Carreau Equation Research Articles

A simple model of plate generation from mantle flow

SUMMARY A simple model of non-Newtoniancreepingflow is used to evaluate classes of rheologieswhichallow viscous mantle flow to becomeplate-like. The model describes shallow-layerlithosphericmotion driven by sources and sinks. The sources represent spreading ridges while the sinks rep-resent subduction zones; the sources and sinks thus also prescribe the poloidal component ofthe surface flow field. The toroidal (strike-slip) component of the flow field is found via thesolution of the Stokes equation with non-Newtonian rheology. As a first basic investigation ofthe model, the horizontal divergence from the two-dimensional rectangular velocity field ofOlson & Bercovici (1991) is used for the source-sink field. The degree to which the inducedfluid flow reproducesthe rectangularplate is used to measure the success of differentrheologiesin generating plate-like flows. Results indicate that power-law rheologies, even in the limit ofvery high power-law indexν, can only produce modest plate-like flow. For example, the ratioof toroidal to poloidal kinetic energy for a source-sink field derived from a square plate is atbest 0.65, whereas a perfect square plate has a ratio of 1.0. Moreover, the power-law rheologyappears to reach an asymptotic limit in its ability to produce plate-like behavior. This impliesthat plate tectonics is unlikely to arise from a power-law rheology even in the limit of very highν. A class of rheologies that yield significantly more promisi ng results arise from the Carreaupseudo-plastic rheology with the power-law index taken to be ν < 0. One rheology in thisclass is the continuum model for stick-slip, earthquake behavior of Whitehead & Gans (1974),which is essentially the Carreau equation with ν = −1. This class of rheologies, referred to asthe stick-slip rheologies, induces a toroidal to poloidal kinetic energy ratio for a square plates’ssource-sink function which can be as high as0.9. The viscosity (or strength) distribution forthis class of rheologies also appears more plate-like, showing fairly uniform high viscosity re-gions (pseudo-plates) and sharply defined low viscosity zones (pseudo-margins). In contrast,even the most nonlinear power-law rheology produces spatially varying high viscosity regionsand relatively smooth low viscosity margins. The greater success of the stick-slip rheologiesin producing plates is attributed to a self-lubricating mechanism in which the transfer of mo-mentum from regions of high shear to low shear is inhibited. In contrast, even in the limit ofinfinite power-law index, a power-law rheology can retard but never prohibit momentum trans-fer. This feature is essential to the sharpening of velocity profiles into plate-like profiles, whichis illustrated with a simple boundary-layertheory.Key words: Plate tectonics, mantle convection, non-Newtonian flow, toroidal-poloidal cou-pling.

Effects of shear thinning on laminar heat transfer behavior in a rectangular duct

The fluid dynamic and heat transfer behavior of laminar non-Newtonian flow through noncircular ducts is of special interest because of their wide range of potential application to compact heat exchangers and in electronics cooling. The present study investigates the heat transfer characteristics for laminar forced convection of inelastic non-Newtonian fluids in a nonuniformly heated rectangular duct where the flow is hydrodynamically developed but thermally developing. The governing equations are solved by a finite volume method. Second-order accurate differencing schemes are employed for both the diffusion and convective terms. The effects of shear thinning, given by the Carreau equation, and the viscous dissipation, characterized by the Brinkman number, are examined via the friction factor, the Nusselt number, and the bulk fluid temperature. The results indicate that when viscous dissipation is present, the heat transfer from the heated surface of the duct is greatly enhanced with increased shear thinning of the fluid while the rate of increase of the bulk fluid temperature with axial distance will significantly decrease. Consequently, the use of a shear-thinning non-Newtonian fluid for heat transfer enhancement appears to be a very promising concept that is worthy of further study.

Effect of aspect ratio on laminar heat transfer behavior of a non-Newtonian fluid in an electronics coldplate

The present note delineates the effect of duct aspect ratio on the heat transfer characteristics for laminar forced convection of inelastic non-Newtonian fluids in a nonuniformly heated rectangular duct, which is an integral part of an electronics coldplate, where the flow is hydrodynamically developed but thermally developing as often encountered in coldplates used in the cooling of phased array radar. The governing equations are solved by a finite volume method. The effects of shear thinning, given by the Carreau equation, and the viscous dissipation, characterized by the Brinkman number, are examined via the the friction factor, the Nusselt number, and the bulk fluid temperature for duct aspect ratios ranging from 1. to 0.2. The results indicate that when viscous dissipation is present, the heat transfer from the heated surface of the duct is not significantly increased as the aspect ratio decrease (i.e., the area of heat transfer increases); however, with a decreasing duct aspect ratio the rate of increase of the bulk fluid temperature with axial distance is significant. Also, the product of the Fanning friction factor and the Reynolds number will not be significantly increased with decreasing aspect ratio for a shear thinning fluid. Consequently, the use of a duct aspect ratio near 1 will provide the best overall design for cooling ducts used in electronics coldplate heat removal systems when the flow is hydrodynamically developed and thermally developing.

Dimensionless Slot Flow Using the Carreau Viscosity Model

In order to find the flow and pressure, the constants for the Carreau Equation must be known for the specific polymer. A method is provided for finding the Carreau Constants from viscosity versus shear data for any polymer

Constitutive equation for modeling mixed extension and shear in polymer solution processing

What is the simplest adequate model of shear thinning and extension thickening of polymer solutions at the intense local deformation rates in coating and related flows? We examine a combination of the Carreau equation of shear sensitivity with an analogous equation of extension sensitivity. The two contributions are mediated by the rate at which the principal straining axes turn with respect to the liquid itself, a relative rotation rate that is frame independent, vanishes in purely extensional flow, and harks back to Lumley’s (1969) idea of persistent molecular strain rate, which was adumbrated by Takserman‐Krozer (1963) and named by Frank and Mackley (1976). We indicate how the new equation can be used to extract viscosity‐in‐extension from the extension‐rich mixed flow between opposed nozzles. We illustrate how the new equation can be applied to the complex mixed flow in slide coating.

Rheological characterization of dextran‐enzymatic synthesis media

AbstractThe bacteria Leuconostoc mesenteroides (strain NRRL‐B512) has been used for the fermentative production of the polysaccharide dextran from sucrose. The polysaccharide production was carried out via a four‐step process involving (1) production of the enzyme dextransucrase (DS), (2) bacteria removal from the culture broth, (3) enzyme purification by gel permeation chromatography, and (4) enzymatic synthesis of dextran in a cell‐free reaction medium.The enzymatic reaction was followed measuring the change in rheological properties of the reaction medium with time at different initial substrate concentrations. The influence of other parameters, such as temperature, addition of acceptor molecules (maltose), and enzyme purity, were also considered.The microbial extracellular polysaccharide is responsible for the non‐newtonian character of the reaction media. At low shear rate values the flow behavior was newtonian, whereas at higher shear rates the reaction media exhibited a marked shear‐thinning behavior. The Carreau equation appears to fit fairly well the experimental data over several decades of shear rate. The reaction media showed elastic behavior as well.

The roles of inertia and shear-thinning in flow of an inelastic liquid through an axisymmetric sudden contraction

The roles of luid inertia and shear-rate dependent viscosity in determining the velocity field in an axisymmetric sudden contraction are assessed by finite-element analysis for a generalized Newtonian fluid with viscosity function given by a Carreau equation. Acting alone, either increasing shear-thinning of the viscosity or increasing fluid inertia suppresses the upstream vortex that surrounds the opening to the small tube. For creeping flows, shear thinning does not produce concavities and off-centered maxima in the axial velocity profile just inside the small tube, even at high Carreau numbers where the velocity field approaches the limiting form for a power-law fluid. Peaks in the axial velocity away from the center of the tube were found only for moderate and high Reynolds numbers and were enhanced by shear thinning, which decreased the viscosity and consequently increased the “local” Reynolds number near the wall of the small tube. The effect of steep velocity gradients near this surface on the accuracy of the finite-element approximations is discussed.