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

TL;DR

This paper derives an analytical solution for symmetric drainage flow of a compressible fluid from a fracture, revealing a slip-like flow rate proportional to channel gap and viscosity, differing from classical Poiseuille flow.

Contribution

It introduces a novel analytical model for compressible fluid drainage in fractures, highlighting a slip-like flow rate and its dependence on fluid and channel properties.

Findings

Flow exhibits a slip-like mass flow rate proportional to channel gap.

Drainage rate is proportional to kinematic viscosity, opposite to Poiseuille flow.

The formula applies to microchannel drainage as well.

Abstract

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.