Pressure-driven transient flows of Newtonian fluids through microtubes with slip boundary

Recent advances in microscale experiments and molecular simulations confirm that slip of fluid on solid surface occurs at small scale, and thus the traditional no-slip boundary condition in fluid mechanics cannot be applied to flow in micrometer and nanometer scale tubes and channels. On the other hand, there is an urgent need to understand fluid flow in micrometer scale due to the emergence of biochemical lab-on-the-chip system and micro-electromechanical system fabrication technologies. In this paper, we study the pressure driven transient flow of an incompressible Newtonian fluid in microtubes with a Navier slip boundary condition. An exact solution is derived and is shown to include some existing known results as special cases. Through analysis of the derived solution, it is found that the influences of boundary slip on the flow behaviour are qualitatively different for different types of pressure fields driving the flow. For pressure fields with a constant pressure gradient, the boundary slip does not alter the interior material deformation and stress field; while, for pressure fields with a wave form pressure gradient, the boundary slip causes the change of interior material deformation and consequently the velocity profile and stress field. We also derive asymptotic expressions for the exact solution through which a parameter is identified to dominate the behaviour of the flow driven by the wave form pressure gradient, and an explicit formulae for the critical slip parameter leading to the maximum transient flow rate is established.