Consider viscous flow over a very small object. Analysis of the equations of motion shows that the inertial terms are much smaller than viscous and pressure terms. Fluid density drops out, and these are called creeping flows. The only important parameters are velocity U, viscosity μ, and body length scale d. For three-dimensional bodies, like spheres, creeping flow analysis yields very good results. It is uncertain, however, if creeping flow applies to two-dimensional bodies, such as cylinders, since even though the diameter may be very small, the length of the cylinder is infinite. Let us see if dimensional analysis can help.
(a) Apply the Pi theorem to two-dimensional drag force F2-D as a function of the other parameters. Be careful: two-dimensional drag has dimensions of force per unit length, not simply force. (b) Is your analysis in part (a) physically plausible? If not, explain why not.
(c) It turns out that fluid density ρ cannot be neglected in analysis of creeping flow over two-dimensional bodies. Repeat the dimensional analysis, this time including ρ as a variable, and find the resulting non-dimensional relation between the parameters in this problem.