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

Abstract

We quantify the transient deformation and breakup of a neutrally-buoyant drop with viscosity λμ∗ immersed in another Newtonian fluid with viscosity μ∗ undergoing oscillatory uniaxial extension at zero Reynolds number. The interfacial tension acting between drop phase and medium phase is γ∗. The drop is initially a sphere of radius a∗. Since the external flow oscillates harmonically with a frequency ω∗, the strength of the imposed flow is characterized by an instantaneous capillary number, Ca=Ca0cos(Det), where Ca0=μ∗ε̇a∗/γ∗ and De=ω∗μ∗a∗/γ∗ is the dimensionless frequency, or Deborah number. Here, ε̇ 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 Ca0 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 Ca0. However, at large Ca0 the drop dynamics exhibits a two time-scale behavior: the drop deforms instantaneously at a fast capillary relaxation time-scale τ∗=μ∗a∗/γ∗, whereas the maximum deformation attained during a cycle of the imposed flow grows at the slow time-scale τ∗Ca0=ε̇(μ∗a∗/γ∗)2. As such, the state of alternance is approached exceedingly slowly at large Ca0. Lastly, we perform calculations for different viscosity ratios and find the drop dynamics to be in qualitative agreement with above observations.