Resolving hydrodynamic singularities through molecular interactions

1
Resolving hydrodynamic singularities through
molecular interactions

• Len Pismen
• Technion, Haifa, Israel
• Supported by Israel Science Foundation
• Minerva Center for Nonlinear Physics of
  Complex Systems

2
Cusp on a free interface
3
Hydrodynamic problem
kinematic condition
boundary conditions
?
tangential stress normal stress
?????? ?
complex flow potential
Stokes solution
boundary conditions
4
Conformal transformation
conformal transformation
???
????? ???
? ?????
Ca???at ? ????? ?
? ?????
5
Hydrodynamic solution
must vanish at q-p/2
defines parameter ? as a function of Cau0????
Ca
?
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Hydrodynamic solution flow pattern
Ca 0.1?
Ca ????
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Flow near the tip inflow or stagnation point?
Ca
log10(??L)
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Anatomy of a cusp
Hydrodynamic picture the interface of zero
thickness terminates in a cusp singularity Shikhmu
rzaevs picture there is an interfacial layer
(1) of unspecified nature its properties change
starting from equilibrium values the cusp widens
into a transition zone (2), and a
surface-tension-relaxation tail (3) becomes a
gradually disappearing internal
interface. Diffuse interface picture A narrow
interfacial zone (1) is close to local
equilibrium vapor condenses near the cusp (2)
bulk fluid relaxes to equilibrium in the
diffusion zone (3)
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Equilibrium density functional theory
Free energy of inhomogeneous fluid
e.g. hard-core potential
Euler Lagrange equation
chemical potential
static equation of state
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Dynamic diffuse interface theory
Cahn Hilliard equation (in Galilean frame)
dynamic equation of state
G mobility
coupling to hydrodynamics
Stokes equation
continuity equation
compressible flow in boundary region
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Multiscale perturbation scheme

• Find an equilibrium density profile
• liquid at z ???? ????? at z ???
• Perturbations
• Weak gradient of chemical potential
• Propagation of interphase boundary
• Disjoining/conjoining potential
• Interfacial curvature
• Expand in scale ratio
• Use solvability condition
• Match to outer solution

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Equilibrium density profile
1D static density functional equation
Q(z)
Density profile for a van der Waals fluid
h3 tail
h3 tail
Gibbs surface (defines the nominal thickness)
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Perturbation scheme inner solution

• Inner equation for chemical potential

Inner chemical potential

• Material balance across the layer

• Dynamic shift of chemical potential

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Interaction of interfaces
liquid
Change of surface tension
h
vapor
liquid
Shift of chemical potential
equilibrium
valid at hgtgtd significant at hd
shifted equilibrium
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1d solutions of the nonlocal equations
h/d
g
h/d
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Variable surface tension
surface tension vs. gap width
g


h lt d
h gt d
h/d
Ca
Cusp appears at a finite Ca it is lower than
Ca in the hydrodynamic solution with the same
tip curvature
Ca(?)
Ca(d)
log10(?L or L/d)
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Effect of variable surface tension velocity
?
?d
velocity
d 108
velocity
Ca??? ???????
?
blow-up near the tip
q
d 108
velocity
velocity
d 108
d 106
Ca??? ???????
q
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Larger Ca? Higher-order singularity
1/3 gt a gt 8/19 0 gt b gt 1/95 close
together
tail close together
a1/3 b0
a8/19 b1/95
tip diverge
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Outer solution gas phase
longitudinal vapor flux
liquid
condensation zone
longitudinal air flux
gas flow
longitudinal condensation flux
liquid
local condensation flux
condensation starts when the gap narrows to
provide necessary driving force
gap h at the entrance of condensation zone
107 m
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Outer solution liquid phase
density depletion
convective diffusion equation ( u velocity
along the cusp)
equilibrium
shifted equilibrium
potential difference driving the condensation flux
Weak but slowly decaying density depletion
downstream
h/d
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Conclusions

• Cusp singularity is resolved trough molecular
  interactions on nanoscale
• Effect of variable surface tension
• Cusp is formed at finite capillary number
• Weak Marangoni flow near the cusp
• Effect of condensation
• Air is removed from the cusp by diffusion
• Weak depletion of liquid density near the cusp
• Relaxation effect
• Weak depletion tail downstream from the cusp