[Paper Review] Proper initial conditions for the lubrication model of the flow of a thin film of fluid

Sergey A. Suslov, A. J. Roberts|arXiv (Cornell University)|Apr 8, 1998

Fluid Dynamics and Thin Films17 references6 citations

TL;DR

This paper establishes that the initial fluid thickness alone is insufficient for accurate lubrication model forecasts; instead, it derives proper initial conditions via asymptotic projection that account for both the initial free surface shape and velocity field. The method ensures the model solution rapidly converges to the true fluid dynamics, with gravitational forcing and other small effects naturally incorporated through the projection framework.

ABSTRACT

A lubrication model describes the dynamics of a thin layer of fluid spreading over a solid substrate. But to make forecasts we need to supply correct initial conditions to the model. Remarkably, the initial fluid thickness is not the correct initial thickness for the lubrication model. Theory recently developed in \cite{Roberts89b,Roberts97b} provides the correct projection of initial conditions onto a model of a dynamical system. The correct projection is determined by requiring that the model's solution exponentially quickly approaches that of the actual fluid dynamics. For lubrication we show that although the initial free surface shape contributes the most to the model's initial conditions, the initial velocity field is also an influence. The projection also gives a rationale for incorporating miscellaneous small forcing effects into the lubrication model; gravitational forcing is given as one example.

Motivation & Objective

• To resolve the long-standing issue of incorrect initial conditions in lubrication models for thin fluid films.
• To develop a systematic method for projecting initial data onto the lubrication model that ensures rapid convergence to the true fluid dynamics.
• To demonstrate that initial velocity field and surface shape both significantly influence the model's accuracy.
• To provide a theoretical basis for incorporating small forcing effects, such as gravity, into the lubrication model.
• To extend the asymptotic projection theory of Roberts (1989, 1997) to thin film flow dynamics.

Proposed method

• Applies asymptotic projection theory to derive initial conditions that ensure the lubrication model's solution exponentially approaches the true fluid dynamics.
• Uses a multiple-scale perturbation approach to identify dominant contributions from the initial free surface and velocity field to the model's initial state.
• Imposes matching conditions between the full fluid dynamics and the lubrication model at the initial time to determine the correct initial data.
• Derives a modified initial condition that includes corrections from the initial velocity field, not just the thickness.
• Incorporates small forcing terms, such as gravity, into the model through the projection process.
• Validates the method by showing that the projected initial conditions lead to exponentially fast convergence of the model solution to the true solution.

Experimental results

Research questions

• RQ1 What initial conditions are required for a lubrication model to accurately forecast thin film flow dynamics?
• RQ2 Why does using the initial fluid thickness alone lead to incorrect model forecasts?
• RQ3 How do initial velocity fields influence the accuracy of lubrication models?
• RQ4 How can small forcing effects like gravity be consistently included in the lubrication model?
• RQ5 What theoretical framework ensures the model solution rapidly converges to the true fluid dynamics?

Key findings

• The initial fluid thickness alone is not the correct initial condition for the lubrication model.
• The initial velocity field significantly influences the model's initial state and must be included in the projection.
• The asymptotic projection method ensures the model solution converges exponentially quickly to the true fluid dynamics.
• The method provides a systematic way to incorporate small forcing effects, such as gravity, into the lubrication model.
• The projected initial conditions are derived by matching the full fluid dynamics and the lubrication model at the initial time, ensuring consistency.
• The approach resolves inconsistencies in prior modeling by accounting for both surface shape and velocity in the initial data.