1. Hydrodynamics of slowly miscible liquids
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Anatoliy Vorobev 09 September 2010

2. Aim
Theoretical model for the dissolution/nucleation dynamics of a slowly miscible multiphase system
(other similar processes: evaporation/ condensation, solidification/meting, polymerization).
mass transfer through the interface
morphological transformations of the interface
dynamic variations of interface’s thickness and surface tension
hydrodynamic flows
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3. Phase-field (diffuse interface) approach
one system of equations is solved
interface is a transitional layer; all quantities experience sharp but continuous change through an interface
concentration is used to trace the interface position
surface tension is mimicked through a volume force

4. Thermodynamic model
specific free energy (Cahn & Hilliard, J. Chem. Phys. 1958):
Classical part (Landau):
Heterogeneous
system, a<0 C1 C2
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Homogeneous
system, a>0
Phase diagram for the IBA-water system

5. Hydrodynamic model. Cahn-Hilliard-Navier-Stokes equations
Mass, species and momentum balances for a binary mixture of two incompressible liquids:
Equations of state:
Lowengrub & Truskinovsky, Proc. R. Soc. Lond. A 1998
Essentials:
Cahn-Hilliard free energy
mass-averaged velocity
- incompressible liquids
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6. - no mass flux through wall
Boundary conditions:
- no-slip
- no mass flux through wall
- thermodynamic equilibrium at liquids-wall interface
Typical time scales:
The idea of the multiple-scale method,
‘expansion’ time
Slow diffusive and convective processes

7. Non-Dimensionalization
Shift of the reference point
Units
density, pressure and energy:
velocity:
time:
chemical potential:

8. Equations in non-dimensional form

9. Non-dimensional parameters
-- Peclet number
-- capillarity number
-- Galileo number
-- Reynolds number

10. Separation of time-scales
The idea of the multiple-scale method,
‘expansion’ time
Slow diffusive and convective processes
We also expand all variables in the series of χ:
And parameters,

11. Averaging
Assume
Here,
Fluctuating part:
Solution:

12. Boussinesq approximation of the Cahn-Hilliard-Navier-Stokes equations
Jacqmin,
J. Comp. Phys. 1999
&
J. Fluid Mech. 2000
Boundary conditions:
Total energy:

13. Conclusions
The Boussinesq approximation of the Cahn-Hilliard- Navier-Stokes equations is strictly derived. This model defines the slow dissolution dynamics of binary mixtures.
Two surface effects:
morphology of the interface is defined by the Korteweg stress, and
the rate of mass transfer through the interface is also restricted by the surface effects.

14. Current work
Phenomenological parameters a, b and ε are to be determined from comparisons of the numerical and real experiments:
Dissolution from a capillary tube
Spinning drop tensiometer z r