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Research Papers: Fundamental Issues and Canonical Flows

Long-Wave Instabilities in a Non-Newtonian Film on a Nonuniformly Heated Inclined Plane

[+] Author and Article Information
I. Mohammed Rizwan Sadiq

Department of Mathematics, Indian Institute of Technology Madras, Chennai 600 036, Indiarizwan@smail.iitm.ac.in

R. Usha1

Department of Mathematics, Indian Institute of Technology Madras, Chennai 600 036, Indiaushar@iitm.ac.in

1

Corresponding author.

J. Fluids Eng 131(3), 031202 (Feb 05, 2009) (17 pages) doi:10.1115/1.3059702 History: Received May 09, 2007; Revised April 23, 2008; Published February 05, 2009

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 American Society of Mechanical Engineers
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Figures

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Figure 1

Schematic representation of a thin power-law film flowing down a nonuniformly heated inclined plane

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Figure 2

Parameter plot as a function of power-law exponent n: (a) The temperature of the plate increases linearly in the downstream direction; (b) the temperature of the plate decreases linearly in the downstream direction

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Figure 3

Critical Reynolds number as a function of Mn for different values of n: (a) β=15 deg(Mn1≃0.13,Mn2≃0.19), (b) β=45 deg, (c) β=60 deg, and (d) β=90 deg

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Figure 4

Neutral stability curves for different angles of inclination: (—) temperature of the plane increasing linearly downstream and (- - -) temperature of the plane decreasing linearly downstream

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Figure 6

Threshold amplitude and nonlinear wave speed in the supercritical stable region for β=60 deg when the plane temperature decreases linearly downstream: (a) plane temperature increases linearly downstream and (b) plane temperature decreases linearly downstream

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Figure 5

Neutral stability curves for different angles of inclination: (a) temperature of the plane increasing linearly downstream and (b) temperature of the plane decreasing linearly downstream; s>0,J2<0 supercritical explosive state, s>0,J2>0 supercritical stable state, s<0,J2<0 subcritical unstable state, and s<0,J2>0 subcritical stable state

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Figure 14

Maximum and minimum amplitudes of waves of a dilatant fluid in the supercritical stable region in an inclined wall (β=60 deg) when temperature decreases linearly downstream for n=1.01, Ren=7.671, and Mn=−0.0157; kc=0.4724, km=0.33403, and ks=0.2362

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Figure 15

Maximum and minimum amplitudes of waves of a dilatant fluid in the supercritical stable region in an inclined wall (β=60 deg) when temperature increases linearly downstream for n=1.01, Ren=7.671, and Mn=0.0157; kc=0.5698, km=0.40291, and ks=0.2849

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Figure 16

Maximum and minimum amplitudes of waves of a dilatant fluid in the supercritical stable region in an inclined wall (β=60 deg) when temperature increases linearly downstream for n=1.1, Ren=3.0238, and Mn=0.00621; kc=0.268, km=0.1895, and ks=0.134

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Figure 17

Evolution of a free surface of (a) pseudoplastic, (b) Newtonian, and (c) dilatant fluids in the supercritical stable region in an inclined wall (β=60 deg) when temperature increases linearly downstream for k=0.3: (a) n=0.99, Ren=9.224, and Mn=0.0189; (b) n=1, Ren=8.42, and Mn=0.0173; (c) n=1.01, Ren=7.671, and Mn=0.0157

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Figure 18

Evolution of a free surface of (a) pseudoplastic, (b) Newtonian, and (c) dilatant fluids in the supercritical stable region in an inclined wall (β=60 deg) when temperature decreases linearly downstream for k=0.3: (a) n=0.99, Ren=9.224, and Mn=−0.0189; (b) n=1, Ren=8.42, and Mn=−0.0173; (c) n=1.01, Ren=7.671, and Mn=−0.0157

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Figure 19

Free surface configurations for a linear decrease in plate temperature for Ren=3.53, when β=45 deg; (—) Mn=−0.01 and (- - - -) Mn=−0.02

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Figure 20

Free surface configurations for a linear increase in plate temperature for Ren=2.8 and Mn=0.02 when β=45 deg

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Figure 7

Evolution of free surface of pseudoplastic and dilatant fluids in the supercritical stable region in an inclined wall (β=60 deg) when temperature decreases linearly downstream for k=0.4: (a) n=0.99, Ren=9.224, and Mn=−0.0189; (b) n=1.01, Ren=7.671, and Mn=−0.0157. (a) kc=0.5722, km=kc/2=0.4046, and ks=kc/2=0.2861; (b) kc=0.4724, km=kc/2=0.33403, and ks=kc/2=0.2362.

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Figure 8

Evolution of free surface of pseudoplastic and dilatant fluids in the supercritical stable region in an inclined wall (β=60 deg) when temperature increases linearly downstream for k=0.4: (a) n=0.99, Ren=9.224, and Mn=0.0189; (b) n=1.01, Ren=7.671, and Mn=0.0157. (a) kc=0.6652, km=kc/2=0.4704, and ks=kc/2=0.3326; (b) kc=0.5698, km=kc/2=0.40291, and ks=kc/2=0.2849.

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Figure 9

Evolution of free surface ((a)–(f)) of a pseudoplastic fluid in the supercritical stable region in an inclined wall (β=60 deg) when temperature increases linearly downstream for k=0.34, n=0.99, Ren=9.224, and Mn=0.0189; kc=0.6652, km=kc/2=0.4704, and ks=kc/2=0.3326

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Figure 10

Evolution of free surface ((a)–(f)) of a pseudoplastic fluid in the supercritical stable region in an inclined wall (β=60 deg) when temperature decreases linearly downstream for k=0.3, n=0.99, Ren=9.224, and Mn=−0.0189; kc=0.5722, km=kc/2=0.4046, and ks=kc/2=0.2861

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Figure 11

Maximum and minimum amplitudes of waves of a pseudoplastic fluid in the supercritical stable region in an inclined wall (β=60 deg) when temperature increases linearly downstream for n=0.99, Ren=9.224, and Mn=0.0189; (1) k=0.34, (2) k=0.4, and (3) k=0.5; kc=0.6652, km=kc/2=0.4704, and ks=kc/2=0.3326

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Figure 12

Evolution of free surface ((a)–(f)) of a dilatant fluid in the supercritical stable region in an inclined wall (β=60 deg) when temperature decreases linearly downstream for k=0.3, n=1.01, Ren=7.671, and Mn=−0.0157; kc=0.4724, km=kc/2=0.33403, and ks=kc/2=0.2362

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Figure 13

Evolution of free surface ((a)–(f)) of a dilatant fluid in the supercritical stable region in an inclined wall (β=60 deg) when temperature increases linearly downstream for k=0.3, n=1.01, Ren=7.671, and Mn=0.0157; kc=0.5698, km=kc/2=0.40291, and ks=kc/2=0.2849

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