Bingham Type Research Articles

Approximation and optimal control for variational–hemivariational inequalities of Bingham type fluid

Approximation and optimal control for variational–hemivariational inequalities of Bingham type fluid

A Quasi-Variational-Hemivariational Inequality for Incompressible Navier-Stokes System with Bingham Fluid

In this paper we examine a class of elliptic quasi-variational inequalities, which involve a constraint set and a set-valued map. First, we establish the existence of a solution and the compactness of the solution set. The approach is based on results for an elliptic variational inequality and the Kakutani-Ky Fan fixed point theorem. Next, we prove an existence and compactness result for a quasi-variational-hemivariational inequality. The latter involves a locally Lipschitz continuous functional and a convex potential. Finally, we present an application to the stationary incompressible Navier-Stokes equation with mixed boundary conditions which model a generalized Newtonian fluid of Bingham type.

ON A NEW CLASS OF MIXED HEMIVARIATIONAL-VARIATIONAL INEQUALITIES

In this paper, we study a new class of mixed hemivariational- variational inequalities in which both the non-smooth convex func­tional and the non-smooth non-convex functional can depend on two arguments. We present solution existence and uniqueness results. Then, we apply the theoretical results on a mixed hemivariational-variational inequality in the study of a stationary incompressible flow of Bingham type fluid subject to non-smooth non-monotone slip boundary condi­tion.

A priori and a posteriori error estimates for hp-FEM for a Bingham type variational inequality of the second kind

A hp -finite element discretization of a Bingham type variational inequality of the second kind is being analyzed. We prove convergence, including guaranteed convergence rates in the mesh size h and polynomial degree p of the discrete FE-solution. The friction functional may be regularized in which case we also prove convergence and convergence rates in the regularization parameter ϵ . Moreover, we derive two families of reliable a posteriori error estimators which are applicable to any “approximation” of the exact solution and not only to the FE-solution and can therefore be coupled with an iterative solver. The exact computable member of each family of a posteriori error estimators with minimal value is proven to satisfy an efficiency estimate. Numerical results underline the theoretical finding.

A Class of Variational\u2013Hemivariational Inequalities for Bingham Type Fluids

In this paper we investigate a new class of elliptic variational–hemivariational inequalities without the relaxed monotonicity condition of the generalized subgradient. The inequality describes the mathematical model of the steady state flow of incompressible fluid of Bingham type in a bounded domain. The boundary condition represents a generalization of the no leak condition, and a multivalued and nonmonotone version of a nonlinear Navier–Fujita frictional slip condition. The analysis provides results on existence of solution to a variational–hemivariational inequality, continuous dependence of the solution on the data, existence of solutions to optimal control problems, and the dependence of the solution on the yield limit. The proofs profit from results of nonsmooth analysis and the theory of multivalued pseudomontone operators.

An inverse problem for Bingham type fluids

In this paper we study a coefficient identification problem described by an elliptic variational-hemivariational inequality with unilateral constraints. The inequality is the weak formulation of the mathematical model of a stationary incompressible flow of Bingham type in a bounded domain. The unknown coefficient is a generalized viscosity function of the fluid. The boundary conditions represent generalizations of the no leak condition and a multivalued and nonmonotone version of a nonlinear Navier–Fujita frictional slip condition. The result on well posedness of the direct problem is established based on the theory of multivalued pseudomonotone operators. The existence to the inverse problem is proved by a Weierstrass type argument.

A Simplified Framework for Modelling Viscoelastic Fluids in Discrete Multiphysics

A simplified modelling technique for modelling viscoelastic fluids is proposed from the perspective of Discrete Multiphysics. This approach, based on the concept of linear additive composition of energy potentials, aims to integrate Smooth Particle Hydrodynamics (SPH) with an equivalent elastic potential tailored for fluid flow simulations. The model was implemented using a particle-based software, explored and thoroughly validated with results from numerical experiments on three different flow conditions. The model was able to successfully capture a large extent of viscoelastic responses to external forcing, ranging from pure viscous flows to creep-dominated Bingham type of behaviour. It is concluded that, thanks to the modularity and tunable characteristics of the parameters involved, the proposed modelling approach can be a powerful simulation tool for modelling or mimicking the behaviour of viscoelastic substances.

On the Weak Solvability Via Lagrange Multipliers for a Bingham Model

We study the stationary flow of an incompressible non-Newtonian fluid of Bingham type, mathematically described by means of a nonlinear boundary value problem governed by PDEs. The variational formulation which we propose is a mixed variational problem with Lagrange multipliers. First, we obtain existence, uniqueness, and stability results into an abstract framework. Then, we discuss the well-posedness of the mechanical model based on the auxiliary abstract results.

Synthesis and characterization of gold (Au): Fullerene (C60)-Poly (vinyl pyrrolidone) nanofluids in an alcoholic medium

Objectives: To synthesize gold (Au) doped fullerene (C60)-Poly (vinyl pyrrolidone)PVP nanofluids in an alcoholic medium. Methods: A simple chemical reduction method was adopted to synthesize Au nanoparticles and then these NPs were doped into the C60-PVP NFs by ultra-sonication. The samples were characterized using spectrophotometer, rheometer and microscope. Findings:We reported Surface Plasmon Resonance enhanced p !p C60(sp2)electron transition in PVP molecules upon insertion of NG into C60 NFs with PVP in butanol. Electron transfer PVP!Au(NG) causes a drastic decrease in the light emission in PVP moieties in a Au:C60-PVP complex. A noticeable red shift of the C=O stretching band of PVP reveals surface interaction between “>C=O and Au-atom. Rheological study of NFs reveals non-Newtonian behavior with an enhanced yield stress and follows a typical Bingham type flow characteristics.High resolution transmission electron micrograph shows formation of Au-C60 metal-non metal NPs of hexagonal shape. Novelty: Decrease in light emission intensity of PVP molecules in presence of Au NPs hints that it could be a candidate for sensing applications. Keywords: Nanogold; nanofluids; surface plasmon resonance; light emission; rheological properties

Effect of solid particles on the lost circulation of drilling fluid: A numerical simulation

Lost circulation usually happens when drill through fractured formation. A clear understanding of the circulation loss mechanism is key to successful drilling operations. More attention has been paid to fracture development and lost circulation materials (LCMs) composition. Much work still needs to be done on how the solid particles affect the overall fluid loss process in rough-walled fractures. The control equations for liquid-solid multiphase flow in a 2D rough-walled stationary fracture were established based on fundamental principles of fluid mechanics. Among these equations, the drilling fluid was treated as a Bingham type non-Newtonian fluid, and the influence of solid particles on fluid flow was analyzed in detail. The finite difference method was adopted to the numerical solution of the equations, and the impact of such working parameters as fracture dimensions, bottomhole pressure differential, physical properties of both fluid and particle on the fluid loss were analyzed. The following conclusions can be drawn from the paper: (1) the presence of solid particles can lower the leakoff rate and accumulative leakoff rate to various degrees; (2) the larger the fracture dimension, the lower the leakoff rate and accumulative leakoff rate; (3) the larger the pressure differential, the higher the leakoff rate and accumulative leakoff rate; (4) the leakoff rate and accumulative leakoff rate will decrease with the increase of particle diameter and its volume concentration; (5) particle sphericity and yield stress have little influence on the leakoff rate and accumulative leakoff rate. (6) The higher the plastic viscosity of drilling fluid, the lower the leakoff rate and accumulative leakoff rate. The limitations of this study, suggested improvement and future direction of current work are also highlighted.

Three-dimensional flows of pore pressure-activated Bingham fluids

We are concerned with a system of partial differential equations (PDEs) describing internal flows of homogeneous incompressible fluids of Bingham type in which the value of activation (the so-called yield) stress depends on the internal pore pressure governed by an advection–diffusion equation. After providing the physical background of the considered model, paying attention to the assumptions involved in its derivation, we focus on the PDE analysis of the initial and boundary value problems. We give several equivalent descriptions for the considered class of fluids of Bingham type. In particular, we exploit the possibility to write such a response as an implicit tensorial constitutive equation, involving the pore pressure, the deviatoric part of the Cauchy stress and the velocity gradient. Interestingly, this tensorial response can be characterized by two scalar constraints. We employ a similar approach to treat stick-slip boundary conditions. Within such a setting we prove long-time and large-data existence of weak solutions to the evolutionary problem in three dimensions.

The Development of New Materials and Modes of Casting Metal Powder Mixtures (MIM Technology)

In the present work, to reduce the cost of production of the parts with MIM technology, proposed to use a new metal powder mixture (feedstock are presented) with the increased size of the powder to 50 microns. New feedstock are presented recommended for use in the manufacture of parts with a wall thickness of 5 mm. Shows the main physical and technological properties of new metal powder mix (feedstock) 40ХМА of steel intended for the manufacture of parts via MIM technology. Given the results of determination of granulometric composition and particle shape. The obtained values of tensile strength, hardness and roughness. The results of determining the dependence of viscosity on shear rate. For simulation of rheological properties of selected elastic-visco-plastic model of Bingham type with nonlinear dependence of viscosity on shear rate and temperature. The example of the calculation of feedstock fill the cavity of the mold. The analysis of defects in green part, associated with an irregular velocity field in the flow channels feedstock form.

Pressure evolution and deformation of confined granular media during pneumatic fracturing.

By means of digital image correlation, we experimentally characterize the deformation of a dry granular medium confined inside a Hele-Shaw cell due to air injection at a constant overpressure high enough to deform it (from 50 to 250 kPa). Air injection at these overpressures leads to the formation of so-called pneumatic fractures, i.e., channels empty of beads, and we discuss the typical deformations of the medium surrounding these structures. In addition we simulate the diffusion of the fluid overpressure into the medium, comparing it with the Laplacian solution over time and relating pressure gradients with corresponding granular displacements. In the compacting medium we show that the diffusing pressure field becomes similar to the Laplace solution on the order of a characteristic time given by the properties of the pore fluid, the granular medium, and the system size. However, before the diffusing pressure approaches the Laplace solution on the system scale, we find that it resembles the Laplacian field near the channels, with the highest pressure gradients on the most advanced channel tips and a screened pressure gradient behind them. We show that the granular displacements more or less always move in the direction against the local pressure gradients, and when comparing granular velocities with pressure gradients in the zone ahead of channels, we observe a Bingham type of rheology for the granular paste (the mix of air and beads), with an effective viscosity μ_{B} and displacement thresholds ∇[over ⃗]P_{c} evolving during mobilization and compaction of the medium. Such a rheology, with disorder in the displacement thresholds, could be responsible for placing the pattern growth at moderate injection pressures in a universality class like the dielectric breakdown model with η=2, where fractal dimensions are found between 1.5 and 1.6 for the patterns.

Existence theorem for homogeneous incompressible Navier–Stokes equation with variable rheology

We look at a homogeneous incompressible fluid with a time and space variable rheology of Bingham type, which is governed by a coupling equation. Such a system is a simplified model for a gas–particle mixture that flows under the influence of gravity. The main application of this kind of model is pyroclastic flows in the context of volcanology. In order to prove long time existence of weak solutions, classical Galerkin approximation method coupled with a priori estimates allows us to partially solve the problem. A difficulty remains with the stress tensor, which must satisfy an implicit constitutive relation. Some numerical simulations of a flow of this type are given in the last section. These numerical experiments highlight the influence of the fluidization phenomenon in the flow.

A bi-projection method for Bingham type flows

We propose and study a new numerical scheme to compute the isothermal and unsteady flow of an incompressible viscoplastic Bingham medium. The main difficulty, for both theoretical and numerical approaches, is due to the non-differentiability of the plastic part of the stress tensor in regions where the rate-of-strain tensor vanishes. This is handled by reformulating the definition of the plastic stress tensor in terms of a projection. A new time scheme, based on the classical incremental projection method for the Newtonian Navier–Stokes equations, is proposed. The plastic tensor is treated implicitly in the first sub-step of the projection scheme and is computed by using a fixed point procedure. A pseudo-time relaxation is added into the Bingham projection whose effect is to ensure a geometric convergence of the fixed point algorithm. This is a key feature of the bi-projection scheme which provides a fast and accurate computation of the plastic tensor. Stability and error analyses of the numerical scheme are provided. The error induced by the pseudo-time relaxation term is controlled by a prescribed numerical parameter so that a first-order estimate of the time error is derived for the velocity field. A second-order cell-centred finite volume scheme on staggered grids is applied for the spatial discretisation. The scheme is assessed against previously published benchmark results for both Newtonian and Bingham flows in a two-dimensional lid-driven cavity for Reynolds number equals 1000. Moreover, the proposed numerical scheme is able to reproduce the fundamental property of cessation in finite time of a viscoplastic medium in the absence of any energy source term in the equations. For a fixed value (100) of the Bingham number, various numerical simulations for a range of Reynolds numbers up to 200000 were performed with the bi-projection scheme on a grid with 10242 mesh points. The effect of this (physical) parameter on the flow behaviour is discussed.

Viscoplastic flow in an extrusion damper

Numerical simulations of the flow in an extrusion damper are performed using a finite volume method. The damper is assumed to consist of a shaft, with or without a spherical bulge, oscillating axially in a containing cylinder filled with a viscoplastic material of Bingham type. The response of the damper to a forced sinusoidal displacement is studied. In the bulgeless case the configuration is the annular analogue of the well-known lid-driven cavity problem, but with a sinusoidal rather than constant lid velocity. Navier slip is applied to the shaft surface in order to bound the reaction force to finite values. Starting from a base case, several problem parameters are varied in turn in order to study the effects of viscoplasticity, slip, damper geometry and oscillation frequency to the damper response. The results show that, compared to Newtonian flow, viscoplasticity causes the damper force to be less sensitive to the shaft velocity; this is often a desirable damper property. The bulge increases the required force on the damper mainly by generating a pressure difference across itself; the latter is larger the smaller the gap between the bulge and the casing is. At high yield stresses or slip coefficients the amount of energy dissipation that occurs due to sliding friction at the shaft-fluid interface is seen to increase significantly. At low frequencies the flow is in quasi steady state, dominated by viscoplastic forces, while at higher frequencies the fluid kinetic energy storage and release also come into the energy balance, introducing hysteresis effects.

Convergence of conforming approximations for inviscid incompressible Bingham fluid flows and related problems

We study approximations by conforming methods of the solution to the variational inequality \({\langle \partial_t u,v-u\rangle + \psi(v) - \psi(u) \ge \langle f,v-u\rangle}\), which arises in the context of inviscid incompressible Bingham type fluid flows and of the total variation flow problem. In the general context of a convex lower semi-continuous functional \({\psi}\) on a Hilbert space, we prove the convergence of time implicit space conforming approximations, without viscosity and for nonsmooth data. Then, we introduce a general class of total variation functionals \({\psi}\), for which we can apply the regularization method. We consider the time implicit regularized, linearized or not, algorithms and prove their convergence for general total variation functionals. A comparison with an analytical solution concludes this study.

Study on the Generalized Darcy's Law for Bingham and Herschel-Bulkley Fluids

Extra-heavy oil, polymer solution and some drilling fluids are typical non-Newtonian Herschel-Bulkley fluids, which behave as sheer-thinning with yield stress. In this paper, the Generalized Darcy's law for Herschel-Bulkley fluids flow in porous media was formulated, by the same way formulating the Generalized Darcy's Law for Bingham fluids. Then, the applications of the two type flow models were compared; Bingham type model was still widely applied due to its conciseness and relatively satisfied accuracy. In addition, the Generalized Darcys Law was revised to describe thixotropic non-Newtonian fluids conceptually.

Occurrence conditions of roll waves for three grain–fluid models and comparison with results from experiments and field observation

For the flow and deposition behaviour of debris flows, phenomena like particle sorting, levee formation and the development of roll waves are expected to be important processes. However, these processes are not well understood and hardly implemented in modelling approaches. In this study, we focus on the development of roll waves and derive advanced criteria separating stable and instable flow regimes for three debris flowmodels. These criteria are expressed using critical Froude numbers. Each of these simple flow models reflect different sources of flow resistance: laminar-viscous stress (Bingham type), dispersive stress due to particle collision (Bagnold type), and a model combining turbulent and dispersive stresses. Subsequently, we compare the predictions from these models with results from laboratory experiments with grain–fluid mixtures in a straight flume and with observations from a debris flow monitoring site at the Lattenbach creek in Austria. The experimental flows match with a turbulent flow model including particle collisions. For the natural flows the connection between models and observations is not clear due to limited field data. The results of our study contribute to an improved determination of critical flow conditions and provide data for model testing.

A regularized–stabilized mixed finite element formulation for viscoplasticity of Bingham type

In this article, we present a new mixed stabilized–regularized finite element formulation in primitive variables, with continuous velocity and discontinuous pressure interpolations for the steady flow of an incompressible fluid of Bingham type. This formulation is based on an augmented Lagrangian regularization technique and a least squares stabilization method. Mathematical analyses are performed for the new formulation in terms of stability, existence and uniqueness of the solution. Optimal orders of convergence are obtained mathematically, improving on classical methods, which also present limitations as regards the values of the yield stress. Numerical results are presented confirming the theory developed here, and they show the robustness of the new method, with stability obtained for the velocity and, especially, for the pressure when the yield stress is very high.