This work introduces a thermodynamically-consistent theoretical framework for lyotropic liquid crystals (LCs) that is based on the GENERIC (General Equation for the Non-Equilibrium Reversible-Irreversible Coupling) formalism. This formalism ensures conservation of energy and production of entropy while coupling concentration, momentum balance, and liquid crystalline order. Starting from a binary nematic–isotropic mixture, a theory is first derived for these key variables, and then extended to multi-component systems. The binary equations are solved numerically using a Julia-based solver that relies on an upwind finite-difference scheme, thereby enabling stable and efficient simulations that are capable of handling multiple time scales while satisfying fundamental mathematical constraints. The results of simulations are shown to be consistent with experimental observations of topological core defects in chromonic LCs, as well as flow-driven droplet shape transformations under Couette and Poiseuille flows. The resulting platform can be applied to simulations of multi-component lyotropic LCs and can be extended to systems with multiple interfaces, active and autonomous materials, and materials subject to external fields.