9 - Dimensional analysis

Summary

Fluid often passes through pores in cell membranes or cell layers. The dimensions are small and the velocities low, so viscous forces dominate (low Reynolds number). Use dimensional analysis, or an approximate method of analysis based on the viscous flow equations, to determine the scaling law that expresses the dependence of the pressure drop across a pore (ΔP) on the flow rate through it (Q). The other parameters that are given include the pore radius, R, and the viscosity of the fluid, μ. The membrane itself should be considered infinitesimally thin so that its thickness does not influence the pressure drop.

Using a stroboscope, it has been observed that freely falling water drops vibrate. The characteristic time for this vibration does not depend on the viscosity of the water (except for very, very small drops). Determine what parameters you expect this vibration time to depend on, and find a relationship between the vibration time and these parameters. Estimate the characteristic vibration time for a water droplet of diameter 2 mm at a temperature of 25° C. (Hint: this time-scale is the same for a droplet inside of a rocket in space as it is for a droplet falling on the Earth.)