Bifurcation of Bingham Streamline Topologies in Rectangular Double-Lid-Driven Cavities

doi:10.4236/jamp.2014.212122

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Journal of Applied Mathematics and Physics, 2014, 2, 1069-1072

Published Online November 2014 in SciRes. http://www.scirp.org/journal/jamp

http://dx.doi.org/10.4236/jamp.2014.212122

How to cite this paper: Zhang, J.Y. (2014) Bifurcation of Bingham Streamline Topologies in Rectangular Double-Lid-Driven

Cavities. Journal of Applied Mathematics and Physics, 2, 1069-1072. http://dx.doi.org/10.4236/jamp.2014.212122

Bifurcation of Bingham Streamline

Topologies in Rectangular

Double-Lid-Driven Cavities

Jianying Zhang

Department of Mathematics, Western Washington University, Bellingham, USA

Email: [email protected]

Received Sep te mber 2014

Abstract

Numerical simulation of the bifurcation of Bingham fluid streamline topologies in rectangular

double-lid-driven cavity, with varying aspect (height to width) ratio A, is presented. The lids on the

top and bottom move at the same speed but in opposite directions so that symmetric flow patterns

are generated. Similar to the Newtonian case, bifurcations occur as the aspect ratio decreases.

Special to Bingham fluids, the non-Newtonian indicator, Bingham number B, also governs the bi-

furcation besides the bifurcation parameter A.

Keywords

Bingham Fluids, Dou b le -Lid-Driven Cavity Flow, Bifu rca tion, No n-Newtonian Fluids

1. Introduction

The lid-driven cavity flow within a rectangular cavity has been studied extensively as a benchmark model not

only for testing the validity of numerical methods but also for understanding the rheology of wide varieties of

complex industrial fluids. The cavity flow for a single moving lid with varying aspect (height to width) ratio, A,

generates corner eddies as A increases [1]. Consequently, Sturges [2 ] considered the symmetric flow with double

(the top and bottom) lids moving at the same speed but in opposite directions and revealed the presence of side

eddies attached to the stationary walls. For

0.9A≥

, the structure of Newtonian fluid flows and a mechanism

for eddy generation as A is increased has been fully investigated in [2]. An extension to a more complex class of

non-Newtonian fluids, namely Bingham fluids, can be found in [3]. On the other hand, for

0.9A≤

, topological

changes of the streamlines in Newtonian fluid flows occur as A decreases due to the break or creation of the

stagnation points [4]. The main purpose of the present work is to extend the existing results to Bingham fluids,

in which the non-Newtonian indicator, Bingham number B, plays a significant role in governing the bifurcation.

2. Constitutive Laws and the Momentum Equations

Viscoplastic fluids are complex fluids with yield stresses. A viscoplastic fluid behaves like a fluid only when the

applied shear stress exceeds the yield stress, otherwise it behaves like a solid. Many multi-component industrial

J. Y. Zhang

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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

γγ

τ ηγγγ

== 

 

(1)

where

( )

η ηγ

=

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

ij ij

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

noted by

ij ij

ττ

. If

( )

τγ τ

→=>



, then the models are viscoplastic, with yield stress

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

if , 0 if .

ij ij

BBB

τγγ τγτ

−



=+>= ≤





 



with

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

ˆY

The dimensionless parameter

ˆˆ

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

( )

τ ηγγ

=

. Setting

1n=

and

returns the Newtonian model,

. Setting

1n=

, we recover the popular Bingham model. Note that

for the Herschel-Bulkley model, if

, then

→∞

→



Consider a Bingham fluid in a rectangular cavity

[] []

1,1 ,AAΩ=−× −

, driven by the symmetric sheer motion

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

( )

u uu=

and the pressure

with the corresponding boundary conditions can be written as

in , for 1,2

ij i

pgi

∂

∂=+Ω =

∂∂

(2)

0 in u∇⋅ =Ω

(3)

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

1 22211

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

(4)

Where

()()

,0, 1ggg== −

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

J. Y. Zhang

1071

law.

It can also be shown via integration by parts that the desired vector field

for the boundary problem (2)-(4)

is the one that minimizes

( )()( )( )

Jvavvj vLv

= +−

(5)

over the admissible set

, which is the collection of all the divergence-free

( )

HΩ

vector fields satisfying

the boundary conditions (4). Here

()( )()( )

nij ij

auvv uv

γ γγ

−

Ω

∫

 

, referred to as the viscous dissipation rate in

some of the literature, is linear in its argument

for general Herschel-Bulkley fluids and bilinear in either of

its argument for Bingham fluids, i. e., when

1n=

. The force term

( )

L vgv

Ω

= −

∫

is linear in its argument,

whereas the yield stress dissipation rate

( )( )

jv Bv

Ω

∫



is nonlinear and non-differentiable in its argument.

The nonlinear and non-dif ferentiable yield stress term is brought in by the discontinuity of the constitutive

law of a viscoplastic fluid. The augmented Lagrange method [6] [7], as an effective numerical technique, re-

solves this difficulty by introducing an auxiliary variable to relax the undesired yield stress term and then adding

an augmented constraint. Consequently, the original problem is decoupled into a series of element-wise optimi-

zation problems, each of which can be solved with standard optimization techniques. This is also the virtue of

the augmented Lagrange method (ALM).

The ALM is implemented based on the variational Equality (5) following the Uzawa type iterations [9]:

• Step 1: Solve an elliptic problem for the velocity. The finite element method is naturally preferred.

• Step 2: Update the pressure based on the incompressible constraint.

• Step 3: Solve element-wise optimization problems for the rate of strain tensor.

• Step 4: Update the Lagrange multiplier corresponding to the augmented constraint.

The detailed numerical implementation of this algorithm can be found in [5] or [9].

4. Numerical Results

In the Newtonian case

( )

0B=

, the flow pattern starts with a single center at the center of the cavity when A is

bigger than 0.318. Then the center changes to a saddle and breaks down to two centers at the critical value A =

0.318. This is referred to as the level 1 bifurcation here. When A decreases to 0.169, the saddle becomes a center,

and the two centers become saddles, then two more centers generate. This is referred to as the level 2 bifurcation

here. The flow pattern keeps changing as A decreases [4].

Numerical simulation on Bingham fluids with various Bingham number B is conducted. Similar bifurcation

process is observed as A drops. However, the yield stress effect of Bingham fluids slows down the bifurcation

process. The larger B is (hence the more non-Newtonian the fluid is), the smaller A is required for the streamline

to bifurcate. A comparison of the bifurcation status in various B values with the Newtonian case is shown in

Table 1 and Table 2, for A = 0.25 and A = 0.15, where the Newtonian fluid is undergo the level 1 and level 2

bifurcations, respectively.

Also unique is the existence of the unyielded regions near the left and right walls in Bingham fluids. They

tend to squeeze the flow pattern inward as B increases. The B dependence of center locations for A = 0.25 are

shown in Table 3.

Table 1. B dependence of bifurcation when A = 0.25.

Saddle C en t er Bifurcation level

B = 0 (Newtonian) 1 2 1

B = 0.2 1 2 1

B = 0.5 1 2 1

B = 1 0 1 0

B = 5 0 1 0

J. Y. Zhang

1072

Table 2. B dependence of bifurcation when A = 0.15.

Saddle C en t er Bifurcation level

B = 0 (Newtonian) 2 3 2

B = 0.2 2 3 2

B = 0.3 1 2 1

B = 2 1 2 1

B =

0 1 0

Table 3. B dependence of the center locations when A = 0.25.

Saddle

Cen ter

Bifurcation level

B = 0 (Newtonian) (0,0) (±0.315,0) 1

B = 0.2 (0,0) (±0.29,0) 1

B = 0.5

(0,0)

(±0.24,0)

5. Conclusion and Future Investigations

We investigate the bifurcation of Bingham fluid flows in rectangular double-lid-driven cavities with varying as-

pect ratio A less than 0.9. The proposed numerical algorithm based on an augmented Lagrange approach with a

mesh adaptive strategy is implemented to render the streamlines and yield surfaces with desired resolution. Due

to the non-Newtonian feature of Bingham fluids, the bifurcation is governed by not only the bifurcation para-

meter A but the Bingham number B as well. The numerical results motivate the bifurcation analysis based on B,

which will be studied in the future.

Acknowledgem ents

The author would like to thank Western Washington University for the Summer Research Grant awarded in the

summer quarter 2014, during which this work was mainly conducted.

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