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Research Papers: Fundamental Issues and Canonical Flows

Pulsatile Poiseuille Flow of a Viscoplastic Fluid in the Gap Between Coaxial Cylinders

[+] Author and Article Information
Irene Daprà1

DICAM,  University of Bologna, 2 via Risorgimento, Bologna, Italy, 40136irene.dapra@unibo.it

Giambattista Scarpi

DICAM,  University of Bologna, 2 via Risorgimento, Bologna, Italy, 40136giambattista.scarpi@unibo.it

1

Corresponding author.

J. Fluids Eng 133(8), 081203 (Aug 23, 2011) (7 pages) doi:10.1115/1.4003926 History: Revised April 04, 2009; Received July 14, 2009; Published August 23, 2011; Online August 23, 2011

Several materials that are of interest in engineering present a yield stress and behave as viscoplastic fluids. This paper investigates numerically the motion of a Bingham fluid between two coaxial cylinders due to a periodic pressure gradient and/or to the periodic displacement of the internal cylinder. The constitutive equation presents a discontinuity at the zero shear rate. To overcome the difficulty, the rheologic law has been regularized using a smooth function based on the error function. The velocity fields have been calculated using an implicit finite difference method. The procedure has been validated, comparing the numerical results with the analytical solution of the same problem for a Newtonian fluid. The nonlinear behavior of the fluid is emphasized, comparing the effects due to the simultaneous action of the pressure gradient and the displacement of the internal wall with the sum of the effects due to the single actions. In all cases, the mean discharge in a period increases. The comparison between the effects of the forcing agents shows that if the dimensionless frequency is less than 10 the increases of the discharge obtained by applying the pulsatile pressure gradient or moving the internal wall are similar. At low frequencies the action of the gradient exceeds that of the moving wall, whereas for higher frequencies the effect of the moving wall increases rapidly because a fixed displacement of the internal cylinder leads to very great values for the velocity of the internal wall.

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Copyright © 2011 by American Society of Mechanical Engineers
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Figures

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Figure 1

Flow geometry; steady velocity profile

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Figure 2

Regularized constitutive equation for Bn=10: from right to left K=100, K=1000, K=10000, and K=100000; thick line denotes ideal Bingham fluid

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Figure 3

Solid plug amplitude versus Bingham number

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Figure 4

Percent increase of discharge ΔQp due to oscillating pressure gradient versus ɛ for some frequencies; Bn=10

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Figure 5

Percent increase of discharge ΔQd due to the displacement of the internal wall versus ɛ for some frequencies; Bn=10

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Figure 6

Increase of percent discharge ΔQp+d due to simultaneous action of oscillating pressure gradient and displacement of internal wall (solid line), and ΔQp+ΔQd (dashed line) versus ɛ for some frequencies; Bn=10

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Figure 7

Plug velocity for Bn=10 as a function of time for cases (1), (2), and (3), and velocity at the middle of the gap for a Newtonian fluid (Bn=0) for case (3) with frequency f=10 and ɛ=0.6

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Figure 8

Discharge for Bn=10 as a function of time for cases (1), (2), and (3) and for a Newtonian fluid (Bn=0) with ɛ=0.6 and frequency f=10

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Figure 9

Power consumption for Bn=10 as a function of time for cases (1), (2), and (3) and for a Newtonian fluid (Bn=0) with ɛ=0.6 and frequency f=10

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Figure 10

Velocity profiles for Bn=10 for cases (2) (solid line) and (3) (dashed line) with ɛ=0.6 and frequency f=40 at some instants

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Figure 11

Velocity profiles for case (3) with ɛ=0.6 and frequency f=40 for Bn=10 (solid line) and for a Newtonian fluid (dashed line) at some instants

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Figure 12

ΔQp (dashed line) and ΔQd (solid line) versus frequency for ɛ=0.6 and for some Bingham numbers

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Figure 13

ΔW (solid lines) and ΔQ (dashed lines) versus frequency for ɛ=0.6 and Bn=10 for cases (1), (2), and (3)

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