Motion nondimensional form

Three independent Reynolds numbers arise when the equations of motion and boundary conditions are cast in nondimensional form the particle Reynolds number based on slip velocity, RCp = 2al p /v a shear Reynolds number, Rej = (2a) S/v and a rotational Reynolds number, Re = 2df (o l. It is assumed that... 

When the equations of motion and boundary conditions are expressed in nondimensional form, the following functional relationship ultimately emerges / u ... 

The governing equations of motion in nondimensional form are then as follows ... 

Before performing averaging, (6.5) must be transformed to the standard form [56]. To that end, some simplifications are necessary. In the following sections, first the equation of motion is simplified and then converted into a nondimensionalized form. Next, a small parameter, e, is introduced and the new dimensionless parameters are ordered to reach an approximate weakly nonlinear equation of motion accurate up to 0(e). 

If the motion is steady when viewed from an inertial reference frame, the term 5v/5i vanishes identically. Providing that the remaining nondimensional terms remain finite as i -> 0, these then reduce to Stokes equations at sufficiently small Reynolds numbers. In dimensional form, Stokes equations are... 

The first step toward transforming the equation of motion to a proper form for averaging is to nondimensionalize it. This is an important step to appropriately order each parameter according to its size. Expanding (6.5), yields... 

---