Non-self-adjoint sixth-order eigenvalue problems arising from clamped elastic thin films on closed domains

TL;DR

The paper addresses a non-self-adjoint sixth-order initial-boundary-value problem arising from bending-dominated infinitesimal deformations of a clamped elastic thin film on a closed domain. It develops a biorthogonal framework by constructing both the EVP and its adjoint, enabling expansions in two biorthogonal eigenfunction sets. A Petrov–Galerkin spectral method is formulated around these bases and shown to converge rapidly on two manufactured-model problems with exact solutions, surpassing the expected sixth-order rate. The results validate the theoretical predictions from Birkhoff–type theory for higher-order non-self-adjoint problems and highlight the method’s robustness for complex boundary-value problems in interfacial hydrodynamics.

Abstract

Sixth-order boundary value problems (BVPs) arise in thin-film flows with a surface that has elastic bending resistance. We consider the case in which the elastic interface is clamped at the lateral walls of a closed trough and thus encloses a finite amount of fluid. For a slender film undergoing infinitesimal deformations, the displacement of the elastic surface from its initial equilibrium position obeys a sixth-order (in space) initial boundary value problem (IBVP). To solve this IBVP, we construct a set of odd and even eigenfunctions that intrinsically satisfy the boundary conditions (BCs) of the original IBVP. These eigenfunctions are the solutions of a non-self-adjoint sixth-order eigenvalue problem (EVP). To use the eigenfunctions for series expansions, we also construct and solve the adjoint EVP, leading to another set of even and odd eigenfunctions, which are orthogonal to the original set (biorthogonal). The eigenvalues of the adjoint EVP are the same as those of the original EVP, and we find accurate asymptotic formulas for them. Next, employing the biorthogonal sets of eigenfunctions, a Petrov--Galerkin spectral method for sixth-order problems is proposed, which can also handle lower-order terms in the IBVP. The proposed method is tested on two model sixth-order BVPs, which admit exact solutions. We explicitly derive all the necessary formulas for expanding the quantities that appear in the model problems into the set(s) of eigenfunctions. For both model problems, we find that the approximate Petrov--Galerkin spectral solution is in excellent agreement with the exact solution. The convergence rate of the spectral series is rapid, exceeding the expected sixth-order algebraic rate.

Figures (10)

• Figure 1: Schematic illustration of the problem of a clamped elastic thin film on a closed domain. The elastic interface (with only out-of-plane bending rigidity and negligible mass) sits atop a viscous fluid of equilibrium height $h_0$ over a closed trough (no fluid flux through the lateral boundaries $x=\pm\ell$). Gravity is oriented in the $-y$ direction. When the interface is perturbed by an infinitesimal (dimensionless) displacement $u(x,t)$, the competition between the flow generated underneath it, its resistance to bending, and gravity sets the dynamics of leveling back to equilibrium, $u\to0$.
• Figure 2:
• Figure 3: The profiles for $m=1,2,3,4$ of the (a) even, and (b) odd eigenfunctions of the adjoint EVP \ref{['eq:6th_EVP_adj']} associated with \ref{['eq:6th_EVP']}. The even eigenfunction $\phi_0^c(x)=1$, which corresponds to $\mu_0^c=0$, for $m=0$ is shown in (a) along with the even ones.
• Figure 4: The spectral expansion of ${\text{d}^2\psi_5^{c}}/{\text{d} x^2} \equiv (\psi^c_5)"(x)$ back into the even eigenfunctions \ref{['eq:efuncs_6_015_even']}, with $M=100$ terms. (a) The profile of the approximation compared to that of the exact representation of $(\psi^c_5)"(x)$. (b) The convergence rate of the absolute value of the spectral coefficients $|\beta_{5m}^c/c_m|$ alongside the asymptotic best fit $400m^{-0.8}$ (for $m>20$).
• Figure 5: The spectral expansion of $x^7$ into the odd eigenfunctions \ref{['eq:efuncs_6_015_odd']}, with $M=100$ terms. (a) The spectral approximation compared to $x^7$ itself. (b) Convergence rate of the absolute value of the spectral coefficients ${|\chi^{\{7\}}_m}/{s_m|}$ alongside the asymptotic best fit $1.85m^{-1}$ (for $m>20$).

Theorems & Definitions (9)

• Definition 3.1
• Definition 3.2
• Proposition 3.1
• Proposition 3.2
• Proposition 3.3
• Proposition 3.4
• proof
• Proposition 3.5
• Proposition 3.6