On the spin coating of viscoplastic fluids

Tsamopoulos, John; Chen, Meng E.; Borkar, Abhay

doi:10.1007/bf00396510

Cited by 37 publications

(14 citation statements)

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“…From mechanical and engineering points of view, the great amount of research in Bingham fluid has been done in the following areas: non-Newtonian fluid in porous media [68,67,11,47], heat transfer [65,66], mudflows resulting from submarine slides [52,42,43], chemical processing [15,54,25,62], electrorheological (ER) fluids [51,48] and other mechanical properties [43,64,24,41,49,17]. From mathematical point of view, Duvaut, Lions, Brézis and Glowinski have laid a mathematical foundation of Bingham fluid in the framework of variational inequalities [4,5,16,32,29].…”

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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“…Its accuracy and its effectiveness have been also verified for various types of flow, e.g. by Smyrnaios and Tsamopoulos [37], Karapetsas and Tsamopoulos [38], Tsamopoulos et al [39], and Chatzimina et al [40]. For all these reasons, we decided in this work to employ the Papanastasiou model, which relates the stress tensor, τ * , to the rate of strain tensor,γ * , by a simple exponential relation:…”

Section: Introductionmentioning

confidence: 96%

Transient displacement of Newtonian and viscoplastic liquids by air in complex tubes

Dimakopoulos

Tsamopoulos

2007

Journal of Non-Newtonian Fluid Mechanics

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“…The Papanastasiou model has been used in transient simulations by Tsamopoulos et al. [20] to study the thinning of a viscoplastic fluid film on a rotating disk in the process of spin coating and by Dimakopoulos and Tsamopoulos [21] to study the displacement of a viscoplastic material by air in straight and suddenly constricted tubes.…”

Section: Introductionmentioning

confidence: 99%