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Research Papers: Fundamental Issues and Canonical Flows

Numerical and Theoretical Investigations of Flow in a Microcone and Plate Viscometer

[+] Author and Article Information
Khaled Bataineh

Department of Mechanical Engineering,
Jordan University of Science and Technology,
P.O. Box 3030,
Irbid 22110, Jordan
e-mail: k.bataineh@just.edu.jo

Contributed by the Fluids Engineering Division of ASME for publication in the JOURNAL OF FLUIDS ENGINEERING. Manuscript received June 15, 2012; final manuscript received March 28, 2014; published online July 24, 2014. Assoc. Editor: Meng Wang.

J. Fluids Eng 136(10), 101201 (Jul 24, 2014) (12 pages) Paper No: FE-12-1296; doi: 10.1115/1.4027321 History: Received June 15, 2012; Revised March 28, 2014

The slip flow of fluid in a microcone-plate viscometer has been considered. The first-order slip boundary condition is applied. A slip flow primary solution corresponding to unidirectional flow has been obtained. By ignoring all the convective terms and assuming a small gap angle, another closed-form solution called zeroth-order solution for slip flow is obtained. Previous investigations of the flow in the cone and plate did not consider the slip conditions at the solid-fluid interfacial boundary. Slip flow is also numerically studied in a microcone and plate viscometer. The accuracy of the two solutions is assessed by comparing their results with numerical solutions obtained by solving the full Navier–Stokes equation. The primary solution does not completely describe the flow in microcone and plate, and some deviations are found. On the other hand, the zeroth-order solution perfectly predicted the slip and no-slip flow in the inner region. A slip factor that takes into account spatial distribution yields a perfect match between the zeroth-order solution and numerical solution. On the other hand, analytical zeroth-order solution for outer region does not agree with the numerical work, and we should rely on the numerical solution. Taking a realistic range of slip length resulting from actual devices, numerical and theoretical results show that the differences in viscosity measurements between considering slip and no-slip are significant.

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Figures

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Fig. 1

Dimensionless slip length versus Knudsen number

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Fig. 2

Schematic of cone and plate viscometer

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Fig. 3

2D axisymmetric numerical model of simulated geometry

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Fig. 4

Comparison between present numerical solution and experimental results of Cheng [34], T0 is torque for the primary solution given as [28] T0 = (2πΩμR3/3α)

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Fig. 5

Comparison between the present numerical solution and measured pressure distribution (Re*= 8.97)

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Fig. 6

Nondimensional strain rate distribution along the ordinate for given radial position. (a) Re = 0.13, (b) Re = 1.3, (c) Re = 4, and (d) Re = 13.

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Fig. 7

Radial velocity component for no-slip Re = 0.13 (Ω = 1000 rad/s)

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Fig. 8

Nondimensional pressure gradient ψ at the plate wall, Re = 0.13 (Ω = 1000 rad/s)

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Fig. 9

Nondimensional swirl velocity vϕ distribution along the abscissa at given ordinate for Re = 0.13, cone radius = 0.5 mm, gap angle = 5 deg. (a) at θ = 87 deg and (b) at plate (θ = 90 deg).

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Fig. 10

Radial velocity component computed based on zeroth-order analytical and numerical solution

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Fig. 11

Radial velocity component computed based on zeroth-order analytical and numerical solution

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Fig. 12

Radial velocity component computed based on zeroth-order analytical and numerical solution. (a) r/R = 0.4, (b) r/R=0.5, (c) r/R=0.6, and (d) r/R=0.8.

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Fig. 13

Radial velocity component computed based on zeroth-order analytical incorporating modified slip factor and numerical solution. (a) r/R=0.4 and (b) r/R=0.8.

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Fig. 14

Radial distribution of dimensionless pressure on the stationary plate

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Fig. 15

Nondimensional strain rate distribution along the ordinate for given radial position. Ω = 1000 rad/s, (a) no-slip, (b) β=0.05, and (c) β=0.1

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Fig. 16

Deviation of the zeroth-order solution from the numerical solution near the edge

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Fig. 17

Comparison between the zeroth-order solution for the outer region with the numerical solution

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