Weakly nonlinear stability analysis of thin power-law liquid film flowing down an inclined plane including the phase change effects at the interface has been investigated. A normal mode approach and the method of multiple scales are employed to carry out the linear stability solution and the nonlinear stability solution for the film flow system. The results show that both the supercritical stability and subcritical instability are possible for condensate, evaporating and isothermal power-law liquid film down an inclined plane. The stability characteristics of the power-law liquid film show that isothermal and evaporating films are unstable for any value of power-law index ‘n’ while there exists a critical value of power-law index ‘n’ for the case of condensate film above which condensate film flow system is always stable. Thus, the results of the present analysis show that the mass transfer effects play a significant role in modifying the stability characteristics of the non-Newtonian power-law fluid flow system. The condensate (evaporating) power-law fluid film is more stable (unstable) than the isothermal power-law fluid film flowing down an inclined plane.

Issue Section:
Technical Papers
Keywords:
film flow, liquid films, mass transfer, non-Newtonian flow, flow instability, interface phenomena, evaporation, film condensation, shear flow
Topics:
Condensed matter, Evaporation, Stability, Liquid films, Film flow
1.
Yih, C. S., 1954, “Stability of Parallel Laminar Flow With a Free Surface,” Proc. 2nd U.S. Congr. Appl. Mech., pp. 623–628.
2.
Landau
,
L. D.
,
1944
, “
On the Problem of Turbulence
,”
C. R. (Dokl.) Acad. Sci. URSS
,
44
, pp.
311
–
314
.
3.
Benjamin
,
T. B.
,
1957
, “
Wave Formation in Laminar Flow Down an Inclined Plane
,”
J. Fluid Mech.
,
2
, pp.
554
–
574
.
4.
Yih
,
C. S.
,
1963
, “
Stability of Liquid Flow Down an Inclined Plane
,”
Phys. Fluids
,
6
, pp.
321
–
334
.
5.
Nusselt
,
W.
,
1916
, “
Die Oberflachen Kondensation des Wasserdamfes
,”
Z. VDI
,
50
, pp.
541
–
546
.
6.
Bankoff
,
S. G.
,
1971
, “
Wave Formation in Laminar Flow Down an Inclined Plane
,”
J. Fluid Mech.
,
14
, pp.
377
–
385
.
7.
Marshall
,
E.
, and
Lee
,
C. Y.
,
1973
, “
Stability of Condensate Flow Down a Vertical Wall
,”
Int. J. Heat Mass Transfer
,
16
, pp.
41
–
48
.
8.
Lin
,
S. P.
,
1975
, “
Stability of a Liquid Down a Heat Incline Plane
,”
Lett. Heat Mass Transfer
,
2
, pp.
361
–
370
.
9.
U¨nsal
,
M.
, and
Thomas
,
W. C.
,
1978
, “
Linear Stability Analysis of Film Condensation
,”
ASME J. Heat Transfer
,
100
, pp.
629
–
634
.
10.
Burelbach
,
J. P.
,
Bankoff
,
S. G.
, and
Davis
,
S. H.
,
1988
, “
Nonlinear Stability of Evaporating/Condensing Liquid Films
,”
J. Fluid Mech.
,
195
, pp.
463
–
494
.
11.
Joo
,
S. W.
,
Davis
,
S. H.
, and
Bankoff
,
S. G.
,
1991
, “
Long-Wave Instabilities of Heated Falling Films: Two-Dimensional Theory of Uniform Layers
,”
J. Fluid Mech.
,
230
, pp.
117
–
146
.
12.
U¨nsal
,
M.
, and
Thomas
,
W. C.
,
1980
, “
Nonlinear Stability of Film Condensation
,”
ASME J. Heat Transfer
,
102
, pp.
483
–
488
.
13.
Hwang
,
C. C.
, and
Weng
,
C. I.
,
1987
, “
Finite-Amplitude Stability Analysis of Liquid Films Down a Vertical Wall With and Without Interfacial Phase Change
,”
Int. J. Multiphase Flow
,
13
, pp.
803
–
814
.
14.
Gupta
,
A. S.
,
1967
, “
Stability of a Visco-Elastic Liquid Film Flowing Down an Inclined Plane
,”
J. Fluid Mech.
,
28
, pp.
17
–
28
.
15.
Dandapat
,
B. S.
, and
Gupta
,
A. S.
,
1997
, “
Solitary Waves on the Surface of a Viscoelastic Fluid Running Down an Inclined Plane
,”
Rheol. Acta
,
36
, pp.
135
–
143
.
16.
Shaqfeh
,
E. S. G.
,
Larson
,
R. G.
, and
Fredrickson
,
G.H.
,
1989
, “
The Stability of Gravity Driven Viscoelastic Film Flow at Low to Moderate Reynolds Number
,”
J. Non-Newtonian Fluid Mech.
,
31
, pp.
87
–
113
.
17.
Joo
,
S. W.
,
1994
, “
The Stability and Nonlinear Flow Developments of a Viscoelastic Draining Film With Shear Thinning
,”
J. Non-Newtonian Fluid Mech.
,
51
, pp.
125
–
140
.
18.
Andersson
,
H. I.
, and
Irgens
,
F.
,
1988
, “
Gravity-Driven Laminar Film Flow of Power-Law Fluids Along Vertical Walls
,”
J. Non-Newtonian Fluid Mech.
,
27
, pp.
153
–
172
.
19.
Hwang
,
C. C.
,
Chen
,
J. L.
,
Wang
,
J. S.
, and
Lin
,
J. S.
,
1994
, “
Linear Stability of Power-Law Liquid Film Flows Down an Inclined Plane
,”
J. Phys. D
27
, pp.
2297
–
2301
.
20.
Lee, J. J., 1996, “Nonlinear Dynamics of a Rapidly Flowing Viscous Fluid Layer Down an Incline,” Ph.D. Thesis, Massachusetts Institute of Technology, Cambridge, MA.
21.
Lin
,
J. S.
, and
Hwang
,
C. C.
,
2000
, “
Finite Amplitude Long-Wave Instability of Power-Law Liquid Films
,”
Int. J. Non-Linear Mech.
,
35
, pp.
769
–
777
.
22.
Hung
,
C. I.
,
Tsai
,
J. S.
, and
Chen
,
C. K.
,
1996
, “
Nonlinear Stability of the Thin Micropolar Liquid Film Flowing Down on a Vertical Plane
,”
ASME Trans. J. Fluids Eng.
,
118
, pp.
498
–
505
.
23.
Nayfeh, A. H., and Mook, D. T., 1979, Nonlinear Oscillations, John Wiley and Sons, New York.
Copyright © 2003
 by ASME
You do not currently have access to this content.