Viscoplastic fluids can hold bubbles/particles stationary by balancing the buoyancy stress with the yield stress. In the present study, we investigate a suspension of bubbles in a yield-stress fluid. More precisely, we compute how much the gas fraction $ϕ$ that could be held trapped in a yield-stress fluid without motion. The key parameter is the yield number which represents the ratio of the yield stress to the buoyancy stress. Here the goal is to shed light on how the bubbles feel their neighbours through the stress field and to compute the critical yield number for a bubble cloud beyond which the flow is suppressed. We perform 2D computations in a full periodic box with randomized positions of the monosized circular bubbles. A large number of configurations are investigated to obtain statistically converged results. We intuitively expect that for higher volume fractions the critical yield number is larger. Not only here do we establish that this is the case, but also we show that short range interactions of bubbles increase the critical yield number even more dramatically for bubble clouds. The results show that the critical yield number is a linear function of volume fraction in the dilute regime. An algebraic expression model is given to approximate the critical yield number (semi-empirically) based on the numerical experiment in the studied range of $0\le ϕ\le 0.31$, together with lower and upper estimates.
The ubiquitous wall slip behavior of viscoplastic fluids renders the analysis of their steady torsional flow data to determine their yield stress and other parameters of their shear viscosity material function challenging. Roughened surfaces to prevent wall slip can be used but such methods frequently result in the fracturing of the sample. Instead, it is shown here that the yield stress of a viscoplastic fluid can be determined from the torque versus apparent shear rate data of steady torsional flow using a single gap. The step change in the slope of the torque versus apparent shear rate data…
The spreading under surface tension and gravity of a droplet of yield-stress fluid over a thin film of the same material is studied. The droplet converges to a final equilibrium shape once the driving stresses inside the droplet fall below the yield stress. Scaling laws are presented for the final radius and complemented with an asymptotic analysis for shallow droplets. Moreover, numerical simulations using the volume-of-fluid method and a regularized constitutive law, and experiments with an aqueous solution of Carbopol are presented.
A planar two-dimensional computational analysis is presented to qualify traditional and fractal vane-in-cup geometries for accurate rheometry of simple viscoplastic fluids with and without slip. Numerical simulations based on an adaptive augmented Lagrangian scheme are used to study the two-dimensional flow field of yield-stress fluids within and around vane tools with N=3 to 24 arms for a wide range of Bingham numbers, B (i.e. the ratio of the yield stress over the characteristic viscous stress). This allows for accurate calculations of the velocity and stress fields around vanes with various…
This study presents the spreading of a single filament of a yield stress (viscoplastic) fluid extruded onto a pre-wetted solid surface. The filaments spread laterally under surface tension forces until they reach a final equilibrium shape when the yield stress dominates. We use a simple experimental setup to print the filaments on a moving surface and measure their final width using optical coherence tomography. Additionally, we present a scaling law for the final width and determine the corresponding pre-factor using asymptotic analysis. We then analyse the level of agreement between the theo…
An axisymmetric squirmer in a Bingham viscoplastic fluid is studied numerically to determine the effect of a yield stress environment on locomotion. The nonlinearity of the governing equations necessitates numerical methods, which is accomplished by solving a variable-viscosity Stokes equation with a Finite Element approach. The effects of stroke modes, both pure and combined, are investigated and it is found that for the treadmill or "neutral" mode, the swimmer in a yield stress fluid has a lower swimming velocity and uses more power. However, the efficiency of swimming reaches its maximum at…
Turbulent flows of viscoplastic fluids at high Reynolds numbers have been investigated recently with direct numerical simulations (DNS) but experimental results have been limited. For this reason, we carry out an experimental study of fully turbulent flows of a yield stress fluid in a rectangular aspect ratio channel with a high-resolution laser doppler velocimetry (LDA) setup. We employ aqueous Carbopol solutions, often considered to be a simple yield stress fluid. We formulate different concentrations to address the effect of the rheology of the fluid on the turbulence statistics at an appro…
Considerations on implementation of the stress-strain constitutive relations applied in Computational Fluid dynamics (CFD) simulation of cardiovascular flows have been addressed extensively in the literature. However, the matter is yet controversial. The author suggests that the choice of non-Newtonian models and the consideration of non-Newtonian assumption versus the Newtonian assumption is very application oriented and cannot be solely dependent on the vessel size. In the presented work, where a renal disease patient-specific geometry is used, the non-Newtonian effects manifest insignifican…
The evolution of the liquid bridge formed between two coalescing sessile yield-stress drops is studied experimentally. We find that the height of the bridge evolves similar to a viscous Newtonian fluid, $h_0\sim t$, before arresting at long time prior to minimizing its liquid/gas interfacial energy. We numerically solve for the final arrested profile shape and find it depends on the fluid's yield stress $τ_y$ and coalescence angle $α$, represented by the Bingham number $τ_y h_{drop} / σ$ modified by the drop's height-width aspect ratio. We present a scaling argument for the bridge's temporal e…
Modeling elasto-viscoplastic free-surface flows with different yield surfaces
Institut Jean Le Rond d’Alembert, Sorbonne Université（索邦大学阿勒姆伯特力学研究所）；CNRS - UMR 7190（法国国家科学研究中心第7190联合研究实验室）；Institute for Geotechnical Engineering, ETH Zürich（苏黎世联邦理工学院岩土工程研究所）；Université Grenoble Alpes（格勒诺布尔阿尔卑斯大学）；INRAE（法国国家农业、食品和环境研究院）；CNRS（法国国家科学研究中心）；IRD（法国国家科研中心）；Grenoble INP（格勒诺布尔国立高等工程学院）；IGE（冰川与地球物理学研究所）
We discuss the applicability of a unified hyperbolic model for continuum fluid and solid mechanics to modeling non-Newtonian flows and in particular to modeling the stress-driven solid-fluid transformations in flows of viscoplastic fluids, also called yield-stress fluids. In contrast to the conventional approaches relying on the non-linear viscosity concept of the Navier-Stokes theory and representation of the solid state as an infinitely rigid non-deformable solid, the solid state in our theory is deformable and the fluid state is considered rather as a "melted" solid via a certain procedure …
Bubbles in complex fluids are often desirable, and sometimes simply inevitable, in the processing of formulated products. Bubbles can rise by buoyancy, grow or dissolve by mass transfer, and readily respond to changes in pressure, thereby applying a deformation to the surrounding complex fluid. The deformation field around a stationary, spherical bubble undergoing a change in radius is simple and localised, thus making it suitable for rheological measurements. This article reviews emerging approaches to extract information on the rheology of complex fluids by analysing bubble dynamics. The foc…
We use computational methods to determine the minimal yield-stress required in order to hold static a buoyant bubble in a yield-stress liquid. The static limit is governed by the bubble shape, the dimensionless surface tension ($γ$) and the ratio of the yield-stress to the buoyancy stress ($Y$). For a given geometry, bubbles are static for $Y > Y_c$, which we determine for a range of shapes. Given that surface tension is negligible, long prolate bubbles require larger yield-stress to hold static compared to oblate bubbles. Non-zero $γ$ increases $Y_c$ and for large $γ$ the yield-capillary numb…
Yield-stress fluids naturally trap small bubbles when their buoyancy applies an insufficient stress to induce local yielding of the material. Under acoustic excitation, trapped bubbles can be driven into volumetric oscillations and apply an additional local strain and stress that can trigger yielding and assist their release. In this paper we explore different regimes of microbubble oscillation and translation driven by an ultrasound field in a model yield-stress fluid, a Carbopol microgel. We first analyse the linear bubble oscillation dynamics to measure the local, high-frequency viscosity o…
Various biological and chemical processes lead to the nucleation and growth of non-wetting fluid bubbles within the pore space of a granular medium, such as the formation of gas bubbles in liquid-saturated lake-bed sediments. In sufficiently soft porous materials, the non-wetting nature of these bubbles can result in the formation of open cavities within the granular solid skeleton. Here, we consider this process through the lens of phase separation, where thermomechanics govern the separation of the non-wetting phase from a fluid-fluid-solid mixture. We construct a phase-field model informed …
The motion of a long gas bubble in a confined capillary tube is ubiquitous in a wide range of engineering and biological applications. While the understanding of the deposited thin viscous film near the tube wall in Newtonian fluids is well developed, the deposition dynamics in commonly encountered non-Newtonian fluids remains much less studied. Here, we investigate the dynamics of a confined bubble moving in shear-thinning fluids with systematic experiments, varying the zero-shear-rate capillary number $Ca_0$ in the range of $O(10^{-3}-10^2)$ considering the zero-shear-rate viscosity. The thi…
In this paper, we have studied the three-dimensional dynamics of two equally sized air bubbles rising in a shear-thinning fluid. We have used the combined level set and volume of fluid (CLSVOF) method to track interface, maintain mass balance and estimate the interface curvature. Additionally, we have incorporated a Sharp Surface Force Method (SSF) for surface tension forces. This method significantly suppressed the spurious velocities commonly observed with the conventional volume of fluid method and the Continuum Surface Force (CSF). The algorithm is implemented in an in-house code called CU…
A numerical method is presented to simulate gas-liquid-solid flows with bubble-particle interaction, including particle collision, sliding, and attachment. Gas-liquid flows are simulated in an Eulerian framework using a volume-of-fluid method. Particle motions are predicted in a Lagrangian framework. Algorithms that are used to detect collision and determine the sliding or attachment of the particle are developed. An effective bubble is introduced to model these bubble-particle interaction. The proposed numerical method is validated through experimental cases that entail the rising of a single…
The oscillatory flows present in an inkjet printhead can lead to strong deformations of the air-liquid interface at the nozzle exit. Such deformations may lead to an inward directed air jet with bubble pinch-off and the subsequent entrainment of an air bubble, which is highly detrimental to the stability of inkjet printing. Understanding the mechanisms of bubble entrainment is therefore crucial in improving print stability. In the present work, we use ultrafast X-ray phase-contrast imaging and direct numerical simulations based on the Volume-of-Fluid method to study the mechanisms underlying t…
