Low Reynolds Number Flow

Newton’s second law of motion requires that the rate of change of momen- tum of a fluid parcel must be balanced by the body force exerted over the parcel volume and by the surface force exerted on the parcel boundary. Under certain conditions, the rate of change of momentum of the parcel is small compared to the body and surface force, and can be neglected without introducing seri- ous error. The flow is then governed by a balance between the volume force and the surface force. Physically, this occurs when the fluid viscosity is high, when the fluid density is small, when the velocity changes rapidly over a small distance yielding a sharp spatial gradient, or when the convection velocity of a fluid parcel is sufficiently small. The formal requirement for fluid inertia to be negligible is that a properly defined Reynolds number is sufficiently small. How small it should be, depends on the particular problem under consideration. In this chapter, we consider a family of flows occurring at small Reynolds numbers and discuss the solution of simplified systems of governing equations that arise by dropping the inertial terms in the equation of motion. The simplification will allow us to address a multitude of physical and engineering problems and derive solutions by a host of analytical and numerical methods.