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Research Papers: Fundamental Issues and Canonical Flows

Stokes Flow Characteristics in a Cylindrical Quadrant Duct With Rotating Outer Wall

[+] Author and Article Information
Zongyong Wang

School of Energy and Power Engineering,
Shenyang University of Chemical Technology,
Shenyang 110142, Liaoning, China
e-mail: syuctwzy@163.com

Jiayu Zhao

School of Energy and Power Engineering,
Shenyang University of Chemical Technology,
Shenyang 110142, Liaoning, China
e-mail: zhaojiayu.456@163.com

Jianhua Wu

School of Energy and Power Engineering,
Shenyang University of Chemical Technology,
Shenyang 110142, Liaoning, China

1Corresponding author.

Contributed by the Fluids Engineering Division of ASME for publication in the JOURNAL OF FLUIDS ENGINEERING. Manuscript received July 17, 2013; final manuscript received April 30, 2014; published online September 4, 2014. Editor: Malcolm J. Andrews.

J. Fluids Eng 136(11), 111202 (Sep 04, 2014) (11 pages) Paper No: FE-13-1433; doi: 10.1115/1.4027586 History: Received July 17, 2013; Revised April 30, 2014

The Stokes flow in a cylindrical quadrant duct with a rotating wall was analytically and numerically studied. Based on mathematics and fluid dynamics theory, the analytical expressions of three velocity components were achieved by solving a Poisson's equation and a biharmonic equation. Especially, a closed-form analytical expression of axial velocity was obtained, which can greatly improve the calculating accuracy and speed in analyzing Stokes flow. The velocity distributions for different Reynolds numbers were investigated numerically to insure the accuracy of the analytical results at low Reynolds numbers and to confirm the error range of the analytic results at higher Reynolds numbers. The conclusion indicates that there exists an infinite sequence of eddies that decrease exponentially in size towards the sectorial vertex. The width of the first eddy region reached 99.4% of the sector radius; the sum of the width of other eddies is only 0.6% of the sector radius, which cannot be easily displayed graphically, while the sequence of eddies contributes to form the chaotic flow. The maximum deviations of the velocity components between the analytical results and simulated ones are all less than 1% when Re < 0.1, which verifies the validity and accuracy of the analytical expressions in the creeping flow regime. The analytical expressions are not only suitable for creeping flow but also for laminar flow with smaller Reynolds number (Re < 50).

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Figures

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Fig. 1

Schematic view of PPM

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Fig. 2

Schematic view of MPPM

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Fig. 3

Physical model and coordinate system

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Fig. 4

Contours of axial velocity

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Fig. 5

Polar coordinate of the cross section

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Fig. 6

Stream function contours with different collocation points: (a) N = 5 and (b) N = 20

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Fig. 7

Stream functions on the circular wall with different collocation points: (a) N = 1, (b) N = 5, and (c) N = 20

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Fig. 8

Contours of radial velocity

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Fig. 9

Contours of azimuthal velocity

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Fig. 10

Contours of pressure

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Fig. 11

Sketch map of eddy regions

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Fig. 12

The first and second vortex area radius

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Fig. 13

The physical model in numerical simulation

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Fig. 14

Contours of velocity components in numerical simulation: (a) axis velocity, (b) radial velocity, and (c) azimuthal velocity

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Fig. 15

Comparison between simulated axis velocity and analytic one for different r: (a) r = 0.4, (b) r = 0.6, and (c) r = 0.8

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Fig. 16

Comparison between simulated radial velocity and analytic one for different r: (a) r = 0.4, (b) r = 0.6, and (c) r = 0.8

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Fig. 17

Comparison between simulated azimuthal velocity and analytic one for different r: (a) r = 0.4, (b) r = 0.6, and (c) r = 0.8

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Fig. 18

Velocity deviation versus Reynolds number

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