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Cessation of Couette and Poiseuille �ows of a Bingham plastic and �nite

stopping times

Maria Chatzimina� Georgios C� Georgiou

Department of Mathematics and Statistics� University of Cyprus

P�O� Box �� Nicosia� CYPRUS

Tel� �� Fax �� E�mail georgios�ucy�ac�cy

Evan Mitsoulis

School of Mining Engineering and Metallurgy

National Technical University of Athens

Heroon Polytechniou �� Zografou� Athens� GREECE

R�R� Huilgol

School of Informatics and Engineering

Flinders University of South Australia

G�P�O� Box �� Adelaide� SA �� AUSTRALIA

Short title� CESSATION OF BINGHAM FLOWS

�

Abstract

We solve the one�dimensional cessation Couette and Poiseuille �ows of a Bingham plastic

using the regularized constitutive equation proposed by Papanastasiou and employing �nite

elements in space and a fully implicit scheme in time� The numerical calculations con�rm pre�

vious theoretical �ndings that the stopping times are �nite when the yield stress is nonzero�

The decay of the volumetric �ow rate� which is exponential in the Newtonian case� is accel�

erated and eventually becomes linear as the yield stress is increased� In all �ows studied� the

calculated stopping times agree very well with the theoretical upper bound estimates�

KEYWORDS � Couette Flow� Poiseuille Flow� Bingham Plastic� Papanastasiou Model� Ces�

sation�

�

� Introduction

In viscometric �ows� one can bring the �uid to a halt by setting the moving boundary to rest�

in the case of Couette �ows� or by reducing the applied pressure gradient to zero in Poiseuille

�ows� In a Newtonian �uid� the corresponding velocity �elds decay to zero in an in�nite

amount of time �� In a Bingham plastic� the velocity �elds go to zero in a �nite time� which

emphasizes the role of the yield stress �� Glowinski � and Huilgol and co�workers ��

provided explicit theoretical �nite upper bounds on the time for a Bingham material to rest

in various �ows� such as the plane and circular Couette �ows and the plane and axisymmetric

Poiseuille �ows� In each case� the theoretical bound depends on the density� the viscosity�

the yield stress� and the least eigenvalue of the Laplacian operator on the �ow domain� More

recently� Huilgol has also derived upper bounds for the cessation of round Poiseuille �ow

of more general viscoplastic �uids�

The objective of the present work is to compute numerically the stopping times and make

comparisons with the theoretical upper bounds provided in the literature for the cessation

of three �ows of a Bingham �uid� �a� the plane Couette �ow� �b� the plane Poiseuille �ow�

and �c� the axisymmetric Poiseuille �ow� Instead of the ideal Bingham�plastic constitutive

equation� we employ the regularized equation proposed by Papanastasiou �� in order to

avoid the need of determining a priori the yielded and unyielded regions in the �ow domain�

It should be noted that preliminary results for the case of the plane Poiseuille �ow can also

be found in Ref� ��

The paper is organized as follows� In section �� we discuss the regularized Papanastasiou

equation for a Bingham plastic� In section �� we present the dimensionless forms of the

governing equations for the three �ows of interest along with the corresponding theoretical

�

stopping times� In section �� we present and discuss representative numerical results for all

�ows� The numerical stopping times agree very well with the theoretical upper bounds� Some

discrepancies are observed only for low Bingham numbers when the growth parameter in the

Papanastasiou model is not su�ciently high� Finally� section contains the conclusions of

this work�

� Constitutive equation

Let u and � denote the velocity vector and the stress tensor� respectively� and �� denote the

rate�of�strain tensor�

�� ru � �ru�T � ��

where ru is the velocity�gradient tensor� and the superscript T denotes its transpose� The

magnitudes of �� and � are respectively de�ned as follows�

��

r�

�II ��

r�

�� and � �

r�

�II� �

r�

��

where II stands for the second invariant of a tensor�

In tensorial form� the Bingham model is written as follows�

��

��

� �

��

��

��

where �� is the yield stress� and � is a constant viscosity�

In any �ow of a Bingham plastic� determination of the yielded �� and unyielded ��

regions in the �ow �eld is necessary� which leads to considerable computational di�culties

in the use of the model� These are overcome by using the regularized constitutive equation

�

proposed by Papanastasiou ��

� �

�� exp��m ��

��

��

where m is a stress growth exponent� For su�ciently large values of the regularization

parameterm� the Papanastasiou model provides a satisfactory approximation of the Bingham

model� while at the same time the need of determining the yielded and the unyielded regions

is eliminated� The model has been used with great success in solving various steady and

time�dependent �ows �see� for example� �� and references therein��

� Flow problems and governing equations

The governing equations along with the boundary and initial conditions of the three time�

dependent� one�dimensional Bingham�plastic �ows of interest are discussed below� The the�

oretical upper bounds for the stopping times are also presented�

�� Cessation of plane Couette �ow

The geometry of the plane Couette �ow is shown in Fig� �a� The steady�state solution is

given by

usx�y� �

��

y

H

�V � ��

where V is the speed of the lower plate �the upper one is kept �xed� and H is the dis�

tance between the two plates� We assume that at t�� the velocity ux�y� t� is given by the

above pro�le and that at t�� the lower plate stops moving� To non�dimensionalize the

x�momentum equation� we scale the lengths by H � the velocity by V � the stress components

by �V�H � and the time by �H�� where � is the constant density of the �uid� With these

scalings� the x�momentum equation becomes

�ux�t

��yx�y

� ��

The dimensionless form of the Papanastasiou model is reduced to

�yx �

�Bn � � exp��M ��

��

��ux�y

� ��

where ��j�ux��yj�

Bn ��H

�V��

is the Bingham number� and

M �mV

H��

is the dimensionless growth parameter�

The dimensionless boundary and initial conditions are as follows�

ux�� t� � � � t � �

ux�� t� � � � t � �

ux�y� �� y � � � y � �

��

� ��

In the case of a Newtonian �uid �Bn�� the analytical solution of the time�dependent �ow�

governed by Eqs� �� and �� is known ��

ux�y� t� ��

�Xk��

�

ksin �ky� e�k

��t � ��

Hence� the �ow ceases theoretically at in�nite time� If the �uid is a Bingham plastic �Bn � ��

however� the �ow comes to rest in a �nite amount of time� as demonstrated by Huilgol et al�

�� who provide the following upper bound for the dimensionless stopping time�

Tf ��

�

��

�

�

jjux�y� ��jj

Bn

��

�

where

jjux�y� ��jj �

Z ��

u�x�y� �� dy

��

�� Cessation of plane Poiseuille �ow

The geometry of the plane Poiseuille �ow is depicted in Fig� �b� The steady�state solution

is given by

usx�y� �

��

��

��p�x

�s�H � y��

� � � � y � y�

��

��p�x

�s�H� � y�� H � y� � y� � y � H

��

where ��p��x�s is the pressure gradient� and

y� ��

��p��x�s

H ��

denotes the point at which the material yields� Note that �ow occurs only if ��p��x�s �

��H � The volumetric �ow rate is given by

Q ��W

��

��p

�x

�sH�

��

�

�

�y�H

��

�

�

�y�H

��

where W is the width of the plates �in the z�direction��

We assume that at t�� the velocity ux�y� t� is given by the steady�state solution �� and that

at t�� the pressure gradient is vanished� or reduced to ��p��x� ��p��x�s� in which

case the �ow is expected to stop� The evolution of the velocity is again governed by the

x�momentum equation� Using the same scales as in the plane Couette �ow� with V denoting

now the mean velocity in the slit� the dimensionless form of the x�momentum equation is

obtained�

�ux�t

� f ��yx�y

� ��

�

where f denotes the dimensionless pressure gradient� The dimensionless form of the consti�

tutive equation is given by Eq� �� The dimensionless steady velocity pro�le becomes�

usx�y� �

��

�� f

s �� y�� y � y�

�� f

s �� y�� Bn �� y� � y� � y � �

��

where

y� �Bn

f s� ��

It turns out that y� is the real root of the cubic equation�

y��

��

�

Bn

�y� � � � � � ��

It is clear that a steady �ow in the channel occurs only if f s � Bn� The dimensionless

boundary and initial conditions for the time�dependent problem read�

�ux�y

�� t� � � � t � �

ux�� t� � � � t � �

ux�y� �� usx�y� � � � y � �

��

� ��

In the case of Newtonian �ow �Bn�� the time�dependent solution is given by �

ux�y� t� ��

�

�Xk��

��k��

��k� ��cos

��k� ��

�y

�exp

��k � ��

�t

��

which indicates that the �ow stops only after an in�nite amount of time� In the case of a

Bingham plastic �Bn � �� Huilgol et al� � provide the following estimate for the stopping

time�

Tf ��

�

��

�

�

jjux�y� ��jj

Bn � f

�� f Bn � ��

The above estimate is valid when f Bn �or� equivalently� when f f s�� otherwise� the

�ow will not stop�

�

�� Cessation of axisymmetric Poiseuille �ow

The geometry of the axisymmetric Poiseuille �ow is depicted in Fig� �c� The steady�state

solution is given by

usz�r� �

��

��

��p�z

�s�R � r��

� � � � r � r�

��

��p�z

�s�R� � r�� R � r� � r� � r � R

��

where ��p��z�s is the pressure gradient� and the yield point is given by

r� ��

��p��z�s

R � ��

The volumetric �ow rate is given by

Q �

��

��p

�z

�sR�

��

�

�

�r�R

��

�

�

�r�R

��

We assume that at t�� the velocity uz�r� t� is given by the steady�state solution and that

at t�� the pressure gradient is vanished� or reduced to ��p��z� ��p��z�s� Scaling

the lengths by the tube radius R� the velocity by the mean velocity V � the pressure and the

stress components by �V�R� and the time by �R�� we obtain the dimensionless form of

the z�momentum equation

�uz�t

� f ��

r

�

�r�r�rz� � ��

where f is the dimensionless pressure gradient� The dimensionless form of the constitutive

equation is given by

�rz �

�Bn � � exp��M ��

��

��uz�r

� ��

where ��j�uz��rj�

Bn ��R

�V� ��

�

and

M �mV

R� ��

The dimensionless steady velocity pro�le takes the form

usz�r� �

��

�� f

s �� r�� r � r�

�� f

s �� r�� Bn �� r� � r� � r � �

��

where r� satis�es

r� ��Bn

f s��

and

r��

��

�

Bn

�r� � � � � � ��

Note that a steady �ow in the tube occurs only if f s � �Bn� The growth of r� with Bn is

illustrated in Fig� �� in which steady�state velocity pro�les calculated for various Bingham

numbers are shown�

The dimensionless boundary and initial conditions read�

�uz�r

�� t� � � � t � �

uz�� t� � � � t � �

uz�r� �� usz�r� � � � r � �

��

� ��

The time�dependent solution for Newtonian �ow �Bn�� is given by �

uz�r� t� � ��Xk��

J��akr�

a�k J��ak�e�a

�

kt � ��

where J� and J� are respectively the zeroth� and �rst�order Bessel functions of the �rst kind�

and ak � k�� are the roots of J�� In the case of a Bingham plastic �Bn � �� Glowinski

��

� provides the following estimate for the stopping time�

Tf ��

��

� � ��

jjuz�r� ��jj

�Bn � f

�� f �Bn � ��

where

jjuz�r� ��jj �

�

Z�

�

u�z�r� �� rdr

��

and �� is the smallest �positive� eigenvalue of the problem�

�

r

d

dr

�rdw

dr

�� w � � � w�� w��

It is easily found that ��a�� where a� is the least root of J��x�� with the corre�

sponding eigenfunction being given by w��x��J��a�x�� Therefore�

Tf ��

a��

� � a��

jjuz�r� ��jj

�Bn � f

�� f �Bn � ��

The estimate �� holds only when f �Bn �or� equivalently� when f f s�� otherwise� the

�ow will not stop�

� Numerical results

Since there are no analytical solutions to the �ows under study� in the case of the Bingham

plastic or the Papanastasiou model� we have used a numerical method� namely the �nite

element method with quadratic �P��C�� elements for the velocity� For the spatial discretiza�

tion� we used the Galerkin form of the momentum equation� For the time discretization� we

used the standard fully�implicit �Euler backward�di�erence� scheme� At each time step� the

nonlinear system of discretized equations was solved using the Newton method� In the case

of Couette �ow� a ��element mesh re�ned near the two plates has been used� In Poiseuille

�ows� the mesh consisted of �� elements and was re�ned near the wall� The code has been

��

tested by solving �rst the Newtonian �ows and making comparisons with the analytical so�

lutions� In all three problems� the agreement between the theory and the calculations was

excellent�

Cessation of plane Couette �ow

Figures �� show the evolution of the velocity for Bn�� Newtonian �uid�� and �� re�

spectively� The growth parameter has been taken to be M�� The numerical solution in

Figure � compares very well with the analytical solution �� for the Newtonian �ow� The

numerical solutions for Bingham �ow �Figs� � and �� show that a small unyielded region�

where the velocity is �at� appears near the moving plate� Note that for high Bingham num�

bers �i�e�� Bn � �� very small time steps �of the order of �� were necessary in order to

get convergence in the early stages of cessation� The size of the unyielded region increases as

the time proceeds� Its left limit initially moves to the right but at higher times starts moving

to the left� as the �ow approaches complete cessation�

Figure � shows the evolution of the volumetric �ow rate�

Q�t� �Z ��

ux�y� t� dy � ��

for various Bingham numbers� These curves show the dramatic e�ect of the yield stress� which

accelerates the cessation of the �ow� In the Newtonian case �Bn�� and for small Bingham

numbers the decay of the volumetric �ow rate is exponential� at least initially� At higher

Bingham numbers� the decay of Q becomes polynomial and eventually linear� The times at

which Q�� and �� are plotted as functions of the Bingham number in Fig� �� The two

times coincide for moderate or large Bingham numbers� which indicates that the �ow indeed

stops at a �nite time� In order to make comparisons with the theoretical upper bound ��

we consider as numerical stopping time the time at which Q�� The comparison between

��

calculations and theory� provided in Fig� �� shows a very good agreement for moderate and

higher Bingham numbers� The small discrepancies observed for low Bingham numbers are

due to the fact that the value of M is not su�ciently high� as discussed below� For very

small Bn� the e�ect of M is not crucial� since the material is practically Newtonian� which

explains why the calculations fall again below the theoretical upper bound� as they should�

Cessation of plane Poiseuille �ow

Figures �� show the evolution of ux�y� t� for Bn�� Newtonian case�� and �� In

Fig� �� we see the evolution of the calculated volumetric �ow rate for various Bingham

numbers� As in plane Couette �ow� the decay of the volumetric �ow rate is exponential

for small Bingham numbers and becomes polynomial at higher Bn values� Figure �� shows

plots of the times required for the volumetric �ow rate to become �� and �� versus the

Bingham number for both plane and round Poiseuille �ows �with M��

Before proceeding to the comparisons with the theoretical estimate �� let us investigate

the e�ect of the growth parameter M on the calculated stopping times� As demonstrated

in Fig� �� which shows results obtained with M�� and M�� the calculated stopping

times are not so sensitive to M when the Bingham number is moderate or high� i�e� Bn � ��

For smaller Bingham numbers� i�e� �� Bn � �� the time required for the volumetric �ow

rate to become �� is reduced as M is increased� For very small Bingham numbers� the

�uid is essentially Newtonian� and therefore the value of M has no e�ect on the calculations�

Hence� in order to get converged results in the range �� Bn � �� the value ofM has to be

increased further� However� our studies showed that when M�� convergence di�culties

are observed of Bn � �� One way to resolve the problem is to reduce the time step� This

might be good for extending the calculations to a slightly higher Bn� Beyond this critical Bn

value� the required time step is very small and the accumulated round�o� errors are so high so

��

that the error in the calculated stopping time is higher than that corresponding to a smaller

value ofM � As a conclusion� decreasing the time step is not the best way to obtain converged

results for M � �� If results for small �but not vanishingly small� Bingham numbers and

large M are necessary� then continuation in M must be used at each time step� According to

our numerical experiments� using such a continuation will increase the computational time

by at least �� times� Since we are not interested in so small values of Bn� such calculations

have not been pursued�

A comparison between theory and calculations is provided in Fig� �� for the case f�� i�e��

when the imposed pressure gradient is set to zero�� Again� the agreement between theory

and computations is excellent for moderate and high Bingham numbers� Again� the small

discrepancies observed for small values of the Bingham number are due to the fact thatM is

not su�ciently high� We have also examined the case in which the imposed pressure gradient

f is not zero� An ideal Bingham plastic stops after a �nite time if f � Bn and reaches a new

steady�state if f � Bn �with the volumetric �ow rate corresponding to the new value of f��

This is not the case with a regularized Bingham �uid� Since M is �nite� the �ow will reach a

new steady�state in which the volumetric �ow rate may be small but not zero� To illustrate

this e�ect� we considered the case in which Bn�� andM�� and carried out simulations for

di�erent values of f � In Fig� ��a� we see the evolution of the volumetric �ow rate for di�erent

values of f � Figure ��b is a zoom of the previous �gure showing that indeed the volumetric

�ow rate reaches a �nite value when f �� This value may be reduced further by increasing

the value of M � The new volumetric �ow rate is plotted against f in Fig� �� Finally� in

Fig� �� we compare the times required to reach Q�� with the theoretical estimate ��

For smaller values of Q� the numerical results move closer to the theoretical curve� but in a

smaller range of f � as it is easily deduced from Fig� �� The deviations between theory and

��

experiment become larger as the value of the imposed pressure gradient is increased� These�

however� can be further reduced by increasing the value of M �

Cessation of axisymmetric Poiseuille �ow

The results for the axisymmetric Poiseuille �ow are very similar to those obtained for the

planar case� Figures �� show the evolution of ux forM�� and Bn�� Newtonian case��

�� and �� In Fig� �� we zoom near the wall in order to see how the velocity pro�le changes

when Bn�� It is clear that a second unyielded region of a smaller size appears near the

wall� in which the velocity is zero�

In Fig� �� we see the evolution of the calculated volumetric �ow rate �scaled by ��

Q�t� �Z ��

uz�r� t� rdr � ��

for M�� and various Bingham numbers� As in plane Poiseuille �ow� the cessation of

the �ow is accelerated as the Bingham number is increased� The calculated stopping times

for Q�� and f�� plotted versus the Bingham number in Fig� �� agree well with the

theoretical estimate �� with the small discrepancies observed for low Bn excepted�

� Conclusions

The Papanastasiou modi�cation of the Bingham model has been employed in order to solve

numerically the cessation of plane Couette� plane Poiseuille� and axisymmetric Poiseuille

�ows of a Bingham plastic� The �nite element calculations showed that the volumetric

�ow rate decreases exponentially for low� polynomially for moderate� and linearly for high

Bingham numbers� Unlike their counterparts in a Newtonian �uid� the corresponding times

for complete cessation are �nite� in agreement with theory� The numerical stopping times are

�

found to be in very good agreement with the theoretical upper bounds provided in Refs� ��

for moderate and higher Bingham numbers� Some minor discrepancies observed for rather

low Bingham numbers can be reduced by increasing the regularization parameter introduced

by the Papanastasiou model�

A noteworthy di�erence between the predictions of the ideal and the regularized Bingham

model is revealed when the imposed pressure gradient is nonzero and below the critical value

at which a nonzero steady�state Poiseuille solution exists� In contrast to the ideal Bingham

�ow� which reaches complete cessation at a �nite time� the regularized �ow reaches a velocity

pro�le corresponding to a small but nonzero volumetric �ow rate� The value of the latter

may be reduced by increasing the value of the regularization parameter M but will always

be nonzero�

Acknowledgement

This research was partially supported by the Research Committee of the University of Cyprus�

References

� T� Papanastasiou� G� Georgiou and A� Alexandrou� Viscous Fluid Flow� CRC Press� Boca

Raton� ��

� R�R� Huilgol� B� Mena� and J�M� Piau� Finite stopping time problems and rheometry of

Bingham �uids� J� Non�Newtonian Fluid Mech� ��

� R� Glowinski� Numerical Methods for Nonlinear Variational Problems� Springer�Verlag�

New York� ��

��

� R�R� Huilgol� and B� Mena� On kinematic conditions a�ecting the existence and non�

existence of a moving yield surface in unsteady unidirectional �ows of Bingham �uids� J�

Non�Newtonian Fluid Mech� ��

R�R� Huilgol� Variational inequalities in the �ows of yield stress �uids including inertia�

Theory and applications� Phys� Fluids ��

� T�C� Papanastasiou� Flows of materials with yield� J� Rheology ��

� M� Chatzimina� G�C� Georgiou� E� Mitsoulis� and R�R� Huilgol� Finite stopping times in

Couette and Poiseuille �ows of viscoplastic �uids� in� Proceedings of the XIVth Int� Cong�

Rheol�� Seoul� Korea� pp� NFF��NF��

� Y� Dimakopoulos and J� Tsamopoulos� Transient displacement of a viscoplastic material

by air in straight and suddenly constricted tubes� J� Non�Newtonian Fluid Mech� ��

��

� E� Mitsoulis and R�R� Huilgol� Entry �ows of Bingham plastics in expansions� J� Non�

Newtonian Fluid Mech� ��

��

FIGURE CAPTIONS

�� Flow problems under study� �a� cessation of plane Couette �ow� �b� cessation of plane

Poiseuille �ow� and �c� cessation of axisymmetric Poiseuille �ow�

�� Steady velocity distributions for various Bingham numbers in round Poiseuille �ow�

M��

�� Evolution of the velocity in cessation of plane Couette �ow of a Newtonian �uid� Com�

parison of the analytical �solid lines� with the numerical �dashed lines� solutions�

�� Evolution of the velocity in cessation of plane Couette �ow of a Bingham �uid with

Bn�� and M��

� Evolution of the velocity in cessation of plane Couette �ow of a Bingham �uid with


�� Evolution of the volumetric �ow rate during the cessation of plane Couette �ow of a

Bingham �uid with M�� and various Bingham numbers�

�� Calculated times for Q�� and �� in cessation of plane Couette �ow of a Bingham

�uid with M��

�� Comparison of the computed stopping time �Q�� in cessation of plane Couette

�ow of a Bingham �uid with the theoretical upper bound� M��

�� Evolution of the velocity in cessation of plane Poiseuille �ow of a Newtonian �uid�

�� Evolution of the velocity in cessation of plane Poiseuille �ow of a Bingham �uid with


��


Bn� and M��



�� Evolution of the volumetric �ow rate during the cessation of plane Poiseuille �ow of a


�� Calculated times for Q�� and �� in cessation of plane and round Poiseuille �ows

of Bingham �uids with M��

�� Calculated times for Q�� and �� in cessation of plane Poiseuille �ow of Bingham

�uids with M�� dashed� and M�� solid��

�� Comparison of the computed stopping time �Q�� in cessation of plane Poiseuille

�ow of a Bingham �uid with the theoretical upper bound� f�� and M��

�� a� Evolution of the volumetric �ow rate for various values of the imposed pressure

gradient f � �b� Zoom of the same plot showing that a �nite volumetric �ow rate is

reached when f � �� plane Poiseuille �ow with Bn�� and M��

�� Volumetric �ow rates reached with the regularized Papanastasiou model versus the

imposed pressure gradient f � plane Poiseuille �ow� Bn�� and M��

�� Comparison of the times required to reach Q�� in cessation of plane Poiseuille �ow

of a regularized Bingham �uid with the theoretical estimate of Huilgol et al� �� for

an ideal Bingham �uid� Bn�� and M��

�� Evolution of the velocity in cessation of round Poiseuille �ow of a Newtonian �uid�

��

�� Evolution of the velocity in cessation of round Poiseuille �ow of a Bingham �uid with



Bn� and M��




Bn�� and M�� zoom near the wall��

�� Evolution of the volumetric �ow rate during the cessation of round Poiseuille �ow of a


�� Comparison of the computed stopping time �Q�� in cessation of round Poiseuille

�ow of a Bingham �uid with the theoretical upper bound� f�� and M��

��

x

yux�y� �� ux�y� t�H

ux��

ux�V

ux��

ux��

t�� t ��

�a�

x

yux�y� �� ux�y� t�H

�

�p�x

� � ��p�x

� �

t�� t ��

�b�

z

ruz�r� �� uz�r� t�R

�

�p�z

� � ��p�z

� �

t�� t ��

�c�

Figure �� Flow problems under study� �a� cessation of plane Couette �ow� �b� cessation ofplane Poiseuille �ow� and �c� cessation of axisymmetric Poiseuille �ow�

�

0

0.5

1

1.5

2

2.5

0 0.2 0.4 0.6 0.8 1

uz

r

Bn�� Newtonian �uid�

Bn��

Bn��

Bn��

Bn�� Solid�

Figure � Steady velocity distributions for various Bingham numbers in round Poiseuille �ow�M��

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

0 0.2 0.4 0.6 0.8 1

ux

y

t��

��

��

��

��

��

�

Figure � Evolution of the velocity in cessation of plane Couette �ow of a Newtonian �uid�Comparison of the analytical �solid lines� with the numerical �dashed lines� solutions�

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

0 0.2 0.4 0.6 0.8 1

ux

y

t��

��

��

��

��

��

Figure �� Evolution of the velocity in cessation of plane Couette �ow of a Bingham �uid withBn� and M��

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

0 0.2 0.4 0.6 0.8 1

ux

y

t��

��

��

��

��

Figure �� Evolution of the velocity in cessation of plane Couette �ow of a Bingham �uid withBn�� and M��

0

0.1

0.2

0.3

0.4

0.5

0.6

0 0.1 0.2 0.3

Q

t

Bn��

��

Figure � Evolution of the volumetric �ow rate during the cessation of plane Couette �ow ofa Bingham �uid with M�� and various Bingham numbers�

0.01

0.1

1

10

0.0001 0.01 1 100

Tf

Bn

Q��

Q��

Figure �� Calculated times for Q�� and �� in cessation of plane Couette �ow of aBingham �uid with M��

�

0.01

0.1

1

10

0.0001 0.01 1 100

Tf

Bn

Theory

Numerical

�Q��

Figure �� Comparison of the computed stopping time �Q�� in cessation of plane Couette�ow of a Bingham �uid with the theoretical upper bound� M��

-0.5

0

0.5

1

1.5

2

0 0.2 0.4 0.6 0.8 1

ux

y

t��

��

��

��

Figure �� Evolution of the velocity in cessation of plane Poiseuille �ow of a Newtonian �uid�

�

-0.5

0

0.5

1

1.5

2

0 0.2 0.4 0.6 0.8 1

ux

y

t��

��

��

��

��

��

��

Figure �� Evolution of the velocity in cessation of plane Poiseuille �ow of a Bingham �uidwith Bn�� and M��

-0.5

0

0.5

1

1.5

2

0 0.2 0.4 0.6 0.8 1

ux

y

t��

��

��

��

��

��


-0.5

0

0.5

1

1.5

2

0 0.2 0.4 0.6 0.8 1

ux

y

t��

��

��

��

��


0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1

Q

t

Bn��

��

Figure �� Evolution of the volumetric �ow rate during the cessation of plane Poiseuille �owof a Bingham �uid with M�� and various Bingham numbers�

�

0.01

0.1

1

10

0.0001 0.01 1 100

Tf

Bn

Q��

��

��

��

PlanePoiseuille�ow

RoundPoiseuille�ow

Figure �� Calculated times for Q�� and �� in cessation of plane and round Poiseuille�ows of Bingham �uids with M��

0.01

0.1

1

10

0.0001 0.01 1 100

Tf

Bn

Q��

Q��

Figure �� Calculated times for Q�� and �� in cessation of plane Poiseuille �ow ofBingham �uids with M�� dashed and M�� solid�

��

0.01

0.1

1

10

0.0001 0.01 1 100

Tf

Bn

Theory

Numerical

�Q��

Figure �� Comparison of the computed stopping time Q�� in cessation of planePoiseuille �ow of a Bingham �uid with the theoretical upper bound f�� and M��

��

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1 1.2

0

0.005

0.01

0.015

0 0.5 1 1.5

Q

t

f��

��

Q

t

f��

��

Figure �� a Evolution of the volumetric �ow rate for various values of the imposed pressuregradient f b Zoom of the same plot showing that a �nite volumetric �ow rate is reachedwhen f � � plane Poiseuille �ow with Bn�� and M��

��

0

0.0005

0.001

0.0015

0 0.2 0.4 0.6 0.8 1

Qf

f

Figure �� Volumetric �ow rates reached with the regularized Papanastasiou model versusthe imposed pressure gradient f plane Poiseuille �ow� Bn�� and M��

0

1

2

3

0 0.2 0.4 0.6 0.8 1

Tf

f

Regularized Bingham �uid �Q��

Estimate for ideal Bingham �uid

Figure �� Comparison of the times required to reach Q�� in cessation of plane Poiseuille�ow of a regularized Bingham �uid with the theoretical estimate of Huilgol et al� �� foran ideal Bingham �uid Bn�� and M��

��

-0.5

0

0.5

1

1.5

2

2.5

0 0.2 0.4 0.6 0.8 1

ux

y

t��

��

��

��

��

��

�

Figure �� Evolution of the velocity in cessation of round Poiseuille �ow of a Newtonian �uid�

-0.5

0

0.5

1

1.5

2

2.5

0 0.2 0.4 0.6 0.8 1

ux

y

t��

��

��

��

��

Figure �� Evolution of the velocity in cessation of round Poiseuille �ow of a Bingham �uidwith Bn�� and M��

��

-0.5

0

0.5

1

1.5

2

2.5

0 0.2 0.4 0.6 0.8 1

ux

y

t��

��

��

��


-0.5

0

0.5

1

1.5

2

2.5

0 0.2 0.4 0.6 0.8 1

ux

y

t��

��

��

��

��


��

-0.05

0

0.05

0.1

0.15

0.2

0.25

0.85 0.9 0.95 1

ux

y

t��

��

��

��

��

��

��

��

Figure �� Evolution of the velocity in cessation of round Poiseuille �ow of a Bingham �uidwith Bn�� and M�� zoom near the wall�

0

0.2

0.4

0.6

0.8

1

0 0.1 0.2 0.3 0.4 0.5

Q

t

Bn��

Figure �� Evolution of the volumetric �ow rate during the cessation of round Poiseuille �owof a Bingham �uid with M�� and various Bingham numbers�

��

0.01

0.1

1

10

0.0001 0.01 1 100

Tf

Bn

Theory

Numerical

�Q��

Figure �� Comparison of the computed stopping time Q�� in cessation of roundPoiseuille �ow of a Bingham �uid with the theoretical upper bound f�� and M��

��

ws one can bring the uid to a halt b - ucy · 2013. 6. 28. · w of a bingham plastic determination...

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