Drag on an Object in Creeping Flow
Since density has vanished from the Navier–Stokes equation, aerodynamic drag on an object in creeping flow is a function only of its speed V, some characteristic length scale L of the object, and fluid viscosity 𝜇 (Fig. 10–12). Use dimensional analysis to generate a relationship for F _{D} as a function of these independent variables.
SOLUTION We are to use dimensional analysis to generate a functional relationship between F _{D} and variables V, L, and 𝜇.
Assumptions 1 We assume Re ≪ 1 so that the creeping flow approximation applies. 2 Gravitational effects are irrelevant. 3 No parameters other than those listed in the problem statement are relevant to the problem.
Analysis We follow the step-by-step method of repeating variables discussed in Chap. 7; the details are left as an exercise. There are four parameters in this problem (n = 4). There are three primary dimensions: mass, length, and time, so we set j = 3 and use independent variables V, L, and 𝜇 as our repeating variables. We expect only one Pi since k = n – j = 4 – 3 = 1, and that Pi must equal a constant. The result is
F_{D}=\text { constant } \cdot \mu V L
Thus, we have shown that for creeping flow around any three-dimensional object, the aerodynamic drag force is simply a constant multiplied by 𝜇VL.
Discussion This result is significant, because all that is left to do is find the constant, which is a function only of the shape of the object.