Johnson--Segalman--Oldroyd model

The Johnson--Segalman--Oldroyd model is a linear relaxation-retardation equation made nonlinear through a Gordon--Schowalter objective derivative. Its slip parameter can produce a rate-dependent steady shear viscosity.

Table of contents

Johnson-Segalman extensional response ( $η_{E} = 3 μ_{0} + \frac{3 α ^{2} G _{0} τ}{( 1 - 2 ατ ϵ ˙ ) ( 1 + ατ ϵ ˙ )}$ )

 0  0

In uniaxial extensional flow, the conformation equations give

A_{xx} = 1/ (1 - 2 ατ \overset{ϵ}{˙})

and

A_{yy} = A_{zz} = 1/ (1 + ατ \overset{ϵ}{˙})

. Their stress difference gives the displayed extensional viscosity. A stable positive steady conformation requires both denominators positive; a first pole marks loss of a bounded steady response. At zero extension rate the Trouton ratio is three relative to the zero-shear viscosity.

Johnson-Segalman steady viscometric functions ( $D = 1 + (1 - α^{2}) τ^{2} \overset{γ}{˙}^{2}$ )

 0  0

For stress

- p I + α G_{0} A + 2 μ_{0} E

and slip derivative

D_{t} A + Ω A - A Ω - α (E A + A E) = - (A - I) / τ

, the steady viscometric functions are

μ = μ_{0} + α^{2} G_{0} τ / D

ψ_{1} = 2 α^{2} G_{0} τ^{2} / D

and

ψ_{2} = - (1 - α) ψ_{1} /2

. Solving the three symmetric xy conformation equations gives

A_{x y} = ατ \overset{γ}{˙} / D

A_{xx} = 1 + (1 + α) τ \overset{γ}{˙} A_{x y}

A_{yy} = 1 - (1 - α) τ \overset{γ}{˙} A_{x y}

and

A_{zz} = 1

Johnson-Segalman negative-slope shear threshold ( $μ_{0} / (α^{2} G_{0} τ) < 1/8$ )

 0  0

For nonzero slip parameter with

∣ α ∣ < 1

, scale shear rate as

x = τ 1 - α^{2} \overset{γ}{˙}

and stress as

S = x [b + 1/ (1 + x^{2})]

, where

b = μ_{0} / (α^{2} G_{0} τ)

. The derivative is

b + (1 - x^{2}) / (1 + x^{2})^{2}

and its least value is

b - 1/8

, attained at

x^{2} = 3

. A decreasing homogeneous constitutive segment therefore exists exactly under the displayed condition. It permits different shear rates at the same stress and motivates shear banding; the local curve alone does not select a band-interface stress.

Gordon--Schowalter derivative

 0  0

For a tensor

σ

, the Gordon--Schowalter derivative used here is

σ □ = σ ▽ + \frac{a}{2} (\dot{γ} σ + σ \dot{γ}),

(1)

which interpolates among objective convected derivatives as the dimensionless slip parameter

a

changes.

Retardation time of an Oldroyd fluid ( $λ_{2}$ )

 0  0

The retardation time controls the delayed strain-rate term in an Oldroyd constitutive equation. Together with the relaxation time it determines the solvent-like high-frequency or high-rate response.