It has previously been demonstrated that the drag experienced by a Poiseuille flow in a channel can be reduced by subjecting the flow to a dynamic regime of blowing and suction at the walls of the channel (also known as ‘transpiration’). Furthermore, it has been found to be possible to induce a ‘bulk flow’, or steady motion through the channel, via transpiration alone. In this work, we derive explicit asymptotic expressions for the induced bulk flow via a perturbation analysis. From this we gain insight into the physical mechanisms at work within the flow. The boundary conditions used are of travelling sine waves at either wall, which may differ in amplitude and phase. Here it is demonstrated that the induced bulk flow results from the effect of convection. We find that the most effective arrangement for inducing a bulk flow is that in which the boundary conditions at either wall are equal in magnitude and opposite in sign. We also show that, for the bulk flow induced to be non-negligible, the wavelength of the boundary condition should be comparable to, or greater than, the height of the channel. Moreover, we derive the optimal frequency of oscillation, for maximising the induced bulk flow, under such boundary conditions. The asymptotic behaviour of the bulk flow is detailed within the conclusion. It is found, under certain caveats, that if the amplitude of the boundary condition is too great, the bulk flow induced will become dependent only upon the speed at which the boundary condition travels along the walls of the channel. We propose the conjecture that for all similar flows, if the magnitude of the transpiration is sufficiently great, the bulk flow will depend only upon the speed of the boundary condition.

The addition of elastic polymers to turbulent liquids is known to produce significant drag reduction. In this study, we prove that the drag in pipe and channel flows of an unforced laminar fluid constitutes a lower bound for the drag of a fluid containing dilute elastic polymers. Further, the addition of elastic polymers to laminar fluids invariably increases drag. This proof does not rely on the adoption of a particular constitutive equation for the polymer force, and would also be applicable to other similar methods of drag reduction, which are also achieved by the addition of certain particles to a flow. Examples of such methods include the addition of surfactants to a flowing liquid and the presence of sand particles in sandstorms and water droplets in cyclones.