Asymptotic analysis of the viscous micro/nano pump at low Reynolds number

Abstract

The steady viscous parabolic flow past an eccentrically placed rotating cylinder is studied in the asymptotic limit of small Reynolds number. It is assumed that the flow around the rotating cylinder undergoes boundary slip described by the Navier boundary condition. This involves a single parameter to account for the slip, referred to as the slip length , and replaces the standard no-slip boundary condition at solid boundaries. The streamlines for > 0 are closer to the body than for = 0, and it is discovered that the loss of symmetry due to the rotation of the cylinder is significantly reduced by the inclusion of slip. This arises as a result of a balance between the rotation velocity and the slip velocity on that portion of the cylinder which rotates opposite to the free-stream flow. Streamline patterns for nonzero eccentricity partially agree with Navier-Stokes simulations of the viscous pump; the small discrepancy is primarily due to the fact that here wall effects are not explicitly considered. Expressions for the frictional drag and the torque on the cylinder are obtained. The expression for the torque agrees well with the lubrication solution for the flow past a rotating cylinder placed symmetrically in a fully developed channel flow. The results presented here may be used to validate numerical schemes developed to study the viscous pump.

Preface to fourth Special Issue on Practical Asymptotics

“…Other topics in the fluid-mechanics discipline included here are high-Reynoldsnumber branching and network flows [4], time-dependent flows through curved pipes [5], low-Reynolds-number flow between two parallel rotating disks [6] and past a rotating cylinder [7], the classical dam-breaking problem [8], and premixed-flame propagation [9]. A few illustrative applications of these works are to blood flow in human arteries, coating processes in industry, or viscous pumps operating on the microscale.…”