Analysis of boundary slip in a flow with an oscillating wall

Molecular dynamics (MD) simulation is used to study slip at the fluid-solid boundary in an unsteady flow based on the Stokes’ second problem. An increase in slip is observed in comparison to the steady flow for shear rates below the critical shear rate of the corresponding steady flow. This increased slip is attributed to fluid inertial forces not represented in a steady flow. An unsteady mathematical model for slip is established, which estimates the increment in slip at the boundary. The model shows that slip is also dependent on acceleration in addition to the shear rate of fluid at the wall. By writing acceleration in terms of shear rate, it is shown that slip at the wall in unsteady flows is governed by the gradient of shear rate and shear rate of the fluid. Nondimensionalizing the model gives a time dependent yet universal curve, independent of wall-fluid properties, which can be used to find the slip boundary condition at the fluid-solid interface based on the information of shear rate, gradient of shear rate of the fluid, and the instant of time during the cycle. A governing nondimensional number, defined as the ratio of phase speed to speed of sound, is identified to help in explaining the mechanism responsible for the transition of slip boundary condition from finite to a perfect slip and determining when this would occur. Phase lag in fluid velocity relative to wall is observed. The lag increases with decreasing time period of wall oscillation and increasing wall hydrophobicity. The phenomenon of hysteresis is seen when looking into the variation of slip velocity as a function of wall velocity and slip velocity as a function of fluid shear rate. The cause for hysteresis is attributed to the unsteady inertial forces of the fluid. The rate of heat generated by viscous shear is compared for an unsteady Stokes’ second problem and simple Couette flow and is shown to be higher for the unsteady flow.