Phan Thien and Tanner (PTT) model

(1)

is the convected derivative defined as

(2)

are relaxation times and moduli, are adjustable parameters.

is the velocity gradient:

(3)

is the rate-of-strain tensor:

(4)

Uniaxial elongation:

Conditions:

(5)

PTT Equation for uniaxial elongation:

(6)

Components:

(7a)

(7b)

(7c)

Remark:

Biaxial extension:

Conditions:

(8)

PTT Equation for biaxial extension :

(9)

Components:

(10a)

(10b)

(10c)

Remark:

Biaxial extension:

Conditions:

(8)

PTT Equation for biaxial extension :

(9)

Components:

10a)

(10b)

(10c)

Phan Thien and Tanner - Nonisothermal (WLF)

(11)

where are reference and current temperature and are adjustable parameters (WLF).

Equation (11) can be written (for biaxial or uniaxial extension):

(12)

[6] Phan-Tien, N., Tanner, R.I., J. Non-Newt. Fluid Mech., 2, 353 (1977)