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Research Papers: Fundamental Issues and Canonical Flows

Starting Poiseuille Flow in a Circular Tube With Two Immiscible Fluids

[+] Author and Article Information
Chiu-On Ng

Department of Mechanical Engineering,
The University of Hong Kong,
Pokfulam Road,
Hong Kong, China
e-mail: cong@hku.hk

C. Y. Wang

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

1Corresponding author.

Contributed by the Fluids Engineering Division of ASME for publication in the JOURNAL OF FLUIDS ENGINEERING. Manuscript received February 26, 2018; final manuscript received July 18, 2018; published online August 20, 2018. Assoc. Editor: M'hamed Boutaous.

J. Fluids Eng 141(3), 031201 (Aug 20, 2018) (7 pages) Paper No: FE-18-1129; doi: 10.1115/1.4040972 History: Received February 26, 2018; Revised July 18, 2018

Starting flow due to a suddenly applied pressure gradient in a circular tube containing two immiscible fluids is solved using eigenfunction expansions. The orthogonality of the eigenfunctions is developed for the first time for circular composite regions. The problem, which is pertinent to flow lubricated by a less viscous near-wall fluid, depends on the ratio of the radius of the core region to that of the tube, and the ratios of dynamic and kinematic viscosities of the two fluids. In general, a higher lubricating effect will lead to a longer time for the starting transient to die out. The time development of velocity profile and slip length are examined for the starting flows of whole blood enveloped by plasma and water enveloped by air in a circular duct. Owing to a sharp contrast in viscosity, the starting transient duration for water/air flow can be ten times longer than that of blood/plasma flow. Also, the slip length exhibits a singularity in the course of the start-up. For blood with a thin plasma skimming layer, the singularity occurs very early, and hence for the most part of the start-up, the slip length is nearly a constant. For water lubricated by air of finite thickness, the singularity may occur at a time that is comparable to the transient duration of the start-up, and hence, an unsteady slip length has to be considered in this case.

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Figures

Grahic Jump Location
Fig. 1

Starting flow in a circular tube of radius R enclosing a core region of radius b. Fluids in the core region (0 ≤ r < b) and the near-wall region (b < r ≤ R) are labeled as fluids 1 and 2, respectively.

Grahic Jump Location
Fig. 2

The leading eigenvalue λ̂1 as a function of the normalized radius b̂ and the dynamic viscosity ratio γ2, where the density ratio (a) σ2 = 1 and (b) σ2 = 100

Grahic Jump Location
Fig. 3

Velocity profiles for starting flow of blood/plasma, ŵ(r̂,t̂). Dashed line denotes the steady limit.

Grahic Jump Location
Fig. 4

Velocity profiles for starting flow of water/air, ŵ(r̂,t̂). Dashed line denotes the steady limit.

Grahic Jump Location
Fig. 5

For starting flow of blood/plasma, the slip length Ŝ (solid line, left axis) and the flow rate Q̂1 (dash-dot-dot line, right axis) as functions of time t̂. Horizontal dashed line is the steady limit of the slip length, and vertical dashed line is the point where the slip length is singular: (a) b̂ = 0.9 and (b) b̂ = 0.8.

Grahic Jump Location
Fig. 6

For starting flow of water/air, the slip length Ŝ (solid line, left axis), and the flow rate Q̂1 (dash-dot-dot line, right axis) as functions of time t̂. Horizontal dashed line is the steady limit of the slip length, and vertical dashed line is the point where the slip length is singular: (a) b̂ = 0.9 and (b) b̂ = 0.8.

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