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Research Papers: Multiphase Flows

Stability Analysis of a Droplet Pinned in Channel Under Gravity

[+] Author and Article Information
Haider Hekiri

School of Aerospace
and Mechanical Engineering,
The University of Oklahoma,
Norman, OK 73019
e-mail: haider@ou.edu

Takumi Hawa

School of Aerospace
and Mechanical Engineering,
The University of Oklahoma,
Norman, OK 73019
e-mail: hawa@ou.edu

Contributed by the Fluids Engineering Division of ASME for publication in the JOURNAL OF FLUIDS ENGINEERING. Manuscript received December 9, 2013; final manuscript received May 2, 2014; published online September 10, 2014. Assoc. Editor: John Abraham.

J. Fluids Eng 137(1), 011301 (Sep 10, 2014) (10 pages) Paper No: FE-13-1717; doi: 10.1115/1.4027600 History: Received December 09, 2013; Revised May 02, 2014

The stability of a two-dimensional, incompressible droplet, with two cylindrical-caps that is held in a channel under gravity, is investigated through the development of an analytical model based on the Young–Laplace relationship. The droplet state is measured by the location of its center of mass, where the center of mass is derived analytically by assuming a circular shape for the droplet cap. The derived analytical expressions are validated through the use of computational fluid dynamics (CFD). When a droplet is suspended under no gravity conditions, there is a critical droplet volume Vcr where asymmetric droplet states appear in addition to the basic symmetric states when the drop volume V > Vcr. When V < Vcr, the symmetric droplet states are stable, and when V > Vcr, the symmetric states are unstable and the asymmetric states are stable. With gravity, the pitchfork bifurcation diagram of the droplet system changes into two separate branches of equilibrium states: The primary branch describes a gradual and stable change of the droplet from a symmetric to asymmetric state as the droplet volume is increased. The secondary branch appears at a modified critical volume Vmcr and describes two additional asymmetric states when V > Vmcr. The large-amplitude states along the secondary branch are stable whereas the small-amplitude states are unstable. There exists a maximum volume on each of the primary and secondary branch where the droplet no longer sustains its weight and where the maximum volume on the primary branch is smaller than the maximum volume on the secondary branch. There is a critical value for the strength of the gravity force, relative to the capillary force, that provides the condition at which a droplet state exists only at the primary branch; the secondary branch is unstable. Analytical solutions show good agreement with CFD results as long as the circular shape assumption of the droplet cap is approximately valid.

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Figures

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Fig. 1

Various droplet states: (a) Droplet caps volume is less than the critical volume V < Vcr. (b) Droplet cap volume is equal to the critical volume, RU = RL and V = Vcr. (c) Droplet cap volume is greater than the critical volume V > Vcr.

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Fig. 2

Geometry and dimension of the simulation domain

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Fig. 3

Cells structure with central cell in the middle

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Fig. 4

Pressure-based solution diagram

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Fig. 5

The absolute value of the CM at the equilibrium state with respect to the number of nodes for V˜ = 0.5 and g¯ = 0

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Fig. 6

Mesh refinements over the corners

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Fig. 7

Comparison of the dimensionless CM obtained from the analytical expression with the simulation results for β = 0

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Fig. 8

Stability chart for various β values

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Fig. 9

Comparison of CFD results (dots) with analysis (solid and dashed lines) at β = 0.106

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Fig. 10

Temporal snapshots of droplet states for β = 0.106 at V˜ = 1.7. (a) Initial state at t¯ = 0 where X¯2_cm = 0.92, (b) intermediate state at t¯ = 0.5, and (c) equilibrium state at t¯ = 3.

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Fig. 11

Temporal snapshots of droplet states for β = 0.106 at V˜ = 3. (a) Initial state at t¯ = 0, where X¯2_cm = 1.78, (b) intermediate state at t¯ = 1.1, and (c) equilibrium state at t¯ = 5.

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Fig. 12

Temporal snapshots of droplet states for β=0.106 at V˜ = 3. (a) Initial state at t¯ = 0, where X¯2_cm = 1.02, (b) intermediate state at t¯ = 0.9, and (c) equilibrium state at t¯ = 3.

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Fig. 13

Temporal snapshots of droplet states for β=0.106 at V˜ = 3 with V˜p_max = 4.29. (a) Initial state at t¯ = 0, (b) intermediate state at t¯ = 0.9, and (c) equilibrium state at t¯ = 2.5.

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Fig. 14

Temporal snapshots of droplet states for β = 0.106 at V˜ = 7 with V˜mcr = 2.25 and V˜s_max = 6.07. (a) Initial state at t¯ = 0, where X¯2_cm = 1.02, (b) intermediate state at t¯ = 2, and (c) detachment at t¯ = 3.

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Fig. 15

Deformation of the droplet state at V˜ = 5.5 and β = 0.106

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Fig. 16

Schematic views of (a) a primary droplet state with an initial small disturbance, (b) an upper side of the secondary branch with an initial small disturbance, and (c) a lower side of the secondary branch with an initial small disturbance

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Fig. 17

Droplet domain labels

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