Local dynamics during pinch-off of liquid threads of power law fluids: Scaling analysis and self-similarity
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...
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Published in | Journal of non-Newtonian fluid mechanics Vol. 138; no. 2; pp. 134 - 160 |
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Main Authors | , |
Format | Journal Article |
Language | English |
Published |
Amsterdam
Elsevier B.V
01.10.2006
Elsevier |
Subjects | |
Online Access | Get full text |
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Summary: | 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. |
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Bibliography: | ObjectType-Article-2 SourceType-Scholarly Journals-1 ObjectType-Feature-1 content type line 23 |
ISSN: | 0377-0257 1873-2631 |
DOI: | 10.1016/j.jnnfm.2006.04.008 |