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Research Papers: Fundamental Issues and Canonical Flows

Developing Region Solution for High Reynolds Number Laminar Flows of Pseudoplastic and Dilatant Fluids in Circular Ducts

[+] Author and Article Information
Massimo Capobianchi

Professor
Mem. ASME
Department of Mechanical Engineering,
Gonzaga University,
502 E. Boone Avenue,
Spokane, WA 99258-0026
e-mail: capobianchi@gonzaga.edu

Patrick McGah

Mem. ASME
Department of Mechanical Engineering,
University of Washington,
Box 352600,
Seattle, WA 98195
e-mail: pmcgah@u.washington.edu

1Corresponding author.

Contributed by the Fluids Engineering Division of ASME for publication in the JOURNAL OF FLUIDS ENGINEERING. Manuscript received August 8, 2015; final manuscript received November 7, 2016; published online February 6, 2017. Assoc. Editor: Kausik Sarkar.

J. Fluids Eng 139(4), 041202 (Feb 06, 2017) (11 pages) Paper No: FE-15-1547; doi: 10.1115/1.4035242 History: Received August 08, 2015; Revised November 07, 2016

This article reports the results of a numerical computation of the length and total pressure drop in the entrance region of a circular tube with laminar flows of pseudoplastic and dilatant fluids at high Reynolds numbers (i.e., approximately 400 or higher). The analysis utilizes equations for the apparent viscosity that span the entire shear rate regime, from the zero to the infinite shear rate Newtonian regions, including the power law and the two transition regions. Solutions are thus reported for all shear rates that may exist in the flow field, and a shear rate parameter is identified that quantifies the shear rate region where the system is operating. The entrance lengths and total pressure drops were found to be bound by the Newtonian and power law values, the former being approached when the system is operating in either the zero or the infinite shear rate Newtonian regions. The latter are approached when the shear rates are predominantly in the power law region but only if, in addition, the zero and infinite shear rate Newtonian viscosities differ sufficiently, by approximately four orders of magnitude or more. For all other cases, namely, when more modest differences in the limiting Newtonian viscosities exist, or when the system is operating in the low- or high-shear rate transition regions, then intermediate results are obtained. Entrance length and total pressure drop values are provided in both graphical form, and in tabular and correlation equation form, for convenient access.

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Figures

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Fig. 1

Schematic flow curves for pseudoplastic and dilatant fluids and definition of the five flow regions: region I—zero shear rate Newtonian region; region II—low shear rate transition region; region III—power law region; region IV—high-shear rate transition region; and region V—infinite shear rate Newtonian region

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Fig. 2

Schematic of the entrance region problem with definition of coordinate axes andcorresponding velocity components. HBL—hydrodynamic boundary layer; HER—hydrodynamic entrance region; and HFDR—hydrodynamic fully developed region.

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Fig. 3

u+ velocity profiles for n = 0.50, Log10(β*) = 2.00, and Log10(R*) = −4.00 at x+= 0.0001, 0.005, 0.02284, 0.05534, 0.075, and 0.1228 (from left to right along the horizontal axis). The profile drawn in heavy-weight line, at x+= 0.1228, is the x+ location where the 99% centerline velocity criterion is met.

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Fig. 4

u+ velocity profiles for n = 1.00 (Newtonian fluid) at x+= 0.0001, 0.005, 0.02284, and 0.05534 (from left to right along the horizontal axis). The profile drawn in heavy-weight line, at x+= 0.05534, is the x+ location where the 99% centerline velocity criterion is met.

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Fig. 5

u+ velocity profiles for n = 1.5, Log10(β*) = −2.00, and Log10(R*) = 4.00 at x+= 0.0001, 0.005, and 0.02284 (from left to right along the horizontal axis). The profile drawn in heavy-weight line, at x+= 0.02284, is the x+ location where the 99% centerline velocity criterion is met.

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Fig. 6

xe+ results using the 99% centerline velocity criterion. From inner to outer curve in each family, n = 0.90, 0.80, 0.70 0.60, and 0.50 for pseudoplastic fluids, and n = 1.10, 1.20, 1.30 1.40, and 1.50 for dilatant fluids. Also shown are horizontal linesat the Newtonian and power law asymptotes, the latter computed at | Log10(β*)|=10 with | Log10(R*)|=20.

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Fig. 7

xe+ results using the 101% f·ReM criterion. From inner to outer curve in each family, n = 0.90, 0.80, 0.70 0.60, and 0.50 for pseudoplastic fluids, and n = 1.10, 1.20, 1.30 1.40, and 1.50 for dilatant fluids. Also shown are horizontal lines at the Newtonian and power law asymptotes, the latter computed at | Log10(β*)|=10 with | Log10(R*)|=20. Numerical values for the points indicated by “” are tabulated in Table 1.

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Fig. 8

fx·ReM¯ results using the 101% f·ReM criterion. From inner to outer curve in each family, n = 0.90, 0.80, 0.70 0.60, and 0.50 for pseudoplastic fluids, and n = 1.10, 1.20, 1.30 1.40, and 1.50 for dilatant fluids. Also shown are horizontal lines at the Newtonian and power law asymptotes, the latter computed at | Log10(β*)|=10 with | Log10(R*)|=20.

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Fig. 9

f·ReM (i.e., HFDR) results. From inner to outer curve in each family, n = 0.90, 0.80, 0.70 0.60, and 0.50 for pseudoplastic fluids, and n = 1.10, 1.20, 1.30 1.40, and 1.50 for dilatant fluids. Also shown are horizontal lines at the Newtonian and power law asymptotes, the latter computed at | Log10(β*)|=10 with | Log10(R*)|=20.

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