Dynamics of a viscous drop under an oscillatory uniaxial extensional Stokes flow

We quantify the transient deformation and breakup of a neutrally-buoyant drop with viscosity lambda mu* immersed in another Newtonian fluid with viscosity mu* undergoing oscillatory uniaxial extension at zero Reynolds number. The interfacial tension acting between drop phase and medium phase is gamma*. The drop is initially a sphere of radius a*. Since the external flow oscillates harmonically with a frequency omega*, the strength of the imposed flow is characterized by an instantaneous capillary number, Ca = Ca-0 cos(Det), where Ca-0 = mu*(epsilon) over dota*/gamma* De = omega*mu*a*/gamma* is the dimensionless frequency, or Deborah number. Here, (epsilon) over dot is the rate of extension in the imposed flow. We utilize boundary-integral computations to calculate the evolution of drop interface as a function of Ca-0 and De, focusing primarily on the case where the drop and surrounding fluid have equal viscosities. The computations suggest two families of behavior for the drop deformation. First, below a critical Deborah number (which we determine to be in the interval 0.375 < De < 1.0), the drop breaks up in a finite time at a critical capillary number that is a function of De. At sufficiently small De the critical capillary number increases linearly with De and the breakup mode is that of "center-pinching''. On increasing De the break-up mode switches to "end-pinching'', and on further increasing De. it appears that the critical capillary number diverges at a critical Deborah number between 0.375 and 1.0. This divergence signals the transition to the second family of behavior where the drop attains a long-time periodic state, or alternance, regardless of Ca-0. However, at large Ca-0 the drop dynamics exhibits a two time-scale behavior: the drop deforms instantaneously at a fast capillary relaxation time-scale tau* = mu*a*/gamma*, whereas the maximum deformation attained during a cycle of the imposed flow grows at the slow time-scale tau* Ca-0 = (epsilon) over dot(mu*a*/gamma*)(2). As such, the state of alternance is approached exceedingly slowly at large Ca-0. Lastly, we perform calculations for different viscosity ratios and find the drop dynamics to be in qualitative agreement with above observations.