Momentum equation surface excess

To close the system of equations for the fluid motion the tangential stress boundary condition and the force balance equation are used. The boundary condition for the balance of the surface excess linear momentum, see equations (8) and (9), takes into account the influence of the surface tension gradient, surface viscosity, and the electric part of the bulk pressure stress tensor. In the lubrication approximation the tangential stress boundary condition at the interface, using Eqs. (17) and (18), is simplified to... 

The inteifacial momentum balance equation (Equation 5.35) will now be written for the case of negligible siuface excess mass and momentum and two Newtonian fluids. The stress tensor in a Newtonian fluid is written following Bird et al. (2002) as pi - pt [Vv + (Vv) ], where I is the identity tensor and the superscript T represents the transpose of a tensor. When the inertial forces are considered as well, the total force on an interface exerted by the ith phase is n (Pi(v - d )v + pj - / [VVj + (Vv )T]). The dot product with n denotes that the forces act on a surface characterized by m. It is in making a force balance on the intraface that the effects of interfacial tension make themselves felt. The balance is known to be (see Chapter 5)... 

The 0(6) jump in x n results from unbalanced tangential stresses. It is proportional to the vorticity [Vx ] produced at the flame and to the rotation V x n of a surface element. We note in particular the dependence on both the heat release q, and the Prandtl number Pr. Finally, equation (13) states that a net normal momentum flux across the flame, results from unbalanced pressure gradients, flame curvature and from the momentum associated with the net excess mass given by (11). 

The quasilaminar sublayer resistance / b describes the excess resistance for the transfer of matter from the atmosphere to the surfaces of the vegetation, that is, the difference between the resistance for matter and the resistance for momentum. It is primarily associated with molecular diffusion through quasi laminar boundary layers. Several parameterizations for Rb have been developed, but that employed by Brook et al. (1999), which like Equations 7.3 and 7.6 is valid for conditions of neutral atmospheric stability, is particularly easy to apply ... 

---