A Nondimensionalization and the Creeping-Flow Equations

To begin, we restrict our attention to isothermal, laminar flow of an incompressible Newtonian fluid, in which all boundaries are solid surfaces, so that we can use the equations of continuity and motion in the forms (2 20) and (2 91), respectively, that is, 

These equations contain two independent dimensional parameters, the density p and the viscosity /x, which are both constant and assumed to be known. For convenience in what follows, we mark the dimensional variables that appear in these equations with a prime u, p, and so on and, in addition, denote the dimensional gradient operator as V.  

The reader may again be curious whether there are consequences of making the wrong choice for c, Uc, or Tc when there is more than one possibility available We will need to discuss this point in some detail, once we see how we intend to use the nondimensional-ized versions of our governing equations (and boundary conditions) in the development of asymptotic approximations. For now, we simply assume that the appropriate choices have been made. Then, for each additional dimensional scale that appears in a particular problem, we get one more dimensionless parameter, in addition to the two that will appear based on 

For present purposes, we assume that there is only a single length, velocity, and time scale available for lc, Uc, and. To express (2-20) and (2-91) in dimensionless form, we introduce dimensionless dependent and independent variables as follows  

The reader may wish to verify that the dimensionless group p Uc /L c has dimensions of force per unit area and is thus appropriate as a choice for the characteristic pressure. Substituting (7-1) into (2-20) and (2-91), as given at the beginning of this chapter, we find 

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