The lubrication flow of a Herschel-Bulkley fluid in a symmetric long channel of varying width, 2h(x), is modeled extending the approach proposed by Fusi et al. [“Pressure-driven lubrication flow of a Bingham fluid in a channel: A novel approach,” J. Non-Newtonian Fluid Mech. 221, 66–75 (2015)] for a Bingham plastic. Moreover, both the consistency index and the yield stress are assumed to be pressure-dependent. Under the lubrication approximation, the pressure at zero order depends only on x and the semi-width of the unyielded core is found to be given by σ(x) = −(1 + 1/n)h(x) + C, where n is the power-law exponent and the constant C depends on the Bingham number and the consistency-index and yield-stress growth numbers. Hence, in a channel of constant width, the width of the unyielded core is also constant, despite the pressure dependence of the yield stress, and the pressure distribution is not affected by the yield-stress function. With the present model, the pressure is calculated numerically solving a...

Pressure-driven flow of a Herschel-Bulkley fluid with pressure-dependent rheological parameters

We investigate a mathematical model describing 3D steady-state flows of Bingham-type fluids in a bounded domain under threshold-slip boundary conditions, which state that flows can slip over solid surfaces when the shear stresses reach a certain critical value. Using a variational inequalities approach, we suggest the weak formulation to this problem. We establish sufficient conditions for the existence of weak solutions and provide their energy estimates. Moreover, it is shown that the set of weak solutions is sequentially weakly closed in a suitable functional space.

/pdf/on-flows-of-bingham-type-fluids-with-threshold-slippage-ansjt4clst.pdf

On Flows of Bingham-Type Fluids with Threshold Slippage

https://hal-upec-upem.archives-ouvertes.fr/hal-01133786/document

Two-dimensional simulation by regularization of free surface viscoplastic flows with Drucker–Prager yield stress and application to granular collapse

The aim of this contribution is to present a new Newton-type solver for yield stress fluids, for instance for viscoplastic Bingham fluids. In contrast to standard globally defined (‘outer’) damping strategies, we apply weighting strategies for the different parts inside of the resulting Jacobian matrices (after discretizing with FEM), taking into account the special properties of the partial operators which arise due to the differentiation of the corresponding nonlinear viscosity function. Moreover, we shortly discuss the corresponding extension to fluids with a pressure-dependent yield stress which are quite common for modelling granular material. From a numerical point of view, the presented method can be seen as a generalized Newton approach for non-smooth problems.

https://www.mathematik.tu-dortmund.de/lsiii/cms/papers/MandalOuazziTurek2015.pdf

Modified Newton Solver for Yield Stress Fluids

We present a two-dimensional model of the primary cementing process for foamed cement slurries. Foamed cement slurries have a number of claimed advantages, but also have a pressure-dependent density and rheology. The rheology is hard to quantify fully over all ranges of foam quality, which compromises the accuracy of models. The density variation is due to expansion/compression of the gas phase along the well, caused by variations in the static pressure. We show that in the absence of careful control, buoyancy-driven instabilities can result in the annulus, as the foamed slurry expands and the density drops below that of the displaced drilling mud. These instabilities appear to be of a classic porous media/Hele-Shaw cell fingering type, triggered by a threshold unstable density difference. We show that these instabilities are amplified by wellbore eccentricity, occurring lower in the well than in a concentric annulus. Our results question the safe usage of foamed cements in primary cementing.

A model for foamed cementing of oil and gas wells

The goal of this article is to provide some essential results for the solution of a regularized viscoplastic frictional flow model adapted from the extensive mathematical analysis of the Bingham model. The Bingham model is a standard for the description of viscoplastic flows and it is widely used in many application areas. However, wet granular viscoplastic flows necessitate the introduction of additional non-linearities and coupling between velocity and stress fields. This article proposes a step toward a frictional coupling, characterized by a dependence of the yield stress to the pressure field. A regularized version of this viscoplastic frictional model is analysed in the framework of stationary flows. Existence, uniqueness and regularity are investigated, as well as finite-dimensional and algorithmic approximations. It is shown that the model can be solved and approximated as far as a frictional parameter is small enough. Getting similar results for the non-regularized model remains an issue. Numerical investigations are postponed to further works.

Analysis of a Regularized Bingham Model with Pressure-Dependent Yield Stress

Advection–dispersive phenomena inside two geometries of porous media, corresponding to a structured and unstructured network of minichannels, are studied from a combination of analytical, numerical, and electrodiffusion techniques. The instantaneous limiting diffusion current, connected to the concentration of electroactive species flowing in the porous matrix, is recorded and measured during flow experimentation. Transport phenomena investigation consists of the step injection of a tracer of higher concentration than the bulk flow in order to characterize the mixing of electrolyte species inside two arrangements of network, so called $$\times $$
 _network and T_network. The experimental results are supported by 2D-numerical simulations performed in the $$\times $$
 _network. A pore model is proposed in order to predict the pore velocity, which is used within the resolution of the diffusion–convection balance. The numerical simulations, based on a second-order finite difference scheme, give rise to a good agreement in terms of mixing index and the methodology employed for the numerical injection concentration appears suitable. The numerical experiments are quite-well representative in laminar regime below the critical Reynolds number (
 $$Re_{\mathrm{crit}}\approx \,200$$
 ). Above $$Re_{\mathrm{crit}}$$
 , the inertial effects are not negligible and the momentum transfer needs to be taken into account. This latter phenomenon is analyzed at the pore-scale in term of local skin friction measured at the channels crossing, then compared and discussed from analytical solutions.

Advection–Dispersive Mechanisms of Electrolyte Species in Porous Matrix

The aim of this paper is to study the flow of a frictional viscoplastic material around a circular cylinder. A Frictional Viscoplastic Law FVL(p0) for this purpose is described and discussed. It is a Bingham law whose threshold depends on hydrostatic pressure p0. Numerical simulations by mixed finite element are used together with a fixed point resolution algorithm. These simulations put in relief different kinematics with respect to reference pressure p0. However, a behavior similar Bingham fluid of threshold Bi(p0) is observed. Thus, the energy cost is compared as that of a Bingham fluid. Moreover, the non-existence of Stokes paradox is postulated. Mots clefs : pâte viscoplastique frictionnelle, fluide a seuil, cylindre.

http://documents.irevues.inist.fr/bitstream/handle/2042/52684/a_IFZCXU1T.pdf?sequence=1

Simulation numérique d'un écoulement viscoplastique frictionnel autour d'un cylindre circulaire

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Analysis of a Regularized Bingham Model with Pressure-Dependent Yield Stress

Advection–Dispersive Mechanisms of Electrolyte Species in Porous Matrix

Simulation numérique d'un écoulement viscoplastique frictionnel autour d'un cylindre circulaire