# zbMATH — the first resource for mathematics

##### Examples
 Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

##### Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
##### Fields
 any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
Stability of liquid film falling down a vertical non-uniformly heated wall. (English) Zbl 1149.76024
Summary: We study the stability of a viscous film flowing down a vertical non-uniformly heated wall under gravity. The wall temperature is assumed linearly distributed along the wall, and the free surface is taken to be adiabatic. A long wave perturbation method is used to derive the nonlinear evolution equation for the falling film. Using the method of multiple scale, the nonlinear stability analysis is performed for travelling wave solution of the evolution equation. The complex Ginzburg-Landau equation is determined to discuss the bifurcation analysis of the evolution equation. The results indicate that the supercritical unstable region increases and the subcritical stable region decreases with the increase in Peclet number. It has been also shown that the spatial uniform solution corresponding to the sideband disturbance may be stable in the unstable region.

##### MSC:
 76E17 Interfacial stability and instability (fluid dynamics) 76E30 Nonlinear effects (fluid mechanics) 76A20 Thin fluid films (fluid mechanics)
Full Text:
##### References:
 [1] Kapitza, P. L.; Kapitza, S. P.: Wave flow of thin layers of a viscous fluid. Zh. tero. Fiz. 19, 105-120 (1949) · Zbl 0029.17603 [2] Yih, C. S.: Stability of liquid flow down an inclined plane. Phys. fluids 6, 321-334 (1963) · Zbl 0116.19102 [3] Benney, D. J.: Long waves on liquid films. J. math. Phys. 45, 150-155 (1966) · Zbl 0148.23003 [4] Lin, S. P.: Finite amplitude side-band stability of viscous film. J. fluid mech. 63, 417-429 (1974) · Zbl 0283.76035 [5] Gjevik, B.: Occurrence of finite-amplitude surface waves on falling liquid films. Phys. fluids 13, 1918-1925 (1970) · Zbl 0217.56102 [6] Nakaya, C.: Long waves on a thin fluid layer flowing down an inclined plane. Phys. fluids 15, 1407-1412 (1975) · Zbl 0325.76038 [7] Chang, H. C.: Onset of nonlinear waves of falling films. Phys. fluids A 1, 1314-1327 (1989) · Zbl 0673.76038 [8] Pumir, A.; Manneville, A.; Pomeau, Y. J.: On solitary waves running down an inclined plane. J. fluid mech. 135, 27-50 (1983) · Zbl 0525.76016 [9] Oron, A.; Gottlieb, O.: Subcritical and supercritical bifurcations of the first and second-order benney equations. J. eng. Math. 50, 121-140 (2004) · Zbl 1073.76038 [10] Lin, S. P.: Stability of a liquid film down a heated inclined plane. Lett. heat mass transfer 2, 361-370 (1975) [11] Goussis, D. A.; Kelly, R. E.: On the thermocapillary instabilities in a liquid layer heated from below. Int J. Heat mass transfer 33, 2237-2245 (1990) [12] Goussis, D. A.; Kelly, R. E.: Surface wave and thermocapillary instabilities in a liquid film flow. J. fluid mech. 223, 24-45 (1991) · Zbl 0717.76038 [13] Scriven, L.; Sternling, C.: On cellular convection driven by surface tension gradients: effects of mean surface tension and surface viscosity. J. fluid mech. 19, 321-340 (1964) · Zbl 0123.42203 [14] Pearson, J.: On convective cells induced by surface tension. J. fluid mech. 4, 489-500 (1958) · Zbl 0082.18804 [15] Joo, S. W.; Davis, S. H.; Bankoff, S. G.: Long-wave instabilities of heated falling films: two dimensional theories of uniform layers. J. fluid mech. 230, 117-146 (1991) · Zbl 0728.76047 [16] Kabov, O. A.: Formation of regular structures in a falling liquid film upon local heating. Thermophys. aeromech. 5, 547-551 (1998) [17] Kalliadasis, S.; Kiyashko, A.; Demekhin, E. A.: Marangoni instability of a thin liquid filmheated from below by a local heat source. J. fluid mech. 475, 377-408 (2003) · Zbl 1081.76028 [18] Trevelyan, P.; Kalliadasis, S.: Wave dynamics on a thin liquid film falling down a heated wall. J. eng. Math. 50, 177-208 (2004) · Zbl 1074.76009 [19] Scheid, B.; Oron, A.; Colinet, P.; Thiele, U.; Legros, J. C.: Onlinear evolution of nonuniformly heated falling liquid films. Phys. fluids 14, 4130-4151 (2002) · Zbl 1185.76322 [20] Miladinova, S.; Slavtchev, S.; Lebon, G.; Legros, J. C.: Long-wave instabilities of non-uniformly heated falling films. J. fluid mech. 453, 153 (2002) · Zbl 1053.76024 [21] Armbruster, D.; Guckenheimer, J.; Holmes, P.: Heteroclinic cycles and modulated traveling waves in system with $O(2)$ symmetry. Physica D 29, 257-282 (1988) · Zbl 0634.34027 [22] Mizushima, J.; Fuzimura, K.: Higher harmonic resonance of two-dimensional disturbances in Rayleigh--Bénard convection. J. fluid mech. 234, 651-667 (1992) · Zbl 0744.76053