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Viscosity of some ļ¬‚uids
Fluid Air (at Benzene Water (at 18 ā—¦ C) Olive oil (at 20 ā—¦ C) Motor oil SAE 50 Honey Ketchup Peanut butter Tar Earth lower mantle 18 ā—¦ C) Viscosity [cP] 0.02638 0.5 1 84 540 2000ā€“3000 50000ā€“70000 150000ā€“250000 3 Ɨ 1010 3 Ɨ 1025

Table: Viscosity of some ļ¬‚uids
Josef MĀ“lek a Non-Newtonian ļ¬‚uids

Viscosity of some ļ¬‚uids Models with variable viscosity Diļ¬€erential type models Rate type models Integral type models Download

Shear dependent viscosity Models with pressure dependent viscosity Models with stress dependent viscosity Models with discontinuous rheology

Models with variable viscosity

General form: T = āˆ’pI + 2Āµ(D, T)D
S

(2.1)

Particular models mainly developed by chemical engineers.

Josef MĀ“lek a

Non-Newtonian ļ¬‚uids

Viscosity of some ļ¬‚uids Models with variable viscosity Diļ¬€erential type models Rate type models Integral type models Download

Shear dependent viscosity Models with pressure dependent viscosity Models with stress dependent viscosity Models with discontinuous rheology

Ostwaldā€“de Waele power law
ĀØ Wolfgang Ostwald. Uber die Geschwindigkeitsfunktion der ViskositĀØt disperser Systeme. I. Colloid Polym. Sci., 36:99ā€“117, a 1925 A. de Waele. Viscometry and plastometry. J. Oil Colour Chem. Assoc., 6:33ā€“69, 1923 Āµ(D) = Āµ0 |D|nāˆ’1 (2.2)

Fits experimental data for: ball point pen ink, molten chocolate, aqueous dispersion of polymer latex spheres

Josef MĀ“lek a

Non-Newtonian ļ¬‚uids

Viscosity of some ļ¬‚uids Models with variable viscosity Diļ¬€erential type models Rate type models Integral type models Download

Shear dependent viscosity Models with pressure dependent viscosity Models with stress dependent viscosity Models with discontinuous rheology

Carreau Carreauā€“Yasuda
Pierre J. Carreau. Rheological equations from molecular network theories. J.
Rheol., 16(1):99ā€“127, 1972 Kenji Yasuda. Investigation of the analogies between viscometric and linear viscoelastic properties of polystyrene ļ¬‚uids. PhD thesis, Massachusetts Institute of Technology. Dept. of Chemical Engineering., 1979 Āµ0 āˆ’ Āµāˆž (1 + Ī± |D|2 ) 2 n nāˆ’1 a

Āµ(D) = Āµāˆž +

(2.3) (2.4)

Āµ(D) = Āµāˆž + (Āµ0 āˆ’ Āµāˆž ) (1 + Ī± |D|a ) Fits experimental data for: molten polystyrene Josef MĀ“lek a Non-Newtonian ļ¬‚uids

Viscosity of some ļ¬‚uids Models with variable viscosity Diļ¬€erential type models Rate type models Integral type models Download

Shear dependent viscosity Models with pressure dependent viscosity Models with stress dependent viscosity Models with discontinuous rheology

Eyring
Henry Eyring. Viscosity, plasticity, and diļ¬€usion as examples of absolute reaction rates. J. Chem. Phys., 4(4):283ā€“291, 1936 Francis Ree, Taikyue Ree, and Henry Eyring. Relaxation theory of transport problems in condensed systems. Ind. Eng. Chem., 50(7):1036ā€“1040, 1958 Āµ(D) = Āµāˆž + (Āµ0 āˆ’ Āµāˆž ) arcsinh (Ī± |D|) Ī± |D| arcsinh (Ī±1 |D|) arcsinh (Ī±2 |D|) Āµ(D) = Āµ0 + Āµ1 + Āµ2 Ī±1 |D| Ī±2 |D| (2.5) (2.6)

Fits experimental data for: napalm (coprecipitated aluminum salts of naphthenic and palmitic acids; jellied gasoline), 1% nitrocelulose in 99% butyl acetate Josef MĀ“lek a Non-Newtonian ļ¬‚uids

Viscosity of some ļ¬‚uids Models with variable viscosity Diļ¬€erential type models Rate type models Integral type models Download

Shear dependent viscosity Models with pressure dependent viscosity Models with stress dependent viscosity Models with discontinuous rheology

Cross

Malcolm M. Cross. Rheology of non-newtonian ļ¬‚uids: A new ļ¬‚ow equation for pseudoplastic systems. J. Colloid Sci., 20(5):417ā€“437, 1965 Āµ(D) = Āµāˆž + Āµ0 āˆ’ Āµāˆž 1 + Ī± |D|n (2.7)

Fits experimental data for: aqueous polyvinyl acetate dispersion, aqueous limestone suspension

Josef MĀ“lek a

Non-Newtonian ļ¬‚uids

Viscosity of some ļ¬‚uids Models with variable viscosity Diļ¬€erential type models Rate type models Integral type models Download

Shear dependent viscosity Models with pressure dependent viscosity Models with stress dependent viscosity Models with discontinuous rheology

Sisko

A. W. Sisko. The ļ¬‚ow of lubricating greases. Ind. Eng. Chem., 50(12):1789ā€“1792, 1958 Āµ(D) = Āµāˆž + Ī± |D|nāˆ’1 Fits experimental data for: lubricating greases (2.8)

Josef MĀ“lek a

Non-Newtonian ļ¬‚uids

Viscosity of some ļ¬‚uids Models with variable viscosity Diļ¬€erential type models Rate type models Integral type models Download

Shear dependent viscosity Models with pressure dependent viscosity Models with stress dependent viscosity Models with discontinuous rheology

Barus

C. Barus. Isotherms, isopiestics and isometrics relative to viscosity. Amer. J. Sci., 45:87ā€“96, 1893 Āµ(T) = Āµref eĪ²(pāˆ’pref ) Fits experimental data for: mineral oils1 , organic liquids2 (2.9)

Michael M. Khonsari and E. Richard Booser. Applied Tribology: Bearing Design and Lubrication. John Wiley & Sons Ltd, Chichester, second edition, 2008 2 P. W. Bridgman. The eļ¬€ect of pressure on the viscosity of forty-four pure liquids. Proc. Am. Acad. Art. Sci., 61(3/12):57ā€“99, FEB-NOV 1926 Josef MĀ“lek a Non-Newtonian ļ¬‚uids

Viscosity of some ļ¬‚uids Models with variable viscosity Diļ¬€erential type models Rate type models Integral type models Download

Shear dependent viscosity Models with pressure dependent viscosity Models with stress dependent viscosity Models with discontinuous rheology

Ellis
Seikichi Matsuhisa and R. Byron Bird. Analytical and numerical solutions for laminar ļ¬‚ow of the non-Newtonian Ellis ļ¬‚uid. AIChE J., 11(4):588ā€“595, 1965 Āµ(T) = Āµ0 1 + Ī± |TĪ“ |nāˆ’1 (2.10)

Fits experimental data for: 0.6% w/w carboxymethyl cellulose (CMC) solution in water, poly(vynil chloride)3

T. A. Savvas, N. C. Markatos, and C. D. Papaspyrides. On the ļ¬‚ow of non-newtonian polymer solutions. Appl. Math. Modelling, 18(1):14ā€“22, 1994 Josef MĀ“lek a Non-Newtonian ļ¬‚uids

Viscosity of some ļ¬‚uids Models with variable viscosity Diļ¬€erential type models Rate type models Integral type models Download

Shear dependent viscosity Models with pressure dependent viscosity Models with stress dependent viscosity Models with discontinuous rheology

Glen

J. W. Glen. The creep of polycrystalline ice. Proc. R. Soc. A-Math. Phys. Eng. Sci., 228(1175):519ā€“538, 1955 Āµ(T) = Ī± |TĪ“ |nāˆ’1 Fits experimental data for: ice (2.11)

Josef MĀ“lek a

Non-Newtonian ļ¬‚uids

Viscosity of some ļ¬‚uids Models with variable viscosity Diļ¬€erential type models Rate type models Integral type models Download

Shear dependent viscosity Models with pressure dependent viscosity Models with stress dependent viscosity Models with discontinuous rheology

Seely

Gilbert R. Seely. Non-newtonian viscosity of polybutadiene solutions. AIChE J., 10(1):56ā€“60, 1964 Āµ(T) = Āµāˆž + (Āµ0 āˆ’ Āµāˆž ) e āˆ’ |TĪ“ |
Ļ„0

(2.12)

Fits experimental data for: polybutadiene solutions

Josef MĀ“lek a

Non-Newtonian ļ¬‚uids

Viscosity of some ļ¬‚uids Models with variable viscosity Diļ¬€erential type models Rate type models Integral type models Download

Shear dependent viscosity Models with pressure dependent viscosity Models with stress dependent viscosity Models with discontinuous rheology

Blatter
Erin C. Pettit and Edwin D. Waddington. Ice ļ¬‚ow at low deviatoric stress. J. Glaciol., 49(166):359ā€“369, 2003 H Blatter. Velocity and stress-ļ¬elds in grounded glaciers ā€“ a simple algorithm for including deviatoric stress gradients. J. Glaciol., 41(138):333ā€“344, 1995 Āµ(T) = 2

A |TĪ“ | +
2 Ļ„0
nāˆ’1 2

(2.13)

Fits experimental data for: ice

Josef MĀ“lek a

Non-Newtonian ļ¬‚uids

Viscosity of some ļ¬‚uids Models with variable viscosity Diļ¬€erential type models Rate type models Integral type models Download

Shear dependent viscosity Models with pressure dependent viscosity Models with stress dependent viscosity Models with discontinuous rheology

Bingham Herschelā€“Bulkley
C. E. Bingham. Fluidity and plasticity. McGrawā€“Hill, New York, 1922 Winslow H. Herschel and Ronald Bulkley. Konsistenzmessungen von Gummi-BenzollĀØsungen. Colloid Polym. Sci., 39(4):291ā€“300, o August 1926 |TĪ“
| > Ļ„ āˆ— |TĪ“ | ā‰¤ Ļ„ āˆ— if and only if TĪ“ = Ļ„ āˆ— if and only if D=0 D + 2Āµ(|D|)D |D|

(2.14)

Fits experimental data for: paints, toothpaste, mango jam
Santanu Basu and U.S. Shivhare. Rheological, textural, micro-structural and sensory properties of mango jam. J. Food Eng., 100(2):357ā€“365, 2010 Josef MĀ“lek a Non-Newtonian ļ¬‚uids

Viscosity of some ļ¬‚uids Models with variable viscosity Diļ¬€erential type models Rate type models Integral type models Download

Rivlinā€“Ericksen ļ¬‚uids

Rivlinā€“Ericksen
R. S. Rivlin and J. L. Ericksen. Stress-deformation relations for isotropic materials. J. Ration. Mech. Anal., 4:323ā€“425, 1955 R. S. Rivlin and K. N. Sawyers. Nonlinear continuum mechanics of viscoelastic ļ¬‚uids. Annu. Rev. Fluid Mech., 3:117ā€“146, 1971 General form: T = āˆ’pI + f(A1 A2 A3 . . . ) (3.1) where A1 = 2D dAnāˆ’1 + Anāˆ’1 L + L Anāˆ’1 An = dt (3.2a) (3.2b)

d where dt denotes the usual Lagrangean time derivative and L is the velocity gradient. Josef MĀ“lek a Non-Newtonian ļ¬‚uids

Viscosity of some ļ¬‚uids Models with variable viscosity Diļ¬€erential type models Rate type models Integral type models Download

Rivlinā€“Ericksen ļ¬‚uids

Criminaleā€“Ericksenā€“Filbey

William O. Criminale, J. L. Ericksen, and G. L. Filbey. Steady shear ļ¬‚ow of non-Newtonian ļ¬‚uids. Arch. Rat. Mech. Anal., 1:410ā€“417, 1957 T = āˆ’pI + Ī±1 A1 + Ī±2 A2 + Ī±3 A2 1 (3.3)

Fits experimental data for: polymer melts (explains mormal stress diļ¬€erences)

Josef MĀ“lek a

Non-Newtonian ļ¬‚uids

Viscosity of some ļ¬‚uids Models with variable viscosity Diļ¬€erential type models Rate type models Integral type models Download

Rivlinā€“Ericksen ļ¬‚uids

Reinerā€“Rivlin

M. Reiner. A mathematical theory of dilatancy. Am. J. Math., 67(3):350ā€“362, 1945 T = āˆ’pI + 2ĀµD + Āµ1 D2 Fits experimental data for: N/A (3.4)

Josef MĀ“lek a

Non-Newtonian ļ¬‚uids

Viscosity of some ļ¬‚uids Models with variable viscosity Diļ¬€erential type models Rate type models Integral type models Download

Maxwell, Oldroyd, Burgers Giesekus Phan-Thienā€“Tanner Johnsonā€“Segalman Johnsonā€“Tevaarwerk

Maxwell
J. Clerk Maxwell. On the dynamical theory of gases. Philos. Trans. R. Soc., 157:49ā€“88, 1867

T = āˆ’pI + S S + Ī»1 S = 2ĀµD dM āˆ’ LM āˆ’ ML dt Fits experimental data for: N/A M =def Josef MĀ“lek a Non-Newtonian ļ¬‚uids

(4.1a) (4.1b)

(4.2)

Viscosity of some ļ¬‚uids Models with variable viscosity Diļ¬€erential type models Rate type models Integral type models Download

Maxwell, Oldroyd, Burgers Giesekus Phan-Thienā€“Tanner Johnsonā€“Segalman Johnsonā€“Tevaarwerk

Oldroyd-B

J. G. Oldroyd. On the formulation of rheological equations of state. Proc. R. Soc. A-Math. Phys. Eng. Sci., 200(1063):523ā€“541, 1950

T = āˆ’Ļ€I + S S + Ī»S = Ī·1 A1 + Ī·2 A1 Fits experimental data for: N/A

(4.3a) (4.3b)

Josef MĀ“lek a

Non-Newtonian ļ¬‚uids

Viscosity of some ļ¬‚uids Models with variable viscosity Diļ¬€erential type models Rate type models Integral type models Download

Maxwell, Oldroyd, Burgers Giesekus Phan-Thienā€“Tanner Johnsonā€“Segalman Johnsonā€“Tevaarwerk

Oldroyd 8-constants
J. G. Oldroyd. On the formulation of rheological equations of state. Proc. R. Soc. A-Math. Phys. Eng. Sci., 200(1063):523ā€“541, 1950 T = āˆ’Ļ€I + S Ī»3 Ī»5 Ī»6 (DS + SD) + (Tr S) D + (S : D) I 2 2 2 Ī»7 (D : D) I = āˆ’Āµ D + Ī»2 D + Ī»4 D2 + 2 (4.4a)

S + Ī»1 S +

(4.4b)

Fits experimental data for: N/A
Josef MĀ“lek a Non-Newtonian ļ¬‚uids

Viscosity of some ļ¬‚uids Models with variable viscosity Diļ¬€erential type models Rate type models Integral type models Download

Maxwell, Oldroyd, Burgers Giesekus Phan-Thienā€“Tanner Johnsonā€“Segalman Johnsonā€“Tevaarwerk

Burgers
J. M. Burgers. Mechanical considerations ā€“ model systems ā€“ phenomenological theories of relaxation and viscosity. In First report on viscosity and plasticity, chapter 1, pages 5ā€“67. Nordemann Publishing, New York, 1939

T = āˆ’Ļ€I + S S + Ī»1 S + Ī»2 S = Ī·1 A1 + Ī·2 A1 Fits experimental data for: N/A

(4.5a) (4.5b)

Josef MĀ“lek a

Non-Newtonian ļ¬‚uids

Viscosity of some ļ¬‚uids Models with variable viscosity Diļ¬€erential type models Rate type models Integral type models Download

Maxwell, Oldroyd, Burgers Giesekus Phan-Thienā€“Tanner Johnsonā€“Segalman Johnsonā€“Tevaarwerk

Giesekus
H. Giesekus. A simple constitutive equation for polymer ļ¬‚uids based on the concept of deformation-dependent tensorial mobility. J. Non-Newton. Fluid Mech., 11(1-2):69ā€“109, 1982

T = āˆ’Ļ€I + S S + Ī»S āˆ’ Ī±Ī»2 2 S = āˆ’ĀµD Āµ

(4.6a) (4.6b)

Fits experimental data for: N/A

Josef MĀ“lek a

Non-Newtonian ļ¬‚uids

Viscosity of some ļ¬‚uids Models with variable viscosity Diļ¬€erential type models Rate type models Integral type models Download

Maxwell, Oldroyd, Burgers Giesekus Phan-Thienā€“Tanner Johnsonā€“Segalman Johnsonā€“Tevaarwerk

Phan-Thienā€“Tanner
N. Phan Thien. Non-linear network viscoelastic model. J. Rheol., 22(3):259ā€“283, 1978 N. Phan Thien and Roger I. Tanner. A new constitutive equation derived from network theory. J. Non-Newton. Fluid Mech., 2(4):353ā€“365, 1977

T = āˆ’Ļ€I + S Y S + Ī»S + Ī»Ī¾ (DS + SD) = āˆ’ĀµD 2 Y =e Fits experimental data for: N/A Josef MĀ“lek a Non-Newtonian ļ¬‚uids

(4.7a) (4.7b) (4.7c)

āˆ’Īµ Ī» Tr S Āµ

Viscosity of some ļ¬‚uids Models with variable viscosity Diļ¬€erential type models Rate type models Integral type models Download

Maxwell, Oldroyd, Burgers Giesekus Phan-Thienā€“Tanner Johnsonā€“Segalman Johnsonā€“Tevaarwerk

Johnsonā€“Segalman
M. W. Johnson and D. Segalman. A model for viscoelastic ļ¬‚uid behavior which allows non-aļ¬ƒne deformation. J. Non-Newton. Fluid Mech., 2(3):255ā€“270, 1977

T = āˆ’pI + S (4.8a) S = 2ĀµD + S (4.8b) S +Ī» dS + S (W āˆ’ aD) + (W āˆ’ aD) S dt = 2Ī·D (4.8c)

Fits experimental data for: spurt
Josef MĀ“lek a Non-Newtonian ļ¬‚uids

Viscosity of some ļ¬‚uids Models with variable viscosity Diļ¬€erential type models Rate type models Integral type models Download

Maxwell, Oldroyd, Burgers Giesekus Phan-Thienā€“Tanner Johnsonā€“Segalman Johnsonā€“Tevaarwerk

Johnsonā€“Tevaarwerk
K. L. Johnson and J. L. Tevaarwerk. Shear behaviour of elastohydrodynamic oil ļ¬lms. Proc. R. Soc. A-Math. Phys. Eng. Sci., 356(1685):215ā€“236, 1977

T = āˆ’pI + S S S + Ī± sinh = 2ĀµD s0 Fits experimental data for: lubricants

(4.9a) (4.9b)

Josef MĀ“lek a

Non-Newtonian ļ¬‚uids

Viscosity of some ļ¬‚uids Models with variable viscosity Diļ¬€erential type models Rate type models Integral type models Download

Kayeā€“Bernsteinā€“Kearsleyā€“Zapas

Kayeā€“Bernsteinā€“Kearsleyā€“Zapas
B. Bernstein, E. A. Kearsley, and L. J. Zapas. A study of stress relaxation with ļ¬nite strain. Trans. Soc. Rheol., 7(1):391ā€“410, 1963 I-Jen Chen and D. C. Bogue. Time-dependent stress in polymer melts and review of viscoelastic theory. Trans. Soc. Rheol., 16(1):59ā€“78, 1972 t

T=
Ī¾=āˆ’āˆž

āˆ‚W āˆ’1 āˆ‚W C+ C dĪ¾ āˆ‚I āˆ‚II

(5.1)

Fits experimental data for: polyisobutylene, vulcanised rubber

Josef MĀ“lek a

Non-Newtonian ļ¬‚uids

Viscosity of some ļ¬‚uids Models with variable viscosity Diļ¬€erential type models Rate type models Integral type models Download

git clone [email protected]:non-newtonian-models git clone [email protected]:bibliography-and-macros

Josef MĀ“lek a

Non-Newtonian ļ¬‚uids

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