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Research Papers: Fundamental Issues and Canonical Flows

Experimental Determination of the Virtual Mass Coefficient for Two Spheres Accelerating in a Power Law Fluid

[+] Author and Article Information
Abbas H. Sulaymon

Department of Environmental Engineering, College of Engineering, Baghdad University, Baghdad, Iraqinas_abbas@yahoo.com

Catherine A. M. E. Wilson

Co-director of the Hydro-Environmental Research Center, Senior Lecturer in Environmental Hydraulics, Cardiff School of Engineering, Cardiff University, Cardiff CF24 3AA, UKwilsonca@cardiff.ac.uk

Abeer I. Alwared

Department of Environmental Engineering, College of Engineering, Baghdad University, Baghdad, Iraqabeerwared@yahoo.com

J. Fluids Eng 132(12), 121204 (Dec 29, 2010) (11 pages) doi:10.1115/1.4003001 History: Received November 27, 2009; Revised October 04, 2010; Published December 29, 2010; Online December 29, 2010

The virtual mass coefficient is determined experimentally for the motion of two spheres side by side and in line in a power law fluid. The velocities of the two accelerating spheres and their separation distance was measured as they accelerated under the action of driving weights through a cylindrical column filled with different concentrations of polyacryamaide solution (0.01%, 0.03%, 0.05%, and 0.07% by weight). For comparison purposes, the experiments were repeated with water. Various densities of spheres and separation distances were examined. Within the range of power law indices (0.61–0.834) and Reynolds numbers (1.1–75) examined, the virtual mass coefficient was found to decrease with an increasing Reynolds number for the two spheres moving side by side, and found to be greater than 0.5 when the spheres were touching each other. As the distance between the spheres increased, the virtual mass coefficient was found to decrease and approached the single sphere value of 0.5 when the distance between the spheres was more than ten radii. When the spheres were in line and touching each other, the virtual mass coefficient was found to be less than 0.5, however, when the distance between the spheres increased, the virtual mass coefficient increased and approached the value of 0.5. The virtual mass coefficient was found to be consistent with the shear thinning behavior; for a given Reynolds number, it increased with an increasing power law index.

Copyright © 2010 by American Society of Mechanical Engineers
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References

Figures

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Figure 3

Change of history force with time for two glass spheres side by side (d=25.6 mm) in a 0.01% w/v PAA solution

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Figure 4

Effect of generated Reynolds number on the virtual mass coefficient for one steel sphere in a 0.01% w/v PAA solution

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Figure 5

Variation of virtual mass coefficient with generated Reynolds number for two spheres side by side in a 0.01% w/v PAA solution

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Figure 6

Variation of virtual mass coefficient and generated Reynolds number for two spheres in line in a 0.01% w/v PAA solution

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Figure 7

Variation of virtual mass coefficient with Reynolds number for two steel spheres in water (a) side by side and (b) in line

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Figure 8

Virtual mass coefficient versus separation distance for two spheres side by side in a 0.03% w/v PAA solution: (a) for steel spheres of diameter 10 mm, (b) for glass spheres of diameter 7.2 mm, and (c) for plastic spheres of diameter 11.4 mm

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Figure 9

Virtual mass coefficient versus separation distance for two spheres in line in a 0.03% w/v PAA solution: (a) for steel spheres of diameter 10 mm, (b) for glass spheres of diameter 7.2 mm, and (c) for plastic spheres of diameter 11.4 mm

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Figure 10

Relationship between virtual mass coefficient and power law index at constant generated Reynolds number and separation distance for two spheres side by side

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Figure 11

Relationship between virtual mass coefficient and power law index at constant generated Reynolds number and separation distance for two spheres in line

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Figure 12

Effect of sphere density for two spheres side by side (l/d=1) in a 0.01% w/v PAA solution on (a) the virtual mass force for two spheres side by side (l/d=1) in a 0.01% w/v PAA solution and (b) the ratio of the virtual mass force (F−VM) to the drag force (F−drag)

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Figure 1

Schematic diagram of the experimental apparatus

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Figure 2

Drag coefficient as a function of generated Reynolds number; data from other studies for non-Newtonian and Newtonian fluids is compared with data from the present study. ∗: Present work; +: Kelessidis (8); ×: Miura (22); ◻Pinelli (23); △: Ford (24); —: Haider (25).

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Figure 13

Comparison of history force to virtual mass force for (a) two glass spheres side by side (d=9.3 mm, l/d=1) in a 0.03% w/v PAA solution and (b) two plastic spheres in line (d=11.4 mm, l/d=1) in a 0.05% w/v PAA solution

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Figure 14

Comparison of history force to virtual mass force in water for (a) two glass spheres side by side (d=9.3 mm, l/d=1) and (b) two plastic spheres (d=11.4 mm, l/d=1) in line in water

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Figure 15

Force ratio versus time, for two plastic spheres (d=11.4 mm, l/d=1) in a 0.05% w/v PAA solution (a) side by side and (b) in line, where FD is the drag force, FH is the history force, and FVM is the virtual mass force

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Figure 16

Force ratio versus time in water for (a) two glass spheres side by side (d=25.6 mm, l/d=1) and (b) two steel spheres in line (d=12 mm, l/d=1), where FD is the drag force, FH is the history force, and FVM is the virtual mass force

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Figure 17

Uncertainty of CVM as a function of Regn for a single sphere and two spheres at different arrangements

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Figure 18

Uncertainty of the CVM as a function of l/a for the two spheres

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