Forcing Functions in Navier-Stokes Equations

Two dimensionless forms of the Navier-Stokes equations are derived for unsteady incompressible flows with movable boundaries. Temporal variations of flow parameters and geometry are often seen when the motions of fluid and solid are interrelated. The time-dependent boundary conditions—both geometrical and fluid mechanical—are cast into the equations, making the problems simpler for computational flow simulation. These paramaters can be viewed as the forcing functions in the Navier-Stokes equations. Two flow problems are used to demonstrate this approach, one for axisymmetrical blood flow in an artery with assisted pumping of a balloon, and the other for two-dimensional flow past a finite plate with oscillatory motion. Both cases involve active and passive (flow-induced) motions of the solid boundaries.