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TECHNICAL PAPERS

# Pressure-Driven Flows of Bingham Plastics Over a Square Cavity

[+] Author and Article Information
Evan Mitsoulis1

School of Mining Engineering and Metallurgy, National Technical University of Athens, Zografou 157 80, Athens, Greecemitsouli@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, Greecenikolaos@vergina.eng.auth.gr

1

Corresponding author.

J. Fluids Eng 128(5), 993-1003 (Mar 09, 2006) (11 pages) doi:10.1115/1.2236130 History: Received December 12, 2005; Revised March 09, 2006

## Abstract

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 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 measured by this apparatus would signify the presence of a normal-stress difference in the fluid.

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Copyright © 2006 by American Society of Mechanical Engineers
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## Figures

Figure 3

Finite elements meshes used in the computations for a pressure-driven flow over a square cavity

Figure 4

Streamlines and contours of kinematic and dynamic variables for a Newtonian fluid flowing under pressure over a square cavity. Left, creeping flow, Re=10−3; right, weak inertial flow, Re=10.

Figure 5

Streamlines and contours of kinematic and dynamic variables for a Newtonian fluid flowing under pressure over a square cavity. Left, moderate inertial flow, Re=100; right, strong inertial flow, Re=1000.

Figure 6

Velocity profiles: (a) u components and (b) v components in vertical and horizontal section, respectively, passing through the geometric center of the cavity (0.5H,−0.5H) for various Re numbers

Figure 1

Shear stress versus shear rate according to the modified Bingham constitutive equation 2 for several values of the exponent m

Figure 2

Schematic diagram of pressure-driven flow over a square cavity, notation, and boundary conditions. The flow is from left to right.

Figure 7

(a) Location of the eye of the main vortex in the square cavity, (b) its intensity, and (c) entrance correction as a function of Re number

Figure 8

Streamlines for Bn=10, 100, 1000, 3000 in creeping pressure-driven flow of a viscoplastic Bingham-Papanastasiou fluid over a square cavity (Re=10−3)

Figure 9

Yielded/unyielded (shaded black) regions for Bn=10, 100, 1000, 3000 in creeping pressure-driven flow of a viscoplastic Bingham-Papanastasiou fluid over a square cavity (Re=10−3)

Figure 10

Entrance correction as a function of the Bn number in creeping pressure-driven flow of a viscoplastic Bingham-Papanastasiou fluid over a square cavity (Re=10−3)

Figure 11

Dimensionless shear stress τxy (a1) and first normal stress difference N1 (b1) profiles along the upper wall for different Bn numbers in creeping pressure-driven flow of a viscoplastic Bingham-Papanastasiou fluid over a square cavity (Re=10−3). Blown-up profiles along the upper wall above the cavity for τxy (a2) and N1 (b2). Note that at (x,y)=(0.5,1.0),N1=τxy=0 (τw0 is the fully developed shear stress value at the inlet wall).

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