Symmetric drainage flow of a compressible fluid from a fracture: analytical solution and slip-like flow rate

Symmetric drainage flow of a compressible fluid from a fracture modeled as a long narrow channel is studied on the basis of linearized compressible Navier-Stokes equations with no-slip condition. The Helmholtz decomposition theorem is used to decompose the velocity field into an irrotational part and a solenoidal part. The irrotational velocity is driven by the fluid's volumetric expansion whilst the role of the solenoidal velocity is to enforce the no-slip condition for the overall velocity and it does not contribute to the mass flow rate. It is found that at large times this no-slip flow exhibits a time-dependent slip-like mass flow rate linearly proportional to the channel gap instead of the cubic power of the gap from the Poiseuille-type of flow. The drainage rate is also proportional to the kinematic viscosity, opposite to Poiseuille-type of flow which has a drainage rate proportional to the inverse of the kinematic viscosity. The same drainage rate formula also applies to drainage flow from a semi-sealed microchannel.