A long-wave evolution equation is derived using an asymptotic analysis, and the linear stability of a viscoelastic film flowing along the direction of parallel grooves over a uniformly heated topography is studied. A numerical approach adopting spectral collocation technique is used to demarcate the stable and unstable flow regimes. The combined influence of thermocapillarity and viscoelasticity on the films stability is analyzed. By accounting the bottom topography comprising longitudinal gutters, the possibilities of regulating the film dynamics under iso- and nonisothermal conditions and/or optimizing design structure of an apparatus for desirable flow outcomes have been focussed. 1. Introduction Falling films are relevant to a broad class of interfacial instability problems over a wide range of length and time scales in various technological setups since they offer small thermal resistance and large contact area at small specific flow rates. This serves as an advantage in many processes involving cooling, condensation, absorption, and evaporation, thereby regulating the transport of mass, momentum, and heat across the liquid-gas and liquid-solid interfaces. For a detailed review of falling film problems, refer to Nepomnyashchy et al. [1], Kalliadasis et al. [2], and the references therein. Although many studies available in literature have focused attention on Newtonian fluids, many fluids used in industrial applications are rheologically complex and non-Newtonian in nature. This has led to the generalization of Navier-Stokes model to satisfy highly nonlinear constitutive laws to arrive at complex partial differential equations, which are one order higher than the Navier-Stokes equations [3–8]. Unlike Newtonian fluids, which respond virtually instantaneously to an imposed deformation rate, viscoelastic fluids respond on a macroscopically large time scale, known as the relaxation time. The viscoelastic fluids, a subclass of microstructure flows, display both elastic (for deformation rates larger than the inverse relaxation time) and viscous (for deformation rates smaller than the magnitude of the inverse relaxation time) characteristics. The stress in this liquid is neither directly proportional to the strain nor to the rate of strain but displays a complex relationship [9]. A lot of work on the flow and heat transfer characteristics of non-Newtonian fluids has also been done in order to control the quality of the end product in many manufacturing and processing industries ([10] and the references therein). This area of fluid dynamics can simulate
References
[1]
A. A. Nepomnyashchy, M. G. Velarde, and P. Colinet, Interfacial Phenomena and Convection, Chapman & Hall/ CRC, Boca Raton, Fla, USA, 2002.
[2]
S. Kalliadasis, C. Ruyer-Quil, B. Scheid, and M. G. Velarde, Falling Liquid Films, Springer, London, UK, 2012.
[3]
J. G. Oldroyd, “On the formulations of rheological equations of state,” Proceedings of the Royal Society A, vol. 200, no. 1063, p. 523, 1950.
[4]
C. Truesdell and W. Noll, The Non-Linear Field Theories of Mechanics, Hanbook of Physics, Springer, Berlin, Germany, 1965.
[5]
K. R. Rajagopal, “Mechanics of non-Newtonian fluids,” in Recent Developments in Theoretical Fluid Mechanics, vol. 291 of Pittman Research Notes in Mathematical Series, Longman Scientific & Technical, Harlow, UK, 1993.
[6]
R. B. Bird, “Useful non-Newtonian models,” Annual Review of Fluid Mechanics, vol. 8, pp. 13–134, 1976.
[7]
R. P. Chhabra and J. F. Richardson, Non-Newtonian Flow in the Process Industries: Fundamentals and Engineering Applications, Butterworth-Heinemann, Oxford, UK, 1999.
[8]
G. {\L}ukaszewicz, Micropolar Fluids: Theory and Applications, Birkh？auser, Basel, Switzerland, 1999.
[9]
J. D. Ferry, Viscoelastic Properties of Polymers, John Wiley & Sons, New York, NY, USA, 1961.
[10]
T. Altan, S. I. Oh, and H. Gegel, Metal Forming: Fundamentals and Applications, American Society of Metals, Metals park, Ohio, USA, 1983.
[11]
K. Walters, “Non-Newtonian effects on an elastico-viscous liquid contained between coaxial cylinders (II),” Quarterly Journal of Mechanics and Applied Mathematics, vol. 13, no. 4, pp. 444–461, 1960.
[12]
H. I. Andersson and E. N. Dahl, “Gravity-driven flow of a viscoelastic liquid film along a vertical wall,” Journal of Physics, vol. 32, pp. 1557–1562, 1999.
[13]
I. M. R. Sadiq and R. Usha, “Linear instability in a thin viscoelastic liquid film on an inclined, non-uniformly heated wall,” International Journal of Engineering Science, vol. 43, pp. 1435–1449, 2005.
[14]
M. C. Lin and C. K. Chen, “Finite amplitude long-wave instability of a thin viscoelastic fluid during spin coating,” Applied Mathematical Modelling, vol. 36, no. 6, pp. 2536–2549, 2012.
[15]
M. A. Sirwah and K. Zakaria, “Nonlinear evolution of the travelling waves at the surface of a thin viscoelastic falling film,” Applied Mathematical Modelling, vol. 37, pp. 1723–1752, 2013.
[16]
H.-C. Chang, “Wave evolution on a falling film,” Annual Review of Fluid Mechanics, vol. 26, pp. 103–136, 1994.
[17]
A. Oron, S. H. Davis, and S. G. Bankoff, “Long-scale evolution of thin liquid films,” Reviews of Modern Physics, vol. 69, no. 3, pp. 931–980, 1997.
[18]
S. W. Joo, S. H. Davis, and S. G. Bankoff, “Long-wave instabilities of heated falling films: two-dimensional theory of uniform layers,” Journal of Fluid Mechanics, vol. 230, pp. 117–146, 1991.
[19]
S. Miladinova, S. Slavtchev, G. Lebon, and E. Toshev, “Long-wave instabilities of non-uniformly heated falling films,” Journal of Fluid Mechanics, vol. 453, pp. 153–175, 2002.
[20]
B. Scheid, Evolution and stability of falling liquid films with thermocapillary effects [Ph.D. thesis], Université Libre de Bruxelles, Brussels, Belgium, 2004.
[21]
I. M. R. Sadiq, R. Usha, and S. W. Joo, “Instabilities in a liquid film flow over an inclined heated porous substrate,” Chemical Engineering Science, vol. 65, no. 15, pp. 4443–4459, 2010.
[22]
M. Vlachogiannis and V. Bontozoglou, “Experiments on laminar film flow along a periodic wall,” Journal of Fluid Mechanics, vol. 457, pp. 133–156, 2002.
[23]
A. Wierschem, C. Lepski, and N. Aksel, “Effect of long undulated bottoms on thin gravity-driven films,” Acta Mechanica, vol. 179, no. 1-2, pp. 41–66, 2005.
[24]
K. Argyriadi, M. Vlachogiannis, and V. Bontozoglou, “Experimental study of inclined film flow along periodic corrugations: the effect of wall steepness,” Physics of Fluids, vol. 18, no. 1, Article ID 012102, 2006.
[25]
L. A. Dávalos-Orozco, “Nonlinear instability of a thin film flowing down a smoothly deformed surface,” Physics of Fluids, vol. 19, no. 7, Article ID 074103, 2007.
[26]
A. Oron and C. Heining, “Weighted-residual integral boundary-layer model for the nonlinear dynamics of thin liquid films falling on an undulating vertical wall,” Physics of Fluids, vol. 20, no. 8, Article ID 082102, 2008.
[27]
T. H？cker and H. Uecker, “An integral boundary layer equation for film flow over inclined wavy bottoms,” Physics of Fluids, vol. 21, no. 9, Article ID 092105, 2009.
[28]
S. J. D. D'Alessio, J. P. Pascal, and H. A. Jasmine, “Instability in gravity-driven flow over uneven surfaces,” Physics of Fluids, vol. 21, no. 6, Article ID 062105, 2009.
[29]
C. Heining, V. Bontozoglou, N. Aksel, and A. Wierschem, “Nonlinear resonance in viscous films on inclined wavy planes,” International Journal of Multiphase Flow, vol. 36, pp. 78–90, 2010.
[30]
C. Heining and N. Aksel, “Effects of inertia and surface tension on a power-law fluid flowing down a wavy incline,” International Journal of Multiphase Flow, vol. 36, no. 11-12, pp. 847–857, 2010.
[31]
S. Saprykin, P. M. J. Trevelyan, R. J. Koopmans, and S. Kalliadasis, “Free-surface thin-film flows over uniformly heated topography,” Physical Review, vol. 75, no. 2, Article ID 026306, 2007.
[32]
N. Tiwari and J. M. Davis, “Stabilization of thin liquid films flowing over locally heated surfaces via substrate topography,” Physics of Fluids, vol. 22, no. 4, Article ID 042106, 2010.
[33]
S. J. D. D'Alessio, J. P. Pascal, H. A. Jasmine, and K. A. Ogden, “Film flow over heated wavy inclined surfaces,” Journal of Fluid Mechanics, vol. 665, pp. 418–456, 2010.
[34]
K. A. Ogden, S. J. D. D'Alessio, and J. P. Pascal, “Gravity-driven flow over heated, porous, wavy surfaces,” Physics of Fluids, vol. 23, Article ID 122102, 2011.
[35]
T. Gambaryan-Roisman and P. Stephan, “Falling films in micro- and minigrooves: Heat transfer and flow stability,” Thermal Science & Engineering, vol. 11, p. 43, 2003.
[36]
K. Helbig, R. Nasarek, T. Gambaryan-Roisman, and P. Stephan, “Effect of longitudinal minigrooves on flow stability and wave characteristics of falling liquid films,” Journal of Heat Transfer, vol. 131, no. 1, Article ID 011601, 8 pages, 2009.
[37]
I. M. R. Sadiq, T. Gambaryan-Roisman, and P. Stephan, “Falling liquid films on longitudinal grooved geometries: integral boundary layer approach,” Physics of Fluids, vol. 24, no. 1, Article ID 014104, 2012.
[38]
I. M. R. Sadiq, “First-order energy-integral model for thin Newtonian liquids falling along sinusoidal furrows,” Physical Review, vol. 85, no. 3, Article ID 036309, 2012.
[39]
C. Heining, T. Pollak, and N. Aksel, “Pattern formation and mixing in three-dimensional film flow,” Physics of Fluids, vol. 24, no. 4, Article ID 042102, 2012.
[40]
K.-T. Kim and R. E. Khayat, “Transient coating flow of a thin non-Newtonian fluid film,” Physics of Fluids, vol. 14, no. 7, pp. 2202–2215, 2002.
[41]
E. I. Mogilevskii and V. Y. Shkadov, “Effect of bottom topography on the flow of a non-newtonian liquid film down an inclined plane,” Moscow University Mechanics Bulletin, vol. 62, no. 3, pp. 76–83, 2007.
[42]
R. Usha and B. Uma, “Long waves on a viscoelastic film flow down a wavy incline,” International Journal of Non-Linear Mechanics, vol. 39, no. 10, pp. 1589–1602, 2004.
[43]
H. Tougou, “Long waves on a film flow of a viscous fluid down an inclined uneven wall,” Journal of the Physical Society of Japan, vol. 44, no. 3, pp. 1014–1019, 1978.
[44]
S. Saprykin, R. J. Koopmans, and S. Kalliadasis, “Free-surface thin-film flows over topography: influence of inertia and viscoelasticity,” Journal of Fluid Mechanics, vol. 578, pp. 271–293, 2007.
[45]
M. Pavlidis, Y. Dimakopoulos, and J. Tsamopoulos, “Steady viscoelastic film flow over 2D topography: I. The effect of viscoelastic properties under creeping flow,” Journal of Non-Newtonian Fluid Mechanics, vol. 165, no. 11-12, pp. 576–591, 2010.
[46]
F. Bouchut and S. Boyaval, “A new model for shallow viscoelastic fluids,” Mathematical Models and Methods in Applied Sciences, vol. 23, no. 8, pp. 1479–1526, 2013.
[47]
E. S. G. Shaqfeh, R. G. Larson, and G. H. Fredrickson, “The stability of gravity driven viscoelastic film-flow at low to moderate reynolds number,” Journal of Non-Newtonian Fluid Mechanics, vol. 31, no. 1, pp. 87–113, 1989.
[48]
R. V. Birikh, V. A. Briskman, M. G. Velarde, and J. C. Legros, Liquid Interfacial Systems: Oscillations and Instability, Marcel Dekker, New York, NY, USA, 2003.
[49]
R. B. Bird, R. C. Armstrong, and O. Hassager, “Dynamics of polymeric liquids,” in Fluid Mechanics, vol. 1, John Wiley & Sons, New York, NY, USA, 1987.
[50]
N. Aksel, “A brief note from the editor on the ‘second-order fluid’,” Acta Mechanica, vol. 157, no. 1–4, pp. 235–236, 2002.
[51]
J. H. Spurk and N. Aksel, Fluid Mechanics, Springer, Berlin, Germany, 2nd edition, 2008.
[52]
B. S. Dandapat and A. Samanta, “Bifurcation analysis of first and second order Benney equations for viscoelastic fluid flowing down a vertical plane,” Journal of Physics, vol. 41, no. 9, Article ID 095501, 2008.
[53]
B. D. Coleman and W. Noll, “An approximation theorem for functionals with applications in continuum mechanics,” Archive for Rational Mechanics and Analysis, vol. 6, no. 1, pp. 355–370, 1960.
[54]
K. C. Porteous and M. M. Denn, “Linear stability of plane poiseuille flow of viscoelastic liquids,” Trans Soc Rheol, vol. 16, pp. 295–308, 1972.
[55]
K. Walters, “The solution flow problems in the case of materials with memory,” Journal of Fluid Mechanics, vol. 1, p. 474, 1962.
[56]
P. L. Bhatnagar, “Comparative study of some constitutive equations characterising non-Newtonian fluids,” Proceedings of the Indian Academy of Sciences, vol. 66, no. 6, pp. 342–352, 1967.
[57]
N. Datta and R. N. Jana, “Flow and heat transfer in an elastico-viscous liquid over an oscillating plate in a rotating frame,” ？stanbul üniversitesi Fen Fakültesi, vol. 43, p. 121, 1978.
[58]
C. W. Macosko, Rheology: Principles, Measurements and Applications, John Wiley & Sons, New York, NY, USA, 1994.
[59]
B. Uma and R. Usha, “Nonlinear stability of thin viscoelastic liquid film down a vertical wall with interfacial phase change,” in Proceedings of the ASME Joint U.S.-European Fluids Engineering Conference, vol. 1, Fora, parts A and B, paper no. FEDSM2002-31035, pp. 1303–1310, Quebec, Canada, July 2002.
[60]
P. Kumar and V. Kumar, “On thermosolutal-convective instability in Walters B heterogeneous viscoelastic fluid layer through porous medium,” American Journal of Fluid Dynamics, vol. 3, no. 1, 2013.
[61]
D. W. Beard and K. Walters, “Elastico-viscous boundary-layer flow I. Two-dimensional flow over a stagnation point,” Proceedings of the Cambridge Philosophical Society, vol. 60, p. 667, 1964.
[62]
H. A. Barnes, J. F. Hutton, and K. Walters, An Introduction to Rheology, Elsevier, Amsterdam, The Netherlands, 1989.
[63]
A. S. Gupta, “Stability of a non-Newtonian liquid film flowing down an inclined plane,” Physics of Fluids, vol. 28, p. 17, 1967.
[64]
B. S. Dandapat and A. S. Gupta, “Solitary waves on the surface of a viscoelastic fluid running down an inclined plane,” Rheologica Acta, vol. 36, no. 2, pp. 135–143, 1997.
[65]
W. Nusselt, “Die Oberfl？？chenkondensation des Wasserdampfes,” Zeitschrift des Vereines Deutscher Ingenieure, vol. 60, p. 541, 1916.
[66]
M. Scholle and N. Aksel, “An exact solution of visco-capillary flow in an inclined channel,” Zeitschrift für Angewandte Mathematik und Physik, vol. 52, no. 5, pp. 749–769, 2001.
[67]
I. M. R. Sadiq and R. Usha, “Thin Newtonian film flow down a porous inclined plane: stability analysis,” Physics of Fluids, vol. 20, no. 2, Article ID 022105, 2008.
[68]
K. A. Smith, “On convective instability induced by surface gradients,” Journal of Fluid Mechanics, vol. 24, p. 401, 1966.
[69]
D. A. Goussis and R. E. Kelly, “On the thermocapillary instabilities in a liquid layer heated from below,” International Journal of Heat and Mass Transfer, vol. 33, p. 2237, 1990.
[70]
S. J. Vanhook, M. F. Schatz, J. B. Swift, W. D. McCormick, and H. L. Swinney, “Long wave-length surface-tension-driven Bénard convection: experiment and theory,” Journal of Fluid Mechanics, vol. 345, pp. 45–78, 1997.
[71]
E. Kamke, Differentialgleichungen: Losungsmethoden Und Losungen, Band 1: Gewohnliche Differentialgleichungen, Akademische Verlagsgesellschaft Geest & Portig, Leipzig, Germany, 1961.
[72]
C. Canuto, M. Y. Hussaini, A. Quarteroni, and T. A. Zang, Spectral Methods in Fluid Dynamics, Springer, New York, NY, USA, 1988.
[73]
L. N. Trefthen, Spectral Methods in MATLAB, SIAM, Philadelphia, Pa, USA, 2000.
[74]
A. Pumir, P. Manneville, and Y. Pomeau, “On solitary waves running down an inclined plane,” Journal of Fluid Mechanics, vol. 135, pp. 27–50, 1983.
[75]
S. V. Alekseenko, V. Y. Nakoryakov, and B. G. Pokusaev, “Wave formation on a vertical falling liquid film,” AIChE, vol. 31, no. 9, pp. 1446–1460, 1985.
[76]
C. Heining and N. Aksel, “Bottom reconstruction in thin-film flow over topography: steady solution and linear stability,” Physics of Fluids, vol. 21, no. 8, Article ID 083605, 2009.
[77]
A. Mazouchi and G. M. Homsy, “Free surfaces stokes flow over topography,” Physics of Fluids, vol. 13, no. 10, pp. 2751–2761, 2001.
[78]
S. Nadeem, N. S. Akbar, T. Hayat, and A. A. Hendi, “Peristaltic flow of Walter's B fluid in endoscope,” Applied Mathematics and Mechanics, vol. 32, no. 6, pp. 689–700, 2011.
[79]
B. Scheid, S. Kalliadasis, C. Ruyer-Quil, and P. Colinet, “Interaction of three-dimensional hydrodynamic and thermocapillary instabilities in film flows,” Physical Review, vol. 78, no. 6, Article ID 066311, 2008.
[80]
N. Amatousse, H. A. Abderrahmane, and N. Mehidi, “Traveling waves on a falling weakly viscoelastic fluid film,” International Journal of Engineering Science, vol. 54, pp. 27–41, 2012.
