Flow of a Carreau Fluid in a Circular Tube

A Carreau fluid is a generalized type of Newtonian fluid in which viscosity depends upon shear rate, .
Consider the laminar flow of a Carreau fluid in a long circular tube of radius and length . The fluid flows under the influence of a pressure difference . This Demonstration plots the velocity profile for a Carreau fluid (blue curve), a Newtonian fluid (orange curve), and a pseudoplastic fluid (red curve) for various values of the relaxation parameter, (see Details for the definition). These profiles are obtained under equal volumetric flow conditions. The velocity near the wall is higher for the Carreau and pseudoplastic fluids than for the Newtonian fluid. This results in higher heat transfer rates due to higher convection. For both the pseudoplastic and the Carreau fluids, the exponent is taken equal to . The consistency index for the pseudoplastic fluid and the viscosity of the Newtonian fluids is computed in order to keep the volumetric flows equal. The infinite-shear viscosity and the zero-shear viscosity of the Carreau fluid are taken equal to 0 and , respectively. As one can find from the equation in Details, there are two limiting cases: (1) large (e.g. ), the profiles for pseudoplastic and Carreau fluids superimpose; and (2) small (e.g. ), the profiles for Newtonian and Carreau fluids are identical.


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According to the Carreau model, first proposed by Pierre Carreau,
For , this reduces to a Newtonian fluid with .
For , we obtain a power-law fluid with .
[1] H. Binous,"Introducing Non-Newtonian Fluid Mechanics Computations with Mathematica in the Undergraduate Curriculum," Chemical Engineering Education, 41(1), 2007 pp. 59–64.
[2] J. O. Wilkes, Fluid Mechanics for Chemical Engineers, Upper Saddle River, NJ: Prentice Hall, 1999.
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