Some frequently used models for non-Newtonian ﬂuids

Josef M´lek a malek@karlin.mff.cuni.cz

Mathematical Institute Charles University

18 March 2011

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

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

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...