Sixth-order parabolic equation on an interval: Eigenfunction expansion, Green's function, and intermediate asymptotics for a finite thin film with elastic resistance

Abstract

A linear sixth-order partial differential equation (PDE) of ``parabolic'' type describes the dynamics of thin liquid films beneath surfaces with elastic bending resistance when deflections from the equilibrium film height are small. On a finite domain, the associated sixth-order eigenvalue problem is self-adjoint for the boundary conditions corresponding to a thin film in a closed trough, and the eigenfunctions form a complete orthonormal set. Using these eigenfunctions, we derive the Green's function for the governing sixth-order PDE on a finite interval and compare it to the known infinite-line solution. Further, we propose a Galerkin spectral method based on the constructed sixth-order eigenfunctions and their derivative expansions. The system of ordinary differential equations for the time-dependent expansion coefficients is solved by standard numerical methods. The numerical approach is applied to versions of the governing PDE with a second-order spatial derivative (in addition to the sixth-order one), which arises from gravity acting on the film. In the absence of gravity, we demonstrate the self-similar intermediate asymptotics of initially localized disturbances on the film surface, at least until the disturbances ``feel'' the finite boundaries, and show that the derived Green's function is an attractor for such solutions. In the presence of gravity, we use the proposed Galerkin numerical method to demonstrate that self-similar behavior persists, albeit for shortened intervals of time, even for large values of the gravity-to-bending ratio.\\[1mm]

Figures (7)

• Figure 1: Schematic of the thin film with elastic resistance being considered. The interface has out-of-plane bending rigidity $B$, negligible in-plane tension, and negligible mass. The viscous fluid beneath the interface has density $\rho_f$ and dynamic viscosity $\mu_f$. The equilibrium height of the film (when the interface is flat) is $h_0$, and the dimensionless small vertical displacement of the film about this level is denoted as $u(x,t)$. The thin film is restricted to a finite domain (closed trough) of axial distance $2\ell$. The gravitational acceleration is $g$ in the $-y$ direction. Figure reproduced and adapted from NectarIvan2023 under the Creative Commons Attribution 3.0 license.
• Figure 2: (a) Time-evolution of the solution $u(x,t)$ of IBVP \ref{['eq:IBVP_6']}, given by Eqs. \ref{['eq:Green_2D_interval_convl']} and \ref{['eq:delta_exp']}, with $u^0(x) = \delta(x)$ for $\text{Bo}=0$. (b) The profiles from (a) under the self-similar rescaling, i.e., as $t^{1/6} u(\zeta)$ vs. $\zeta = x/t^{1/6}$. The thick dashed black curve indicated with $\mathcal{G}$ is the self-similar attractor \ref{['eq:attractor_delta']} found via the infinite-line Green's function. (c,d) The equivalent of (b) but for the numerical solution $u(x,t)$ of IBVP \ref{['eq:IBVP_6']} with $u^0(x) = \delta(x)$ for $\text{Bo}=1000$ and $3000$, respectively. For all plots, the spectral series was truncated at $M=50$ (a total of 51 terms).
• Figure 3: Snapshots of solutions $u(x,t)$ of IBVP \ref{['eq:IBVP_6']} with $u^0(x) = \delta(x)$ for different $\text{Bo}$ and (a) $t=10^{-8}$ and (b) $t=10^{-5}$. The green (bright) and red (dark) solid lines are the solutions for $\text{Bo}=3000$ and $\text{Bo}=1000$, respectively, obtained numerically from the Galerkin expansion \ref{['eq:u_x_expansion_formula']}, whereas the black dashed curve is the case $\text{Bo}=0$ obtained from the exact series solution \ref{['eq:Green_2D_interval_convl']} with coefficients \ref{['eq:delta_exp']}. For all presented cases, the spectral series was truncated at $M=50$.
• Figure 4: (a) Time-evolution of the solution $u(x,t)$, given by Eqs. \ref{['eq:Green_2D_interval_convl']} and \ref{['eq:odd_step_defn']}, for $u^0(x) = H_\mathrm{odd}(x)$ with $\text{Bo}=0$. (b) The profiles from (a) under the self-similar rescaling $u \mapsto u(\zeta)$, $\zeta = x/t^{1/6}$. The thick dashed black curve indicated with $\mathcal{H}$ is the self-similar attractor \ref{['eq:attractor_2D_line_odd_step']} obtained using the convolution formula \ref{['eq:Green_2D_line_convl']} based on the inifinte-line Green function. (c,d) The equivalent of (b), but for the numerical solution $u(x,t)$ of IBVP \ref{['eq:IBVP_6']} with $u^0(x) = H_\mathrm{odd}(x)$ for $\text{Bo}=1000$ and $3000$, respectively.
• Figure 5: (a) Dominant eigenvalue $r_1^c$ (symbols) of the matrix $\bm{\mathrm{A}^c}$ as a function of $\text{Bo}$ and the best-fit line $r^c_1(\text{Bo}) = -(\lambda_{1}^c)^6 - 10.043915648\,\text{Bo}$, where $\lambda_{1}^c = 3.66606496814$NectarIvan2023. (b) Dominant eigenvalue $r_1^s$ (symbols) of the matrix $\bm{\mathrm{A}^s}$ as a function of $\text{Bo}$ and the best-fit line $r^s_1(\text{Bo}) = -(\lambda_{1}^s)^6 - 2.483518333\,\text{Bo}$, where $\lambda_{1}^s = 2.07175679767$NectarIvan2023. For both panels, the spectral series was truncated at $M=50$.