J. Y. Zhang

fluids are viscoplastic, such as hair gel, mud, cement, paint, processed food and so on. Flow behavior in viscop-

lastic fluids is a significant and extensively studied topic in processing industry.

Viscoplastic fluids are generalized Newtonian fluids, a class of non-Newtonian fluids. For such fluids, the rate

of strain

and the deviatoric stress

are related through a constitutive equation of form

( )

with 2

ij ij

ij ij

γγ

τ ηγγγ

==

(1)

where

is termed the effective viscosity. In the present work, a subscript form such as

is always

used to indicate a two tensor and its tensor norm is denoted by a non-subscript form such as

defined as

with the Einstein's notation adopted. Hence the second invariant of the deviatoric stress is de-

noted by

. If

, then the models are viscoplastic, with yield stress

.

A typical viscoplastic model is the Herschel-Bulkley model with the following scaled constitutive relation:

1

if , 0 if .

n

ij ij

BBB

τγγ τγτ

γ

−

=+>= ≤

with

being the power-law index. This is an extension of the power-law model to a fluid with a yield stress,

The dimensionless parameter

termed the Bingham number, denotes the ratio of yield stress to

viscous stress. Here

represents the kinematic viscosity,

and

are the reference spatial and velocity

scales, respectively.

For the Herschel-Bulkley model, the effective viscosity is defined from

. Setting

and

returns the Newtonian model,

. Setting

, we recover the popular Bingham model. Note that

for the Herschel-Bulkley model, if

, then

as

.

Consider a Bingham fluid in a rectangular cavity

, driven by the symmetric sheer motion

through the top and bottom lids. The non-dimensionalized momentum equations for the velocity

and the pressure

with the corresponding boundary conditions can be written as

in , for 1,2

ij i

ij

pgi

xx

τ

∂

∂=+Ω =

∂∂

(2)

(3)

()() ()()( )( )

1 22211

1,1,,,0, ,1, ,1uyuyu xAu xAuxAuxA− ==−==−=−=

(4)

Where

is the scaled gravitational acceleration.

Effective numerical algorithm shall be designed and implemented to render the streamlines and yield surfaces

in this Bingham cavity flow with desired resolution.

3. The Augmented Lagrange Method (A LM )

Theoretically, viscoplastic fluids are generalized Newtonian fluids governed by discontinuous constitutive laws,

which implicitly define yield surfaces as interfaces separating the solid and the fluid regions in the correspond-

ing fluids. Due to the unknown shapes and locations of the yield surfaces, the viscous terms in the momentum

equations modeling viscoplastic fluid flows cannot be explicitly expressed, which makes the simulation of vis-

coplastic fluid flows rather difficult. A detailed review and discussion of the existing numerical approaches can

be found in [5 ]. To keep the actual viscoplatic feature of the fluid of interest, we are in favor of the variational

approach [6] [7] in the presented work.

The variational reformulation and its application to viscoplastic fluid flows date back to the pioneer work of

Duvaut and Lions [8], in which a desired flow motion is captured by solving an equivalent variational inequality

whose minimizer set is proven to be the solution set of the momentum equations with the associated constitutive