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Long-wave instabilities in a non-Newtonian film on a nonuniformly heated inclined plane
Published in
2009
Volume: 131
   
Issue: 3
Pages: 0312021 - 03120217
Abstract
A thin liquid layer of a non-Newtonian film falling down an inclined plane that is subjected to nonuniform heating has been considered. The temperature of the inclined plane is assumed to be linearly distributed and the case when the temperature gradient is positive or negative is investigated. The film flow is influenced by gravity, mean surface tension, and thermocapillary forces acting along the free surface. The coupling of thermocapillary instability and surface-wave instabilities is studied for two-dimensional disturbances. A nonlinear evolution equation is derived by applying the long-wave theory, and the equation governs the evolution of a power-law film flowing down a nonuniformly heated inclined plane. The linear stability analysis shows that the film flow system is stable when the plate temperature decreases in the downstream direction while it is less stable for increasing temperature along the plate. Weakly nonlinear stability analysis using the method of multiple scales has been investigated and this leads to a secular equation of the Ginzburg-Landau type. The analysis shows that both supercritical stability and subcritical instability are possible for the film flow system. The results indicate the existence of finite-amplitude waves, and the threshold amplitude and nonlinear speed of these waves are influenced by thermocapillarity. The nonlinear evolution equation for the film thickness is solved numerically in a periodic domain in the supercritical stable region, and the results show that the shape of the wave is influenced by the choice of wave number, non-Newtonian rheology, and nonuniform heating. Copyright © 2009 by ASME.
About the journal
JournalJournal of Fluids Engineering, Transactions of the ASME
ISSN00982202
Open AccessNo
Concepts (42)
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    FILM FLOWS
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    FINITE AMPLITUDE WAVES
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    Free surfaces
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    GINZBURG-LANDAU
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    Inclined planes
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    LONGWAVE INSTABILITY
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    LONGWAVE THEORY
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    METHOD OF MULTIPLE SCALE
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    Non-newtonian
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    NON-NEWTONIAN RHEOLOGY
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    NONLINEAR EVOLUTION EQUATION
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    NONUNIFORM HEATING
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    PERIODIC DOMAINS
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    PLATE TEMPERATURE
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    Power-law
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    SECULAR EQUATIONS
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    STABLE REGION
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    SUBCRITICAL INSTABILITY
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    SUPER-CRITICAL
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    SUPERCRITICAL STABILITY
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    Temperature gradient
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    THERMO-CAPILLARY FORCES
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    THERMOCAPILLARITY
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    THERMOCAPILLARY INSTABILITY
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    THIN LIQUID LAYER
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    THRESHOLD AMPLITUDE
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    WAVE NUMBERS
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    WEAKLY NON-LINEAR STABILITIES
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    Biology
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    Capillary flow
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    Differential equations
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    Heating
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    Linear stability analysis
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    Liquids
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    Machinery
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    Nonlinear analysis
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    Nonlinear equations
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    Surface tension
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    System stability
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    Two dimensional
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    Waves
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    Non newtonian flow