On the imposition of friction boundary conditions for the numerical simulation of Bingham fluid flows

Fortin, A.; Côté, D.; Tanguy, Philippe A.

doi:10.1016/0045-7825(91)90234-w

Cited by 43 publications

(21 citation statements)

References 7 publications

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“…167), Fortin and Glowinski [21] and Glowinski and Le Tallec [33]. For more recent work, we refer to [22,57]. It appears that no error estimate has been proved for this method with application to Bingham fluid.…”

Section: Discussionmentioning

confidence: 99%

Error estimates for the numerical approximation of time-dependent flow of Bingham fluid in cylindrical pipes by the regularization method

Zhang

2003

Numerische Mathematik

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The flow of a Bingham fluid in a cylindrical pipe can give rise to free boundary problems. The fluid behaves like a viscous fluid if the shear stress, expressed as a linear function of the shear rate, exceeds a yield value, and like a rigid body otherwise. The surfaces dividing fluid and rigid zones are the free boundaries. Therefore the solution for such highly nonlinear problems can in general only be obtained by numerical methods. Considerable progress has been made in the development of numerical algorithms for Bingham fluids [2,20,27,29,30,32]. However, very little research can be found in the literature regarding the rate of convergence of the numerical solution to the true continuous solution, that is the error estimate of these numerical methods. Error estimates are a critically important issue because they tells us how to control the error by appropriately choosing the grid sizes and other related parameters. This paper concerns the error estimates of a unsteady Bingham fluid modeled as a variational inequality due to Duvaut-Lions [16] and Glowinski [30]. The difficulty both in the analytical and numerical treatment of the mathematical model is due to the fact that it contains a nondifferentiable term. A common technique, called the regularization method, is to replace the non-differentiable term by a perturbed differentiable term which depends on a small regularization parameter . The regularization method effectively reduces the variational inequality to an equation (a regularized problem) which is much easier to cope with. This paper has achieved the following. (1) Error estimates are derived for a continuous time Galerkin method in suitable norms. (2) We give an estimate of the difference between the true solution and the regularized solution in terms of . (3) Some regularity properties for both regularized solution and the true solution are proved. (4) The error 154 Y. Zhangestimates for full discretization of the regularized problem using piecewise linear finite elements in space, and backward differencing in time are established for the first time by coupling the regularization parameter and the discretization parameters h and t. (5) We are able to improve our estimates in the one-dimensional case or under stronger regularity assumptions on the true solution. The estimates for the one-dimensional case are optimal and confirmed by numerical experiment. The estimates from (4) and (5) provide very important information on the measure of the error and give us a powerful mechanism to properly choose the parameters h, t and in order to control the error (see Corollary 4.4). The above estimates extend the error bounds derived in Glowinski, Lions and Trémolières [32] (chapter 5, pp. 348-404) for the stationary Bingham fluid to the time-dependent one, which is the main contribution of this paper. (2000): 35k85, 65M15, 65M60, 76A05, 76M10 Mathematics Subject Classification

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Section: Discussionmentioning

confidence: 99%

Error estimates for the numerical approximation of time-dependent flow of Bingham fluid in cylindrical pipes by the regularization method

Zhang

2003

Numerische Mathematik

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“…Even though for such liquids the effect of possible sliding on solution behavior is large and important [13], it is always possible to find a solution under sticking conditions. The situation is very similar in the case of non-Newtonian liquids and solids, except that there are many more observations that sliding can occur in such flows [14][15][16]. Nevertheless, it is always possible to find a solution under sticking conditions for liquids/solids whose yield stress approaches infinity as the equivalent strain rate approaches infinity.…”

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confidence: 87%

Viscoplasticity with a saturation stress: distinguishing features of the model

Alexandrov

Mishuris

2006

Arch Appl Mech

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Alexandrov, S; Mishuris, G. Viscoplasticity with a saturation stress: distinguishing features of the model. Archives of Applied Mechanics. 2007, 77(1), 35-47Viscoplastic models including a saturation stress are considered. The existence of the saturation stress significantly changes the mathematical structure of solutions near maximum friction surfaces (surfaces where the friction stress is equal to the local shear yield stress). The main features of solutions based on such theories are: (a) sliding must occur at the maximum friction surfaces under certain conditions, (b) the velocity field may be singular in the vicinity of maximum friction surfaces. The objective of the present paper is to study these features of solutions. The mathematical structure obtained is considered to be advantageous for a class of materials and may lead to a convergence of viscoplastic solutions to the corresponding rigid perfectly plastic solutions. It seems that the latter is of importance for the construction of a unified theory that could describe the material behavior in the range from rate-independent plasticity to viscoplasticity. In the present paper, the study of the main features of the model is based on the exact closed-form solution to the problem of flow between two coaxial rotating cylinders. In the case of sliding, in addition to the aforementioned features, the asymptotic behavior of the velocity field in the vicinity of the maximum friction surface is found for a class of constitutive laws.Peer reviewe

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“…Besides, the above perfect-bonding assumption (10), which could prove somewhat unrealistic in some situations, should be modified by incorporating a more appropriate condition allowing for possible slippage at the soil-substratum interface, as proposed by Fortin et al [19] in the context of Bingham flows numerical simulations. Such a condition will be introduced as follows.…”

Section: Statement Of the Evolution Problem Of A Liquefied Soil Massmentioning

confidence: 98%

Minimum principle and related numerical scheme for simulating initial flow and subsequent propagation of liquefied ground

Montassar¹,

Buhan²

2005

Num Anal Meth Geomechanics

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SUMMARYThe problem of predicting the evolution of liquefied ground, modelled as a viscoplastic material, is addressed by combining a minimum principle for the velocity field, which characterizes such an evolution, and a time step integration procedure. Two different numerical schemes are then presented for the finite element implementation of this minimum principle, namely, the regularization technique and the decomposition-co-ordination method by augmented Lagrangian. The second method, which proves more accurate and efficient than the first, is finally applied to simulate the incipient flow failure and subsequent spreading of a liquefied soil embankment subject to gravity. The strong influence of liquefied soil residual shear strength on reducing the maximum amplitude of the ground displacement is particularly emphasized in such an analysis.

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On the imposition of friction boundary conditions for the numerical simulation of Bingham fluid flows

Cited by 43 publications

References 7 publications

Error estimates for the numerical approximation of time-dependent flow of Bingham fluid in cylindrical pipes by the regularization method

Error estimates for the numerical approximation of time-dependent flow of Bingham fluid in cylindrical pipes by the regularization method

Viscoplasticity with a saturation stress: distinguishing features of the model

Minimum principle and related numerical scheme for simulating initial flow and subsequent propagation of liquefied ground

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