Controlling Active lyotropic Liquid Crystal (LC) experiments, typically an active nematic coexisting with an isotropic fluid and molecular motors providing activity, remains a challenge due to unclear relationships between their multiple component concentrations. This prompts questions about the general conditions and effective parameters transforming their chaotic dynamics into coherent motion. As a result, continuum theory and numerical simulations become essential tools for predicting and guiding such experiments.

In this thesis, we utilize the out-of-equilibrium GENERIC framework to derive a thermodynamically consistent and novel theory for lyotropic liquid crystals. As a first step, we focus on a two-component system, an LC and an isotropic fluid, treating the time-evolution equations as the sum of energetic and entropic contributions. This approach couples the momentum and energy balances with concentration dependency and the liquid crystalline ordering.

Next, we introduce LiquidCrystalGLBT.jl, an open-source Julia-based solver designed for managing these concentration-dependent nematohydrodynamic equations. This hybrid solver combines an upwind finite difference scheme with the Galerkin Lattice Boltzmann method. In our acronym, 'G' indicates the use of the out-of-equilibrium GENERIC framework to derive our equations, while 'LB' signifies the implementation of the Lattice Boltzmann Method through the Trixi.jl package, the 'T' in our name.

We showcase three different scenarios resulting from this methodology, each with increasing complexity and incorporating insights from its predecessor:

1) A study of a binary mixture in 2D at equilibrium, resembling chromonic LC data with +/- 1/2 topological defects. 2) The introduction of hydrodynamics by incorporating a controlled velocity profile (Couette or parabolic) and tracking the shape transition of an axial LC droplet immersed in an isotropic environment. 3) The addition of biochemical activity, resulting in turbulent simulations that flow naturally, similar to microtubule or actin systems.

Finally, we discuss ongoing efforts to extend our model into a multi-component system. This extension, to the best of our knowledge, has not been published before. This work concludes with the hope that the insights gained from the systematic study of the binary system will inform the application of the multi-component equations to more complex scenarios. Importantly, this theoretical study, though brief, would not have been possible without the foundational analysis of the binary system.

In short, this work addresses the challenge of achieving stable and efficient simulations for lyotropic liquid crystals, contributing significantly to our understanding of their dynamic behavior. Furthermore, this project has the potential to advance predictions of lyotropic LCs in 3D, potentially elucidating the functionality of tissues in a liquid crystalline state, guiding the artificial design of biological tissues, and paving the way for new LC applications in bioengineering, sensing, and other emerging fields.