Evan Mitsoulis
1
e-mail: mitsouli@metal.ntua.gr
S. Marangoudakis
M. Spyratos
Th. Zisis
School of Mining Engineering and Metallurgy,
National Technical University of Athens,
Zografou 157 80,
Athens, Greece
Nikolaos A. Malamataris
Department of Mechanical Engineering,
Technological Educational Institution of Western
Macedonia,
Kozani 501 00,
Macedonia, Greece
e-mail: nikolaos@vergina.eng.auth.gr
Pressure-Driven Flows of
Bingham Plastics Over a Square
Cavity
Pressure-driven flows over a square cavity are studied numerically for Bingham plastics
exhibiting a yield stress. The problem is encountered whenever pressure measurements
are made by a drilled-hole based pressure transducer. The Bingham constitutive equation
is used with an appropriate modification proposed by Papanastasiou, which applies
everywhere in the flow field in both yielded and practically unyielded regions. Newtonian
results are obtained for a wide range of Reynolds numbers 0 Re 1000 for the cavity
vortex position and intensity, and the excess pressure drop (entrance correction) in the
system. To reduce the length of the computational domain for highly convective flows, an
open boundary condition has been implemented at the outflow. For viscoplastic fluids the
emphasis is on determining the extent and shape of yielded/unyielded regions along with
the cavity vortex shape, size, and intensity for a wide range of Bingham numbers 0
Bn . The entrance correction is found to be an increasing sigmoidal function of
the Bn number, reaching asymptotically the value of zero. It is shown that for viscoplastic
fluids not exhibiting normal stresses in shear flow (lack of viscoelasticity), the hole
pressure is zero opposite the center of the hole. Thus, any nonzero pressure hole mea-
sured by this apparatus would signify the presence of a normal-stress difference in the
fluid. DOI: 10.1115/1.2236130
Keywords: yield stress, Bingham model, cavity flow, recirculation, inertia flow,
viscoplasticity, pressure-hole error
1 Introduction
An important class of non-Newtonian materials exhibits a yield
stress, which must be exceeded before significant deformation can
occur. A list of several materials exhibiting yield was given in a
seminal paper by Bird et al. 1. The models presented for such
so-called viscoplastic materials included the Bingham, Herschel-
Bulkley, and Casson models. Analytical solutions were provided
for the Bingham plastic model in simple flow fields. Since then a
renewed interest has developed among several researchers for the
study of these materials in nontrivial flows see recent review by
Barnes 2.
Because of the irregularity inherent in models with a yield
stress, several methods have been proposed in the literature. The
augmented Lagrangian method ALM3, based on variational
inequalities, is well suited for such models as evidenced in the
works by Fortin et al. 4 and Huilgol and Panizza 5. On the
other hand, a variety of regularization methods have also been
proposed and implemented with good results 6,7.
A popular approach to regularize the ideal Bingham model has
been the exponential modification proposed by Papanastasiou 8.
With this model, Abdali et al. 9 solved the benchmark entry flow
problem of Bingham plastics through planar and axisymmetric 4:1
contractions, and the exit flow problem, and determined the extru-
date swell. That work showed the evolution of the phenomenon of
viscoplasticity as a dimensionless yield stress or Bingham number
Bn increased from the Newtonian to the fully plastic limit. Fur-
ther work involved non-isothermal viscoplastic simulations in en-
try and exit die flows of a propellant dough with the Herschel-
Bulkley model 10.
In the case of inertial flows, the expansion benchmark problem
has been the object of a number of studies 11–16. Different
models of viscoplasticity have been used, and the effect of Rey-
nolds Re and Bingham Bn numbers has been fully addressed.
An interesting problem of relevance to the rheological commu-
nity is the pressure-driven Poiseuille flow over a cavity, which is
used for pressure measurements by drilling a hole in the conduit
and positioning a pressure transducer in the hole. In the context of
viscoelastic fluids, this gives rise to the “pressure-hole error”
17,18, which can be used to measure normal stress effects. Re-
sults for viscoelastic fluids have been obtained 19,20 with good
agreement between experiments and simulations. However, no
work appears to have been done for viscoplastic fluids.
In the present work, we wish to examine this rheological bench-
mark problem of a pressure-driven flow over a square cavity for a
wide range of Bingham numbers 0 Bn 3000 and Reynolds
numbers 0 Re 1000. The results include both the yielded/
unyielded regions as well as kinematic and dynamic quantities,
such as vortex size and intensity and excess pressure losses en-
trance correction as functions of Bn and Re numbers.
2 Mathematical Modeling
The flow is governed by the usual conservation equations of
mass and momentum for an incompressible fluid under isother-
mal, laminar flow conditions. These are:
Mass: · u ¯ =0 1
Momentum: u ¯ · u ¯ =- p + ·
2
where u ¯ is the velocity vector, p is the scalar pressure, and
is the
extra stress tensor.
The relevant Reynolds number Re for laminar flow is a mea-
sure of the convective forces compared to the viscous forces, and
is defined as 11,12:
1
Corresponding author.
Contributed by the Fluids Engineering Division of ASME for publication in the
JOURNAL OF FLUIDS ENGINEERING. Manuscript received December 12, 2005; final
manuscript received March 9, 2006. Assoc. Editor: Dennis Siginer.
Journal of Fluids Engineering SEPTEMBER 2006, Vol. 128 / 993 Copyright © 2006 by ASME