A theoretical and numerical study of complex sliding flows of yield-stress fluids is presented. Yield-stress fluids are known to slide over solid surfaces if the tangential stress exceeds the sliding yield stress. The sliding may occur due to various microscopic phenomena such as the formation of an infinitesimal lubrication layer of the solvent and/or elastic deformation of the suspended soft particles in the vicinity of the solid surfaces. This leads to a 'stick-slip' law which complicates the modelling and analysis of the hydrodynamic characteristics of the yield-stress fluid flow. In the present study, we formulate the problem of sliding flow beyond one-dimensional rheometric flows. Then, a numerical scheme based on the augmented Lagrangian method is presented to attack these kind of problems. Theoretical tools are developed for analysing the flow/no-flow limit. The whole framework is benchmarked in planar Poiseuille flow and validated against analytical solutions. Then two more complex physical problems are investigated: slippery particle sedimentation and pressure-driven sliding flow in porous media. The yield limit is addressed in detail for both flow cases. In the particle sedimentation problem, method of characteristics - slipline method - in the presence of slip is revisited from the perfectly plastic mechanics and used as a helpful tool in addressing the yield limit. Finally, flows through model and randomized porous media are studied. The randomized configuration is chosen to capture more sophisticated aspects of the yield-stress fluid flows in porous media at the yield limit - channelization. 

We study the dynamics of a neutrally buoyant rigid sphere carried by an elastoviscoplastic fluid in a pressure-driven channel flow numerically. The yielding to flow is marked by the yield stress which splits the flow into two main regions: the core unyielded region and two sheared yielded regions close to the walls. The particles which are initially in the plug region are observed to translate with the same velocity as the plug without any rotation/migration. Keeping the Reynolds number fixed, we study the effect of elasticity (Weissenberg number) and plasticity (Bingham number) of the fluid on the particle migration inside the sheared regions. In the viscoelastic limit, in the range of studied parameters (low elasticity), inertia is dominant and the particle finds its equilibrium position between the centreline and the wall. The same happens in the viscoplastic limit, yet the yield surface plays the role of centreline. However, the combination of elasticity and plasticity of the suspending fluid (elastoviscoplasticity) trigger particle-focusing: in the elastoviscoplastic flow, for a certain range of Weissenberg numbers (≈0.5), isolated particles migrate all the way to the centreline by entering into the core plug region. This behaviour suggests a particle-focusing process for inertial regimes which was not previously found in a viscoelastic or viscoplastic carrying fluid. 

The stability of neutrally and non-neutrally buoyant particles immersed in a plane Poiseuille flow of a yield-stress fluid (Bingham fluid) is addressed numerically. Particles being carried by the yield-stress fluid can behave in different ways: they might (i) migrate inside the yielded regions or (ii) be transported without any relative motion inside the unyielded region if the yield stress is large enough compared to the buoyancy stress and the other stresses acting on the particles. Knowing the static stability of particles inside a bath of quiescent yield-stress fluid (Chaparian & Frigaard, J. Fluid Mech., vol. 819, 2017, pp. 311-351), we analyse the latter behaviour when the yield-stress fluid Poiseuille flow is host to two-dimensional particles. Numerical experiments reveal that particles lose their stability (i.e. break the unyielded plug and sediment/migrate) with smaller buoyancy compared to the sedimentation inside a bath of quiescent yield-stress fluid, because of the inherent shear stress in the Poiseuille flow. The key parameter in interpreting the present results is the position of the particle relative to the position of the yield surface in the undisturbed flow (in the absence of any particle): the larger the portion of a particle located inside the undisturbed sheared regions, the more likely is the particle to be unstable. Yet, we find that the core unyielded plug can grow locally to some extent to contain the particles. This picture holds even for neutrally buoyant particles, although they are strictly stable when they are located wholly inside the undisturbed plug. We propose scalings for all cases.

A numerical and theoretical study of yield-stress fluid flows in two types of model porous media is presented. We focus on viscoplastic and elastoviscoplastic flows to reveal some differences and similarities between these two classes of flows. Small elastic effects increase the pressure drop and also the size of unyielded regions in the flow which is the consequence of different stress solutions compare to viscoplastic flows. Yet, the velocity fields in the viscoplastic and elastoviscoplastic flows are comparable for small elastic effects. By increasing the yield stress, the difference in the pressure drops between the two classes of flows becomes smaller and smaller for both considered geometries. When the elastic effects increase, the elastoviscoplastic flow becomes time-dependent and some oscillations in the flow can be observed. Focusing on the regime of very large yield stress effects in the viscoplastic flow, we address in detail the interesting limit of 'flow/no flow': yield-stress fluids can resist small imposed pressure gradients and remain quiescent. The critical pressure gradient which should be exceeded to guarantee a continuous flow in the porous media will be reported. Finally, we propose a theoretical framework for studying the 'yield limit' in the porous media.

Elastoviscoplastic fluids are a class of yield-stress fluids that behave like neoHookean (or viscoelastic) solids when the imposed stress is less than the yield stress whereas after yielding, their behaviour is described by a viscoplastic fluid with an additional elastic history. This exceptional behaviour has been recently observed by many yield stress fluids in rheometric tests such as waxy crude oil, Carbopol gel, etc. Moreover, interesting phenomena have been evidenced experimentally such as the presence of a negative wake and a loss of fore-aft symmetry about a settling particle which are predominantly related to the elastic behaviour of yield-stress fluids (i.e., coupling of elasticity and plasticity). Here, we present a numerical scheme based on the so-called augmented Lagrangian method for numerical simulation of elastoviscoplastic fluid flows. The method is benchmarked by two rheometric flows: Poiseuille and circular Couette flows for which analytical solutions are derived. Moreover, anisotropic adaptive mesh procedure (which was previously introduced for viscoplastic fluid flows by Saramito and Roquet, Comput. Meth. Appl. Mech. Eng., vol. 190, 2001, pp. 5391-5412) is coupled to obtain a fine resolution of the yield surfaces. Finally, the presented method is applied to study more complex flows: elastoviscoplastic fluid flow in a wavy channel.