A similarity parameter for capillary flows

Abstract

A similarity parameter for quasi-steady fluid flows advancing into horizontal capillary channels is presented. This parameter can be interpreted as the ratio of the average fluid velocity in the capillary channel to a characteristic velocity of quasi-steady capillary flows. It allows collapsing a large data set of previously published and recent measurements spanning five orders of magnitude in the fluid velocity, 14 different fluids, and four different geometries onto a single curve and indicates the existence of a universal prescription for such flows. On timescales longer than the characteristic time it takes for the flow to become quasi-steady, the one-dimensional momentum equation leads to a non-dimensional relationship between the similarity parameter and the penetration depth that agrees well with most measurements. Departures from that prescription can be attributed to effects that are not accounted for in the one-dimensional theory. This work is dedicated to Professor Harvey Lam on the occasion of his recovery.

One-dimensional mathematical and numerical modeling of liquid dynamics in a horizontal capillary

“…This condition implies that Bond, Capillary, and Weber numbers are small, that is Bo = 4ρgR 2 /γ 1, Ca = µ 0 U/γ 1, and W e = 2ρRU 2 /γ 1, where ρ and γ are the liquid density and surface tension, respectively, and g is the acceleration due to gravity. The Bond number, which is defined as the ratio of gravitational to surface tension forces, can be used to monitor if gravity is introducing two-dimensional effects into the problem, see Polzin and Choueiri [24] for a full discussion on this topic. The ratio between viscous and capillary forces is given by the capillary number.…”

Multiphase flow in micro- and nanochannels

“…For example, a slug will move from the wide part to the narrow part when the channel diameter changes [30] ( Fig. 2A).…”

Imbibition in geometries with axial variations

“…The quasi-steady movement of liquids through capillary tubes, due to the wetting of the liquid on the walls, was described a century ago (see Bell & Cameron 1906;Lucas 1918;Washburn 1921): for a capillary tube of constant section, the position of the meniscus obeys diffusive dynamics 2 = Dt, where represents the distance through which the liquid has moved in the time t and D is a coefficient that depends on the characteristics of the tube and the liquid. This robust result, often casually referred to as 'Washburn's law', also applies, at least approximately, to the dynamics of imbibition of porous media such as filter paper, packed beds of granular materials, dry soils (see Dullien 1979), as well as microtextured surfaces (see Bico, Tordeux & Quéré 2001;Courbin et al 2007), etc. Numerous studies on this topic of capillary invasion have been performed, including investigations of the influence of the geometry of the channels: the shape of a uniform cross-section (see Krotov & Rusanov 1999;Polzin & Choueiri 2003), the stepped capillary tube, i.e. a succession of different but uniform cross-sections (see Erickson, Li & Park 2002;Polzin & Choueiri 2003;Young 2004) and V-shaped open grooves (see Romero & Yost 1996;Rye, Yost & O'Toole 1998;Weislogel & Lichter 1998;Dussaud, Adler & Lips 2003;Warren 2004).…”

“…This robust result, often casually referred to as 'Washburn's law', also applies, at least approximately, to the dynamics of imbibition of porous media such as filter paper, packed beds of granular materials, dry soils (see Dullien 1979), as well as microtextured surfaces (see Bico, Tordeux & Quéré 2001;Courbin et al 2007), etc. Numerous studies on this topic of capillary invasion have been performed, including investigations of the influence of the geometry of the channels: the shape of a uniform cross-section (see Krotov & Rusanov 1999;Polzin & Choueiri 2003), the stepped capillary tube, i.e. a succession of different but uniform cross-sections (see Erickson, Li & Park 2002;Polzin & Choueiri 2003;Young 2004) and V-shaped open grooves (see Romero & Yost 1996;Rye, Yost & O'Toole 1998;Weislogel & Lichter 1998;Dussaud, Adler & Lips 2003;Warren 2004). But, to the best of our knowledge, all of these studies concern, at least locally (for the stepped capillary tube geometry) or totally, channels with uniform cross-sections.…”