Yield-stress fluid deposition in circular channels

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 $2/3$ ($h\sim RCa^{2/3}$), where $R$ is the radius of the circular tube and $Ca$ is the capillary number, comparing the viscous and capillary effects. This law, known as Bretherton’s law, is valid only 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, a Herschel–Bulkley fluid, in the limit where the yield stress is of a similar order of magnitude to the capillary pressure and for $0.1<Ca<1$. To understand our observations, we use scaling arguments to extend the analytical expression of Bretherton’s law to YSFs 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 fluids, 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 fluids. 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 the 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, nonlinear effects must be taken into account.