Local dynamics during pinch-off of liquid threads of power law fluids: Scaling analysis and self-similarity

Abstract

Pinch-off dynamics of liquid threads of power law fluids surrounded by a passive ambient fluid are studied theoretically by fully two-dimensional (2-D) computations and one-dimensional (1-D) ones based on the slender-jet approximation for 0 < n ≤ 1 , where n is the power law exponent, and 0 ≤ O h ≤ ∞ , where O h ≡ μ 0 / ρ σ R is the Ohnesorge number and μ 0 , ρ , σ , and R stand for the zero-deformation-rate viscosity, the density, the surface tension, and the initial thread radius, to develop a comprehensive understanding of breakup which has heretofore been lacking. Under the assumption that the thread shape at breakup is slender, Doshi et al. [J. Non-Newtonian Fluid Mech. 113 (2003) 1] showed that inertial, viscous, and capillary forces must remain in balance as the minimum thread radius h min ⁡ → 0 and that in this inertial-viscous power law (IVP) regime, where O h = 1 , the radial length h, the axial length z, and the axial velocity v must scale with time to breakup τ as h ∼ τ n , z ∼ τ 1 − n / 2 , and v ∼ τ − n / 2 . Doshi et al. further deduced that in the viscous power law (VP) regime, in which a pinching thread undergoes creeping flow and O h = ∞ , h ∼ τ n , z ∼ τ δ , where 0.175 ≤ δ is the axial scaling exponent that rises as n falls, and v ∼ τ δ − 1 . Doshi et al. recognized that the slenderness assumption is violated when n falls below a certain value. The critical value of n is 2/3 in the IVP regime and, as shown by Renardy and Renardy [J. Non-Newtonian Fluid Mech. 122 (2004) 303], 0.54 in the VP regime. When viscous force is indentically zero ( O h = 0 ), it has been known for some time that in this potential flow (PF) regime thread shapes at breakup are non-slender and overturned, and that h ∼ τ 2 / 3 , z ∼ τ 2 / 3 , and v ∼ τ − 1 / 3 . Here, the 2-D computations are used to show that the scaling exponents of radial and axial lengths are equal and that h ∼ τ n , z ∼ τ n , and v ∼ τ n − 1 when n ≤ 0.54 in creeping flow, which is henceforward referred to as the non-slender viscous power law (NSVP) regime. For Newtonian fluids ( n = 1 ) , the creeping flow and the potential flow regimes are transitory, and a pinching thread of a high (low) viscosity fluid must ultimately transition to a final asymptotic regime in which inertial, viscous, and capillary forces all diverge but remain in balance as pinch-off nears. Here, the 2-D computations are used to demonstrate that pinching threads of power law fluids exhibit remarkably richer response compared to their Newtonian counterparts. When O h = 1 and n < 2 / 3 , the 2-D computations show that a thread of a power law fluid asymptotically thins according to the potential flow (PF) scaling law as if it were an inviscid fluid and that its profile is non-slender and overturned in the vicinity of the pinch-point. When O h > 1 , the 2-D computations reveal that a thinning thread transitions from the VP to the IVP regime when n > 2 / 3 in accordance with the 1-D results but a thinning thread transitions from the VP to the PF regime when 0.54 < n ≤ 2 / 3 and from the NSVP regime to the PF regime when n ≤ 0.54 . When O h < 1 , the 2-D computations show that a thinning thread transitions from the PF to the IVP regime when n > 2 / 3 in accordance with the 1-D results but a thinning thread remains in the PF regime until breakup when n < 2 / 3 . Moreover, when O h ≪ 1 and n > 2 / 3 , the 2-D computations show that the interface overturns first before the thread transitions from the PF to the IVP regime. When O h < 1 and n > 2 / 3 , the transition between the PF and the IVP regimes is shown to occur when the minimum thread radius h min ⁡ ∼ O h 2 / ( 3 n − 2 ) . Scaling exponents and self-similar thread shapes and axial velocity profiles obtained from the 2-D computations are shown to be in excellent agreement with the 1-D results when thread shapes at breakup are slender.