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

Abstract

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

Progress in numerical simulation of yield stress fluid flows

“…A first weakness of the regularization approach is the lack of general convergence results of the solution with ε, denoted as (σ ε , u ε ) of the regularized problem to the original solution (σ, u) when ε −→ 0. Glowinski et al (1981, p. 370) showed a convergence result for the velocity field u ε , in the case of the Bingham model (n = 1) and for the one-dimensional Poiseuille flow where the domain of computation Ω is a circular pipe section (see Zhang (2003) for some generalizations). There is no convergence results available concerning the corresponding stress deviator σ ε and, from numerical experiences, there is no evidence that this quantity converges to the solution σ associated to the unregularized problem (Frigaard and Nouar, 2005;Putz et al, 2009): the velocity vector converged while decreasing ε whereas no convergence of the stress tensor is observed.…”

Semi-discrete stabilized finite element methods for Navier–Stokes equations with nonlinear slip boundary conditions based on regularization procedure

“…Since j ε (η) is differentiable, then according to (16) and the standard convex analysis, it is easy to show that the regularized problems (17) and (18) are also equivalent to the following variational problems, respectively, :…”

“…From the early 1970s to now, although there exist a large number of papers about the finite element approximations to the variational inequality problem, such as for the obstacle problem [3][4][5][6][7][8][9][10][11][12] and for the Bingham flow problem [13][14][15][16] and for the plate contact problem [17,18] and references cited therein, but the finite element approximations to the variational inequality problems associated with the incompressible Navier-Stokes equations have not studies as so much. Recently, Ayadi et al [19] study the error estimates of finite element approximation to Stokes problem with Tresca friction conditions and derive the optimal orders by the use of the multiplier.…”

Fully Discrete Finite Element Methods for Two-Dimensional Bingham Flows

“…Although there have much works on the numerical simulation for steady and time-dependent Bingham fluid flow, such as [7][8][9][10][11][12][13][14] and references cited therein, the error estimates for the finite element approximation have not been studied as much [15,16]. Roughly speaking, the convergence order for the velocity approximation is (ℎ 1/2 ).…”

“…For example, if Ω is a disc, Glowinski [1] improved the error estimate to (ℎ| ln ℎ| 1/2 ) in terms of the explicit formulation of the velocity. If the flow is axisymmetric, the optimal error estimate can be derived by Zhang [16] in terms of the accurate estimate to the nondifferentiable term. On the other hand, the solution of Bingham fluid flow is of the poor regularity; thus it does not lead us to employ the high-order finite element approximation, such as 2 − 1 finite element for the approximation of the velocity and the pressure.…”