The problem of unsteady suction from a high-Reynolds-number cross-flow into a slot is considered in the case where the suction is driven by a time-dependent slot pressure. The model uses linearised asymptotics based on a small parameter that defines the suction strength. An integro differential equation is derived for the mass flow into the slot and this is solved for various time-dependent slot pressures of practical interest. Closed-form expressions are also found for the shape of the shear layer dividing the external flow from the fluid in the slot. For a step function change in the slot pressure, a non-monotonic decay to the steady solution is observed, and for an oscillatory slot pressure there is a phase lag between the slot pressure and the mass flow. For rapidly changing slot pressures it is shown that slot injection can occur, even when the slot pressure remains below the free-stream pressure.
suction injection film cooling Hilbert transforms thin aerofoil theory.