Analysis and numerical analysis of the Helmholtz-Korteweg equation

TL;DR

The paper analyzes the nematic Helmholtz–Korteweg equation, a time-harmonic model for wave propagation in nematic calamitic fluids with anisotropic Korteweg stresses, proving well-posedness for nonresonant wave numbers and developing a convergent $H^2$-conforming finite element discretization enforced by Nitsche boundary conditions. The authors establish a robust operator-theoretic framework based on weak T-coercivity, derive a discrete eigenproblem to control boundary terms, and prove a best-approximation error bound for the discrete solution under a small-$\beta$ regime, with a mesh-threshold condition tied to $\lambda_h^{(i_*)} < k^2$. Numerical experiments in 2D demonstrate quasi-optimal convergence, anisotropic wave propagation due to nematic ordering, and the potential for tunable acoustic resonators by manipulating the director field $\mathbf{n}$. These results provide a rigorous foundation for simulating and designing nematic-based acoustic devices, including validating key experimental observations such as anisotropic propagation speeds. The work advances both the mathematical theory and numerical methodology for anisotropic Korteweg-type fluids and paves the way for computational design of tunable acoustic components using nematic materials.

Abstract

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.

Figures (5)

• Figure 1: The convergence of the $H^2$-norm of the error for the nematic Helmholtz--Korteweg equation for different values of $k$ (top row) and the corresponding manufactured solution (bottom row),where we display the real part of the condensation wave $u(\mathbf{x})$, i.e. $p(\mathbf{x},t) = \rho_0\left(1+\Re\{u(\mathbf{x}) e^{-i \omega t}\}\right)$, with $\rho(\mathbf{x},t)$ being the density at position $\mathbf{x}$ and time $t$ and $\rho_0$ the mean density of the fluid at rest.
• Figure 2: The propagation of a symmetric Gaussian pulse by the nematic Helmholtz--Korteweg equation with sound soft boundary conditions when $\beta = 0$ (a) and when $\beta=1\cdot 10^{-2}$ and $\mathbf{n}$ is aligned with the $x$-axis (b), the diagonal (c) and the $y$-axis (d). The parameters here are $k=40$, $\alpha=10^{-2}$.
• Figure 3: The propagation of a symmetric Gaussian pulse by the nematic Helmholtz--Korteweg equation with impedance boundary conditions when $\beta = 0$ (a) and when $\beta=1\cdot 10^{-2}$ and $\mathbf{n}$ is aligned with the $x$-axis (b), the diagonal (c) and the $y$-axis (d). The parameters here are $k=40$, $\alpha=10^{-2}$.
• Figure 4: The anisotropic propagation of a planar acoustic wave in the modified Mullen--Lüthi--Stephen experiment (top) and the corresponding nematic director $\mathbf{n}$ (bottom). The parameters are $k=40$, $\alpha=10^{-4}$ and $\beta= 10^{-4}$.
• Figure 5: (a) The eigenmode associated with the discrete eigenvalue \ref{['eq:numex:lambda']} for $\mathbf{n} = (1,0)$. (b) The scattered wave generated by incoming plane wave $\exp\left(i \mathbf{d}\cdot \mathbf{x}\right)$ with $\mathbf{d}=(0,\sqrt{Re(\lambda_h)})$ for a fixed nematic director $\mathbf{n} = (1,0)$ (c) The scattered wave generated by the same incoming plane for a fixed nematic director $\mathbf{n} = (0,1)$ respectively. Notice the difference in scale: the middle figure has scale $10^7$ times larger.

Theorems & Definitions (35)

• Definition 2.1
• Definition 2.2: weak T-coercivity
• Theorem 3.1
• proof
• Lemma 3.2
• proof
• Remark 3.3: Smallness assumption on $\beta$
• Theorem 3.4
• proof
• Lemma 3.5