We consider the viscous limit of a plane channel miscible displacement flow of two generalized Newtonian fluids when buoyancy is significant. The channel is inclined close to horizontal. A lubrication/thin-film approximation is used to simplify the governing equations and a semi-analytical solution is found for the flux functions. We show that there are no steady travelling wave solutions to the interface propagation equation. At short times the diffusive effects of the interface slope are dominant and there is a flow reversal, relative to the mean flow. We are able to find a short-time similarity solution governing this initial counter-current flow. At longer times the solution behaviour can be predicted from the associated hyperbolic problem (where diffusive effects are set to zero). Each solution consists of a number N ≥ 1 of steadily propagating fronts of differing speeds, joined together by segments of interface that are stretched between the fronts. Diffusive effects are always present in the propagating fronts. We explore the effects of viscosity ratio, inclinations and other rheological properties on the front height and front velocity. Depending on the competition of viscosity, buoyancy and other rheological effects, it is possible to have single or multiple fronts. More efficient displacements are generally obtained with a more viscous displacing fluid and modest improvements may also be gained with slight positive inclination in the direction of the density difference. Fluids that are considerably shear-thinning may be displaced at high efficiencies by more viscous fluids. Generally, a yield stress in the displacing fluid increases the displacement efficiency and yield stress in the displaced fluid decreases the displacement efficiency, eventually leading to completely static residual wall layers of displaced fluid. The maximal layer thickness of these static layers can be directly computed from a one-dimensional momentum balance and indicates the thickness of static layer found at long times.

Buoyancy-dominated displacement flows in near-horizontal channels: the viscous limit

We present results of a primarily experimental study of buoyant miscible displacement flows of a yield stress fluid by a higher density Newtonian fluid along a long pipe, inclined at angles close to horizontal. We focus on the industrially interesting case where the yield stress is significantly larger than a typical viscous stress in the displacing fluid, but where buoyancy forces may be significant. We identify two distinct flow regimes: a central-type displacement regime and a slump-type regime for higher density ratios. In the central-type displacement flows, we find non-uniform static residual layers all around the pipe wall with long-wave variation along the pipe. In the slump-type displacement we generally detect two propagating displacement fronts. A fast front propagates in a thin layer near the bottom of the pipe. A much slower second front follows, displacing a thicker layer of the pipe but sometimes stopping altogether when buoyancy effects are reduced by spreading of the front. In the thin lower layer the flow rate is focused which results in large effective Reynolds numbers, moving into transitional regimes. These flows are frequently unsteady and the displacing fluid can channel through the yield stress fluid in an erratic fashion. We show that the two regimes are delineated by the value of the Archimedes numbers (equivalently, the Reynolds number divided by the densimetric Froude number), a parameter which is independent of the imposed flow rate. We present the phenomenology of the two flow regimes. In simplified configurations, we compare computational and analytical predictions of the flow behaviour (e.g. static layer thickness, axial velocity) with our experimental observations.

Incomplete fluid-fluid displacement of yield stress fluids in near-horizontal pipes: Experiments and theory

We implemented a finite element-front tracking method (FE-FTM) to understand the drop dynamics in microfluidic applications. We investigated the effect of viscoelasticity of both drop and medium. The Oldroyd-B model was used for viscoelastic fluid, and DEVSS-G/SUPG/matrix logarithm algorithms were applied to improve numerical stability. We first verified the reliability of the algorithm by comparing with previous results under simple shear flow. The results were in good agreement with previous reports in a wide range of parameters such as capillary number (Ca) and Deborah number (De). Then we applied the algorithm to a 5:1:5 planar contraction – narrow channel – expansion flow which is typical microfluidic flows. One of the goal of this study is to explore the effect of viscoelasticity of drop and medium on drop deformation, and to propose the strategy to control the drop shape. When a Newtonian drop was suspended in a viscoelastic medium, in the narrow channel region, we observed an ‘ellipsoid-like drop’ in contrast to a ‘bullet-like drop’ which is the typical shape for the case of a Newtonian drop in Newtonian medium. When a viscoelastic drop is suspended in Newtonian medium, the extent of ‘drop swell’ to the cross-stream direction was enhanced at the exit of the narrow channel. We explained these phenomena in terms of the normal stress difference developed in the viscoelastic fluid. The present study shows that the viscoelasticity plays a significant role on drop dynamics and drop shape in particular. We expect this study will be helpful to understand the drop dynamics in microchannel flow, and provide useful information in manipulating drops in complicated geometries such as microfluidic channels.

Numerical study on the effect of viscoelasticity on drop deformation in simple shear and 5:1:5 planar contraction/expansion microchannel

Recent years have seen a flurry of research on yield stress fluids, approached from different perspectives (physicochemical, rheological and fluid mechanical), considering different length scales and timescales, using a range of tools: experimental, computational and analytical. In this context, it has been common to denigrate the simple models of yield stress fluids for their various acknowledged deficiencies and proffer improved models. Here, we push back, in defence of the century-old simplicity, exploring what is new and useful and what these models have given in the past few decades.

Simple yield stress fluids

We study steady and pulsating displacement flows of a Bingham fluid by a Newtonian fluid, along a plane channel. For sufficiently large yield stress a static residual wall layer can result during the displacement. The flow is parameterised by the Reynolds number (Re), the Bingham number (B) and the viscosity ratio (M). Perhaps intuitively, thicker layers are found with larger M and at lower Re. The residual layer is formed on the advective timescale of the displacement but drains on a slower timescale governed by M. For larger M truly stationary layers are only found for large t when the layer has thinned sufficiently to become static. Increased Re results in increased energy production locally around the finger. For large enough Re the energy production can play a significant role in yielding the fluid. As the energy production rate increases it also becomes focused around the corner or shoulder region of the front, and spreads axially along the initial part of the residual layer. This causes fluid to yield increasingly far behind the front and allows for the layer to thin. As B increases the static layer tends to decrease (see also [1] , [2] ). At small Re the static layer thickness appears to be independent of M. At large Re the layer thickness is dependent on M and decreases asymptotically to a constant value as B → ∞. For pulsating displacement flow rates, Q(t) = 2(1 + Asin  ωt) : A ∈ [0, 1) we study two ranges: ωRe ≪ 2π and ωRe ≫ 2π. For the viscous regime (ωRe ≪ 2π) a pseudo-steady 1D model predicts that the residual layer should remain static for 3(1 + Asin  ωt)

Static wall layers in plane channel displacement flows

An elliptic mesh generation technique, with the Galerkin Finite Element Method is used to compute the free surface of the two-phase flow problem of a gas displacing a non-Newtonian material in a capillary tube. Two classes of non-Newtonian materials were investigated: a power-law shear-thinning liquid and a visco-plastic material with the viscosity function proposed by Papanastasiou [T.C. Papanastasiou, Flows of materials with yield-stress, J. Rheol. 31 (1987) 385–404]. The results were given as a function of a non-Newtonian capillary number and a rheological dimensionless parameter: the behavior index in the shear-thinning liquid case and a dimensionless yield-stress (equivalent to Bingham number) in the visco-plastic material. The goal of the present work is to study flow patterns, configuration of the interface between the two phases, and fraction of the mass of non-Newtonian material deposited at the wall, as functions of the dimensionless numbers cited. Some general results of experimental and numerical works found in literature were reproduced with a quite good agreement and a wider range of the dimensionless numbers was investigated. In both classes of materials studied, as the displaced fluid departs from Newtonian behavior, the fraction of the mass deposited on the tube wall decreases and the shape of the interface becomes flatter. It was offered a plausible explanation for this counter-intuitive result related to visco-plastic fluid, i.e. an apparent increase of its viscosity (increasing the dimensionless yield-stress number), induces a decrease of the layer thickness left behind. Concerning flow patterns, it was possible to identify ranges, dependent on rheological properties and capillary number, where the transition between bypass flows and fully recirculating flows occurs. In the Newtonian case all the flow patterns predicted by Taylor [G.I. Taylor, Deposition of a viscous fluid on the wall a tube, J. Fluid Mech. 10 (1961) 161–165] were well captured. In the visco-plastic and pseudo-plastic case it was found an interesting type of intermediate flow regime which is not present in the Newtonian transition.

Dione A. de Sousa

Papers

Numerical investigation on gas-displacement of a shear-thinning liquid and a visco-plastic material in capillary tubes