Drainage flow of a viscous compressible fluid from a small capillary with a sealed end

Volumetric expansion driven drainage flow of a viscous compressible fluid from a small capillary with a sealed end is studied in the low Mach number limit using the linearized compressible Navier-Stokes equations with no-slip condition. Density relaxation, oscillation and decay as well as the velocity field are investigated in detail. It is shown that fluid drainage is controlled by the slow decay of the standing acoustic wave inside the capillary; and the acoustic wave retards the density diffusion by reducing the diffusion coefficient of the density envelope equation by one half. Remarkably the no-slip flow exhibits a slip-like mass flow rate. The period-averaged mass flow rate at the exit (drainage rate) is found proportional to the fluid's kinematic viscosity via the density diffusion coefficient and the average drainage speed is independent of the capillary radius. These findings are valid for arbitrarily small capillaries as long as the continuum assumption holds and they are in stark contrast to the classical lubrication based theory. Generalization to a capillary with a sound absorbing end is achieved by a simple model. The reported results offer new insights to the nature of slow viscous compressible flows in very small capillaries.