OVERVIEW

For a fluid that is stationary or translates as a rigid body the continuity equation is satisfied in a trivial manner and the Navier Stokes equation reduces to a first-order differential equation expressing a balance between the body force, the gradient of the pressure, and possibly the fictitious force due to the acceleration and rotation of the frame of reference. This simplified force balance may be integrated using elementary methods subject to appropriate boundary conditions to yield the pressure distribution within the fluid. The integration produces a scalar constant whose value is determined by specifying the level of the pressure at some point on a boundary. The implementation of this procedure and its applications will be discussed in Section 4.1 for stationary fluids as well as for fluids executing steady and unsteady rigid-body motion.

The interfacial boundary conditions discussed in Section 3.5 require that the magnitude of the normal component of the traction, which is equal to the pressure, undergo a discontinuity across the interface that is balanced by surface tension. Accordingly, the interface must assume a shape that is compatible with the pressure distribution within the fluids as well as with the boundary conditions for the contact angle or for the location of a three-phase contact line. The consequent mathematical implication is that the pressure distributions within the fluids on either side of the interface may not be computed independently, but must be found simultaneously with the interfacial shape so that all boundary conditions are fulfilled. To this end,...