Dynamics of a drop at a fluid interface under shear

We analyze the dynamics of a two-dimensional drop lying on a fluid interface, sometimes called a liquid lens, subjected to simple shear flow. The three fluids, the drop and the two external fluids, meet at a triple point (or a triple line in three dimensions). A requirement for steady drop shapes is that the triple points are stationary. This leads to a flow topology different than that of a freely suspended drop. Results are substantiated with numerical results using a level set method for interface evolution and treatment of triple points. Possible implications for new drop instabilities are also discussed.

Figure 1

Interfacial drop in shear flow.

Figure 2

Schematic diagram of the dependence of drop configuration on interfacial tensions. The drop is stable at the interface in the region between the three curves. The system has states $W1$, $W2$, and $W3$ outside this region, and states $D1$, $D2$, and $D3$ in the limit that one interfacial tension $\to 0$ with the other two being equal. Asymmetric conformations occur at da and wa. Symmetric conformations occur at sa, sb, and sc.

Figure 3

Computed streamlines in the small deformation limit for a simple drop in shear flow. The drop interface is the largest circular streamline.

Figure 4

Possible classes of flow in the vicinity of an interfacial drop. Interfaces (circular for the drop and horizontal between the bulk fluids) are only for illustration and do not represent actual interfacial shapes.

Figure 5

Computed streamlines (near the drop) and interfacial shapes (over the entire system) for symmetric conformations: (a) ${\kappa }_{321}={\kappa }_{312}=0.6,\text{Ca}=0.125$ $sa$ conformation. (b) ${\kappa }_{321}={\kappa }_{312}=1,\text{Ca}=0.075,$ $sb$ conformation. (c) ${\kappa }_{321}={\kappa }_{312}=4,\text{Ca}=0.141,$ $sc$ conformation.

Figure 6

Computed streamlines for asymmetric drops. (a) $wa,{\kappa }_{312}=1,{\kappa }_{321}=1.8,\mathrm{Ca}=0.019$; (b) $da,{\kappa }_{312}=0.6,{\kappa }_{321}=1,\mathrm{Ca}=0.33$.

Figure 7

Instabilities for several equilibrium conformations. (a) $sb,{\kappa }_{312}={\kappa }_{321}=1;$ (b) $da,{\kappa }_{312}=0.1,{\kappa }_{321}=1;$ (c) $wa,{\kappa }_{312}=1,{\kappa }_{321}=1.8;$ (d) $sc,{\kappa }_{312}={\kappa }_{321}=10$.