Viscosity of some fluids
Fluid Air (at Benzene Water (at 18 ◦ C) Olive oil (at 20 ◦ C) Motor oil SAE 50 Honey Ketchup Peanut butter Tar Earth lower mantle 18 ◦ C) Viscosity [cP] 0.02638 0.5 1 84 540 2000–3000 50000–70000 150000–250000 3 × 1010 3 × 1025
Table: Viscosity of some fluids
Josef M´lek a Non-Newtonian fluids
Viscosity of some fluids Models with variable viscosity Differential 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 = −pI + 2µ(D, T)D
S
(2.1)
Particular models mainly developed by chemical engineers.
Josef M´lek a
Non-Newtonian fluids
Viscosity of some fluids Models with variable viscosity Differential 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
Ostwald–de Waele power law
¨ Wolfgang Ostwald. Uber die Geschwindigkeitsfunktion der Viskosit¨t disperser Systeme. I. Colloid Polym. Sci., 36:99–117, a 1925 A. de Waele. Viscometry and plastometry. J. Oil Colour Chem. Assoc., 6:33–69, 1923 µ(D) = µ0 |D|n−1 (2.2)
Fits experimental data for: ball point pen ink, molten chocolate, aqueous dispersion of polymer latex spheres
The Essay on International Oil and National Security
When oil fueled the navies of both Britain and the United States in World War I, it began the tremendous impact oil would have on the world. When British and French forces were threatened with oil shortages, the US gave the much needed help. Without the same assistance, Germany had to shorten critical operations. Since military hardware were all dependent on oil for power it became important that ...
Josef M´lek a
Non-Newtonian fluids
Viscosity of some fluids Models with variable viscosity Differential 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 Carreau–Yasuda
Pierre J. Carreau. Rheological equations from molecular network theories. J.
Rheol., 16(1):99–127, 1972 Kenji Yasuda. Investigation of the analogies between viscometric and linear viscoelastic properties of polystyrene fluids. PhD thesis, Massachusetts Institute of Technology. Dept. of Chemical Engineering., 1979 µ0 − µ∞ (1 + α |D|2 ) 2 n n−1 a
µ(D) = µ∞ +
(2.3) (2.4)
µ(D) = µ∞ + (µ0 − µ∞ ) (1 + α |D|a ) Fits experimental data for: molten polystyrene Josef M´lek a Non-Newtonian fluids
Viscosity of some fluids Models with variable viscosity Differential 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 diffusion as examples of absolute reaction rates. J. Chem. Phys., 4(4):283–291, 1936 Francis Ree, Taikyue Ree, and Henry Eyring. Relaxation theory of transport problems in condensed systems. Ind. Eng. Chem., 50(7):1036–1040, 1958 µ(D) = µ∞ + (µ0 − µ∞ ) arcsinh (α |D|) α |D| arcsinh (α1 |D|) arcsinh (α2 |D|) µ(D) = µ0 + µ1 + µ2 α1 |D| α2 |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 M´lek a Non-Newtonian fluids
Viscosity of some fluids Models with variable viscosity Differential 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 fluids: A new flow equation for pseudoplastic systems. J. Colloid Sci., 20(5):417–437, 1965 µ(D) = µ∞ + µ0 − µ∞ 1 + α |D|n (2.7)
The Review on Logic Model Development Guide
Logic Model Development Guide Introduction If you don’t know where you’re going, how are you gonna’ know when you get there? –Yogi Berra In line with its core mission – To help people help themselves through the practical application of knowledge and resources to improve their quality of life and that of future generations – the W.K. Kellogg Foundation has made program evaluation a priority. As ...
Fits experimental data for: aqueous polyvinyl acetate dispersion, aqueous limestone suspension
Josef M´lek a
Non-Newtonian fluids
Viscosity of some fluids Models with variable viscosity Differential 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 flow of lubricating greases. Ind. Eng. Chem., 50(12):1789–1792, 1958 µ(D) = µ∞ + α |D|n−1 Fits experimental data for: lubricating greases (2.8)
Josef M´lek a
Non-Newtonian fluids
Viscosity of some fluids Models with variable viscosity Differential 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., 45:87–96, 1893 µ(T) = µref eβ(p−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 effect of pressure on the viscosity of forty-four pure liquids. Proc. Am. Acad. Art. Sci., 61(3/12):57–99, FEB-NOV 1926 Josef M´lek a Non-Newtonian fluids
Viscosity of some fluids Models with variable viscosity Differential 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 flow of the non-Newtonian Ellis fluid. AIChE J., 11(4):588–595, 1965 µ(T) = µ0 1 + α |Tδ |n−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 flow of non-newtonian polymer solutions. Appl. Math. Modelling, 18(1):14–22, 1994 Josef M´lek a Non-Newtonian fluids
Viscosity of some fluids Models with variable viscosity Differential type models Rate type models Integral type models Download
The Term Paper on What is the ‘covering law’ model of explanation?
Carl Hempel’s “covering law” model of explanation states essentially that an explanation for an event can be drawn from a set of general laws or, in the case of the social sciences, universal hypotheses. Hempel claims the study of history is not generally associated with the search for general laws governing historical events. However, history is a discipline within which the theory of “covering ...
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):519–538, 1955 µ(T) = α |Tδ |n−1 Fits experimental data for: ice (2.11)
Josef M´lek a
Non-Newtonian fluids
Viscosity of some fluids Models with variable viscosity Differential 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):56–60, 1964 µ(T) = µ∞ + (µ0 − µ∞ ) e − |Tδ |
τ0
(2.12)
Fits experimental data for: polybutadiene solutions
Josef M´lek a
Non-Newtonian fluids
Viscosity of some fluids Models with variable viscosity Differential 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 flow at low deviatoric stress. J. Glaciol., 49(166):359–369, 2003 H Blatter. Velocity and stress-fields in grounded glaciers – a simple algorithm for including deviatoric stress gradients. J. Glaciol., 41(138):333–344, 1995 µ(T) = 2
A |Tδ | +
2 τ0
n−1 2
(2.13)
Fits experimental data for: ice
Josef M´lek a
Non-Newtonian fluids
Viscosity of some fluids Models with variable viscosity Differential 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 Herschel–Bulkley
C. E. Bingham. Fluidity and plasticity. McGraw–Hill, New York, 1922 Winslow H. Herschel and Ronald Bulkley. Konsistenzmessungen von Gummi-Benzoll¨sungen. Colloid Polym. Sci., 39(4):291–300, o August 1926 |Tδ
| > τ ∗ |Tδ | ≤ τ ∗ if and only if Tδ = τ ∗ if and only if D=0 D + 2µ(|D|)D |D|
The Essay on Viscosity and Newtonian Liquid
In this practical, viscometry and rheometry are introduced. Viscometry is the measurement of viscosity while rheometry refers to the experimental techniques used to determine the rheological properties of materials, that is the quantitative and qualitative relationships between deformations and stresses and their derivatives. The definition of viscosity was put on a quantitative basis by Newton, ...
(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):357–365, 2010 Josef M´lek a Non-Newtonian fluids
Viscosity of some fluids Models with variable viscosity Differential type models Rate type models Integral type models Download
Rivlin–Ericksen fluids
Rivlin–Ericksen
R. S. Rivlin and J. L. Ericksen. Stress-deformation relations for isotropic materials. J. Ration. Mech. Anal., 4:323–425, 1955 R. S. Rivlin and K. N. Sawyers. Nonlinear continuum mechanics of viscoelastic fluids. Annu. Rev. Fluid Mech., 3:117–146, 1971 General form: T = −pI + f(A1 A2 A3 . . . ) (3.1) where A1 = 2D dAn−1 + An−1 L + L An−1 An = dt (3.2a) (3.2b)
d where dt denotes the usual Lagrangean time derivative and L is the velocity gradient. Josef M´lek a Non-Newtonian fluids
Viscosity of some fluids Models with variable viscosity Differential type models Rate type models Integral type models Download
Rivlin–Ericksen fluids
Criminale–Ericksen–Filbey
William O. Criminale, J. L. Ericksen, and G. L. Filbey. Steady shear flow of non-Newtonian fluids. Arch. Rat. Mech. Anal., 1:410–417, 1957 T = −pI + α1 A1 + α2 A2 + α3 A2 1 (3.3)
Fits experimental data for: polymer melts (explains mormal stress differences)
Josef M´lek a
Non-Newtonian fluids
Viscosity of some fluids Models with variable viscosity Differential type models Rate type models Integral type models Download
Rivlin–Ericksen fluids
Reiner–Rivlin
M. Reiner. A mathematical theory of dilatancy. Am. J. Math., 67(3):350–362, 1945 T = −pI + 2µD + µ1 D2 Fits experimental data for: N/A (3.4)
Josef M´lek a
Non-Newtonian fluids
Viscosity of some fluids Models with variable viscosity Differential type models Rate type models Integral type models Download
Maxwell, Oldroyd, Burgers Giesekus Phan-Thien–Tanner Johnson–Segalman Johnson–Tevaarwerk
Maxwell
J. Clerk Maxwell. On the dynamical theory of gases. Philos. Trans. R. Soc., 157:49–88, 1867
T = −pI + S S + λ1 S = 2µD dM − LM − ML dt Fits experimental data for: N/A M =def Josef M´lek a Non-Newtonian fluids
(4.1a) (4.1b)
The Term Paper on David Garrick And Samuel Johnson A Fading Friendship
DAVID GARRICK AND SAMUEL JOHNSON: A FADING FRIENDSHIP This paper purposes to treat of the relationship between David Garrick and Samuel Johnson. David Garrick becomes famous in the acting scene, whereas Samuel Johnson succeeds as a writer. The lives of these two men have their positive moments, and, moments that reflect their hardships. The relationships of the two men begin with their childhood ...
(4.2)
Viscosity of some fluids Models with variable viscosity Differential type models Rate type models Integral type models Download
Maxwell, Oldroyd, Burgers Giesekus Phan-Thien–Tanner Johnson–Segalman Johnson–Tevaarwerk
Oldroyd-B
J. G. Oldroyd. On the formulation of rheological equations of state. Proc. R. Soc. A-Math. Phys. Eng. Sci., 200(1063):523–541, 1950
T = −πI + S S + λS = η1 A1 + η2 A1 Fits experimental data for: N/A
(4.3a) (4.3b)
Josef M´lek a
Non-Newtonian fluids
Viscosity of some fluids Models with variable viscosity Differential type models Rate type models Integral type models Download
Maxwell, Oldroyd, Burgers Giesekus Phan-Thien–Tanner Johnson–Segalman Johnson–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):523–541, 1950 T = −πI + S λ3 λ5 λ6 (DS + SD) + (Tr S) D + (S : D) I 2 2 2 λ7 (D : D) I = −µ D + λ2 D + λ4 D2 + 2 (4.4a)
S + λ1 S +
(4.4b)
Fits experimental data for: N/A
Josef M´lek a Non-Newtonian fluids
Viscosity of some fluids Models with variable viscosity Differential type models Rate type models Integral type models Download
Maxwell, Oldroyd, Burgers Giesekus Phan-Thien–Tanner Johnson–Segalman Johnson–Tevaarwerk
Burgers
J. M. Burgers. Mechanical considerations – model systems – phenomenological theories of relaxation and viscosity. In First report on viscosity and plasticity, chapter 1, pages 5–67. Nordemann Publishing, New York, 1939
T = −πI + S S + λ1 S + λ2 S = η1 A1 + η2 A1 Fits experimental data for: N/A
(4.5a) (4.5b)
Josef M´lek a
Non-Newtonian fluids
Viscosity of some fluids Models with variable viscosity Differential type models Rate type models Integral type models Download
Maxwell, Oldroyd, Burgers Giesekus Phan-Thien–Tanner Johnson–Segalman Johnson–Tevaarwerk
Giesekus
H. Giesekus. A simple constitutive equation for polymer fluids based on the concept of deformation-dependent tensorial mobility. J. Non-Newton. Fluid Mech., 11(1-2):69–109, 1982
T = −πI + S S + λS − αλ2 2 S = −µD µ
(4.6a) (4.6b)
Fits experimental data for: N/A
Josef M´lek a
The Essay on Walter Johnson A Pitcher
In the beginning there were players like Ty Cobb who hit .300 for 23 consecutive years, and the 'flying dutchman' Honus Wagner. Pitchers like the 'christian gentleman' Christy Mathewson, and the winningest pitcher in history Cy Young. In the years when the only Yankees were the people in the north and there was an upstart franchise called the American League there was a pitcher, his name was ...
Non-Newtonian fluids
Viscosity of some fluids Models with variable viscosity Differential type models Rate type models Integral type models Download
Maxwell, Oldroyd, Burgers Giesekus Phan-Thien–Tanner Johnson–Segalman Johnson–Tevaarwerk
Phan-Thien–Tanner
N. Phan Thien. Non-linear network viscoelastic model. J. Rheol., 22(3):259–283, 1978 N. Phan Thien and Roger I. Tanner. A new constitutive equation derived from network theory. J. Non-Newton. Fluid Mech., 2(4):353–365, 1977
T = −πI + S Y S + λS + λξ (DS + SD) = −µD 2 Y =e Fits experimental data for: N/A Josef M´lek a Non-Newtonian fluids
(4.7a) (4.7b) (4.7c)
−ε λ Tr S µ
Viscosity of some fluids Models with variable viscosity Differential type models Rate type models Integral type models Download
Maxwell, Oldroyd, Burgers Giesekus Phan-Thien–Tanner Johnson–Segalman Johnson–Tevaarwerk
Johnson–Segalman
M. W. Johnson and D. Segalman. A model for viscoelastic fluid behavior which allows non-affine deformation. J. Non-Newton. Fluid Mech., 2(3):255–270, 1977
T = −pI + S (4.8a) S = 2µD + S (4.8b) S +λ dS + S (W − aD) + (W − aD) S dt = 2ηD (4.8c)
Fits experimental data for: spurt
Josef M´lek a Non-Newtonian fluids
Viscosity of some fluids Models with variable viscosity Differential type models Rate type models Integral type models Download
Maxwell, Oldroyd, Burgers Giesekus Phan-Thien–Tanner Johnson–Segalman Johnson–Tevaarwerk
Johnson–Tevaarwerk
K. L. Johnson and J. L. Tevaarwerk. Shear behaviour of elastohydrodynamic oil films. Proc. R. Soc. A-Math. Phys. Eng. Sci., 356(1685):215–236, 1977
T = −pI + S S S + α sinh = 2µD s0 Fits experimental data for: lubricants
(4.9a) (4.9b)
Josef M´lek a
Non-Newtonian fluids
Viscosity of some fluids Models with variable viscosity Differential type models Rate type models Integral type models Download
Kaye–Bernstein–Kearsley–Zapas
Kaye–Bernstein–Kearsley–Zapas
B. Bernstein, E. A. Kearsley, and L. J. Zapas. A study of stress relaxation with finite strain. Trans. Soc. Rheol., 7(1):391–410, 1963 I-Jen Chen and D. C. Bogue. Time-dependent stress in polymer melts and review of viscoelastic theory. Trans. Soc. Rheol., 16(1):59–78, 1972 t
T=
ξ=−∞
∂W −1 ∂W C+ C dξ ∂I ∂II
(5.1)
Fits experimental data for: polyisobutylene, vulcanised rubber
Josef M´lek a
Non-Newtonian fluids
Viscosity of some fluids Models with variable viscosity Differential type models Rate type models Integral type models Download
git clone :non-newtonian-models git clone :bibliography-and-macros
Josef M´lek a
Non-Newtonian fluids
