Oldroyd

Bingham plastics are fluids which remain rigid under the application of shear stresses less than a yield stress, Ty, but flow like a. simple Newtonian fluid once the applied shear exceeds this value. Different constitutive models representing this type of fluids were developed by Herschel and Bulkley (1926), Oldroyd (1947) and Casson (1959). 

The Oldroyd-type differential constitutive equations for incompressible viscoelastic fluids can in general can be written as (Oldroyd, 1950)... 

A frequently used example of Oldroyd-type constitutive equations is the Oldroyd-B model. The Oldroyd-B model can be thought of as a description of the constitutive behaviour of a fluid made by the dissolution of a (UCM) fluid in a Newtonian solvent . Here, the parameter A, called the retardation time is de.fined as A = A (r s/(ri + s), where 7]s is the viscosity of the solvent. Hence the extra stress tensor in the Oldroyd-B model is made up of Maxwell and solvent contributions. The Oldroyd-B constitutive equation is written as... 

Oldroyd, J. G., 1947. A rational formulation of the equations of plastic flow for a Bingham solid. Proc. Camb. Philos. Soc. 43, 100-105. 

The first finite element schemes for differential viscoelastic models that yielded numerically stable results for non-zero Weissenberg numbers appeared less than two decades ago. These schemes were later improved and shown that for some benchmark viscoelastic problems, such as flow through a two-dimensional section with an abrupt contraction (usually a width reduction of four to one), they can generate simulations that were qualitatively comparable with the experimental evidence. A notable example was the coupled scheme developed by Marchal and Crochet (1987) for the solution of Maxwell and Oldroyd constitutive equations. To achieve stability they used element subdivision for the stress approximations and applied inconsistent streamline upwinding to the stress terms in the discretized equations. In another attempt, Luo and Tanner (1989) developed a typical decoupled scheme that started with the solution of the constitutive equation for a fixed-flow field (e.g. obtained by initially assuming non-elastic fluid behaviour). The extra stress found at this step was subsequently inserted into the equation of motion as a pseudo-body force and the flow field was updated. These authors also used inconsistent streamline upwinding to maintain the stability of the scheme. 

In this section the discretization of upper-convected Maxwell and Oldroyd-B models by a modified version of the Luo and Tanner scheme is outlined. This scheme uses the subdivision of elements suggested by Marchal and Crochet (1987) to generate smooth stress fields (Swarbrick and Nassehi, 1992a). 

The elastic part of the extra stress for upper-eonvected Maxwell (UCM) and Oldroyd-B fluids is written as (see Chapter 1)... 

Olagunju, D.O. and Cook, L. P., 1993. Secondary flows in cone and plate flow of an Oldroyd-B fluid. J. Non-Newtonian Fluid Mech. 46, 29-47. 

Many industrially important fluids cannot be described in simple terms. Viscoelastic fluids are prominent offenders. These fluids exhibit memory, flowing when subjected to a stress, but recovering part of their deformation when the stress is removed. Polymer melts and flour dough are typical examples. Both the shear stresses and the normal stresses depend on the history of the fluid. Even the simplest constitutive equations are complex, as exemplified by the Oldroyd expression for shear stress at low shear rates ... 

The relaxation times, and desctibe the times required to reheve stress on the cessation of strain and to reheve strain on the cessation of stress, respectively. The fuU Oldroyd tensor requires knowledge of eight material properties. 

Oldroyd, David R. Some neo-Platonic and Stoic influences on mineralogy in the sixteenth and seventeenth centuries. Ambix 21, no. 2-3 (Jul-Nov 1974) 128-156. 

Oldroyd, David R. An examination of G.E. Stahl s philosophical Principles of universal chemistry. Ambix 20 (1973) 36-52. 

Oldroyd S. M. (1971). Biochemical investigations on various forms of some Papilio species. Entomologist 104 111-123. 

Butler IP, Archer C, Rickard D, Vance D, Oldroyd A (2003) Fe isotope fractionation during Fe(II) monosulfide precipitation from aqueous Fe solutions at pH 8 and ambient temperature. Geochim Cosmochim Acta... 

Doe BR (1994) Zinc, copper, and lead in mid-ocean ridge basalts and the source rock control on Zn/Pb in ocean-ridge hydrothermal deposits. Geochim Cosmochim Acta 58 2215-2223 Ehrlich S, Butler I, Halicz L, Rickard D, Oldroyd A, Matthews A (submitted) Experimental study of copper isotope fractionation between aqueous Cu(II) and covellite, CuS. Chem Geol Finney LA, O Halloran TV (2003) Transition metal speciation in the cell insights from the chemistry of metal ion receptors. Science 300 931-936... 

Shannon IJ, Maschmeyer T, Sankar G, Thomas JM, Oldroyd RD, Sheehy M, MadUl D, Waller AM, Townsend RP (1997) Catal Lett 44 23 Schoemann BJ (1997) Zeolites 18 97... 

Ehrlich S, Butler I, Halicz L, Rickard D, Oldroyd A, Matthews A (2004) Experimental study of the copper isotope fractionation between aqueous Cu(II) and covellite, CuS, Chem Geol 209 259-269... 

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