Some flow equations with yield stresses

6 Some flow equations with yield stresses 

The most popular equations that have been used to describe liquids with yield stresses are the Bingham, Casson and Herschel-BuUdey (sometimes called the generalised Bingham) models, i.e. 

The Bingham model is the most non-Newtonian example of the Sisko model, which is itself a simplification of the Cross (or Carreau model) under the appropriate conditions [3]. 

A simple but versatile model that can cope with a yield stress, yet retain the proper extreme of a finite zero and infinite shear-rate viscosity is the Cross model with the exponent set to unity, so 

This behaviour is then very similar to figures 1 to 10, where we note the large drop in viscosity for only a moderate increase in stress for all the results shown. Above the yield stress where r r o, and Ay 1 and it is easy to show that the equation simplifies to 

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For Newtonian fluids the dynamic viscosity is constant (Equation 2-57), for power-law fluids the dynamic viscosity varies with shear rate (Equation 2-58), and for Bingham plastic fluids flow occurs only after some minimum shear stress, called the yield stress, is imposed (Equation 2-59). 

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. 

Originally, the concept of fluid boundary layer was presented by Prandtl [123]. Prandtl s idea was that for flow next to a solid boundary a thin fluid layer (i.e., a boundary layer) develops in which friction is very important, but outside this layer the fluid behaves very much like a ffictionless fluid. To define a demarcation line between these two flow regions the thickness of the boundary layer, 6, is arbitrarily taken as the distance away from the surface where the velocity reaches 99 % of the free stream velocity (e.g., [55], p. 192 [107], p. 12 [114], p. 545). To proceed giving a thorough description of the equations used for turbulent flows, we need some results from the semi-empirical turbulent boundary layer flow analysis. For a generalized shear flow in the vicinity of a flat horizontal solid wall, the boundary layer flow can be described in Cartesian coordinates. The stress, —Ojty.eff, associated with direction y normal to the wall is apparently dominant, thus the stream-wise Reynolds averaged momentum equation yields ... 

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