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Nonlinear dynamics of long-wave Marangoni convection in a binary mixture with the Soret effect
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10.1063/1.4807599
/content/aip/journal/pof2/25/5/10.1063/1.4807599
http://aip.metastore.ingenta.com/content/aip/journal/pof2/25/5/10.1063/1.4807599
View: Figures

Figures

Image of FIG. 1.
FIG. 1.

Spatial profiles and phase plane portraits for (τ, ξ) and Γ(τ, ξ) (from left to right, respectively) of the regimes observed: a standing wave for = 0.5, Σ = 0.5, = 1.05 (first row from the top), traveling waves for = 1.5, Σ = 1.5, = 1.05 (second row), = 2.5, Σ = 2.5, = 1.05 (third row), and = 3.5, Σ = 3.5, = 0.999 (fourth row). In the panels showing spatial profiles, solid, dashed and dotted curves correspond to subsequent times from the earliest to the latest, respectively.

Image of FIG. 2.
FIG. 2.

Minima and maxima of the film thickness (τ, ξ) versus the fundamental wavenumber of the computational domain for = 9. “+” denotes a one-hump traveling wave, “×”—a pair of one-hump counterpropagating traveling waves, “*”—a one-hump standing wave, “ ”—two-hump counterpropagating traveling waves, “■”—three-hump counterpropagating traveling waves, “○”—four-hump counterpropagating traveling waves, “•”—quadruplicated counterpropagating traveling waves, “△”—a quadruplicated standing wave, “▲”—an eight-hump chaotic traveling wave, “▽”—a triplicated traveling wave, “▼”—triplicated counterpropagating traveling waves, “◊”—a triplicated standing wave, and “♦”—a six-hump chaotic traveling wave. The data points where the same type of a limit regime is observed are connected by a line in order to guide a reader's eye.

Image of FIG. 3.
FIG. 3.

Spatial profiles for (τ, ξ) and Γ(τ, ξ) and phase plane portraits (from top to bottom, respectively) of the regimes observed: a traveling wave for = 0.5 (left column), a pair of counterpropagating traveling waves for = 0.36 (middle column), and a standing wave for = 0.24 (right column). In the panels showing spatial profiles, solid, dashed and dotted curves correspond to subsequent times from the earliest to the latest, respectively.

Image of FIG. 4.
FIG. 4.

Subsequent Poincaré samples of (τ, ξ) for = 0.36, with the sampling period = 44.25. The rightmost curve corresponds to the earliest time.

Image of FIG. 5.
FIG. 5.

Spatial profiles for (τ, ξ) and Γ(τ, ξ), phase plane portraits, and spatial power spectra (from top to bottom, respectively) of the regimes observed: two-hump counterpropagating traveling waves for = 0.16 (left column), three-hump counterpropagating traveling waves for = 0.1 (middle column), and four-hump counterpropagating traveling waves for = 0.09 (right column). In the panels showing spatial profiles and those for the power spectra, solid, dashed and dotted curves correspond to subsequent times from the earliest to the latest, respectively.

Image of FIG. 6.
FIG. 6.

Temporal Fourier spectra of |(τ; 2)| (solid line) and |(τ; 4)| (dashed line) in the case of four-hump counterpropagating traveling waves observed at = 0.09. The spectral peaks occur at ν = ν + ν for integer and .

Image of FIG. 7.
FIG. 7.

Spatial profiles for (τ, ξ) and Γ(τ, ξ), phase plane portraits, and spatial power spectra (from top to bottom, respectively) of the regimes observed: a pair of counterpropagating quadruplicated traveling waves for = 0.08 (left column), a quadruplicated standing wave for = 0.06 (middle column), and an eight-hump chaotic propagating wave for = 0.04 (right column). In the panels showing spatial profiles and those for the power spectra, solid, dashed and dotted curves correspond to subsequent times from the earliest to the latest, respectively.

Image of FIG. 8.
FIG. 8.

The time evolution of the norm of the difference between two solutions Γ and Γ with close initial conditions in the case of four-hump counterpropagating traveling waves observed for = 0.09 (bottom curve) and chaotic propagating waves observed for = 0.06 (top curve).

Image of FIG. 9.
FIG. 9.

Minima and maxima of the film thickness (τ, ξ) versus the fundamental wavenumber of the computational domain for = 13. “+” denotes a traveling wave, “×”—a pair of counterpropagating traveling waves. The data points where the same type of a limit regime is observed are connected.

Image of FIG. 10.
FIG. 10.

Spatial profiles for (τ, ξ) and Γ(τ, ξ), and highest spatial Fourier harmonic evolution of the counterpropagating traveling waves at = 0.777. The solutions obtained based on the method of lines with = 300, Δτ ⩽ 10 (first row) and = 600, Δτ ⩽ 10 (third row); based on Newton-Kantorovich method with = 300, Δτ = 10 (second row) and = 600, Δτ = 10 (fourth row).

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/content/aip/journal/pof2/25/5/10.1063/1.4807599
2013-05-29
2014-04-20
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752b84549af89a08dbdd7fdb8b9568b5 journal.articlezxybnytfddd
Scitation: Nonlinear dynamics of long-wave Marangoni convection in a binary mixture with the Soret effect
http://aip.metastore.ingenta.com/content/aip/journal/pof2/25/5/10.1063/1.4807599
10.1063/1.4807599
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