Entry and exit flows of Bingham fluids

Entry

and exit flows of Bingham

fluids

S. S. Abdali and Evan Mitsoulis

Department

of Chemical Engineering, University Ottawa, Ontario KIN 9B4, Canada

of Ottawa,

N. C. Markatos

Computational Fluid Dynamics Unit, Department of Chemical Engineering, National Technical University of Athens, Zographou, Athena 15773, Greece (Received 17 July 1991; accepted 18 November 1991) Synopsis Entry and exit flows through extrusion dies are studied numerically for Bingham fluids exhibiting a yield stress.A constitutive equation proposedby Papanastasiouis used,which applieseverywherein the flow field in both yielded and practically unyieldcd regions. The emphasisis on determining the extent and shapeof unyielded/yielded regionsalong with the extrudate swell (contraction) for planar and axisymmetricdies.The resultsfor pressureare usedto determine the excesspressurelossesthat give rise to entrance,exit, and the total end (or Bagley) correction. INTRODUCTION A plastic material exhibits little or no deformation up to a certain level of stress, called the yield stress. Above this yield stress the material flows. These materials are often called Bingham plastics after Bingham (1922), who first described paint in this way in 1919. Paint, slurries, pastes, and food substances like margarine, mayonnaise, and ketchup are good examples of Bingham plastics. A list of several materials exhibiting yield was given in a seminal paper by Bird et al. (1983), who have also provided an initial analysis of such materials in simple flow fields. Since then a renewed interest has developed among several researchers for the study of Bingham fluids as evidenced by a series of papers in the literature [see Lipscomb and Denn ( 1984), Gartling and @ 1992 by The Society of Rheology, Inc. J. Rbeol. 36(2), February 1992 0148~6055/92/020389-19$04.00

389

390

ABDALI, MITSOULIS, AND MARKATOS

Phan-Thien (1984), O’Donovan and Tanner (1984), Keentok et al. (1985), Dzuy and Boger (1985), Beris et al. (1985), Papanastasiou (1987), Beverly and Tanner (1989), and Ellwood et al. (1990)]. To model the stress-deformation behavior, several constitutive relations have been proposed and different yield criteria have been used [see, e.g., discussion by Ellwood et al. (1990)]. The existence of a true yield stress has been questioned by several investigators. Quite recently, Papanastasiou ( 1987) proposed a novel constitutive equation for materials with yield, where a material parameter controls the exponential growth of stress and which is valid for both yielded and unyielded areas. It was shown by Papanastasiou (1987) and Ellwood et al. (1990) that this equation closely approximates the ideal Bingham plastic and it provides a better approximation to real data of such materials. In the present work, we shall use Papanastasiou’s constitutive equation to examine the entry and exit flows of Bingham fluids through extrusion dies. The emphasis will be on finding the extent and shape of the yielded/unyielded zones by using the criterion that the material flows and deforms significantly only when the magnitude of the extra stress tensor exceeds the yield stress. The determination of the extrudate shape will also be carried out for planar and circular dies. Finally, excess pressure losses will be determined as a function of a dimensionless yield stress or Bingham number. MATHEMATICAL

MODELING

The flow is governed by the usual conservation equations of mass and momentum for an incompressible fluid under isothermal, creeping flow conditions (Re-0) [Bird et al. (1983)]. The constitutive equation that relates 7 to the deformation is a modified Bingham equation proposed by Papanastasiou (1987) and is written as

where p is a constant viscosity, 7v is the apparent yield stress, and m is a stress growth exponent. The magnitude 1p 1of the rate-of-strain tensor 7 = VV + Vviiis given by (2)

LAGRANGIAN-UNSTEADY

BINGHAM FLOWS

391

3.0

2.5

2.0

1.5

1 .o

0.5

0.0 0.0

2.0

4.0

6.0

8.0

SHEAR RATE, i

10.0

12.0

(a-‘)

FIG. 1. Shear stress vs shear rate according to the modified Bingham constitutive equation (3) proposed by Papanastasiou ( 1987) for several values of the exponent m.

where II 7 is the second invariant of 7. Equation ( 1) approximates the von Mises criterion for relatively big exponent m (m > lOO), and holds uniformly in yielded and unyielded regions. The one-dimensional analog of Eq. ( 1) in simple shear flow gives 712=

CL+ +1 ,+,2, 1

--xp(

-mlh21)1

f12*

(3)

I

where 712 and i/t2 are the shear stress and shear rate, respectively. Figure 1 shows a graphical representation of Eq. (3) for different values of the exponent m. Clearly, for m > 100 the above equation mimics the ideal Bingham plastic. To track down yielded/unyielded regions, we shall employ the criterion that the material flows (yields) only when the magnitude of the extra stress tensor 171 exceeds the yield stress T,,, i.e.,

ABDALI, MITSOULIS, AND MARKATOS

392

yielded: unyielded:

]7 = & I‘;1 rr

(44 (4b)

Note that previous work by Papanastasiou ( 1987) employed ge criterion of the second invariant of half the rate-of-strain tensor D = $7 exceeding an arbitrarily small value of 0.001. This led to erroneous yielded/unyielded regions, as pointed out by Beverly and Tanner ( 1989), who, however, used very coarse grids and were not able to capture well the extent and shape of these regions. In the present work, we try to resolve this matter adequately. We also believe (as was shown by numerical trials) that using as a criterion the value of yield stress, which is not arbitrary and is also a large number, reduces the uncertainty of using values very close to zero, which may be numerical noise. For materials with yield stress, it is appropriate to introduce a dimensionless yield stress $, defined by Papanastasiou (1987) as

where H is a characteristic length (half the channel width or radius R) and V,V is a characteristic speed, taken by Papanastasiou ( 1987) as the average velocity of a corresponding Newtonian liquid with viscosity p at the same pressure gradient. However, this dimensionless number again introduces dependence on a Newtonian counterpart. Bid et al. (1983) suggest the Bingham number Bi=-,

*P

PVB

where D is the diameter or channel width (2H) and V, is the average velocity of the Bingham fluid. In both cases,the Newtonian fluid corresponds to + = Bi = 0. However, at the other extreme of an unyielded solid, Bi-. CC,while r; reaches a dimensionless pressure gradient defined by

LAGRANGIAN-UNSTEADY

BINGHAM FLOWS

393

In the present study both numbers will be used to provide comparisons with the previous works by Papanastasiou (1987) and Ellwood et al. (1990). METHOD OF SOLUTION The conservation and constitutive equations along with the appropriate boundary conditions [Papanastasiou (1987)] are solved by the finite element method. The primary variables are the two velocities and pressure (u-v-p formulation). Streamlines are obtained a posteriori by solving the Poisson equation for the stream function. Although a Newton-Raphson iterative scheme would be recommended for the solution of the nonlinear system of equations, here we have used a direct substitution scheme (Picard method), sometimes with under-relaxation to accelerate convergence. This avoids calculating the Jacobian which can be time consuming, and it also enjoys a wider convergence range. The solution process starts from the Newtonian field (r,,=O), which is used to obtain a first approximation. Iterations are performed until convergence for the current yield stress is achieved (usually when the norm-of-error