very near the sidewall. For comparison, when h = 27.2, Θ = 139 near onset and Θ = 143 at ɛ = 0.08 where the bimodal pattern ceases to be stable. For even larger ɛ, the state with Θ 160 mentioned in the previous paragraph (open diamonds in Figs and 2.15) is encountered beyond a hysteretic secondary bifurcation ε ε ε h h Fig Expanded view of the central section of the bifurcation diagram shown in Fig The data points in the left part were obtained with increasing, and those in the right portion with decreasing ɛ. The symbols are as in Fig The bimodal region is shaded. At fields below h L1 we enter the normal-roll regime. Here it was difficult to study secondary bifurcations in cell 2 because of the disorder in the patterns which was discussed in the section on chaos (see Fig. 2.11). Nonetheless, the three images in Fig for h = 20.7 indicate that there is a transition from normal to oblique rolls near ɛ = Although the angle of obliqueness never becomes very large, it does seem that there is

26 Guenter Ahlers a Lifshitz line going from h L2 to lower fields in the ɛ h plane. However, since its location is somewhat uncertain and since the oblique angle is so close to that of normal rolls, this line was not shown in Fig It was also found in cell 1, which gave somewhat more clear evidence for the bifurcation because only a single mode existed throughout that cell. Images for h = 17.8 are shown in Fig Very similar results were obtained in cell 1 for H = 205, 253, and 327 gauss (h = 14.4, 17.8, 23.0), with transitions to oblique rolls at ɛ 0.07, ɛ 0.06, and ɛ 0.05, respectively. a b c Fig Images of convection patterns for cell 2 and H = 363 Gauss (h = 20.7). a): ɛ = b): ɛ = c): ɛ = Fig Images of convection patterns for cell 1 and H = 253 Gauss (h = 17.8). Left: ɛ = Right: ɛ = After Ref Homeotropic Alignment and a Vertical Magnetic Field 3.1 General Remarks In this interesting case, the director of the conduction state is oriented parallel to the magnetic field and the heat flow. As for Rayleigh-Bénard convection in an isotropic fluid, the conduction state has rotational symmetry in the horizontal plane. Thus, patterns of arbitrary angular orientation should form unless the boundary conditions of the experiment select a particular direction. An interesting possibility is to deliberately break the rotational symmetry by introducing a small horizontal magnetic field in addition to the vertical one. This should lead to the selection of a preferred direction and to patterns which, near threshold, might be accessible to theoretical analysis. It is apparent that this system is potentially very rich for the study of pattern formation. It has only just begun to be exploited for this purpose.

27 Experiments on Thermally Driven Convection 3.2 Heating from below Theoretical Predictions In this case, the first instability should be a Hopf bifurcation, that is the disturbances which first acquire a positive growth rate should be timeperiodic. [6, 7] As the magnetic field is increased, the threshold for convection is predicted to shift to larger values. [26, 27] This was confirmed by early experimental work on this system. [27] For sufficiently high fields the primary bifurcation is predicted to be to a stationary state of convection. [26] There is a codimension-two point where the two bifurcation lines meet. The situation is somewhat similar to binary-mixture convection, [41] which has been studied extensively in recent years. As in the case of planar alignment, the quantitative aspects of the instabilities are determined by the four dimensionless parameters σ, F, R, and h. However, in their definitions as well as in the definitions of the vertical diffusion time t v and the Fréedericksz field (Eqs. 2.1 through 2.6), we need to replace [26] κ by κ, k 11 by k 33, and γ 1 by α 2 /2 (for the values of these parameters for 5CB, see Appendix B). For 5CB at 26 C we have σ = 272 and F = 460. A linear stability analysis was carried out by several investigators. [7, 60 62] A very detailed analysis of this case was presented recently by Feng, Decker, Pesch, and Kramer (FDPK). [26] These authors also provided a weakly nonlinear analysis, and we shall briefly describe their results. For low fields, FDPK predict that the first instability will be a subcritical Hopf bifurcation. The critical Rayleigh number R c (H) varies typically from about 1500 at small fields to about 3400 for h 50. The details of R c (H) depend upon σ and F, and have to be computed for each particular case. The wavevector which first becomes unstable is predicted to vary from about 3.2 to 4.6 as the field increases from h = 0 to h 50. The Hopf frequency is expected to be between about 12 and 2 over this range. It would not be too helpful to be more specific here since the details of all these parameters depend upon σ and F. At h = h ct 50 (assuming typical parameter values for MBBA), the Hopf bifurcation line meets a stationary bifurcation at a codimension-two point. At this point, the Hopf frequency is predicted to be finite (close to 2) and there is a discontinuity of about 10% in the wavevector. The stationary bifurcation for h > h ct initially is also predicted to be subcritical, but for h > 63 (for typical MBBA parameters) it is expected to become supercritical. At h = h ct 63, the coefficient of the cubic term in a Ginzburg-Landau equation vanishes and a tricritical point is predicted for the stationary bifurcation branch.

28 Guenter Ahlers Experimental Results Early measurements for this system were made by Guyon, Pieranski, and Salan [27] (GPS). These authors used the NLC MBBA. Their sample had a thickness d = 5 mm, yielding H F, 15 Gauss. It had a circular cross section, and a diameter of 54 mm. [63] At half-height, several thermocouples were mounted in the fluid to monitor the local temperature. A heater wire near the thermocouples also traversed the sample. It is difficult to say whether these intrusive devices had an influence on the hydrodynamics. The temperature stability of the water baths above and below the sample was of the order of 0.01 C. GPS measured the onset of convection by monitoring the response of their thermocouples to a temperature perturbation induced by a heat pulse delivered by the heater wire. If this response grew (decayed) as a function of time, the threshold of their system had (had not) been exceeded. They were also able to determine a characteristic frequency from the thermocouple response during the transients which led to the convecting nonlinear state. The results for T c are qualitatively consistent with the theoretical results [26] for the laterally infinite system. The magnitude of T c at a given field was within 10 or 20 % of the theoretical value. T c increased with H up to H 580 Gauss(h 33), and then decreased again. The maximum was interpreted [26] as the predicted codimension-two point which for the laterally infinite system is expected at h ct = 51, although it occurred at a rather low field. The measurements also provided clear evidence for hysteresis at the primary bifurcation. The measured Hopf frequency had a maximum near h = 13, whereas the theory predicts the maximum to occur near h = 32. The frequency was generally of the same size as the one given by the theory, but at the highest field values h 33 it was still much larger than expected for h = h ct. We conclude that these experiments clearly established a number of central features of the bifurcation. These include its subcritical nature and the time-periodic behavior of the growing perturbations of the conduction state. However, at the quantitative level there are substantial differences between the experiments and the theory for the laterally infinite system. Using the apparatus described in Appendix A, measurements were made recently in our laboratory [17] using a circular cell with d = 3.94 mm and r = 41.9 mm (cell 4), corresponding to Γ r/d = The fluid was 5CB. For this system, t v = 136 s and H F, = 21.1 Gauss. Monodomain homeotropic samples were prepared before each experimental run as described in Appendix A. The heat current was increased in small steps while the top temperature was held fixed at 19 C, until convection occurred. The heat current then was decreased again in small steps until convection ceased. At each heat-current value, the bottom-plate temperature was measured at one-minute intervals for two hours ( 53t v ). The temperature measurements and the heat current were used to determine the Nusselt number, which is given by Eq. 2.7 with κ replaced by κ. While the cur-

29 Experiments on Thermally Driven Convection rent was steady, images of the convection pattern were acquired by the computer-interfaced CCD camera. Nusselt Number T ( o C ) Fig Nusselt number measurements for h = 9.4 (H = 200 Gauss), using cell 4. Open (filled) circles were obtained with increasing (decreasing) steps in T R c or R s h 2 = ( H / H F ) 2 Fig Solid circles: Critical Rayleigh numbers for the onset of convection as a function of h 2. Open circles: Rayleigh numbers at the saddle node where convection ceased when the heat current was lowered. Figure 3.1 shows N as a function of T for H = 200 Gauss (h 9.4). Surprisingly, N decreased below one when convection started. This can be understood because the conductivity of a sample with parallel alignment, in which Q is perpendicular to ˆn (λ ), is much less than the conductivity of the homeotropic case (λ ) (see Fig. 1.1). [15] The direct hydrodynamic contribution to the heat flux is smaller than the decrease in the heat flux due to the deviations of the director from parallel alignment caused by the flow. As the current decreased, the conduction state was reached at a value of T equal to T s < T c, showing the predicted and previously observed [27] hysteresis. For small fields (H < 250G), the conduction state reached from the convecting state had a conductivity less than λ, corresponding

30 Guenter Ahlers to N < 1, because the hydrodynamic flow experienced by the sample had introduced defects which reduced the average conductivity below λ. At the field value of this experiment, the elimination of defects from the sample occurred on a time scale which was much longer than the duration of the experiment. The visual appearance of the conduction state reached after convection is interesting. It had the appearance of curdled milk, with the clusters of non-homeotropic alignment corresponding to the curds suspended in a nearly-clear background fluid of homeotropic alignment Avg Fig Top two rows: a sequence of images from the same run at 200 Gauss(h = 9.4), taken with constant external conditions. The time elapsed since the start of the run (in units of t v = 136 s) is given in the top left corner of each image. Bottom row: Square root of the structure factor of two of the images shown above, and the average of the structure factor of 250 images spanning a time interval of 1123t v. From data like those in Fig. 3.1, values of the critical temperature difference T c and of the temperature difference at the saddle node T s were determined with an uncertainty of about 0.5 %. The corresponding Rayleigh numbers are shown in Fig. 3.2 as a function of h 2 (solid circles: T c, open circles: T s ). The solid line in the figure is the theoretical pre-

31 Experiments on Thermally Driven Convection diction, [26] evaluated for the properties of 5CB at the mean temperature of the experiment. As can be seen, the agreement with the measurements is excellent. The small deviations at large h are probably caused by excessive variations of the fluid properties over the temperature interval of the measurement when the temperature of the cell bottom is rather close to T NI. There are as yet no predictions for R s. It is interesting that R s is only about 10% below R c. Nonetheless, a calculation may turn out to be difficult because there is already severe distortion of the originally homeotropic director field by the fluid flow, as evidenced by the defects encountered after the conduction state is reached once more. The pattern which evolves beyond the bifurcation is extremely interesting. The first two rows of Fig. 3.3 show typical images of the flow field immediately above the convective threshold for h = 9.4. By examining relatively rapid time sequences of images, it was found that, on the time scale of the inverse of the expected Hopf frequency, the convection rolls were steady rather than travelling or standing waves. This is not in contradiction to the predicted Hopf bifurcation because the subcritical nature of the bifurcation leads to a finite-amplitude state at threshold whereas the theory pertains to an infinitesimal perturbation of the conduction state. A similar situation is encountered in binary-mixture convection, where for a range of values of the separation ratio the convection rolls are steady when T = T c even though small perturbations of the conduction state are travelling waves. In our experiment, there unfortunately was no way to determine the frequency of small-amplitude transients as had been done by GPS. On a much longer time scale, the pattern evolved continuously. This is illustrated by the images in Fig. 3.3, which are from a single experimental run with constant external conditions. They were taken at the times indicated in each image, in units of t v = 136 s, which had elapsed since an arbitrary origin at which the pattern already had been equilibrated for some time. Even in runs of much longer duration (up to two weeks or 9000t v ) no steady state was reached. The nature of the pattern did not change noticeably over the field range 5 < h < 16 covered by the experiments, although no quantitative studies as a function of h have been carried out. It appears that the patterns are disordered both in space and in time, providing an example of spatio-temporal chaos. It is interesting to note that the nature of this chaotic state is quite different from the one which occurs for parallel alignment at low fields and which is illustrated in Fig Whereas Fig illustrates a chaotic time dependence of the envelope of the pattern (amplitude chaos), the homeotropic case consists of a temporal evolution of defects and of the relative orientation of convection rolls and might be referred to as defect or phase chaos. The bottom row of Fig. 3.3 gives the modulus of the Fourier transform. The transforms were base on the central parts of the images, by using the

32 Guenter Ahlers filter function W (r) = cos 2 [(π/2)(r/r 0 )] for r < r 0 and W (r) = 0 for r > r 0. Here r 0 was set equal to 85% of the sample radius. The transforms for t = 449 and 1033 show that the nature of the pattern changed dramatically with time. The rightmost image in the bottom row of Fig. 3.3 (labeled Avg ) shows the square root of the time average of the square of the modulus of the Fourier transform [i.e. of the structure factor S(k)]. The average involved 250 images taken over a total time period of 1123t v (nearly two days). It is seen to contain contributions at all angles, consistent with the idea of a statistically stationary process of non-periodic pattern evolution and with the expected rotational symmetry of the system. S ( k ) (arb. units) S (Θ) (arb. units ) k 1 (b) 0.5 (a) Θ / π Fig a): The azimuthally averaged structure factor S(k) as a function of the modulus k of the wavevector. b): the radially averaged structure factor S(Θ) as a function of the azimuthal angle Θ. Figure 3.4a shows the azimuthal average S(k) of the temporal average of the structure factor for the run described above, i.e. of the lower right image in Fig Both the fundamental and the second harmonic (corresponding to a roll width of half a wavelength) are well developed, but the higher harmonics are so weak as to be unobservable on the scale of the figure. The characteristic wavenumber of the pattern is about 3.4. This is fairly close to the theoretically predicted wavenumber for the mode which first acquires a positive growth rate; but since the observed state is one of finite amplitude, this agreement is not particularly significant. Figure 3.4b shows the average over k of S(k) as a function of the azimuthal angle Θ [the average over k was computed only in the vicinity of the fundamental peak of S(k)]. Although there is a discernable maximum near Θ 0.75, the angular distribution is really quite uniform. Any remaining structure might well disappear if data

33 Experiments on Thermally Driven Convection were averaged over longer time periods. On the other hand, it could also be indicative of a slight asymmetry in the experimental cell Fig A temporal succession of images during the transient leading from conduction to convection when T was raised slightly above T c. The field was h = 9.4. The numbers are the elapsed time, in units of t v, since the threshold was exceeded. It is instructive to examine the transients which lead from the conduction to the convecting state. This is done in Fig Here the number in each image gives the time, in units of t v, which has elapsed since T was raised slightly (1%) above T c. At t = 32.6, there is still no evidence of convection; but at t = 36.4, there are noticeable fluctuations in the image which correspond to hydrodynamic flow. At t = 40.3, some of these fluctuations have grown to macroscopic amplitudes, and a front of convection is invading the quiescent fluid (t = 43.5). This creates a state of nearly-straight parallel convection rolls (t = 53). However, these straight rolls turn out to be unstable to a zig-zag instability. The zig-zag disturbance can be seen to grow at t = 68.4 and In the end, this instability leads to the spatially and temporally disordered pattern shown for t = 163 and in Fig Thus, we see that a secondary instability led to a chaotic state rather than to a

34 Guenter Ahlers new time-independent pattern. This phenomenon most likely is analogous to the one encountered in very early experiments on spatio-temporal chaos using liquid helium [64, 65], where ordinary RB convection became chaotically time dependent, probably because the skewed-varicose instability [66] was crossed (one cannot be absolutely sure about this because in the early work there was no flow visualization). 3.3 Heating from above Heating a NLC with homeotropic alignment in a vertical magnetic field from above leads to an instability because of the heat-focusing mechanism [3] discussed in Sect. 1.1, even though the overall density gradient is stabilizing. This case was investigated in the pioneering work of Pieranski et al., [9] and by Salan and Guyon (SG). [35] A number of theoretical investigations have been performed, [3, 9, 67, 68] with the most detailed and recent one provided by Feng et al. (FDPK). [26] The quantitative aspects of the problem are determined by the same four dimensionless parameters σ, F, R, and h which were quoted in Sect SG used two cells of diameter 52 mm and of circular cross section, and of thickness 1.0 and 0.7 mm. Homeotropic alignment was achieved by surface treatment with lecithin. The fluid was MBBA, with the bottom temperature held fixed at 20 C. In zero applied field, they found T c = 5.3 and 15.5 C respectively. Superficially consistent with Eq. 2.3, the ratio 15.5/5.3 = 2.92 is almost precisely equal to the ratio of the cube of the sample thicknesses. However, for a quantitative comparison the Rayleigh numbers R should be evaluated at the temperature at the midplane of the sample. [69] This would require the kind of detailed evaluation of the fluid properties which is presented for 5CB in Appendix B. Apparently this has not been done for MBBA. The values of R c corresponding to the observed T c are of order 10, i.e. as small as in the case of planar alignment. For d = 1 mm, FDPK calculate T c = 4.89 C. This differs from the measurement by 8%. This difference seems to be within the uncertainties of the fluid properties, especially of the thermal expansion coefficient which enters into R (see Eq. 2.3). As a function of the vertical magnetic-field strength H z, T c varied linearly with Hz 2. [35] This is the behavior expected generally for a property which should be invariant under the transformation H to H. The slope of T c (H 2 ) is consistent with semi-quantitative theoretical analysis, [70] but a detailed comparison with the quantitative theory (FDPK) apparently has not been carried out. So far the measurements have been restricted to rather small fields, corresponding to h 2. Even for the linear properties of the system, much remains unexplored experimentally. A particularly interesting prediction of FDPK is that there is a competition between the instabilities of two different modes of the

35 Experiments on Thermally Driven Convection system. One is symmetric, and the other antisymmetric with respect to reflection through the midplane of the cell. At small H, the symmetric mode is the first one to acquire a positive growth rate as T c is increased. It has a wavenumber which, for h = 0, is about equal to 2.8. With increasing h, it is predicted to grow to about 4.0 when h 3, and to remain constant for larger h. At h 22 (for the properties of MBBA), the antisymmetric mode is expected to be the first to grow. It has a wavenumber (in the horizontal direction) which is about equal to 7, and thus should be easily distinguished from the symmetric mode. The point at which the two marginal curves cross is a codimension-two point where interesting resonant interactions between these two modes are expected to occur. In the experiments of SG, no hysteresis was found. This is consistent with the weakly nonlinear calculations of FDPK, which yield a supercritical bifurcation. For small H, the stable pattern close to but above onset consisted of squares, [9, 35] in agreement with the nonlinear calculations of FDPK. An exception was the thinner of the two cells, which led to hexagons above onset. However, in that case T c was rather large and non-boussinesq effects [69] are believed to be responsible for the observed pattern. A particularly interesting aspect of the experiments of SG are the results obtained when a horizontal field H x was applied to the homeotropic sample. In the experiment, T c decreased linearly with Hx 2. At a critical value of H x, T c vanished. At this point, a line of bifurcations in a non-equilibrium system meets a thermodynamic transition, namely the Fréedericksz transition at H F. For 0 < H x < H F, the horizontal field breaks the rotational symmetry of the homeotropic case in a vertical field, and leads to new pattern-formation phenomena. The pattern which first became unstable consisted of rolls. Beyond a secondary bifurcation, rectangles (consisting of two sets of rolls of different wavenumbers at 90 to each other) became stable. These qualitative observations are consistent with the nonlinear calculations of FDPK, but quantitative information about the secondary bifurcation lines (such as for the rhombi in the planar case, see Fig. 2.15) is lacking at this time. 4 Two-Phase Convection 4.1 Theoretical Predictions As discussed in Sect. 1.1, the governing equations for two-phase convection in isotropic fluids were derived by Busse and Schubert. [10] The theory envisions a situation where the two phases have essentially equal and temperature-independent properties, except for the density difference which is taken to be small compared to ρ. In that case there are three dimensionless parameters which determine the instability. [10] They are

36 Guenter Ahlers and R β = C p q P = ρ ρ 2 d 2 κν R α = αgd3 νκ q C p, (4.1) ( T αt ) NIgd C p q C p, (4.2a) ( dt dp dt ) 1 NI. (4.3a) dp Here p is the pressure, q is the latent heat per unit mass, ρ is the density discontinuity at T NI, and the remaining symbols have the same meaning as in Appendix B. It turns out that the term αt NI gd/c p in Eq. 4.2a is always small in laboratory systems, although its equivalent may become significant in geophysical applications. For 5CB it is about K, and can be neglected compared to any measurable values of T. Then R β becomes R β = T, (4.2b) (q/c p ) which is simply the dimensionless temperature difference across the cell when temperature is scaled by q/c p. Equation 4.3a also simplifies for typical laboratory situations because the slope of the phase-transition line dt NI /dp is much smaller than the variation of temperature with the (hydrostatic) pressure in the sample. Neglecting dt NI /dp and using dt/dp = ( T/d)(dz/dp) with dz/dp = (ρg) 1, we have P = ρ ρ d 3 κν q g c T. (4.3b) For 5CB, the physical properties vary with temperature, particularly for T near T NI which is involved here. However, within 10% or so R α, R β, and P are constant. To this approximation we have R α d 3 with d in cm (here we took ν = α 4 /2ρ with α 4 equal to the average value at T NI in the two phases). Although the heat capacity varies considerably in the transition region, we have approximately C p 2.0 J/gK near T NI. Thus q/c p 0.78 K sets the scale of T. For P, we get P d 3 / T with d 3 / T in cm 3 /K. The theoretical predictions are for the stability boundaries as a function of the interface position z. It is easy to show that the interface location (measured from the cell bottom in units of d) is given by z = I I /(I I + I N ) (4.5a) with I I = TNI T B λdt (4.5b)

37 and I N = Experiments on Thermally Driven Convection Tt T NI λ dt. (4.5c) In the last equation it is assumed that the nematic layer has planar alignment. In the two-phase region the value of z varies from 0 to 1, but its formal definition can be extended to the pure phases. When z < 0, the entire sample is nematic. For z > 1, the entire sample is isotropic. If the two phases had equal conductivities, Eq. 4.5a would simplify to z (T b T NI )/(T b T t ). (4.5d) Busse and Schubert [10] made explicit analytic predictions of the stability boundaries and the critical wavenumbers for the case R α = R β = 0, using free-slip boundary conditions at the top and bottom of the sample. From the preceding paragraphs it is clear that this is not a good approximation for 5CB. They also considered the cases R α R β = 0 and R β R α = 0 and provided some general discussions for arbitrary R α and R β. However, a specific prediction quantitatively applicable to the present case of interest is difficult to extract form this early theoretical work. The important qualitative result of the theory is that the more dense phase can be stably stratified above a less dense one if the temperature gradient is sufficiently large. As discussed in Sect. 1.1, the reason for this is that a steep thermal gradient suppresses interface fluctuations because the interface is constrained to be located at the position where the temperature equals the transition temperature. For the actual fluid properties of 5CB and for rigid no-slip boundaries, detailed numerical solutions of the equations of Busse and Schubert have been carried out recently by Tschammer et al. [71], and those results will be compared below with the experiment. To my knowledge, there have been no nonlinear calculations for twophase convection. General considerations suggest that hexagonal patterns should appear when z differs substantially from 1/2 because then the reflection symmetry about the sample midplane is broken by the existence of the interface. One would expect the flow direction in the hexagon centers to reverse as z is changed from z < 1/2 to z > 1/2. For z 1/2, the system retains the reflection symmetry if we neglect the difference in the properties of the two phases, and thus rolls would be expected at intermediate values of z. To a large extent this corresponds to the experimental findings to be discussed in the next section. One might hope that it could be captured to some extent in an amplitude-equation description where the coefficient of the symmetry-breaking (quadratic) term depends on z.

38 Nusselt Number Guenter Ahlers 4.2 Experimental results There have been very few laboratory investigations relevant to this interesting problem. Qualitative observations of hexagons and rolls in a nematicisotropic two-phase system were made by Salan and Guyon. [35] A more extensive investigation of the patterns which form as a consequence of this instability was conducted by Fitzjarrald. [34] Recent quantitative measurements of the bifurcation lines and of the pattern-formation phenomena were conducted in our laboratory. [14, 16, 71] Here we summarize the recent results Τ ( ο C ) Fig Nusselt number measurements for cell 3 with various interface positions z. The values of z are indicated near each data set. z < 0 (z > 1) corresponds to a single-phase nematic (isotropic) sample. After Ref. 16. As in the other investigations reported in this review, the measurements consisted of Nusselt-number determinations and of flow visualizations. In Ref. 14, cell 2 (d = cm) was used, and the temperature T t at the top of the sample was held fixed while the bottom temperature T b was increased. This work provided an important confirmation of the theoretical prediction that a denser phase could be stably stratified above a less dense one. It yielded information about the bifurcation lines and excellent pattern visualizations, but these were somewhat difficult to relate to theoretical predictions because z changed as T was changed. Using Eq. 4.5a, estimates of z have now been made and some of the results are shown in Fig. 4.3a below, but the results for the bifurcation lines probably are not as accurate as those in the second set of experiments. In the second generation of experiments, [16, 71] cell 3 (d = cm) was used. Both the top and the bottom temperature were controlled, and

39 Experiments on Thermally Driven Convection from point to point they were both varied in such a way as to hold z constant according to Eq. 4.5a. Figure 4.1 shows Nusselt-number measurements [16] for cell 3 and for several values of z. One can see clearly that there are two bifurcation points for a range of z. Specifically for z = 0.15, 0.22, 0.67, and 0.80 there is convection at small T, no convection (N = 1) at intermediate T, and convection again at large T. The instability at small T is attributable to the mechanism associated with the interface (see Sect. 1.1). At large T, the instability is caused predominantly by the Rayleigh-Bénard mechanism (of course in detail there is an interaction between these two). The region at intermediate values of T which corresponds to N = 1 is the re-entrant region where the dense phase is stably stratified above the less dense one. Over the range 0.22 < z < 0.68, this re-entrant range does not exist and convection driven by the interface instability evolves continuously into Rayleigh-Bénard convection. T c ( deg C ) 10 5 conduction convection conduction Interface Position z Fig The critical temperature difference T c as a function of the interface position z for cell 3. For z < 0 (z > 1) there is no interface, and the entire sample is nematic (isotropic). The dash-dotted lines in these single-phase regions are based on the properties at the mean sample temperature as given in Appendix B, and on R c = The dotted line is the theoretical estimate [10] of T c based on the approximation R α = R β = 0 which is not very good for 5CB. The open circles are numerical evaluations by Tschammer et al. [71] of the complete equations given in Ref. 10. The solid circles are the experimental results from Ref. 16. In Fig. 4.2 the bifurcation points [16] for cell 3 determined from Nusselt number measurements like those in Fig. 4.1 are shown as solid circles. The vertical dashed lines show the boundaries of the two-phase region. As already demonstrated for cell 2 by Fig. 2.3, the size and variation of T c in the single-phase regions z < 0 (nematic) and z > 1 (isotropic) can be explained in terms of the temperature-dependent fluid properties near the phase transition. The dash-dotted lines in these regions were obtained

40 Guenter Ahlers from the properties at the mean sample temperature as given in Appendix B, and by setting R c in Eq. 2.3 equal to 1708 as is appropriate for the isotropic phase as well as for the planar nematic sample in a large field. The agreement with the data in both phases is excellent, and well within what might have been expected on the basis of the uncertainties in the fluid properties. 6 T ( deg C ) 4 2 ( a ) T ( deg C ) Vertical Position z ( d ) ( b ) Interface Position z Fig A survey of the patterns which have been encountered. a) is for cell 2, and b) for cell 3. The plusses connected by solid lines are the measured bifurcation points. The vertical dashed lines are the boundaries of the two-phase region. The dash-dotted lines correspond to R c = 1708 in the single-phase regions. The types of patterns, with reference to examples shown in Fig. 4.4, are as follows. Solid squares: parallel rolls (4.4a). Open circles: white cellular flow (4.4b, 4.4c). Solid triangles: parallel rolls with defects (4.4d). Exes: oblique rolls with defects (4.4e, 4.4f, 4.4g). Open diamonds: Irregular circular patterns (4.4n, 4.4o, 4.4p, 4.4q). Open triangles: disordered rolls. Solid circles: black cellular flow (4.4h, 4.4i, 4.4k, 4.4l). Open squares: normal rolls (4.4m). After Ref. 16.

41 Experiments on Thermally Driven Convection Even though it is not really applicable to the experimental system, we show as the dotted line in the two-phase region (0 < z < 1) an evaluation of the prediction of Busse and Schubert for R α = R β = 0 and for free-slip boundary conditions. It is consistent with the data for z close to 0 and 1. As expected, it does not describe the physical system for intermediate values of z. An evaluation of the complete equations of Busse and Schubert, using the properties of 5CB given in Appendix B and rigid boundary conditions, was carried out by Tschammer et al. [71] Their results are shown by the open circles in the two-phase region. They agree rather well with the experiment and capture the main physical ingredient missed by the approximation R α = R β = 0, namely that the two-phase instability continuously evolves into the Rayleigh-Bénard instability as parameters are varied. a b c d e f g h i k l m n o p q Fig Examples of patterns encountered in cell 2. They are for the following values of T in K and z in units of d: a.) 5.27, 0.071; b.) 4.69, 0.149; c.) 0.92, 0.482; d.) 3.94, 0.228; e.) 4.61, 0.321; f.) 5.12, 0.380; g.) 1.62, 0.539; h.) 2.74, 0.600; i.) 4.92, 0.768; k.) 5.04, 0.841; l.) 5.99, 0.865; m.) 7.67, 0.894; n.) 0.99, 0.514; o.) 0.99, 0.514; p.) 0.97, 0.505; q.) 1.09, After Ref. 16.