G Double-Diffusive Convection

The basic starting point is again the governing equations (12-160), (12-161), and (12-164). To these equations, we must add a species transport equation for our solute 

D is the species diffusivity. In deriving the Bousinesq approximation of the Navier-Stokes equations, we must remember that the material properties now depend both on the temperature and the solute concentration, e.g., 

The dimensionless parameters that appear are the Prandtl and Schmidt numbers, 

As in the case of the Rayleigh-Benard problem, there is a steady-state solution of these equations. 

To determine the stability of this stationary state, we consider the fate of an arbitrary, fully 3D initial perturbation of the base state (12-252), namely, 

---

Ozsoy E, Top Z, White G, Murray JW (1991) Double diffusive intrusions, mixing and deep sea convection processes in the Black Sea. In Izdar E, Murray JW (eds) The Black Sea oceanography. NATO/ASI series. Kluwer Academic, Dordrecht, p 17... 

In filtration, the particle-collector interaction is taken as the sum of the London-van der Waals and double layer interactions, i.e. the Deijagin-Landau-Verwey-Overbeek (DLVO) theory. In most cases, the London-van der Waals force is attractive. The double layer interaction, on the other hand, may be repulsive or attractive depending on whether the surface of the particle and the collector bear like or opposite charges. The range and distance dependence is also different. The DLVO theory was later extended with contributions from the Born repulsion, hydration (structural) forces, hydrophobic interactions and steric hindrance originating from adsorbed macromolecules or polymers. Because no analytical solutions exist for the full convective diffusion equation, a number of approximations were devised (e.g., Smoluchowski-Levich approximation, and the surface force boundary layer approximation) to solve the equations in an approximate way, using analytical methods. 

---