V_ L, (3.9)
VP L
Equation 3.8 will later be used in conjunction with the dimensionless fall velocity parameter. Before similitude can occur between model and prototype in regards to the dimensionless fall velocity parameter, values for the fall velocity have to be calculated for the model and prototype condition. Stokes derived a formula for calculating the fall velocity from the Navier-Stokes equation. He solved this equation by assuming a falling particle in water has two forces acting on it, i.e weight and the force of drag. Using this basis, there are three basic steps in solving the fall velocity. First, if the inertia and body forces terms are neglected, then the Navier-Stokes Equation in conjunction with continuity results in equation 3.10.
V2p = 0 (3.10)
Next, for steady state flow past a sphere, the boundary conditions utilized result in an equation for the force of drag on the particle (Equation 3.11).
FD = 6IRV U. (3.11)
Finally, equating the weight of the particle and the drag force and assuming a terminal velocity U., so that U., = W, results in equation 3.12.
FD = W (3.12)
Solving this equality, the fall velocity is defined as:
- =1 D D2 Y-YI) (3.13)
18 v