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Research Papers: Fundamental Issues and Canonical Flows

# Steady Viscous Flow Between Two Porous Disks With Stretching Motion

[+] Author and Article Information
Tiegang Fang

Mechanical and Aerospace Engineering Department,
North Carolina State University,
911 Oval Drive–3246 EBIII,
Campus Box 7910,
Raleigh, NC 27695
e-mail: tfang2@ncsu.edu

Xin He

Center for Combustion Energy and
State Key Laboratory of Automotive
Safety and Energy,
Tsinghua University,
Beijing 100084, China
e-mail: hexin.tsinghua@gmail.com

1Corresponding author.

Contributed by the Fluids Engineering Division of ASME for publication in the JOURNAL OF FLUIDS ENGINEERING. Manuscript received June 28, 2014; final manuscript received June 4, 2015; published online August 10, 2015. Assoc. Editor: Samuel Paolucci.

J. Fluids Eng 138(1), 011201 (Aug 10, 2015) (7 pages) Paper No: FE-14-1337; doi: 10.1115/1.4030805 History: Received June 28, 2014

## Abstract

In this work, an exact solution to the steady-state Navier–Stokes (NS) equations is presented for viscous flows between two stretchable disks with mass transpiration effects. The governing momentum equations were converted into an ordinary differential equation by a similarity transformation technique. The similarity equation was solved numerically and the effects of Reynolds number and the mass transpiration parameter were investigated. At very low Reynolds numbers (i.e., R $→$ 0), a creeping flow was observed with a parabolic radial velocity profile and a cubic function profile for the vertical velocity. With the increase of the Reynolds number, the flow shows a boundary layer behavior near the wall with a constant velocity core flow in the centerline region between the two disks for mass suction or lower mass injection. The effects of the mass transpiration on the flow are quite different and interesting. With strong suction, the radial profiles also show boundary layer type characteristics with a core flow. But for large mass injection, the radial velocity approaches to a linear profile under higher Reynolds number. These results are a rare case of an exact solution to the NS equations and are useful as a benchmark problem for the validation of three-dimensional (3D) numerical computation code.

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## Figures

Fig. 1

The velocity profiles in the (a) vertical and (b) radial directions for different Reynolds numbers without mass transpiration

Fig. 2

The velocity profiles in the (a) vertical and (b) radial directions for different Reynolds numbers with mass suction λ = 2

Fig. 3

The velocity profiles in the (a) vertical and (b) radial directions for different Reynolds numbers with mass injection λ = -5

Fig. 4

The velocity profiles in the (a) vertical and (b) radial directions for different mass suctions strength with Reynolds number R = 0.1

Fig. 5

The velocity profiles in the (a) vertical and (b) radial directions for different mass suctions strength with Reynolds number R = 0.2

Fig. 6

The velocity profiles in the (a) vertical and (b) radial directions for different mass injections strength with Reynolds number R = 0.1

Fig. 7

The velocity profiles in the (a) vertical and (b) radial directions for different mass injections strength with Reynolds number R = 2

Fig. 8

The velocity profiles in the (a) vertical and (b) radial directions for low mass injection with Reynolds number R = 2

Fig. 9

(a) Wall shear stress, H"(1), and (b) radial pressure parameter, β, as a function of the Reynolds number for different mass transpiration parameters

Fig. 10

(a) Wall shear stress, H"(1), and (b) radial pressure parameter, β, as a function of the mass transpiration parameter for different Reynolds numbers

Fig. 11

Centerline radial velocity as a function of the mass transpiration parameter for different Reynolds numbers

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