[Paper Review] Asymptotic regimes in elastohydrodynamic and stochastic leveling on a viscous film

Christian Pedersen, John Niven+3 more|arXiv (Cornell University)|Feb 27, 2019

Fluid Dynamics and Thin Films47 references17 citations

TL;DR

This study investigates elastohydrodynamic and stochastic leveling of an elastic plate on a viscous film, identifying two distinct asymptotic regimes governed by the ratio of bump height to film thickness. Using scaling analysis, numerical simulations, and experiments, it reveals a crossover between t⁻¹/⁴ and t⁻¹ scaling laws in the relaxation dynamics, depending on whether the bump is much larger or smaller than the film thickness, with excellent agreement between theory and simulation in both elastic bending and thermal fluctuation regimes.

ABSTRACT

An elastic sheet that deforms near a solid substrate in a viscous fluid is a situation relevant to various dynamical processes in biology, geophysics and engineering. Here, we study the relaxation dynamics of an elastic plate resting on a thin viscous film that is supported by a solid substrate. By combining scaling analysis, numerical simulations and experiments, we identify asymptotic regimes for the elastohydrodynamic leveling of a surface perturbation of the form of a bump, when the flow is driven by either the elastic bending of the plate or thermal fluctuations. In both cases, two distinct regimes are identified when the bump height is either much larger or much smaller than the thickness of the pre-wetted viscous film. Our analysis reveals a distinct crossover between the similarity exponents with the ratio of the perturbation height to the film height.

Motivation & Objective

• To understand how the relaxation dynamics of a perturbed elastic plate on a viscous film depend on the geometric ratio of bump height to film thickness.
• To determine whether distinct asymptotic scaling regimes emerge under elastic bending forces versus thermal fluctuations.
• To establish a theoretical framework that predicts the crossover between these regimes using scaling analysis and lubrication theory.
• To validate the theoretical predictions through numerical simulations and experimental data.
• To quantify the role of thermal fluctuations in the stochastic leveling process and identify universal scaling behavior.

Proposed method

• Formulation of a lubrication-based model for thin viscous film flow beneath an elastic plate, incorporating bending forces and thermal noise.
• Numerical solution of the dimensionless evolution equation for the film thickness profile using finite difference methods with stochastic noise terms.
• Application of scaling analysis to derive asymptotic scaling laws for the time evolution of the bump height, assuming area conservation and noise scaling ∼(tx)⁻¹/².
• Derivation of a characteristic time scale τΓ = 6μhi³Ri³b/(kBTTAε⁴) that governs stochastic leveling dynamics.
• Use of dimensionless variables to collapse numerical data across different initial conditions, confirming self-similarity and universal scaling.
• Comparison of numerical results with theoretical predictions to validate the absence of adjustable parameters in the derived scaling law.

Experimental results

Research questions

• RQ1 How does the relaxation dynamics of an elastic plate on a viscous film change as the bump height varies relative to the film thickness?
• RQ2 What are the distinct asymptotic scaling regimes in elastohydrodynamic leveling, and how do they depend on the driving mechanism—elastic bending or thermal fluctuations?
• RQ3 Is there a universal crossover behavior between these regimes, and can it be described by a single theoretical framework?
• RQ4 How do thermal fluctuations influence the leveling dynamics, and what scaling laws govern the stochastic regime?
• RQ5 Can the theoretical scaling laws be validated quantitatively against numerical simulations and experimental data?

Key findings

• Two distinct asymptotic regimes are identified: one where the bump height is much larger than the film thickness, leading to t⁻¹/⁴ scaling; and another where the bump is much smaller, leading to t⁻¹ scaling.
• The crossover between these regimes occurs when the average bump height is approximately equal to the film thickness, i.e., ⟨h₀(t)⟩/ε ≈ 1.
• The theoretical scaling law ⟨h₀(t)⟩/ε [1 + ⟨h₀(t)⟩/ε]³ ∼ τΓ/t accurately predicts the time evolution of the bump height across all regimes with no adjustable parameters.
• Numerical simulations show excellent agreement with the theoretical prediction, confirming the validity of the derived scaling law for both elastic bending and stochastic dynamics.
• Rescaling of the bump height profiles in each regime collapses them onto a universal shape, confirming self-similarity in the leveling process.
• The prefactor in the scaling law is found to be close to unity, indicating strong quantitative agreement between theory and simulation.