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Research Papers: Flows in Complex Systems

The Effect of Slip on the Discharge From Partially Filled Circular and Fully Filled Lens and Figure 8 Shaped Pipes

[+] Author and Article Information
Samuel Irvine

Institute of Fundamental Sciences,
Massey University,
Palmerston North 4474, New Zealand
e-mail: sammanam1993@gmail.com

Luke Fullard

Institute of Fundamental Sciences,
Massey University,
Palmerston North 4474, New Zealand
e-mail: L.Fullard@Massey.ac.nz

1Corresponding author.

Contributed by the Fluids Engineering Division of ASME for publication in the JOURNAL OF FLUIDS ENGINEERING. Manuscript received November 16, 2015; final manuscript received April 2, 2016; published online July 15, 2016. Assoc. Editor: Praveen Ramaprabhu.

J. Fluids Eng 138(11), 111104 (Jul 15, 2016) (6 pages) Paper No: FE-15-1841; doi: 10.1115/1.4033373 History: Received November 16, 2015; Revised April 02, 2016

In this work, we examine the effect of wall slip for a gravity-driven flow of a Newtonian fluid in a partially filled circular pipe. An analytical solution is available for the no-slip case, while a numerical method is used for the case of flow with wall slip. We note that the partially filled circular pipe flow contains a free surface. The solution to the Navier–Stokes equations in such a case is a symmetry of a pipe flow (with no free surface) with the free surface as the symmetry plane. Therefore, we note that the analytical solution for the partially filled case is also the exact solution for fully filled lens and figure 8 shaped pipes, depending on the fill level. We find that the presence of wall slip increases the optimal fill height for maximum volumetric flow rate, brings the “velocity dip” closer to the free surface, and increases the overall flow rate for any fill. The applications of the work are twofold; the analytical solution may be used to verify numerical schemes for flows with a free surface in partially filled circular pipes, or for pipe flows in lens and figure 8 shaped pipes. Second, the work suggests that flows in pipes, particularly shallow filled pipes, can be greatly enhanced in the presence of wall slip, and optimal fill levels must account for the slip phenomenon when present.

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Figures

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Fig. 1

The solution for flow in a partially filled circular pipe is a symmetry of the flow in a fully filled lens or figure 8 shaped pipe

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Fig. 2

Comparison of the analytical and numerical solution for the centerline velocity in pipes for four values of α

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Fig. 3

Flow contours for analytical no-slip and numerical small slip solutions for various values of α

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Fig. 4

Flow discharge as a function of α (fill level) for no-slip, small, and medium slip

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Fig. 5

Fill level for maximum discharge as a function of the slip parameter β

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Fig. 6

Discharge for small slip normalized by the no-slip case

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Fig. 7

Position below the free surface of maximum velocity, also known as the velocity dip phenomenon

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