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Research Papers: Fundamental Issues and Canonical Flows

# Stokes Flow Through a Periodically Grooved Tube

[+] Author and Article Information
Chiu-On Ng1

Department of Mechanical Engineering, University of Hong Kong, Pokfulam Road, Hong Kong SARcong@hku.hk

C. Y. Wang

Department of Mathematics, Michigan State University, East Lansing, MI 48824

1

Corresponding author.

J. Fluids Eng 132(10), 101204 (Oct 21, 2010) (8 pages) doi:10.1115/1.4002654 History: Received June 01, 2010; Revised September 21, 2010; Published October 21, 2010; Online October 21, 2010

## Abstract

This is an analytical study on Stokes flow through a tube of which the wall is patterned with periodic transverse grooves filled with an inviscid gas. In one period of the pattern, the fluid flows through an annular groove and an annular rib subject to no-shear and no-slip boundary conditions, respectively. The fluid may penetrate the groove to a certain depth, so there is an abrupt change in the cross section of flow through the two regions. The problem is solved by the method of domain decomposition and eigenfunction expansions, where the coefficients of the expansion series are determined by matching velocities, stress, and pressure on the domain interface. The effective slip length and pressure distributions are examined as functions of the geometrical parameters (tube radius, depth of fluid penetration into grooves, and no-shear area fraction of the wall). Particular attention is paid to the limiting case of flow through annular fins on a no-shear wall. Results are generated for the streamlines, resistance, and pressure drop due to the fins. It is found that the wall condition, whether no-shear or no-slip, will be immaterial when the fin interval is smaller than a certain threshold depending on the orifice ratio.

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## Figures

Figure 1

Flow through a circular tube with a transversely grooved wall, where the radial and axial components of velocity are (u,w). The radial and axial coordinates (r,z1) are for region I, and (r,z2) are for region II. The length dimensions are normalized with respect to half the period of the wall pattern. The liquid-gas interface (r=R1 in region I) is a no-shear surface, and the liquid-solid interface (R<r<R1, z1=±a in region I and r=R in region II) is a no-slip surface.

Figure 2

Effective slip length δ as a function of the no-shear area fraction a and the tube radius R, where b=0 (solid) and b=0.1 (dashes). The dotted lines on the left ends and the open circles on the right ends of the curves for b=0 are, respectively, the small-R and large-R asymptotic limits given by Eqs. 46,47.

Figure 3

Effective slip length δ as a function of the no-shear area fraction a and the depth of fluid penetration into grooves b, where R=0.5 (solid) and R=1 (dashes). The dotted line is for a finned tube with a=1, R=0.5, and a no-slip wall using the model of Wang (2).

Figure 4

(a) Axial distributions of the section-mean pressure p¯ as a function of the no-shear area fraction a and the depth of fluid penetration into grooves b, where R=0.5. Corresponding distributions of the centerline pressure pc are shown for (b) b=0 and (c) b=0.5. Note that 0<z<a is region I and a<z<1 is region II.

Figure 5

Streamlines and pressure contours for flow through a finned section in a tube with a no-shear wall, where R/R1=1/2 and (a) R1=1 and (b) R1=2. The stream function and pressure are, respectively, even and odd in z.

Figure 6

Resistance of a finned tube σ, as defined in Eq. 49, as a function of the separation distance of fins R1−1 and the orifice ratio R/R1, where the solid lines are for a no-shear wall and the dashed lines are for a no-slip wall (2)

Figure 7

A parameter group for pressure drop due to a single fin in a tube R3Δp/Q as a function of the orifice ratio R/R1, where the solid line is for a no-shear wall, the dashed line is for a no-slip wall (2), and the dotted line is by the approximation formula 52 of Sisavath (10). Inset: axial distributions of the centerline pressure for no-shear (solid) and no-slip (dashes) walls, where the orifice ratio R/R1=2/3.

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