Tube flow of non‐Newtonian polymer solutions: PART I. Laminar flow and rheological models

Abstract

The correlation must be considered only provisional, however, because it does not simplify to the limiting case of extremely dilute solutions. Plausible extensions based on models of steady-flow non-Newtonian viscosity behavior, and other possible correlation schemes utilizing viscoelastic fluid properties, are briefly discussed.Three kinds of mechanical behavior of polymers and polymer solutions have been studied experimentally: the shear-dependent (non-Newtonian ) viscosity 7); the normal stress (Weissenberg) effect, described by the normal stress coefficient i; and the transient response to small amplitude displacements which can be described by the complex viscosity t*. Recently it has been proposed (24, 25, 26) that the transient response of the normal stresses should also receive experimental attention.One of the primary aims of rheology is to formulate constitutive equations capable of describing all of the above phenomena and their interrelations. To this end some very general and elegant equations have been proposed. Despite their all-inclusiveness they may not necessarily offer the most judicious description of systems encountered in engineering applications. For many applications an adequate description of a viscoelastic fluid is provided by the generalized Newtonian model ( 5 , 18)in which q is a function of the second invariant of the viscous momentum flux tensor T or the rate-of-deformationThe Ton-Newtonian viscosity 7 appearing in Equation (1) can usually be curve fitted with a function containing a small number of constant parameters. One then states that the mechanical behavior of the fluid is characterized by specifying values of these parameters. The use of Equation (1) with a curve-fit function for 71 has been studied rather extensively at this and other laboratories for several fluids and flow systems; the following facts seem to emerge regarding the above method:1. The method seems to be adequate for describing steady state laminar flow in a variety of systems; that is when the model parameters are determined in one geome- try, the same parameters are able to describe the flow of the same fluid in a different geometry (17, 10, 19, 20, 23).2. The method seems to be adequate for describing unsteady state laminar flow provided that the dominant time constant of the system is larger than the dominant time constant for the fluid ( 2 ) .3. The method provides a small number of parameters which can be used for the preparation of dimensionless correlations for steady flow in geometrically complex systems (20, 21, 22).4. The parameters in the non-Newtonian viscosity function may be useful for correlating some turbulent flow phenomena (12).5. The method is probably quite adequate for solving steady state heat transfer problems (1 ) .6. The generalized Newtonian model is easily adapted to the use of variational methods for obtaining analytical solutions to complex flow problems (13, 3, 9, 2 7 ) .Hence, although the generalized Newtonian models cannot possibly describe normal stresses or small amplitude...

Theoretical studies of non-Newtonian and Newtonian fluid flow through porous media

“…A similar correlation of the apparent viscosity for polymer solutions was proposed by Meter ( Meter and Bird, 1964), -- where a and z , are constants. Equation 2.8 has been used to investigate flow problems of polymer solutions in porous media (Lake, 1987;and Carnilleri et al, 1987).…”

“…Because of the complexity of pore geometries in a porous medium, Darcy's law has to be used to obtain any meaningful insights into the physics of flow in porous media. Some (Meter and Bird, 1964) to describe the more realistic shear-thinning behavior of polymer solutions. Meter and Bird (1964) presented a practical procedure to determine the four parameters in Equation 2.8 by fitting experimental non-Newtonian viscosity data.…”

Rivulet flow of generalized Newtonian fluids

“…An example of a generalized Newtonian fluid whose viscosity is usually expressed in the form μ = μ(τ ) is an Ellis fluid, sometimes also known as a Meter fluid (see, for example, Refs. [44][45][46][47]), given by…”

Shear Rate Determination from Pore-Scale Flow Fields

“…Even though these approaches become more and more available, the most commonly used approach are semi-empirical correlations for the Darcy-scale effective shear rate (Delshad et al 2008). These semi-empirical correlations are based on the Cannella or Blake-Kozeny equation (Meter and Bird 1964;Cannella et al 1988;Sorbie 1991) which was originally developed for capillary bundles. They are semi-empirical and contain empirical factors such as the shear rate coefficient, often also termed the "C factor", which in the framework of the correlation is a constant.…”