Reflections on inflections

C. Saengow; A.J. Giacomin; P.H. Gilbert; C. Kolitawong

Korea-Australia Rheology Journal, Vol.27, No.4, 267-285, November, 2015

DOI10.1007/s13367-015-0027-7 Export Citation

Reflections on inflections

C. Saengow^{1, 2}, A.J. Giacomin^{1, 3, †}, P.H. Gilbert^{1, 4}, and C. Kolitawong²

¹Chemical Engineering Department, Polymers Research Group, Korea
²Mechanical and Aerospace Engineering Department, Polymer Research Center, King Mongkut’s University of Technology North Bangkok, Bangkok, Thailand 10800
³Mechanical and Materials Engineering Department, Queen’s University, Kingston, Ontario, Canada K7L 3N6
⁴, Korea

E-mail:

In plastics processing, the single most important rheological property is the steady shear viscosity curve: the logarithm of the steady shear viscosity versus the logarithm of the shear rate. This curve governs the volumetric flowrate through any straight channel flow, and thus governs the production rate of extruded plastics. If the shear rate is made dimensionless with a characteristic time for the fluid (called the Weissenberg number, Wi≈1, then we can readily identify the end of the Newtonian plateau of a viscosity curve with the value Wi≈1. Of far greater importance, however, is the slope at the point where the viscosity curve inflects, (n.1), where n is called the shear power-law index. This paper explores the physics of this point and related inflections, in the first and second normal stress coefficients. We also discuss the first and second inflection pairing times, and . First, we examine the generalized Newtonian fluid (Carreau model). Then, we analyze the more versatile model, the corotational Oldroyd 8-constant model, which reduces to many simpler models, for instance, the corotational Maxwell and Jeffreys models. We also include worked examples to illustrate the procedure for calculating inflection points and power-law coefficients for all three viscometric functions, and η(γ), Ψ1 (γ) and Ψ2 (γ).

Keywords:steady shear flow;viscosity curve;viscosity inflection;normal stress coefficients;normal stressdifferences

[References]

Abramowitz M, Stegun IA, 1970, Handbook of mathematical functions-with formulas, graphs, and mathematical tables, National Bureau of Standards, Washington DC.
Baird DG, Collias DI, 1998, Polymer processing: principles and design, John Wiley & Sons, New York.
Baird DG, Collias DI, 2014, Polymer processing: principles and design, 2nd ed., John Wiley & Sons, New York.
Bird RB, Armstrong RC, Hassager O, 1977, Dynamics of Polymeric Liquids, Vol. 1, 1st ed., Wiley, New York.
Bird RB, Armstrong RC, Hassager O, 1987, Dynamics of Polymeric Liquids, Vol. 1, 2nd ed., Wiley, New York.
Bird RB, Stewart WE, Lightfoot EN, 2007, Transport Phenomena, Revised 2nd ed., Wiley & Sons, New York.
Bird RB, Stewart WE, Lightfoot EN, Klingenberg DJ, 2015, Introductory Transport Phenomena, Wiley & Sons, New York.
Camilleri CJ, Jones JR, Zeitschrift fur angewandte Mathematik und Physik, 17, 78 (1966)
Huppler JD, Ashare E, Holmes JA, Transac. Soc. Rheo., 11, 159 (1967)
Jeffreys H, 1924, The Earth: Its Origin, History and Physical Constitution, Cambridge, London.
Jeffreys H, 1929, The Earth: Its Origin, History and Physical Constitution, 2nd ed., Cambridge, London.
Jones JR, J. Mecanique, 3, 79 (1964)
Jones JR, J. Mecanique, 4, 121 (1965)
Jones JR, Jones RS, J. Mecanique, 5, 375 (1966)
Jones RS, J. Mecanique, 6, 443 (1967)
Leider PJ, Ind. Eng. Chem. Fundam., 13, 342 (1974)
Leider PJ, Bird RB, Ind. Eng. Chem. Fundam., 13, 336 (1974)
Oldroyd JG, Proc. Roy. Soc. London, 245, 278 (1958)
Saengow C, Giacomin AJ, Kolitawong C, J. Non-Newton. Fluid Mech., 223, 176 (2015)
Saengow C, Giacomin AJ, Kolitawong C, 2015b, Extruding Plastic Pipe from Eccentric Dies, PRG Report No.011, QU-CHEE-PRG-TR--2015-11, Polymers Research Group, Chem. Eng. Dept., Queen’s University, Kingston, Canada.

[1] Abramowitz M, Stegun IA, 1970, Handbook of mathematical functions-with formulas, graphs, and mathematical tables, National Bureau of Standards, Washington DC.

[2] Baird DG, Collias DI, 1998, Polymer processing: principles and design, John Wiley & Sons, New York.

[3] Baird DG, Collias DI, 2014, Polymer processing: principles and design, 2nd ed., John Wiley & Sons, New York.

[4] Bird RB, Armstrong RC, Hassager O, 1977, Dynamics of Polymeric Liquids, Vol. 1, 1st ed., Wiley, New York.

[5] Bird RB, Armstrong RC, Hassager O, 1987, Dynamics of Polymeric Liquids, Vol. 1, 2nd ed., Wiley, New York.

[6] Bird RB, Stewart WE, Lightfoot EN, 2007, Transport Phenomena, Revised 2nd ed., Wiley & Sons, New York.

[7] Bird RB, Stewart WE, Lightfoot EN, Klingenberg DJ, 2015, Introductory Transport Phenomena, Wiley & Sons, New York.

[8] Camilleri CJ, Jones JR, Zeitschrift fur angewandte Mathematik und Physik, 17, 78 (1966)

[9] Huppler JD, Ashare E, Holmes JA, Transac. Soc. Rheo., 11, 159 (1967)

[10] Jeffreys H, 1924, The Earth: Its Origin, History and Physical Constitution, Cambridge, London.

[11] Jeffreys H, 1929, The Earth: Its Origin, History and Physical Constitution, 2nd ed., Cambridge, London.

[12] Jones JR, J. Mecanique, 3, 79 (1964)

[13] Jones JR, J. Mecanique, 4, 121 (1965)

[14] Jones JR, Jones RS, J. Mecanique, 5, 375 (1966)

[15] Jones RS, J. Mecanique, 6, 443 (1967)

[16] Leider PJ, Ind. Eng. Chem. Fundam., 13, 342 (1974)

[17] Leider PJ, Bird RB, Ind. Eng. Chem. Fundam., 13, 336 (1974)

[18] Oldroyd JG, Proc. Roy. Soc. London, 245, 278 (1958)

[19] Saengow C, Giacomin AJ, Kolitawong C, J. Non-Newton. Fluid Mech., 223, 176 (2015)

[20] Saengow C, Giacomin AJ, Kolitawong C, 2015b, Extruding Plastic Pipe from Eccentric Dies, PRG Report No.011, QU-CHEE-PRG-TR--2015-11, Polymers Research Group, Chem. Eng. Dept., Queen’s University, Kingston, Canada.