Hydrodynamics of slowly miscible liquids Please use the dd month yyyy format for the date for example 11 January 2008. The main title can be one or two lines long. Anatoliy Vorobev 09 September 2010

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 If using a school logo, make sure that if you have a long page title, it does not encroach on the logo. Allow about 2cm around the logo. Run the page title onto two lines if necessary.

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

Thermodynamic model specific free energy (Cahn & Hilliard, J. Chem. Phys. 1958): Classical part (Landau): Heterogeneous system, a<0 C1 C2 When using more than one image, use images that are roughly the same size and proportions. You can scale the images to make the sizes appear more even. Avoid stretching images - hold down the shift key when you drag to resize to avoid altering the proportions. Homogeneous system, a>0 Phase diagram for the IBA-water system

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 When using more than one image, use images that are roughly the same size and proportions. You can scale the images to make the sizes appear more even. Avoid stretching images - hold down the shift key when you drag to resize to avoid altering the proportions.

- 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

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

Equations in non-dimensional form

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

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,

Averaging Assume Here, Fluctuating part: Solution:

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

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.

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