Search for: All records

The flow of a thin viscous liquid layer under an elastic film arises in natural processes, such as magmatic intrusions between rock strata, and industrial applications, such as coating surfaces with cured polymeric films. We study the linear dynamics of small perturbations to the equilibrium state of the film in a closed trough (finite film). Specifically, we are interested in the spreading (early-time) and leveling (late-time) dynamics as the film adjusts to equilibrium, starting from different initial perturbations. We consider both smooth and non-smooth spatially symmetric and localized initial conditions (perturbations). We find the exact series solutions for the film height, using the sixth-order complete orthonormal eigenfunctions associated with the posed initial-boundary-value problem derived in our previous work [Papanicolaou N C and Christov I C 2023J. Phys.: Conf. Ser.2675012016]. We show that the evolution of the perturbations begins with the spreading of the localized perturbations, followed by their mutual interaction as they spread, and finally, interactions with the confining lateral boundaries of the domain as the perturbations level. In particular, we highlight how the leading eigenvalues of the problem determine the scalings of certain figures of merit with time. 

Sixth-order boundary value problems (BVPs) arise in thin-film flows with a surface that has elastic bending resistance. To solve such problems, we first derive a complete set of odd and even orthonormal eigenfunctions — resembling trigonometric sines and cosines, as well as the so-called “beam” functions. These functions intrinsically satisfy boundary conditions (BCs) of relevance to thin-film flows, since they are the solutions of a self-adjoint sixth-order Sturm–Liouville BVP with the same BCs. Next, we propose a Galerkin spectral approach for sixth-order problems; namely the sought function as well as all its derivatives and terms appearing in the differential equation are expanded into an infinite series with respect to the derived complete orthonormal (CON) set of eigenfunctions. The unknown coefficients in the series expansion are determined by solving the algebraic system derived by taking successive inner products with each member of the CON set of eigenfunctions. The proposed method and its convergence are demonstrated by solving two model sixth-order BVPs.