Yield-stress fluid deposition in circular channels

Abstract

Since the pioneering works of Taylor and Bretherton, the thickness h of the film deposited behind a long bubble invading a Newtonian fluid is known to increase with the Capillary number power 2/3 (h ∼ RCa 2/3 ), where R is the radius of the circular tube and the Capillary number, Ca, comparing the viscous and capillary effects. This law, known as Bretherton law, is only valid in the limit of Ca < 0, 01 and negligible inertia and gravity. We revisit this classical problem when the fluid is a Yield-Stress Fluid (YSF) exhibiting both a yield stress and a shear-thinning behaviour. First, we provide quantitative measurement of the thickness of the deposited layer for Carbopol HerschelBulkley fluid in the limit where the yield-stress is of similar order of magnitude as the capillary pressure and for 0.1 < Ca < 1. To understand our observation, we use scaling arguments to extend the analytical expression of Bretherton's law to YSF in circular tubes. In the limit of Ca < 0, 1, our scaling law, in which the adjustable parameters are set using previous results concerning non-Newtonian fluid, successfully retrieves several features of the literature. First, it shows that (i) the thickness deposited behind a Bingham YSF (exhibiting a yield stress only) is larger than for a Newtonian fluid and (ii) the deposited layer increases with the amplitude of the yield stress. This is in quantitative agreement with previous numerical results concerning Bingham fluid. It also agrees with results concerning pure shear-thinning fluids in the absence of yield stress : the shear-thinning behaviour of the fluid reduces the deposited thickness as previously observed. Last, in the limit of vanishing velocity, our scaling law predicts that the thickness of deposited YSF converges towards a finite value, which presumably depends on the microstructure of the YSF, in agreement with previous research on the topic performed in different geometries. For 0.1 < Ca < 1,the scaling law fails to describe the data. In this limit, non-linear effects must be taken into account.

On the stability of the production of bubbles in yield-stress fluid using flow-focusing and T-junction devices

“…Indeed, when the flow accelerates the slip layer should be destructured and the no-slip boundary conditions must be recovered; (iii) for Bi −1 > 1 the film thickness seems to saturate toward an almost constant value. This value roughly corresponds to what is predicted for YSF with no-slip boundary conditions 33 , which is expected if the slip layer is destructured. In Figure 8, we also compare the measurements of the film thickness obtained for a single plug on a smooth surface (glass) to the results obtain for a train of plugs in STL channels (circumferantial roughness of typical size 50 µm).…”

“…These results highlight that the deposition of YSF on the wall of the channel is governed by Bi −1 . On the one hand, for Bi −1 > 1, the YSF is fluidized and the thickness of the deposited film increases with the velocity as observed recently in different geometries 26,33,36,37,46 . This efficient deposition process induces the rapid thinning of the plugs separating adjacent bubbles leading to their collapse.…”

“…Values of the film thickness h, obtained in the steady or unsteady experiments are shown on Figure 7 in a smooth glass capillary with R = 513 µm along with data obtained in the same geometry but for no-slip boundary conditions (using treated capillaries) 33 . The data obtained in the two different configurations (steady and unsteady experiments) collapse on the same figure, thus validating the unsteady measurement protocol.…”

Long bubbles in tubes filled with viscoplastic fluid

“…Previous computations and experiments for the propagation of bubbles down tubes filled with flowing viscoplastic fluid have been given by [10,11,12,13]. The existing computations deal with relatively large Capillary number, outside of the regime of validity of Bretherton's lubrication-style theory, and so there is minimal overlap between our results and these previous studies.…”

Motion of an oil droplet through a capillary with charged surfaces

“…The theory of Bretherton and various extensions thereof (e.g. Giavedoni & Saita (1997); Hazel & Heil (2002); Heil (2001); Laborie et al (2017); Ro & Homsy (1995); Severino et al (2003); Ubal et al (2008)) have numerous applications in fields such as foam dynamics (Cantat et al 2004;Green et al 2006;Reinelt & Kraynik 1990;Saugey et al 2006), microfluidics (Cantat 2013) or gas-assisted injection of a polymeric liquid into a mould (Gauri & Koelling 1999). One important application however is in oil recovery.…”