The energy balance in modulated plane Poiseuille flow

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Abstract
Plane Poiseuille flow in which the pressure gradient has a small amplitude time-periodic component in addition to a constant component is considered. The velocity field close to the boundaries, arising from a small amplitude high frequency disturbance to the flow, is calculated to second order in the modulation amplitude. The energy-transfer integral for the disturbance is then calculated to the same order. It is found that, if the thickness of the disturbance shear wave relative to that of the modulation shear wave is greater than ½, the modulation inhibits energy transfer into the disturbance and so stabilizes the flow.

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