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Transport in Porous Media

, Volume 84, Issue 1, pp 109–132 | Cite as

Radial Capillary Transport from an Infinite Reservoir

  • Michael ConrathEmail author
  • Nicolas Fries
  • Ming Zhang
  • Michael E. Dreyer
Article

Abstract

Radial capillary transport occurs, for example, when wine spreads in the tablecloth ink in paper, rain drops in textiles, or dye into yarn. It is of technical relevance for propellant and other liquid transport in space. We present a theoretical and experimental study on the more basic situation when liquid spreads radially from an infinite reservoir. Our theoretical model predicts both outward and inward radial transport in a porous screen. While the outward wicking is fed by a circular wick in the center, the inward wicking is fed by a ring-like wick from the outside. For both cases, an analytical solution is obtained in terms of time versus radius as well as radius versus time aided by the Lambert W function. In the experiments, we use four different filter papers combined with three cylindrical wicks for outward wicking and one ring wick for inward wicking, respectively. The wicking process is recorded by a digital camera. Afterward, the resulting image series are evaluated with Matlab routines to detect the wicking front line. From the wetted area, we derive the mean radius versus time. Beside radially outward and inward wicking, we consider also experimental reference data from horizontal and vertical wicking in a strip.

Keywords

Wicking Radial capillary transport Porous medium Lucas–Washburn Imbibition 

Nomenclature

\({\phi}\)

Porosity of the screen

μ

Dynamic viscosity (kg/ms)

π1

Dimensionless parameter

π2

Dimensionless parameter

π3

Dimensionless parameter

ρ

Density of liquid (kg/m3)

ρh

Regression coefficient (horizontal strip exp.)

ρv

Regression coefficient (vertical strip exp.)

σ

Surface tension (N/m)

θ

Contact angle

a

Viscous wicking constant (m2/s)

ah

a found in horizontal strip experiment (m2/s)

av

a found in vertical strip experiment (m2/s)

\({\overline{a}}\)

Mean value of a (m2/s)

Δa

Error of a (m2/s)

b

Inertial wicking constant (m2/s2)

Ca

Capillary number

Δh

Height difference between screen and reservoir (m)

H

Height (=thickness) of the screen (m)

K

Permeability of the screen (m2)

L

Characteristic length (m)

Oh

Ohnesorge number

r

Radial coordinate (m)

r0

Initial radius of the wetted spot (wick radius) (m)

R

Radius of the wetted spot (m)

Rs

Mean static pore radius (m)

R*

Dimensionless spot radius

s

Distance between screen and camera (m)

t

Time (s)

t*

Dimensionless time

v

Velocity of the front line (m/s)

\({\dot{V}}\)

Liquid flow rate (m3/s)

\({\dot{V^\ast}}\)

Dimensionless liquid flow rate

w

Strip width in horizontal reference experiment (m)

W(..)

Lambert W function

x0

Initial meniscus position in the strip experiments (m)

X

Meniscus position in the strip experiments (m)

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Copyright information

© Springer Science+Business Media B.V. 2009

Authors and Affiliations

  • Michael Conrath
    • 1
    Email author
  • Nicolas Fries
    • 1
  • Ming Zhang
    • 1
  • Michael E. Dreyer
    • 1
  1. 1.Center of Applied Space Technology and Microgravity (ZARM)University of BremenBremenGermany

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