On the stability of plane Couette-Poiseuille flow with uniform cross-flow

We present a detailed study of the linear stability of plane Couette-Poiseuille flow in the presence of a cross-flow. The base flow is characterised by the cross flow Reynolds number, $R_{inj}$ and the dimensionless wall velocity, $k$. Squire's transformation may be applied to the linear stability equations and we therefore consider 2D (spanwise-independent) perturbations. Corresponding to each dimensionless wall velocity, $k\in[0,1]$, two ranges of $R_{inj}$ exist where unconditional stability is observed. In the lower range of $R_{inj}$, for modest $k$ we have a stabilisation of long wavelengths leading to a cut-off $R_{inj}$. This lower cut-off results from skewing of the velocity profile away from a Poiseuille profile, shifting of the critical layers and the gradual decrease of energy production. Cross-flow stabilisation and Couette stabilisation appear to act via very similar mechanisms in this range, leading to the potential for robust compensatory design of flow stabilisation using either mechanism. As $R_{inj}$ is increased, we see first destabilisation and then stabilisation at very large $R_{inj}$. The instability is again a long wavelength mechanism. Analysis of the eigenspectrum suggests the cause of instability is due to resonant interactions of Tollmien-Schlichting waves. A linear energy analysis reveals that in this range the Reynolds stress becomes amplified, the critical layer is irrelevant and viscous dissipation is completely dominated by the energy production/negation, which approximately balances at criticality. The stabilisation at very large $R_{inj}$ appears to be due to decay in energy production, which diminishes like $R_{inj}^{-1}$. Our study is limited to two dimensional, spanwise independent perturbations.Comment: Accepted for publication in Journal of Fluid Mechanic

Clouds of bubbles in a viscoplastic fluid

Viscoplastic fluids can hold bubbles/particles stationary by balancing the buoyancy stress with the yield stress - the key parameter here is the yield number, the ratio of the yield stress to the buoyancy stress. In the present study, we investigate a suspension of bubbles in a yield-stress fluid. More precisely, we compute how much is the gas fraction that could be held trapped in a yield-stress fluid without motion. 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 two-dimensional 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 ≤ φ ≤ 0.31,, together with lower and upper estimates

Flow onset for a single bubble in a yield-stress fluid

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 and the ratio of the yield stress to the buoyancy stress . For a given geometry, bubbles are static for 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 with oblate bubbles. Non-zero increases and for large the yield-capillary number determines the static boundary. In this limit, although bubble shape is important, bubble orientation is not. Two-dimensional planar and axisymmetric bubbles are studied

Flow onset for a single bubble in a yield-stress fluid

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 and the ratio of the yield stress to the buoyancy stress . For a given geometry, bubbles are static for 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 with oblate bubbles. Non-zero increases and for large the yield-capillary number determines the static boundary. In this limit, although bubble shape is important, bubble orientation is not. Two-dimensional planar and axisymmetric bubbles are studied

Particle settling in yield stress fluids: Limiting time, distance and applications

International audienceWe examine the problem of a single heavier solid particle settling in a yield stress fluid that behaves as a classical Bingham plastic. The flow configuration we are interested in is the transient dynamics from a particle settling in a Newtonian fluid to a Bingham plastic. Depending on the magnitude of the yield stress (or dimensionlessly the Bingham number), the particle and the surrounding fluid may return to rest in a finite time or reach another steady but lower settling velocity. At the analytical level, we write the total kinetic energy decay of the system. We evidence the existence of a critical Bingham number beyond which motion is suppressed and derive upper bounds for the finite stopping time as well as the maximum path length. These estimates can be obtained in 2D only while the extension to 3D remains an open question. At the numerical level, we design a robust and efficient Lagrange multiplier based algorithm that enables us to compute actual finite time decay. The algorithm combines an Augmented Lagrangian outer loop to treat the exact Bingham law to a Distributed Lagrange Multiplier/Fictitious Do- main inner loop to account for freely-moving particles. We show that the ability to compute the balance between net weight (weight plus buoyancy) and yield stress resistance is the key point. The algorithm is implemented together with a Finite Volume/Staggered Grid algorithm in the numerical platform Peli- GRIFF. We investigate 2D configurations with the following particle shape: (i) a circular disc and (ii) a 2:1 rectangle

Nonlinear stability of the Bingham Rayleigh-Benard Poiseuille flow

International audienceA nonlinear stability analysis of the Rayleigh-Bé}nard Poiseuille flow is performed for a yield stress fluid. Because the topology of the yielded and unyielded regions in the perturbed flow is unknown, the energy method is used, combined with classical functional analytical inequalities. We determine the boundary of a region in the $(Re, Ra)$-plane where the perturbation energy decreases monotonically with time. For increasing values of Reynolds numbers, we show that the energy bound for Ra varies like $(1-\frac{Re}{Re_{EN}} )$, where $Re_{EN}$ is the energy stability limit of isothermal Poiseuille flow. It is also shown that $Re_{EN}\sim 120 \sqrt{B}$ when $B \rightarrow \infty$

Yielding to stress: Recent developments in viscoplastic fluid mechanics

International audienceThe archetypal feature of a viscoplastic fluid is its yield stress: If the material is not sufficiently stressed, it behaves like a solid, but once the yield stress is exceeded, the material flows like a fluid. Such behavior characterizes materials common in industries such as petroleum and chemical processing, cosmetics, and food processing and in geophysical fluid dynamics. The most common idealization of a viscoplastic fluid is the Bingham model, which has been widely used to rationalize experimental data, even though it is a crude oversimplification of true rheological behavior. The popularity of the model is in its apparent simplicity. Despite this, the sudden transition between solid-like behavior and flow introduces significant complications into the dynamics, which, as a result, has resisted much analysis. Over recent decades, theoretical developments, both analytical and computational, have provided a better understanding of the effect of the yield stress. Simultaneously, greater insight into the material behavior of real fluids has been afforded by advances in rheometry. These developments have primed us for a better understanding of the various applications in the natural and engineering sciences