Confined viscoplastic flows with heterogeneous wall slip

The steady, pressure-driven flow of a Herschel-Bulkley fluid in a microchannel is considered, assuming that different power-law slip equations apply at the two walls due to slip heterogeneities, allowing the velocity profile to be asymmetric. Three different flow regimes are observed as the pressure gradient is increased. Below a first critical pressure gradient G ₁, the fluid moves unyielded with a uniform velocity, and thus, the two slip velocities are equal. In an intermediate regime between G ₁ and a second critical pressure gradient G ₂, the fluid yields in a zone near the weak-slip wall and flows with uniform velocity near the stronger-slip wall. Beyond this regime, the fluid yields near both walls and the velocity are uniform only in the central unyielded core. It is demonstrated that the central unyielded region tends towards the midplane only if the power-law exponent is less than unity; otherwise, this region rends towards the weak-slip wall and asymmetry is enhanced. The extension of the different flow regimes depends on the channel gap; in particular, the intermediate asymmetric flow regime dominates when the gap becomes smaller than a characteristic length which incorporates the wall slip coefficients and the fluid properties. The theoretical results compare well with available experimental data on soft glassy suspensions. These results open new routes in manipulating the flow of viscoplastic materials in applications where the flow behavior depends not only on the bulk rheology of the material but also on the wall properties.     

Unsteady shear flow of a viscoplastic material

A viscoplastic, or yield-stress, liquid occupies the space between two infinite parallel plates. Initially the whole system is at rest. The lower plate is suddenly jerked into motion with given speed or shear stress, while the upper plate is kept fixed. The flow consists of two regions; (1) a lower sheared region bounded above by the yield surface, (2) an upper unyielded region bounded below by the yield surface. The yield surface propagates to the upper plate as time proceeds. We first consider the equivalent one plate problem of flow over a jerked plate, and find similarity solutions and small time asymptotic solutions for prescribed shear and speed cases respectively. These solutions are used as initial solutions for the two plate case. The motion of the yield surface and the time taken for the entire material to yield are investigated. The problems considered here are two dimensional representations of some control devices, for example the light duty clutch, which consists of two corotating, coaxial discs separated by a layer of electrorheological material. In this application it is useful to know the time taken for all the material to yield.     

Transient squeeze flow of viscoplastic materials

The transient, axisymmetric squeezing of viscoplastic materials under creeping flow conditions is examined. The flow of the material even outside the disks is followed. Both cases of the disks moving with constant velocity or under constant force are studied. This time-dependent simulation of squeeze flow is performed for such materials in order to determine very accurately the evolution of the force or the velocity, respectively, and the distinct differences between these two experiments, the highly deforming shape and position of all the interfaces, the effect of possible slip on the disk surface, especially when the slip coefficient is not constant, and the effect of gravity. All these are impossible under the quasi-steady state condition used up to now. The exponential constitutive model, suggested by Papanastasiou, is employed. The governing equations are solved numerically by coupling the mixed finite element method with a quasi-elliptic mesh generation scheme in order to follow the large deformations of the free surface of the fluid. As the Bingham number increases, large departures from the corresponding Newtonian solution are found. When the disks are moving with constant velocity, unyielded material arises only around the two centers of the disks verifying previous works in which quasi-steady state conditions were assumed. The size of the unyielded region increases with the Bingham number, but decreases as time passes and the two disks approach each other. Their size also decreases as the slip velocity or the slip length along the disk wall increase. The force that must be applied on the disks in order to maintain their constant velocity increases significantly with the Bingham number and time and provides a first method to calculate the yield stress. On the other hand, when a constant force is applied on the disks, they slow down until they finally stop, because all the material between them becomes unyielded. The final location of the disk and the time when it stops provide another, probably easier, method to deduce the yield stress of the fluid.     

Unsteady circular Couette flow of a Bingham plastic with the Augmented Lagrangian Method

Numerical simulations are undertaken for unsteady flows of an ideal Bingham fluid in a circular Couette viscometer. The main difficulties in such simulations are caused by the non-differentiability of the constitutive equation and the need to determine the position and shape of the yield surface separating the yielded zones from the unyielded ones. In this work, these difficulties are overcome by using a numerical method based on variational inequalities, i.e. the augmented Lagrangian/Uzawa method. The start-up and cessation of circular Couette flows of a Bingham fluid are solved numerically assuming that only one of the cylinders is rotating. An improved theoretical upper bound for the stopping time in the case of cessation is derived. The numerical estimates for the stopping time compare well with the theoretical bounds. Moreover, with the adopted method the evolution of the velocity profiles and the locations of yielded/unyielded surfaces are accurately calculated. In flow cessation, we observe an interesting effect, namely the appearance of a small unyielded region adjoined to the outer cylinder shortly before cessation.     

Numerical simulations of complex yield-stress fluid flows

Viscoplasticity is characterized by a yield stress, below which the materials will not deform and above which they will deform and flow according to different constitutive relations. Viscoplastic models include the Bingham plastic, the Herschel-Bulkley model and the Casson model. All of these ideal models are discontinuous. Analytical solutions exist for such models in simple flows. For general flow fields, it is necessary to develop numerical techniques to track down yielded/unyielded regions. This can be avoided by introducing into the models a regularization parameter, which facilitates the solution process and produces virtually the same results as the ideal models by the right choice of its value. This work reviews several benchmark problems of viscoplastic flows, such as entry and exit flows from dies, flows around a sphere and a bubble and squeeze flows. Examples are also given for typical processing flows of viscoplastic materials, where the extent and shape of the yielded/unyielded regions are clearly shown. The above-mentioned viscoplastic models leave undetermined the stress and elastic deformation in the solid region. Moreover, deviations have been reported between predictions with these models and experiments for flows around particles using Carbopol, one of the very often used and heretofore widely accepted as a simple “viscoplastic” fluid. These have been partially remedied in very recent studies using the elastoviscoplastic models proposed by Saramito.     

Interfacial instability of turbulent two-phase stratified flow: Pressure-driven flow and non-Newtonian layers

We derive a flat-interface model to describe the flow of two horizontal, stably stratified fluids, where the bottom layer exhibits non-Newtonian rheology. The model takes into account the yield stress and power-law nature of the bottom fluid. In the light of the large viscosity contrast assumed to exist across the fluid interface, and for large pressure drops in the streamwise direction, the possibility for the upper Newtonian layer to display fully developed turbulence must be considered, and is described in our model. We develop a linear-stability analysis to predict the conditions under which the flat-interface state becomes unstable, and pay particular attention to characterizing the influence of the non-Newtonian rheology on the instability. Increasing the yield stress (up to the point where unyielded regions form in the bottom layer) is destabilizing; increasing the flow index, while bringing a broader spectrum of modes into play, is stabilizing. In addition, a second mode of instability is found, which depends on conditions in the bottom layer. For shear-thinning fluids, this second mode becomes more unstable, and yet more bottom-layer modes can become unstable for a suitable reduction in the flow index. One further difference between the Newtonian and non-Newtonian cases is the development of unyielded regions in the bottom layer, as the linear wave on the interface grows in time. These unyielded regions form in the trough of the wave, and can be observed in the linear analysis for a suitable parameter choice.     

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

In this note, we present some results concerning the gas displacement of power-law liquids and visco-plastic Papanastasiou’s materials improving the understanding of the problem considered in Sousa et al. (2007) [1]. Specifically, we present: the fraction of mass attached to the wall for a viscosity-thickening power-law fluid, different transition patterns between by pass and full-recirculating flow regimes, expressions for the critical (in the sense proposed by Soares and Thompson (2009) [2]) fraction of residual mass as a function of the rheological parameter of interest, and fields of yielded and unyielded zones for the visco-plastic material.     

Numerical simulation of Taylor Couette flow of Bingham fluids

The Bingham fluid flow between two concentric cylinders is studied using numerical simulation. The cylinders are assumed to rotate independently, and with an imposed axial sliding. The flow field is decomposed with linearity arguments of the base circular Couette shear flow and corresponding deviation field. The numerical methods are based on the expression of the deviation field in terms of complete sets of orthogonal functions and Chebyshev series. The Galerkin projection method is used with the pressure term being eliminated. The Adams Bashforth scheme is adopted for time marching. The results show that the vortices are squeezed toward the inner cylinder due to the effect of yield stress. When the outer cylinder is held stationary, the yield stress plays a role in weakening the vortex flow. However, for the co-rotation situation, the vortex flow is initially strengthened with an increase of yield stress, and then weakened as the yield stress is raised large enough. The annular unyielded regions emerge and stick to the outer cylinder. In case of Taylor Couette flow with an imposed axial sliding, a spiral vortex flow is visible with spiral unyielded region being obtained.     

Numerical study of the Bingham squeeze film problem

In studies of the flow of a Bingham fluid in a parallel-plate plastometer there has been disagreement about whether or not a yield surface exists, and if it does exist what shape the yield surface has.The present authors have re-exemined the problem using a finite element program and have concluded that a small plug of unyielded fluid exists adjacent to the centre of the plates. This result has been verified by replacing the unyielded plug with a solid body.     

Elastic and viscous effects on flow pattern of elasto-viscoplastic fluids in a cavity

Inertialess flows of elasto-viscoplastic fluids inside a leaky cavity are numerically analyzed using the finite element technique, with the goal of understanding the influence of both the elastic and viscous effects on the topology of the yield surfaces of an elasto-viscoplastic material. Assuming that the collapse of the material microstructure is instantaneous, a mechanical model is composed of the governing equations of mass and momentum for incompressible fluids, and associated with a hyperbolic equation for the extra-stress tensor based on the Oldroyd-B model (Nassar et al., 2011). The main feature of the model is the consideration of the viscosity and relaxation time as functions of the strain rate to allow the shear-thinning of the viscosity and to restrict the elastic effects to the unyielded regions of the material. The numerical simulations are performed through a three-field Galerkin least-squares-type method in terms of the extra-stress tensor and the pressure and velocity fields. The results indicate that the material yield surfaces are strongly influenced by the interplay between the elastic and viscous effects, in accordance with recent experimental visualization of elasto-viscoplastic flows.     

Numerical simulations of cessation flows of a Bingham plastic with the augmented Lagrangian method

The augmented Lagrangian/Uzawa method has been used to study benchmark one-dimensional cessation flow problems of a Bingham fluid, such as the plane Couette flow, and the plane, round, and annular Poiseuille flows. The calculated stopping times agree well with available theoretical upper bounds for the whole range of Bingham numbers and with previous numerical results. The applied method allows for easy determination of the yielded and unyielded regions. The evolution of the rigid zones in these unsteady flows is presented. It is demonstrated that the appearance of an unyielded zone near the wall occurs for any non-zero Bingham number not only in the case of a round tube but also in the case of an annular tube of small radii ratio. The advantages of using the present method instead of regularizing the constitutive equation are also discussed.     

Viscoplastic flow around a cylinder kept between parallel plates

Numerical simulations have been undertaken for the creeping pressure-driven flow of a Bingham plastic past a cylinder kept between parallel plates. Different gap/cylinder diameter ratios have been studied ranging from 2:1 to 50:1. The Bingham constitutive equation is used with an appropriate modification proposed by Papanastasiou, which applies everywhere in the flow field in both the yielded and practically unyielded regions. The emphasis is on determining the extent and shape of yielded/unyielded regions along with the drag coefficient for a wide range of Bingham numbers. The present results extend previous simulations for creeping flow of a cylinder in an infinite medium and provide calculations of the drag coefficient around a cylinder in the case of wall effects.     

Non-isothermal flow of a Bingham fluid with pressure and temperature dependent viscosity

In this paper we investigate the non-isothermal flow of a Bingham fluid whose viscosity and yield stress depend on temperature and pressure. We consider two situations: in the first one we assume that the buoyancy effects are dominant and influence the development and evolution of the unyielded plug. In this case the governing equations are obtained via the Oberbeck–Boussinesq approximation which is derived using a perturbative approach. We show that within this approximation the heat generated by viscous friction can be safely neglected. In the second situation we assume that the frictional heating effects are dominant and influence the flow via the viscosity and yield stress that depend on temperature. For both situations we investigate the simple unidirectional flow between plates subjected to given thermal conditions. We derive the equations for the steady fully developed flow and we determine the exact position of the yield surfaces separating the yielded and the unyielded domain. We also show some plots to assess the effects due to the dependence of the rheological parameters on the temperature and pressure.     

Free surface effects in squeeze flow of Bingham plastics

New results for the squeeze flow of Bingham plastics show the shape of the free surface in quasi-steady-state simulations, and its effect on the yielded/unyielded regions and the squeeze force. The present simulation results are obtained for both planar and axisymmetric geometries as in our previous paper [A. Matsoukas, E. Mitsoulis, Geometry effects in squeeze flow of Bingham plastics, J. Non-Newtonian Fluid Mech. 109 (2003) 231–240] and for aspect ratios ranging from 0.01 to 1. Bigger aspect ratios produce more free surface movement relative to the disk radius or plate length, but less movement relative to the gap. Planar geometries give more free surface movement than axisymmetric ones. Viscoplasticity serves to reduce the free surface movement and its deformation. In some cases of planar geometries and big aspect ratios, unyielded regions appear at the free surface, while the small unyielded regions near the center of the disks or plates are not affected. Including the free surface in the calculations of the squeeze force adds a small percentage to the values depending on aspect ratio and Bingham number. The previously fitted easy-to-use equations are corrected to account for that effect.     

Plastic flow at a crack tip. Energy failure criterion and its relationship to the J integral

Plastic flow at the tip of a crack moving in an elastic body is considered using the theory of an ideally rigid-plastic body. The material at the crack tip is treated as a body consisting of an elastic outer region and a rigid-plastic inner region. It is shown that this representation is energetically justified for small plastic regions. The distribution of the specific dissipation of the work of internal forces and deformations along the particle trajectory at the crack tip is obtained. The relationship between the specific dissipation of the work of internal forces and the J integral under plane-strain conditions is established.     

Peristaltic flow of a Bingham fluid in a channel

We study the peristaltic transport of a Bingham fluid in a channel with small aspect ratio whose walls behave as a periodic traveling wave. The governing equations in the unyielded phase are obtained writing the integral formulation for the momentum balance. As shown in Fusi et al. (2015), this approach allows to overcome the so-called “lubrication paradox” which may arise in the thin film approximation. We consider the case in which the inlet flux is prescribed and the one in which the flow is driven by a given pressure drop. In both cases the solution of the problem is determined solving a nonlinear integral equation for the yield surface. We perform some numerical simulations to illustrate the behavior of the yield surface, assuming that the traveling wave describing the peristaltic motion has a sinusoidal shape.     

Stable two-layer flows at all Re; visco-plastic lubrication of shear-thinning and viscoelastic fluids

Multi-fluid flows are frequently thought of as being less stable than single phase flows. Consideration of different non-Newtonian models can give rise to different types of hydrodynamic instability. Here we show that with careful choice of fluid rheologies and flow paradigm, one can achieve multi-layer flows that are linearly stable for Re = ∞. The basic methodology consists of two steps. First we eliminate interfacial instabilities by using a yield stress fluid in one fluid layer and ensuring that for the base flow configurations studied we maintain an unyielded plug region at the interface. Secondly we eliminate linear shear instabilities by ensuring a strong enough Couette component in the second fluid layer, imposed via the moving interface. We show that this technique can be applied to both shear-thinning and visco-elastic fluids.     

Unsteady simple shear flow in a viscoplastic fluid: comparison between analytical and numerical solutions

In this paper, an unsteady flow of a viscoplastic fluid for simple shear flow geometry is solved numerically using two regularizing functions to overcome the discontinuity for zero shear rate of the Bingham constitutive law. The adopted models are the well-known Papanastasiou relation and one based on the error function. The numerical results are compared with the analytical solution of the same problem obtained by Sekimoto (J Non-Newton Fluid Mech 39:107–113, 1991). The analysis of the results emphasizes that the errors are much smaller in the yielded than in the unyielded region. The models approximate closer the ideal Bingham model as the regularization parameters increase. The differences between the models tend to vanish as the regularization parameters are at least greater than 10⁵.     

Heat transfer characteristics in flow around a spherically blunted cone at incidence and gas injection from a blunted surface

New methods of controlling thermal regimes in a high-enthalpy spatial flow around a body are considered. They are related to gas injection from the blunted surface and heat overflow in the material of the shell. The effect of injection is analyzed for different thermal conductivities. It is shown that highly heat-conducting materials can be successfully used to decrease the maximum temperatures at the windward side due to intense heat removal to the region of a porous spherical bluntness. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 40, No. 4, pp. 162–169, July–August, 1999.     

The reflection phenomena of quasi-vertical transverse waves in piezoelectric medium under initial stresses

The propagation of plane vertical transverse waves at an interface of a semi-infinite piezoelectric elastic medium under the influence of the initial stresses is discussed. The free surface of the piezoelectric elastic medium is considered to be adjacent to vacuum. We assumed that the piezoelectric material is anisotropic of the type of a transversely isotropic crystals (hexagonal crystal structure, class 6 mm). For an incident of vertical transverse plane wave, four types (two for the displacement and two for the electric potential) of reflected plane waves, called quasi-longitudinal (qP) and quasi-shear vertical (qSV) waves are shown to be exist. The relations governing the reflection coefficients of these reflected waves for various boundary conditions (mixed-free-fixed) are derived. It has been shown analytically that reflected coefficients of (qP) and (qSV) waves depend upon the angle of incidence, the parameters of electric potential, the material constants of the medium as well as the initial stresses presented in the medium. The numerical computations of reflection coefficients for different values of initial stresses have been carried out by computer for aluminum nitride (AlN) as an example and the results are given in the form of graphs. Finally, particular cases are considered in the absence of the initial stresses and the electric potential. Some of earlier studies have been compared to the special cases and shown good agreement with them.