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Numerical Modelling

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Fluid Mechanics of Viscoplasticity

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Abstract

This chapter is divided into two parts, with Part I describing two powerful numerical techniques, viz., the augmented Lagrangian method (ALM) and the operator-splitting method, to simulate the flows of Bingham fluids using the finite element method. In Part II, the finite difference method is used exclusively. First of all, it is employed to solve the flow problem of a viscoplastic fluid in a concentric annulus, using a shooting method. Next, it is applied to simulate the flow of a Bingham fluid in a pipe of square cross-section, and the lid driven, thermally influenced flow of this fluid in a square cavity. The last two problems are solved by means of the particle velocity based lattice Boltzmann method (PVLBM). The numerical modelling of flows of viscoplastic fluids requires two tests of accuracy. The first one measures the convergence of the iterations. The second one is the level of tolerance in delineating the yielded zone from the unyielded zone. These matters are discussed in this chapter. A successful numerical method must be easy to implement, be robust and deliver the desired level of accuracy; it must reproduce results found through other simulations. The ALM, the operator-splitting method and PVLBM meet these criteria.

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Notes

  1. 1.

    Uzawa’s papers deal with finding the maximum of concave functions subject to constraints; see Chaps. 3, 5, 7 and 10 in [6]. The algorithm for solving the saddle point problem appears in this chapter.

  2. 2.

    For a very readable introduction to constrained problems, see Nocedal and Wright [7]. See Chaps. 12 and 17 therein for the quoted results.

  3. 3.

    The proof given here follows that in Chap. 4, Sect. 19 in [3].

  4. 4.

    In viscoelastic fluid mechanics, splitting the constitutive equation into a viscous and an elastic part was conceived and applied to Oldroyd-B fluids in 1977; see [16].

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Authors and Affiliations

  1. College of Science and Engineering, Flinders University, Adelaide, SA, Australia

    Raja R. Huilgol

  2. Department of Mathematics and Statistics, University of Cyprus, Nicosia, Cyprus

    Georgios C. Georgiou

Authors
  1. Raja R. Huilgol
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  2. Georgios C. Georgiou
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Correspondence to Raja R. Huilgol .

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Huilgol, R.R., Georgiou, G.C. (2022). Numerical Modelling. In: Fluid Mechanics of Viscoplasticity. Springer, Cham. https://doi.org/10.1007/978-3-030-98503-5_10

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