Two finite-difference schemes for calculation of Bingham fluid flows in a cavity

Muravleva, Ekaterina; Olshanskii, Maxim A.

doi:10.1515/rjnamm.2008.035

Cited by 15 publications

(15 citation statements)

References 28 publications

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“…This pair of spaces satisfies the LBB condition (2.1). Another is the extension of the well-known MAC FD scheme [13] for the case of non-Newtonian flows as described in [23]. Both discretizations use the uniform grid with mesh-size h.…”

Section: Numerical Resultsmentioning

confidence: 99%

An Iterative Method for the Stokes-Type Problem with Variable Viscosity

Grinevich¹,

Olshanskii²

2009

SIAM J. Sci. Comput.

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Abstract. This paper concerns an iterative technique for solving discretized Stokes-type equations with varying viscosity coefficient. We build a special block preconditioner for the discrete system of equations and perform an analysis revealing its properties; the theoretical analysis is based on the weighted Nečas inequality. The subject of this paper is motivated by numerical solution of incompressible non-Newtonian fluid equations. In particular, the general analysis is applied to the linearized equations of the regularized Bingham model of viscoplastic fluid. Numerical experiments show that the suggested preconditioner leads to an iterative method insensitive to the variation of mesh size and the regularization parameter of the fluid model.

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Section: Numerical Resultsmentioning

confidence: 99%

An Iterative Method for the Stokes-Type Problem with Variable Viscosity

Grinevich¹,

Olshanskii²

2009

SIAM J. Sci. Comput.

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“…There are different ways to discretize (1.1), examples are a MAC discretization on a staggered grid and collocated finite difference methods [27,30], finite volume [36] or LBB-stable finite elements [13]. In this paper we consider Galerkin finite element discretization methods, although the approach is essentially independent of a specific discretization method.…”

Section: Finite Element Approximationmentioning

confidence: 99%

A mixed formulation of the Bingham fluid flow problem: Analysis and numerical solution

Aposporidis

Haber

Olshanskii

et al. 2011

Computer Methods in Applied Mechanics and Engineering

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In this paper we introduce a mixed formulation of the Bingham fluid flow problem. We consider both the original and a regularized version of the problem, where a parameter ε is introduced, forcing the entire domain to be formally a fluid region. In general, common solvers for the regularized problem experience performance degradation when the parameter ε gets smaller. The method studied here introduces an auxiliary tensor variable and shows enhanced numerical properties for small values of ε. Good performance is also observed for the non-regularized case. The well posedness for the regularized problem and the equivalence -at the continuous level -between the original (primitive variables) and the mixed formulation are demonstrated. We analyze properties of linearized problems that are relevant for the convergence of numerical solvers. A finite element method for the mixed formulation is discussed. Numerical results confirm the predicted better performances of the mixed formulation comparing to the primitive variables formulation. A comparison to a non regularized solver based on the augmented Duvaut-Lions-Glowinski formulation of the problem is carried out as well.

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“…The system was discretized using a homogeneous MAC scheme (see [4,33]). In this scheme, a semi-staggered grid is utilized, i.e., the values of the two velocity components are taken on the grid points, while the pressure is considered at the center of each square cell; see Figure 2.…”

Section: Examplementioning

confidence: 99%

“…To obtain compatibility between the variables, the discrete dual quantity is also considered on the grid points. Following [33,Section 3.2], for the discretization of the Laplacian we use the nine-point approximation:…”

Section: Examplementioning

confidence: 99%

A duality based semismooth Newton framework for solving variational inequalities of the second kind

Reyes¹,

Hintermüller²

2011

Interfaces Free Bound.

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In an appropriate function space setting, semismooth Newton methods are proposed for iteratively computing the solution of a rather general class of variational inequalities (VIs) of the second kind. The Newton scheme is based on the Fenchel dual of the original VI problem which is regularized if necessary. In the latter case, consistency of the regularization with respect to the original problem is studied. The application of the general framework to specific model problems including Bingham flows, simplified friction, or total variation regularization in mathematical imaging is described in detail. Finally, numerical experiments are presented in order to verify the theoretical results.

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Two finite-difference schemes for calculation of Bingham fluid flows in a cavity

Cited by 15 publications

References 28 publications

An Iterative Method for the Stokes-Type Problem with Variable Viscosity

An Iterative Method for the Stokes-Type Problem with Variable Viscosity

A mixed formulation of the Bingham fluid flow problem: Analysis and numerical solution

A duality based semismooth Newton framework for solving variational inequalities of the second kind

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