Symmetric drainage flow of a compressible fluid from a fracture

Analytical solution and slip-like flow rate

Di Shen, Kangping Chen

Research output: Contribution to journalArticle

Abstract

Symmetric drainage flow of a compressible fluid from a thin fracture modeled as a long narrow 2D channel is studied on the basis of linearized compressible Navier-Stokes equations with the 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 for the Poiseuille-type of flow. The drainage rate is also proportional to the kinematic viscosity, opposite to Poiseuille-type of flow, which produces 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 thin fracture.

Original languageEnglish (US)
Pages (from-to)15-21
Number of pages7
JournalMechanics Research Communications
Volume99
DOIs
StatePublished - Jul 1 2019

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slip flow
compressible fluids
drainage
Drainage
flow velocity
Flow rate
Fluids
slip
mass flow rate
kinematics
Viscosity
viscosity
Navier-Stokes equation
Navier Stokes equations
theorems
velocity distribution
Decomposition
decomposition
expansion
fluids

Keywords

  • Compressible flow
  • Microchannel flow
  • Navier-Stokes equations
  • No-slip condition

ASJC Scopus subject areas

  • Civil and Structural Engineering
  • Materials Science(all)
  • Condensed Matter Physics
  • Mechanics of Materials
  • Mechanical Engineering

Cite this

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abstract = "Symmetric drainage flow of a compressible fluid from a thin fracture modeled as a long narrow 2D channel is studied on the basis of linearized compressible Navier-Stokes equations with the 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 for the Poiseuille-type of flow. The drainage rate is also proportional to the kinematic viscosity, opposite to Poiseuille-type of flow, which produces 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 thin fracture.",
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author = "Di Shen and Kangping Chen",
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AU - Shen, Di

AU - Chen, Kangping

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N2 - Symmetric drainage flow of a compressible fluid from a thin fracture modeled as a long narrow 2D channel is studied on the basis of linearized compressible Navier-Stokes equations with the 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 for the Poiseuille-type of flow. The drainage rate is also proportional to the kinematic viscosity, opposite to Poiseuille-type of flow, which produces 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 thin fracture.

AB - Symmetric drainage flow of a compressible fluid from a thin fracture modeled as a long narrow 2D channel is studied on the basis of linearized compressible Navier-Stokes equations with the 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 for the Poiseuille-type of flow. The drainage rate is also proportional to the kinematic viscosity, opposite to Poiseuille-type of flow, which produces 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 thin fracture.

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