0.9 Flow with free surface

We have already encountered free surfaces in systems such as the drainage of a liquid along a wall. In this case the free surface was a material surface and the boundary condition was that of continuity of pressure and shear stress. The same boundary conditions would be used for wind-driven waves on water and the shape of the vortex formed when water drains from a bathtub. The dimensionless numbers of importance are the Reynolds number $N_{\mathrm{Re}}=\rho UL/\mu$ , Froude number $N_{Fr}=U^2/gL$ , and gravity number $N_G=\frac{\rho g{L}^2}{\mu U}=\frac{N_{\mathrm{Re}}}{N_{Fr}}$ . These mentioned systems are of a macroscopic scale compared to surface forces and rheology and thus surface tension, surface elasticity, and surface viscosity were not significant. However, when the system dimensions become about 1 cm or less surface forces are no longer negligible and play an important role in the shape of the interface and in transport processes. The capillary number $N_{Ca}=\mu U/\sigma$ and Bond number $N_{Bo}=\rho g{L}^2/\sigma$ introduced in Chapter 6 become important dimensionless groups that quantify the ratio of viscous/capillary and gravity/capillary forces. As the dimensions decrease to about 1 mm we are in the range of capillary phenomena where surface tension and contact angles become important (e.g., the rise of a wetting liquid in a small capillary). As the dimensions decrease to $1\mu m$ we are in the colloidal regime and not only is capillarity a dominant effect but also particles have spontaneous motion due to Brownian motion and thin films display optical interference as in the color of soap films. When the dimensions decrease to the range of 1 nm, it is necessary to include surface forces due to electrostatic, van der Waals, steric and hydrogen bonding effects to describe the thermodynamics and hydrodynamics of the fluid interfaces. At this scale the phases can no longer be assumed to be homogenous right up to the interface. The overlap of the inhomogeneous regions next to the interfaces results in forces that either attract or repel the interfaces.

Boundary conditions at a fluid-fluid interface

Analysis of macroscopic systems usually assume the fluid-fluid interface to simply be a surface of discontinuity in the density and viscosity of the bulk phases with no discontinuity in stress, (i.e., continuous pressure and shear stress across the surface). If there is no significant mass transfer, the surface is also a material surface and thus follows the motion of the fluid particles at the surface.

Systems with a length scale about a centimeter or less and having fluid-fluid interfaces can no longer neglect the discontinuity in stress across the interface. A momentum balance across the interface is needed to describe the stress at the boundary. Also, if the system has surface-active components that affect the surface tension and/or surface viscosity, then a material balance is also needed to determine the composition of the interfacial region. A general treatment of the momentum balance at fluid-fluid interfaces is given in Chapter 10 of Aris, Vectors, Tensors and the Basic Equations of Fluid Mechanics . We summarize the results here assuming no slip at the surface and a Newtonian surface constitutive relation. The terms in the momentum balance are given on the left side and its description is given on the right side.