Past exam of the mathematics course of the University of Cambridge / 2003 / iii / Paper 71 / 2 / a

Approximate the gap by steady Couette flow, neglecting edge effects and additional losses from membrane deformation. Its velocity gradient is $U/h$, so the shear stress is $\eta U/h$. The force on the gliding bacterium and the rate of viscous dissipation areThe same power follows by integrating $\eta (U/h{)}^2$ over gap volume $Ah$. In SI units, $A=10^{-10}{\mathrm{m}}^2$, $h=10^{-7}\mathrm{m}$, and $U=10^{-6}\mathrm{m}{\mathrm{s}}^{-1}$. ThusAn energy needs a duration or distance: $E=10^{-18}\Delta t\mathrm{J}$ for $\Delta t$ in seconds, equivalently $E=F_{d}s$ for travel distance $s$. Moving one micrometre takes one second and dissipates $10^{-18}\mathrm{J}$. This is the mechanical cost of the sheared film. A numerical biological efficiency would additionally require the motor's input power and the losses associated with its peristaltic actuation.