Shapes and stability of viscous rotating drops in a compressional/extensional flow

The deformation, stationarity, and stability of a drop rotating axisymmetrically in an immiscible viscous fluid, under the influence of an externally imposed flow, are studied. The ambient fluid, slightly heavier or lighter than the drop, is under simultaneous rotating and compressional or extensional forcing flow at infinity. The problem is formulated and solved via an integral equation having unknown surface velocity, considering low Reynolds number with equal viscosity inside the drop and of the ambient fluid. Numerical simulations are carried out by using the boundary integral method. The drop stationarity is discussed for a variety of Bond numbers, Bo, and capillary numbers, Ca, for the simultaneous action of both flow fields. The critical bounds of Ca for the stability of stationary flat and extended shapes of the drop were established for the considered range of Bo.

Figure 1

A comparison of stationary drop deformation at stable states calculated a solid-body rotation by using the ordinary differential equations [Eqs. (13, 14, 15)] with the deformation obtained by using boundary integral method, [Eqs. (16)–(18)] when $\text{Ca}=1$.

Figure 2

Representation of the evolution of the sum of normal velocities at the major and minor axes of the drop [Eq. (19)], with time for varying capillary numbers

Figure 3

Variation of Taylor deformation parameter and corresponding stationary axisymmetrical shapes of drops for $\text{Bo}=0.0$ with various Ca. Negative Ca denotes compressional cases and the positive ones denote the extensional case.

Figure 4

Variation of Taylor deformation parameter and drop shapes with Ca for the case $\text{Bo}=-10,-5,5,10$ (solid lines) compared with the case $\text{Bo}=0$ (dashed lines).

Figure 5

Variation of stable deformation region with respect to capillary numbers for various Bond numbers in the interval $[-10,10]$.

Figure 6

Physical phase plane depicting the boundaries of the region of the stable stationary shapes of drops and the corresponding shapes at such critical states of drop stability for various Bond numbers. Bold shapes show the extreme case at $\text{Bo}=0$.