A geometric diffuse-interface method for droplet spreading

Abstract

This paper exploits the theory of geometric gradient flows to introduce an alternative regularization of the thin-film equation valid in the case of large-scale droplet spreading-the geometric diffuse-interface method. The method possesses some advantages when compared with the existing models of droplet spreading, namely the slip model, the precursor-film method and the diffuse-interface model. These advantages are discussed and a case is made for using the geometric diffuse-interface method for the purpose of numerical simulations. The mathematical solutions of the geometric diffuse interface method are explored via such numerical simulations for the simple and well-studied case of large-scale droplet spreading for a perfectly wetting fluid-we demonstrate that the new method reproduces Tanner's Law of droplet spreading via a simple and robust computational method, at a low computational cost. We discuss potential avenues for extending the method beyond the simple case of perfectly wetting fluids.

Keywords: contact-line flows; diffuse-interface method; geometric mechanics.

Figure 1.

Schematic description of the fluid mechanical problem of droplet spreading, as derived from the Navier–Stokes equations in the lubrication limit. (Online version in colour.)

Figure 2.

Schematic description of the fluid mechanical problem of droplet spreading, showing the inner and outer regions of the problem. (Online version in colour.)

Figure 3.

Validation of the numerical method (3.4). Model parameters: α = 0.2, L = 2π, ϵ = 10⁻³. Simulation parameters: Δt = 10⁻³, N = 300. (Online version in colour.)

Figure 4.

(a) Space–time diagram showing the evolution of the diffuse surface height $\overline{h}(x,t)$. (b) Snapshot of the free-surface height $\overline{h}(x,t)$ at t = 50. The snapshot also shows the location of the macroscopic contract line x_m. Model parameter: α = 0.05. Numerical parameters: L = 2π, N = 500 gridpoints, Δt = 10⁻². (Online version in colour.)

Figure 5.

Contact-line evolution based on the numerical simulation, showing a power-law behaviour at late times x_m(t) ∼ t^p, with p = 0.135. (Online version in colour.)

Figure 6.

Space–time diagram in similarity variables showing the evolution of the diffuse surface height $t^{-1/7}\overline{h}$. (Online version in colour.)

Figure 7.

Comparison of the solution of the regularized problem (3.3) in similarity variables at t = 50 with the numerical solution of the unregularized problem f²f″′ = ηf/7. Unadorned solid line: regularized problem. Line with circles: unregularized problem. (Online version in colour.)

Figure 8.

Plot of $\overline{h}(x,t=50)$ showing the spatial structure of the solution in the tail, for |x| ≫ x_m. (Online version in colour.)

Figure 9.

Snapshot of the sharp free-surface height h(x, t) at t = 50. Numerical parameters as before. (Online version in colour.)

Figure 10.

Convergence study: effect of varying N at fixed Δt = 10⁻². The snapshots in (a) are taken at t = 100. (Online version in colour.)

Figure 11.

Convergence study: effect of varying Δt on the plot of the macroscopic contact line position x_m(t). (Online version in colour.)

Figure 12.

Sensitivity analysis. (a) Effect on results of varying α, with $K={(1-{\alpha }^2{\mathrm{\partial }}_x^2)}^{-1}$ (b) Effect on results of varying the kernel function, at fixed α = 0.05. In each case, L = 2π, N = 500 and Δt = 10⁻². (Online version in colour.)