Analysis and numerical analysis of the Helmholtz-Korteweg equation

We analyse the nematic Helmholtz-Korteweg equation, a variant of the classical Helmholtz equation that describes time-harmonic wave propagation in calamitic fluids in the presence of nematic order. A prominent example is given by nematic liquid crystals, which can be modeled as nematic Korteweg fluids - that is, fluids whose stress tensor depends on density gradients and on a nematic director describing the orientation of the anisotropic molecules. These materials exhibit anisotropic acoustic properties that can be tuned by external electromagnetic fields, making them attractive for potential applications such as tunable acoustic resonators. We prove the existence and uniqueness of solutions to this equation in two and three dimensions for suitable (nonresonant) wave numbers and propose a convergent discretisation for its numerical solution. The discretisation of this problem is nontrivial as it demands high regularity and involves unfamiliar boundary conditions; we address these challenges by using high-order conforming finite elements and enforcing the boundary conditions with Nitsche's method. We illustrate our analysis with numerical simulations in two dimensions.