We report experimental results for convection near onset in a thin layer of a homeotropically aligned nematic liquid crystal heated from below as a function of the temperature difference $\Delta T$ and the applied vertical magnetic field $H.\mathrm{}$ When possible, these results are compared with theoretical calculations. The experiments were done with three cylindrical cells of aspect ratios [(radius)/(height)] $\Gamma =10.6,$ 6.2, and 5.0 over the field range $8\lesssim h\equiv {H/H}_F\lesssim 80\hspace{0.5em}{(H}_F=20.9,$ 12.6, and 9.3 G are the Fréedericksz fields for the three cells). We used the Nusselt number $\mathcal{N}$ (effective thermal conductivity) to determine the critical Rayleigh number $R_c$ and the nature of the transition. We analyzed digital images of the flow patterns to study the dynamics and to determine the mean wave numbers of the convecting states. For h less than a codimension-two field $h_{\mathrm{ct}}\simeq 46$ the bifurcation is subcritical and oscillatory, with traveling- and standing-wave transients. Beyond $h_{\mathrm{ct}}$ the bifurcation is stationary and subcritical until a tricritical field $h_t=57.2\mathrm{}$ is reached, beyond which it is supercritical. We analyzed the patterns to obtain the critical wave number $k_c$ and, for $h<\mathrm{}{h}_{\mathrm{ct}},$ the Hopf frequency ${\omega }_c.$ In the subcritical range we used the early transients towards the finite-amplitude state for this purpose. The bifurcation sequence as a function of h found in the experiment confirms the qualitative aspects of the theoretical predictions. Even quantitatively the measurements of $R_c,$ $k_c,$ and ${\Omega }_c$ are reproduced surprisingly well considering the complexity of the system. However, the value of $h_{\mathrm{ct}}$ is about 10% higher than the predicted value and the results for $k_c$ are systematically below the theory by about 2% at small h and by as much as 7% near $h_{\mathrm{ct}}.$ At $h_{\mathrm{ct}},$ $k_c$ is continuous within the experimental resolution whereas the theory indicates a 7% discontinuity. The theoretical tricritical field $h_t^{\mathrm{th}}=51\mathrm{}$ is somewhat below the experimental one. The fully developed flow above $R_c$ for $h<\mathrm{}{h}_{\mathrm{ct}}$ has a very slow chaotic time dependence that is unrelated to the Hopf frequency. For $h_{\mathrm{ct}}<h<\mathrm{}{h}_t$ the subcritical stationary bifurcation also leads to a chaotic state. The chaotic states persist upon reducing the Rayleigh number below $R_c,$ i.e., the bifurcation is hysteretic. Above the tricritical field $h_t,$ we find a bifurcation to a time independent pattern which within our resolution is nonhysteretic. However, in this field range, there is a secondary hysteretic bifurcation that again leads to a chaotic state observable even slightly below $R_c.$ We discuss the behavior of the system in the high-field limit, and show that at the largest experimental field values $R_c$ and $k_c$ are within 6% and 1%, respectively, of their values for an infinite field.

• Received 22 April 1998

DOI:https://doi.org/10.1103/PhysRevE.58.5885