Lattice-Boltzmann Methodology.

We utilize a lattice-Boltzmann (LB) method^18-22 to numerically simulate the capillary instability of a confined thread. LB is a computationally efficient approach for modeling multicomponent fluid systems. The approach is to consider a typical volume element of fluid to be composed of a collection of particles that are represented in terms of a particle velocity distribution function at each point in space where fluid particles collide with each other as they move under applied forces. Macroscopic variables such as density and velocity are obtained by taking appropriate moments of the velocity distribution function. It has been shown that this formalism leads to a velocity and density field that is a solution of both the Navier-Stokes and continuity equations. The specific multicomponent LB model utilized for this paper is based on the Shan-Chen approach²² with further modifications found in Martys and Douglas.¹⁸

For the present work, we utilize a parallel plate geometry having periodic boundary conditions. A cylindrical thread is introduced, centered between the parallel plates such that its z axis is parallel to the plates. The cylindrical thread is embedded in a second fluid. No-slip boundary conditions are maintained at the fluid/wall interface by using a second-order bounce-back algorithm. To introduce a perturbation to the thread, a small localized body force is applied near the fluid/fluid interface for a few lattice Boltzmann time steps. The widths of the thread in the perpendicular (y) and parallel (x) directions are determined as a function of axial position and time.

Two viscosity ratios were considered. First, we consider the case p = 1 (viscous matched) because for computational reasons related to the LB method, significant computer memory can be saved, allowing for much larger simulations than for p 1. Here, the system dimension is 800 (in units of lattice spacing) in the z direction and 740 in x. The gap spacing, H, for the three simulations is such that H/D₀ is equal to 2.1, 1.45, and then 1.25 for the three simulations, where D₀ = 160. This system size was required in order to have an adequate numerical resolution for the case of H/D₀ = 1.25. In a second set of runs we used the same viscosity ratio as the experiments p = 0.25 with the following system dimensions: 180 units in z, 88 in x and H, such that H/D₀ = 2.1 and 1.4 where D₀ = 40. Time is given in the dimensionless quantity qt.

One important difference between the experiment and the simulation concerns the wavelength of the instability. In the experiment, the thread is essentially free to find the fastest growing wavelength, whereas in the simulation the length of the thread is finite (for reasons of computational resources). The wavelength of the distortion can only take values that are L/n, where L is the length of the simulation box in the z direction and n is an integer. For the case of n = 1, then 2R₀/ = 0.7, which is somewhat larger than what one obtains by considering the fastest growing mode 2R₀/ = 0.59 at p = 1. This discrepancy means that comparisons are qualitative in nature.