Contents lists available at ScienceDirect International Journal of Non–Linear Mechanics journal homepage: www.elsevier.com/locate/nlm On the physics of viscoplastic fluid flow in non-circular tubes Mario F. Letelier a , Dennis A. Siginer a,b, ⁎ , Cristian Barrera Hinojosa a a Centro de Investigación en Creatividad y Educación Superior & Departmento de Ingeniería Mechánica, Universidad de Santiago de Chile, Santiago, Chile b Department of Mathematics and Statistical Sciences & Department of Mechanical, Energy and Industrial Engineering, Bostwana International University of Science and Technology, Palapye, Bostwana ARTICLE INFO Keywords: Viscoplastic Bingham Hershey-Bulkley Non-circular tube Plug and stagnant zones ABSTRACT Flow of Bingham plastics through straight, long tubes is studied by means of a versatile analytical method that allows extending the study to a large range of tube geometries. The equation of motion is solved for general non- circular cross-sections obtained via a continuous and one-to-one mapping called the shape factor method. In particular the velocity field and associated plug and stagnant zones in tubes with equilateral triangular and square cross-section are explored. Shear stress normal to equal velocity lines, energy dissipation distribution and rate of flow are determined. Shear-thinning and shear-thickening effects on the flow, which cannot be accounted for with the Bingham model, are investigated using the Hershey-Bulkley constitutive formulation an extension of the Bingham model. The existence and the extent of undeformed regions in the flow field in a tube with equilateral triangular cross-section are predicted in the presence of shear-thinning and shear-thickening as a specific example. The mathematical flexibility of the analytical method allows the formulation of general results related to viscoplastic fluid flow with implications related to the design and optimization of physical systems for viscoplastic material transport and processing. 1. Introduction Knowledge of the flow of viscoplastic materials is relevant in many contexts such as flow of paints, pastes, suspensions in complex geometries with industrial applications, coating and mining, foodstuffs processing, cosmetic, and pharmaceutical and construction industries, ceramics extrusion, blood and other biological fluid flows, semi-solid materials and in some natural flows such as mud, lava displacements and debris flow. In all these applications as well as natural phenomena the rate of flow, the velocity distribution and the energy dissipation are important flow variables to determine. One model of viscoplastic fluid widely used is Bingham model for its capacity of predicting useful results in most areas of interest. The Bingham model becomes non- linear for flow configurations different from parallel axisymmetric or unbounded parallel surfaces and, moreover, requires careful interpre- tation and analysis of mathematical results, which are meaningful only when all physically relevant conditions between stress and rate of deformation are met. Other types of non- Newtonian bounded flows, such as viscoelastic fluid flows in tubes, may be fully described physically over the whole flow region by means of mathematical or numerical results derived from the constitutive and linear momentum balance equations. But this is not the case for viscoplastic flows described by the Bingham model since it predicts physically mean- ingless results in some zones that must be identified and characterized as plug zones and stagnant zones where there is no deformation. This is not explicitly predicted by the Bingham model, and must be deduced from conditions associated with the yield stress, tube contour and the related physical considerations. Understanding the dynamics of the formation of dead regions for instance is important to the design of extrusion geometries. It is quite difficult to model viscoplastic fluid flow and design operating systems in most real-life contexts. In particular the determination of the location and shape of the boundary separating the yielded and unyielded masses of the fluid must be part of the solution of the initial boundary value problem. Several authors have addressed in the past the analysis of the flow of Bingham fluids in conduits and related geometries. The ground- breaking work of Russian researchers in the sixties set the tone for the research direction for decades to come. Safronchik [1–3] and Mosolov and Mjasnikov [4–6] conducted fundamental investigations on the propagation and the location of the yield surface and its properties and the plug and dead regions in the flow, respectively. The channel flow of a Bingham plastic with a given initial velocity distribution and a time dependent pressure gradient is investigated in [1] to determine the subsequent velocity field and the location of the yield surface. A highly complicated equation for the velocity dependent on the location of the interface between the plug zone and the flowing mass of the fluid is http://dx.doi.org/10.1016/j.ijnonlinmec.2016.09.012 Received 19 September 2016; Accepted 23 September 2016 ⁎ Corresponding author at: Centro de Investigación en Creatividad y Educación Superior & Departmento de Ingeniería Mechánica, Universidad de Santiago de Chile, Santiago, Chile. E-mail addresses: dennis.siginer@usach.cl, siginerd@biust.ac.bw (D.A. Siginer). International Journal of Non–Linear Mechanics 88 (2017) 1–10 0020-7462/ © 2016 Elsevier Ltd. All rights reserved. Available online 04 October 2016 crossmark