- Introduction
- Power-law model
- Bingham model
- Simulation of turbulent non-Newtonian flow
- Sources of further information

PHOENICS provides two representative options for the simulation of inelastic time-independent non-Newtonian fluids, namely the Power-law and Bingham models. The power-law model is also known as the Ostwald-de Waele model.

Pseudoplastic and dilitant fluids are described by the power-law model. The former are fluids for which the rate of increase in shear stress with velocity gradient decreases with increasing velocity gradient. Dilitant fluids are those for which the rate of increase in shear stress with velocity gradient increases as the velocity gradient is increased.

Many purely viscous fluids encountered in polymer processing operations and thermal processing of liquid foods conform to the power-law model within engineering accuracy. Other examples of power-law fluids include rubber solutions, adhesives, polymer solutions or melts, and biological fluids.

A Bingham fluid is a fluid for which the imposed stress must exceed a critical yield stress to initiate motion. Examples of fluids which behave as, or nearly as, Bingham plastics include water suspensions of clay, sewage sludge, some emulsions and thickened hydrocarbon greases, and slurries of uranium oxide in nuclear reactors.

For both incompressible Newtonian fluids and the Power-law and Bingham fluids, the stress-strain relationship may be written as:

TAUij = - EMUL*Dij (1.1)

where TAUij is the stress tensor, Dij is the symmetrical rate-of- deformation tensor, and EMUL is the coefficient of dynamic viscosity.

For Newtonian fluids, EMUL depends on local pressure and temperature but not on TAUij or Dij.

For the Bingham and Power-law
fluids, EMUL is a function of Dij and/or TAUij, as well as of temperature and pressure. In
PHOENICS it is the kinematic viscosity ENUL, *i.e.* EMUL divided by density,
which is specified, rather than the dynamic
viscosity EMUL.

The empirical functions employed to calculate ENUL in PHOENICS for Bingham and Power -law fluids are given below.

For a power-law fluid, the non-Newtonian scalar kinematic viscosity ENUL is given by:

ENUL = (ENULA * LGEN1**[0.5*(ENULB-1.)])/RHO1 equation (2.1)

where: RHO is the local fluid density; ENULA is the fluid consistency index at a reference temperature; ENULB is the power-law or flow- behaviour index; and LGEN1 is the magnitude of the total rate of strain, given by

LGEN1 = 0.5 * ( Dij : Dij ) equation (2.2)

where : denotes the double-dot scalar product of two tensors. In PHOENICS, LGEN1 is simply the generation function which is stored in the F-array location identified by the integer variable LGEN1.

For ENULB=1.0, the power-law model reduces to a Newtonian fluid model. For ENULB <1 the fluid behaves as a Pseudoplastic, so that ENUL decreases with increasing rate of shear. For ENULB> 1 the fluid behaves as a dilitant, so that ENUL increases with increasing rate of shear.

The power-law model may be activated in PHOENICS by introducing the following PIL commands in the Q1 file:

ENUL=GRND4

ENULA= consistency index

ENULB= flow behaviour index

STORE(VISL)

The calculation of strain-rate squared LGEN1 is activated automatically via GREX3 where GENK=T, but can also be controlled by DUDX, DUDY etc. In addition, printout LGEN1 may be effected when a variable named GENK is stored in the Q1 file.

This option is a general one which is available for use with both Newtonian and non-Newtonian simulations ( see the Encyclopaedia entry entitled 'TURBULENCE-ENERGY GENERATION').

The model may also be activated for use in conjugate-heat-transfer applications by setting ENUL=GRND10 and using GRND4 in the "props" file to select the power-law option for ENUL.

Alternatively, the option may be selected via the Q1 file rather than the "props" file, e.g. the following commands would be equivalent to using ENUL=GRND4 with ENULA=0.02 and ENULB=0.25.

ENUL=GRND10;STORE(VISL)

STORE(PRPS);FIINIT(PRPS)=33

** mat no. rho enul cp kond expan

CSG10=Q1

MATFLG=T;NAMAT=1

33 1. GRND4 1. 1. 0

0.02 0.25

In the foregoing example the settings made for the tabulated entries rho, cp, kond and expan are merely dummies.

For the Bingham model, ENUL is given by:

ENUL = (ENULA + TAUO/SQRT(LGEN1))/RHO equation (3.1)

when TAU <TAUO; and by

ENUL = infinity equation(3.2)

when TAU < TAUO. ENULA is the rigidity coefficient or plastic viscosity, TAUO is the yield stress, and TAU is the magnitude of stress tensor, given by:

TAU = 0.5 * [ TAUij : TAUij ] equation (3.3)

which here, is also given by:

TAU = TAUO + ENULA*SQRT(LGEN1) equation (3.4)

Note that LGEN1 is given by equation (2.2) above.

The Bingham model may be activated in PHOENICS by introducing the following PIL commands in the Q1 file:

ENUL=GRND5

ENULA= rigidity coefficient or plastic viscosity

ENULB= yield stress

STORE(VISL)

As with the power-law option, the calculation of LGEN1 is activated automatically. When STORE(BTAU) appears in the Q1 file, the local values of TAU as defined by equation (3.4) may be printed in the RESULT file or viewed via PHOTON and AUTOPLOT. Similar to the power- law option, the model may also be activated for use in conjugate- heat-transfer applications by setting ENUL=GRND10 and using GRND5 in the "props" file to select the Bingham option for ENUL.

For the simulation of the turbulent flow of Bingham or Power-law fluids with wall functions, the user is advised that the use of standard wall functions in these flows is probably questionable, and more accurate results are likely to be obtained via the use of a low-Reynolds-number turbulence model ( see the Encyclopaedia entry "LOW_REynolds number turbulence models" ) or from an enhanced wall- function treatment.

Relevant "help-file" entries are :- ENUL.

The FORTRAN coding for the non-Newtonian laws may be found in subroutine GXVISL which resides in the file gxknvsl.htm.

Several Q1's may be found in the PHOENICS input library which demonstrate and validate laminar and turbulent flow of viscous non-Newtonian fluids. (Use super-search for non-newtonian)

A.H Skelland, 'Non-Newtonian flow and heat transfer', John Wiley, New York, Book Publication, (1967).

G.W. Govier and K. Aziz, 'The flow of complex mixtures in pipes', R.E.Kreiger Pub. Co., Huntington, New York, (1977).

R.B. Bird, W.E. Stewart and E.N. Lighfoot, ' Transport Phenomena', John Wiley, New York, Book Publication, (1960).

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