Flagellated bacteria such as Escherichia coli and Bacillus subtilis exhibit effective mechanisms for swimming in fluids and exploring the surrounding environment. In isotropic fluids such as water, the bacteria change swimming direction through the run-and-tumble process. Lyotropic chromonic liquid crystals (LCLCs) have been introduced recently as an anisotropic environment in which the direction of preferred orientation, the director, guides the bacterial trajectories. In this work, we describe the behavior of bacteria B. subtilis in a homeotropic LCLC geometry, in which the director is perpendicular to the bounding plates of a shallow cell. We demonstrate that the bacteria are capable of overcoming the stabilizing elastic forces of the LCLC and swim perpendicularly to the imposed director (and parallel to the bounding plates). The effect is explained by a finite surface anchoring of the director at the bacterial body; the role of surface anchoring is analyzed by numerical simulations of a rod realigning in an otherwise uniform director field. Shear flows produced by a swimming bacterium cause director distortions around its body, as evidenced both by experiments and numerical simulations. These distortions contribute to a repulsive force that keeps the swimming bacterium at a distance of a few micrometers away from the bounding plates. The homeotropic alignment of the director imposes two different scenarios of bacterial tumbling: one with an 180°r eversal of the horizontal velocity and the other with the realignment of the bacterium by two consecutive 90°turns. In the second case, the angle between the bacterial body and the imposed director changes from 90°to 0°and then back to 90°; the new direction of swimming does not correlate with the previous swimming direction. with a relatively flat rigid polyaromatic core and polar groups at the periphery [11][12][13]. In water, these molecules aggregate face-to-face in order to minimize the areas of unfavorable contact with water. Unlike their surfactantbased micellar and thermotropic counterparts, the LCLCs are not toxic to biological organisms [14].Recent experiments demonstrate that the prevailing direction of swimming is parallel to the directorn, i.e. to the average direction of LCLC orientation ( º -n n, = |ˆ| n 1) [7,8]. The orientational order of the LCLC environment can be controlled by temperature, the concentration of liquid crystal organic molecules, external electromagnetic fields and by surface alignment of the director [9,11,12]. The dispersion of swimming bacteria in LCLC, also called a living liquid crystal [8], offers new opportunities to control the dynamic behavior of the bacteria.The studies of swimming bacteria in LCLCs have been performed mostly for sandwich-type cells, in which the LCLC is confined between two glass plates, with the director being uniformly aligned along a certain direction in the plane of the cell (planar alignment). It has been shown that rod-like flagellated bacteria prefer to swim along the director [7,8,15,16]. It is ass...

We use the method of Γ-convergence to study the behavior of the Landau-1 de Gennes model for a nematic liquid crystalline film in the limit of vanishing thickness. In this asymptotic regime, surface energy plays a greater role and we take particular care in understanding its influence on the structure of the minimizers of the derived two-dimensional energy. We assume general weak anchoring conditions on the top and the bottom surfaces of the film and the strong Dirichlet boundary conditions on the lateral boundary of the film. The constants in the weak anchoring conditions are chosen so as to enforce that a surface-energy-minimizing nematic Q-tensor has the normal to the film as one of its eigenvectors. We establish a general convergence result and then discuss the limiting problem in several parameter regimes.

We study tensor-valued minimizers of the Landau-de Gennes energy functional on a simply-connected planar domain Ω with noncontractible boundary data. Here the tensorial field represents the second moment of a local orientational distribution of rod-like molecules of a nematic liquid crystal. Under the assumption that the energy depends on a single parameter-a dimensionless elastic constant ε > 0-we establish that, as ε → 0, the minimizers converge to a projectionvalued map that minimizes the Dirichlet integral away from a single point in Ω. We also provide a description of the limiting map.

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We use a variational principle to derive a mathematical model for a nematic electrolyte in which the liquid crystalline component is described in terms of a second-rank order parameter tensor. The model extends the previously developed director-based theory and accounts for presence of disclinations and possible biaxiality. We verify the model by considering a simple but illustrative example of liquid crystal-enabled electro-osmotic flow around a stationary dielectric spherical particle placed at the center of a large cylindrical container filled with a nematic electrolyte. Assuming homeotropic anchoring of the nematic on the surface of the particle and uniform distribution of the director on the surface of the container, we consider two configurations with a disclination equatorial ring and with a hyperbolic hedgehog, respectively. The computed electro-osmotic flows show a strong dependence on the director configurations and on the anisotropies of dielectric permittivity and electric conductivity of the nematic, characteristic of liquid crystal-enabled electrokinetics. Further, the simulations demonstrate space charge separation around the dielectric sphere, even in the case of isotropic permittivity and conductivity. This is in agreement with the induced-charge electro-osmotic effect that occurs in an isotropic electrolyte when an applied field acts on the ionic charge it induces near a polarizable surface.

We carry out an asymptotic analysis of a variational problem relevant in the studies of nematic liquid crystalline films when one elastic constant dominates over the others, namelyHere u : Ω → R 2 is a vector field, 0 < ε 1 is a small parameter, and L > 0 is a fixed constant, independent of ε. We identify a candidate for the Γ-limit E 0 , which is a sum of a bulk term penalizing divergence and an Aviles-Giga type wall energy involving the cube of the jump in the tangential component of the S 1 -valued nematic director. We establish the lower bound and provide the recovery sequence for this candidate within a restricted class. Then we consider a set of variational problems for E 0 arising for a choice of domain geometries and boundary conditions. We demonstrate that the criticality conditions for E 0 can be expressed as a pair of scalar conservation laws that share characteristics. We use the method of characteristics to analytically construct critical points of E 0 that we observe numerically.

We use the method of Γ-convergence to study the behavior of the Landau-de Gennes model for a nematic liquid crystalline film attached to a general fixed surface in the limit of vanishing thickness. This paper generalizes the approach in [1] where we considered a similar problem for a planar surface. Since the anchoring energy dominates when the thickness of the film is small, it is essential to understand its influence on the structure of the minimizers of the limiting energy. In particular, the anchoring energy dictates the class of admissible 1

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