The effect of first- and second-order slip condition on oscillatory flows: Several exact solutions

In last two decades, several studies have been conducted to understand the molecular mechanism of slip of fluids (both liquids and gases) at a solid wall. Most of these studies have dealt with steady flow. Recently, Thalakkottor and Mohseni have shown by molecular dynamics simulations that in unsteady flow there is an additional slip above the slip observed in the steady flow. They have developed an unsteady slip flow model by extending the Maxwell's slip theory for steady flows to encapsulate unsteady flows. The model indicates that the slip velocity of a fluid in unsteady flow is also a function of the acceleration of the fluid in addition to its shear rate. Thus the slip velocity is both a function of first- and second-order derivatives normal to the wall multiplied by different coefficients dependent upon the nature of the fluid (liquid or gas). In this paper, we consider several unsteady flows - the Stokes flow, the Couette flow, and flow in a channel due to an oscillatory wall. In addition, we consider unsteady flow due to an impulsively started flat plate. Exact solutions are obtained by using the most general form of slip boundary condition that contains both first and second-order derivatives normal to the boundary. The coefficients that multiply these derivatives can be determined by knowledge of the nature of the fluid. It should be noted that these solutions can also be reduced to steady flows with slip with higher-order slip boundary condition that includes both first- and second-order derivatives normal to the wall.