Simple viscous flows: From boundary layers to the renormalization group

The seemingly simple problem of determining the drag on a body moving through a very viscous fluid has, for over 150 years, been a source of theoretical confusion, mathematical paradoxes, and experimental artifacts, primarily arising from the complex boundary layer structure of the flow near the body and at infinity. The extensive experimental and theoretical literature on this problem is reviewed, with special emphasis on the logical relationship between different approaches. The survey begins with the development of matched asymptotic expansions, and concludes with a discussion of perturbative renormalization-group techniques, adapted from quantum field theory to differential equations. The renormalization-group calculations lead to a new prediction for the drag coefficient, one which can both reproduce and surpass the results of matched asymptotics.

Figure 1

(Color online) Comparing experiment with state of the art theoretical predictions for a sphere (111; 55) (right) and a cylinder (80; 74; 32) (left).

Figure 2

(Color online) Schematic for flow past a sphere or cylinder.

Figure 3

(Color online) Early measurements of the drag on a sphere (45).

Figure 4

(Color online) Maxworthy’s accurate measurements of the drag on a sphere (80) contrasted with previous experiments (45).

Figure 5

(Color online) Summary of experimental and numerical studies of $C_D$ for a sphere (80; 74; 32).

Figure 6

(Color online) Tritton’s measurements of the drag on a cylinder (111).

Figure 7

(Color online) Summary of measurements of the drag on a cylinder (39; 55; 111).

Figure 8

(Color online) Drag on a sphere, experiment vs Oseen theory (80; 74; 32). The Stokes’ solution [Eq. (29)] is shown at the bottom for reference. In these coordinates, it is defined by the line $y=0$.

Figure 9

(Color online) Drag on a cylinder, experiment vs Oseen theory (111; 55).

Figure 10

(Color online) Drag on a sphere, experiment vs matched asymptotic theory. Experimental and numerical results are plotted as in Fig. 8.

Figure 11

(Color online) Drag on a cylinder, experiment vs matched asymptotic theory (111; 55).

Figure 12

(Color online) Drag on a cylinder, comparing uniformly valid calculations and matched asymptotics results (111; 55).

Figure 13

(Color online) Drag on a sphere, experiment vs theory (80; 74; 32).

Figure 14

(Color online) Drag on cylinder, comparing RG predictions to other theories at low $R$.

Figure 15

(Color online) Drag on a cylinder, comparing RG predictions to other theories at higher $R$.

Figure 16

(Color online) Drag on a sphere, comparing RG to other theories (80; 74; 32).

Figure 17

(Color online) Drag on a sphere, comparing RG at larger $R$ (80; 74; 32).