Generalized Reynolds-Orr Energy Equation with Wall Slip

The Reynolds-Orr energy equation is generalized to include the velocity slip effect at the walls, for investigating the energy gain or loss in the disturbed slip flow. Taken the Poiseuille flow, typical prototype of parallel shear flows as an example, it is found that for the very weak effect of wall slip, the disturbance energy being transferred from the basic flow overcomes the viscous dissipation, resulting in the growth of disturbance energy and the destabilizing role of wall slip. Otherwise, the viscous dissipation overcomes the energy production resulting in the decay of the disturbance energy and the stabilizing role of wall slip.