1. Direct numerical simulations of droplet emulsions in sliding bi-periodic frames using the level-set method See Jo Kim(sjkim1@andong.ac.kr), Wook Ryol Hwang* School of Mechanical Engineering, Andong National University * School of Mechanical and Aerospace Engineering, Gyeongsang National University

2. Objective Rheology and flow-induced microstructural development in droplet emulsions in viscoelastic fluids by direct numerical simulations  A large number of small drops suspended freely in a viscoelastic fluid.  Fully coupled viscoelastic flow simulation with drops under sliding bi-periodic flows.  A well-defined sliding bi-periodic domain concept with drops is necessary.

3.  2D, Circular disk-like drops, negligible inertia.  Inertialess drops in viscoelastic fluids in a sliding bi-periodic frame under simple shear.  Sliding bi-periodic frame of simple shear flow

4. This problem represents a regular configuration of an infinite number of such a configuration in the unbounded domain Question 1: How to find INTERFACES ? Question 2: How to apply INTERFACIAL TENSION ?

5. Question 1: How to find INTERFACES ? Interface Tracking – Mesh Moves with Interface: Deformation characteristics of spherical bubble collapse in Newtonian fluids near the wall using the Finite Element Method with ALE formulation

6.  Bowyer-Waston Algorithm Andong National University Advanced Material Processing Lab.

7. Andong National University Advanced Material Processing Lab.

8. Node number : 437, element number : 814. Boundary Mark Andong National University Advanced Material Processing Lab.

9.  Mesh Generation for Two-Phase Fluid Systems  Graphic Display by OpenGL (a) Show Number of Node (b) Show Number of Element(c) Show Number of Material Andong National University Advanced Material Processing Lab.

10. Normal Stress Balance : Shear Stress Balance : Local Mean Curvature: Interfacial Boundary Conditions by Interface Tracking N T R Liquid Droplet

11. Question 1: How to find INTERFACES ? Interface Capturing – Fixed Meshes across Interface: VOF: Level Set Method: Diffuse Interface:

12. Interface capturing based on a fixed mesh. Evolution Equation of the interface in terms of Level Set Function. Interface Capturing by Level Set Method.

14. Continuum Surface Stress (CSS):  Interfacial tension is treated as a body force  Interfacial tension is treated as an additional stress Continuum Surface Force (CSF): Interfacial Boundary Conditions by Interface Capturing

15. Governing Equations Computational Domain (Oldroyd-B) B.C. on computational domain Γ : (Sliding bi-periodic frame constraints)

16. Finite Element Formulation Modification of combined weak formulation of Glowinski et al for right-ring description of particles and sliding bi-periodic frame constraints 1.Both fluid and particle domains are described by the fluid problem; 2. Force-free, torque-free, rigid-body motion is satisfied weakly with the constraint on the particle boundary only; 3.Sliding bi-periodicity is applied weakly through the constraints of the sliding bi-periodic frame; 4.The weak form has been coupled with the DEVSS/DG scheme to solve emulsions in a viscoelastic fluid. 5.The weak form has been coupled with the DG scheme to solve the Level set function.

19. A single particle of radius r=0.2 in a sliding bi-periodic frame of size 1 x 1 in a Newtonian fluid with Regular configuration of an infinite number of drops of the same size in an unbounded domain Drops do not translate, but rotate with deformation. Good example for study of rheology of emulsion. A Single Drop in Newtonian Fluid

20. The pressure contour and streamline Convergence to steady shape of deformed drop

21. Distance function and drop deformation

22. Time-dependent bulk suspension properties Convergence to steady oscillation  bulk normal stress is zero for Particle-Newtonian medium system  bulk normal stress is not zero for Drop-Newtonian medium system  possibility of viscoelastic effects even for Drop-Newtonian medium system

23. Two Drops in Newtonian Fluid Two symmetrically located particles of radius 0.2 in a sliding bi-periodic frame Of size 1 x 1 in a Newtonian fluid with

24. Distance function and drop deformation

26. Multiple Droplets in Newtonian Fluid

27. 1. Direct numerical methods of drop emulsions in a viscoelastic fluid has been developed and implemented. 2. Incorporation with the Level set scheme for interfacial tension of droplet. 3. Deformation phenomena were observed for a single droplet, and multiple droplets. 4. Bulk normal stress is not zero for Drop- Newtonian medium system. Conclusions