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Technical Brief

Starting Flow in a Channel With Two Immiscible Fluids

[+] Author and Article Information
C. Y. Wang

Department of Mathematics;Department of Mechanical Engineering,
Michigan State University,
East Lansing, MI 48824
e-mail: cywang@mth.msu.edu

Contributed by the Fluids Engineering Division of ASME for publication in the JOURNAL OF FLUIDS ENGINEERING. Manuscript received March 3, 2017; final manuscript received July 12, 2017; published online September 7, 2017. Assoc. Editor: Praveen Ramaprabhu.

J. Fluids Eng 139(12), 124501 (Sep 07, 2017) (7 pages) Paper No: FE-17-1130; doi: 10.1115/1.4037495 History: Received March 03, 2017; Revised July 12, 2017

The starting flow due to a sudden pressure gradient in a channel containing two layers of different fluids is studied for the first time. The necessary eigenvalues and eigenfunctions, including orthogonality, for the composite regions are developed. Infinite series analytic solution is obtained for the starting transient. The properties of the instantaneous velocity profiles depend on the thickness ratio of the layers, the viscosity ratio, and the density ratio. Starting times are determined for the important cases of air over water and oil over water. The bulk flow is greatly increased when there exists a low-viscosity layer buffeting the channel wall. An important conclusion is that, in general, Navier's partial slip condition cannot be applied to unsteady starting flows.

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Figures

Grahic Jump Location
Fig. 1

Upper (lighter) layer and lower (heavier) layer driven by a horizontal pressure gradient

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Fig. 2

Velocity profiles for air over water: (a) b = 0.1, from bottom: t = 0.005, 0.01, 0.02, 0.03, 0.05, 0.1, ∞; (b) b = 0.5, frombottom: t = 0.002, 0.01, 0.02, 0.05, 0.5, ∞; and (c) b = 0.9, from bottom: t = 0.1, 0.5, 1, 2, 3, 5, 10, ∞

Grahic Jump Location
Fig. 3

Velocity profiles for oil over water: (a) b = 0.1, from bottom: t = 0.02, 0.05, 0.1, 0.2, 0.5, ∞; (b) b = 0.5, from bottom: t = 0.02, 0.05, 0.1, 0.2, 0.5, ∞; and (c) b = 0.9, from bottom: t = 0.1, 0.2, 0.5, 1, 2, 5, ∞

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Fig. 4

(a) Normalized slip length for air over water, b = 0.9. Dashed line is the steady-state S = 19.80. (b) Normalized slip length for air over water, b = 0.95. Dashed line is the steady-state S = 3.466.

Grahic Jump Location
Fig. 5

(a) Normalized slip length for oil over water, b = 0.1. Dashed line is the steady-state S = 2.215. (b) Normalized slip length for oil over water, b = 0.05. Dashed line is the steady-state S = 0.881.

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