Direct numerical simulations of droplet emulsions in sliding bi-periodic frames using the level-set method

Numerical Simulations of the Impact and Spreading of a Particulate Drop on a Solid Substrate

“…Jeong et al [1] presented the weak form for capillary spreading of a particle-laden droplet using (i) the standard velocity-pressure formulation for the Stokes flow, (ii) the combined weak formulation of Glowinski et al [14] for implicit treatment of freely suspended particles, and (iii) the level-set method with the CSS formulation [17] for the interface description between droplet and fluid. In the present work, an additional treatment is necessary to incorporate the inertia and convective terms in the Navier-Stokes equation for the description of impact and subsequent spreading behaviors.…”

“…Although (15) is expressed in terms of the explicit Euler method, what is actually employed is the TVD-RK3 (total variance diminishing 3rd order Runge Kutta) method by applying this explicit Euler method three times for the single time step advance [17,21].…”

“…As the level-set function evolves over time, the distance function property of (4) is no longer satisfied and the level-set function must therefore be reinitialized to restore the distance function property, because the accuracy of the distance function is crucial for the evaluation of the interfacial stress τ s . In this work, we reinitialize the level-set function by a direct geometrical redistancing method, following the work of Kim and Hwang [17]. We search for line segments along the interface on the subdivided element boundary and introduce a separate function, the distance function, whose function value is the shortest distance from any nodal position to the line segments along the interface.…”

“…We search for line segments along the interface on the subdivided element boundary and introduce a separate function, the distance function, whose function value is the shortest distance from any nodal position to the line segments along the interface. Details of the geometric construction of the distance function can be found in Kim and Hwang [17]. For the discretization of the weak form, we use a uniform regular quadrilateral element with the continuous biquadratic interpolation for the velocity u, a discontinuous linear interpolation for the pressure p, a discontinuous biquadratic interpolation for the level-set function ϕ, and the collocation point method the Lagrangian multipliers for the rigid-ring constraint.…”

“…If the level-set function is seriously distorted as time evolves, the level-set function must be redistributed to satisfy the distance function property (4) and we employ a direct geometric redistancing scheme developed by Kim and Hwang [17] for this purpose. This level-set description allows a simple treatment of the interfacial condition (3) by introducing the smeared-out interfacial stress τ s with the delta function δ(ϕ) based on the level-set function, which is called the continuous surface stress (CSS) formulation [18]:…”