Abstract
The long-wave theory is used to model the thin film flow of a generalized second-grade fluid (GSGF) down a tilted plate with a bump topography. The derived single non-linear partial differential equation for the film thickness describes the surface wave generated by the bump, which disturbs the uniform flow. The model involves the non-Newtonian and geometrical parameters that investigate the wave’s shape and amplitude. The model equation is strongly non-linear due to the GSGF’s constitutive equations, and it is solved numerically using the finite volume method, where the flux function is approximated implicitly using the upwind scheme. The simulation reveals that the bump creates the surface wave, it splits and propagates, and its shape and size are influenced by the bump’s height and the non-Newtonian fluid properties.
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References
Bandelli R (1995) Unsteady unidirectional flows of second grade fluids in domains with heated boundaries. International journal of non-linear mechanics 30(3):263–269. https://doi.org/10.1016/0020-7462(94)00051-B
Bandelli R, Rajagopal K (1994) On the falling of objects in non-newtonian fluids. Annali dell’Università di Ferrara 40(1):71–95. https://doi.org/10.1007/BF02834513
Benharbit A, Siddiqui A (1992) Certain solutions of the equations of the planar motion of a second grade fluid for steady and unsteady cases. Acta Mechanica 94(1–2):85–96. https://doi.org/10.1007/BF01177007
Benney DJ (1966) Longwave on liquid films. J Maths Phys 45:150–155
Bird RR, Armstrong RC, Hassager O (1977) Dynamics of Polymeric Liquids, Volume 1: Fluid Mechanics. Wiley, New York
Chakraborty S, Sheu TWH, Ghosh S (2019) Dynamics and stability of a power-law film flowing down a slippery slope. Physics of Fluids 31(1):013102. https://doi.org/10.1063/1.5078450
Dandapat B, Gupta A (1978) Long waves on a layer of a visco-elastic fluid down an inclined plane. Rheologica Acta 17(5):492–499. https://doi.org/10.1007/BF01534276
Datt C, Kansal M, Snoeijer JH (2022) A thin-film equation for a viscoelastic fluid, and its application to the landau-levich problem. Journal of Non-Newtonian Fluid Mechanics 305:104816. https://doi.org/10.1016/j.jnnfm.2022.104816
Di Federico V (1998) Permanent waves in slow free-surface flow of a herschel-bulkley fluid. Meccanica 33(2):127–137. https://doi.org/10.1023/A:1004300716125
Dunn JE, Fosdick RL (1974) Thermodynamics, stability, and boundedness of fluids of complexity 2 and fluids of second grade. ArchRation Mech Anal 56(3):191–252. https://doi.org/10.1007/BF00280970
Fetecău C, Zierep J (2001) On a class of exact solutions of the equations of motion of a second grade fluid. Acta Mechanica 150(1–2):135–138. https://doi.org/10.1007/BF01178551
Fetecău C, Fetecău C, Zierep J (2002) Decay of a potential vortex and propagation of a heat wave in a second grade fluid. International journal of non-linear mechanics 37(6):1051–1056. https://doi.org/10.1016/S0020-7462(01)00028-2
Fosdick R, Rajagopal K (1979) Anomalous features in the model of “second order fluids". Arch Ration Mech Anal 70:145–152. https://doi.org/10.1007/BF00250351
Gupta A (1967) Stability of a visco-elastic liquid film flowing down an inclined plane. Journal of fluid mechanics 28(1):17–28. https://doi.org/10.1017/S0022112067001879
Gupta G, Massoudi M (1993) Flow of a generalized second grade fluid between heated plates. Acta Mechanica 99(1):21–33. https://doi.org/10.1007/BF01177232
Hayat T, Khan M, Siddiqui A et al (2004) Transient flows of a second grade fluid. International Journal of Non-Linear Mechanics 39(10):1621–1633. https://doi.org/10.1016/j.ijnonlinmec.2002.12.001
Hung CI, Tsai JS, Chen CK (1996) Nonlinear stability of the thin micropolar liquid film flowing down on a vertical plate. J Fluids Eng 118(3):498–505. https://doi.org/10.1115/1.2817786
Liu K, Mei CC (1989) Slow spreading of a sheet of bingham fluid on an inclined plane. J Fluid Mech 207:505–529. https://doi.org/10.1017/S0022112089002685
Magdalena I, Wiryanto LH (2020) Wave generation on an inclined open channel with a bump. GEOMATE Journal 19(76):126–133. https://doi.org/10.21660/2020.76.42744
Mahesh T, Panda S (2023) Longwave modeling of thin film flow of a generalized second-grade fluid down a slanted plate. International Journal of Non-Linear Mechanics 149:104327. https://doi.org/10.1016/j.ijnonlinmec.2022.104327
Man CS (1992) Nonsteady channel flow of ice as a modified second-order fluid with power-law viscosity. Arch Ration Mech Anal 119(1):35–57. https://doi.org/10.1007/BF00376009
Man CS, Sun QX (1987) On the significance of normal stress effects in the flow of glaciers. J Glaciol 33(115):268–273. https://doi.org/10.3189/S0022143000008832
Manshoor B, Salleh H, Hariri A et al (2020) Thin film flow of non-newtonian third grade fluid down an inclined plane by variation of parameter method. J Compl Flow 1(1):29–32
Massoudi M, Phouc TX (2004) Flow of a generalized second grade non-newtonian fluid with variable viscosity. Continuum Mech Thermodyn 16:529–538. https://doi.org/10.1007/s00161-004-0178-0
Massoudi M, Phuoc TX (2012) Remarks on constitutive modeling of nanofluids. Adv Mech Eng 4:927580. https://doi.org/10.1155/2012/927580
Massoudi M, Vaidya A (2008) On some generalizations of the second grade fluid model. Nonlinear Analysis: Real World Applications 9(3):1169–1183. https://doi.org/10.1016/j.nonrwa.2007.02.008
Miladinova S, Lebon G, Toshev E (2004) Thin-film flow of a power-law liquid falling down an inclined plate. Jr non-Newtonain Fluid Mechanics 122:69–78. https://doi.org/10.1016/j.jnnfm.2004.01.021
Mukhopadhyay A, Chattopadhyay S (2018) Long wave instability of thin film flowing down an inclined plane with linear variation of thermophysical properties for very small biot number. International Journal of Non-Linear Mechanics 100:20–29. https://doi.org/10.1016/j.ijnonlinmec.2018.01.005
Oron A, Davis SH, Bankoff SG (1997) Long-scale evolution of thin liquid films. Reviews of Modern Physics 69(3)
Patra KK, Panda S (2019) Formation of the capillary ridge on the free surface dynamics of second-grade fluid over an inclined locally heated plate. Zeitschrift für Naturforschung A 74(12):1099–1108. https://doi.org/10.1515/zna-2019-0126
Perazzo CA, Gratton J (2003) Thin film of non-newtonian fluid on an incline. Phys Rev E 67(1):016307. https://doi.org/10.1103/PhysRevE.67.016307
Rajagopal K (1981) The flow of a second order fluid between rotating parallel plates. Journal of Non-Newtonian Fluid Mechanics 9(1–2):185–190. https://doi.org/10.1016/0377-0257(87)87015-5
Rajagopal K (1982) A note on unsteady unidirectional flows of a non-newtonian fluid. International Journal of Non-Linear Mechanics 17(5):369–373. https://doi.org/10.1016/0020-7462(82)90006-3
Rajagopal K (1984) On the creeping flow of the second-order fluid. Journal of Non-Newtonian Fluid Mechanics 15(2):239–246. https://doi.org/10.1016/0377-0257(84)80008-7
Rajagopal K, Gupta A (1984) An exact solution for the flow of a non-newtonian fluid past an infinite porous plate. Meccanica 19:158–160. https://doi.org/10.1007/BF01560464
Rivlin RS, Ericksen JL (1997) Stress-deformation relations for isotropic materials. Collected Papers of RS Rivlin pp 911–1013. https://doi.org/10.1007/978-1-4612-2416-7_61
Ruyer-Quil C, Manneville P (1998) Modeling film flows down inclined planes. Eur Phys J B 6:277–292. https://doi.org/10.1007/s100510050550
Ruyer-Quil C, Manneville P (2000) Improved modeling of flows down inclined planes. Eur Phys J B 15:357–369. https://doi.org/10.1007/s100510051137
Sadiq IM, Usha R (2008) Thin newtonian film flow down a porous inclined plane: Stability analysis. Phys Fluids 20:022105. https://doi.org/10.1063/1.2841363
Santra B, Dandapat BS (2009) Thin film flow over a non-linear stretching sheet. Z Angew Math Phys ZAMP 60:688–700. https://doi.org/10.1007/s00033-008-8015-0
Shen S, Shen M, Sun S (1989) A model equation for steady surface waves over a bump. Journal of Engineering Mathematics 23:315–323. https://doi.org/10.1007/BF00128905
Slattery JC (1999) Advanced transport phenomena. Cambridge University Press, Cambridge,. https://doi.org/10.1017/CBO9780511800238
Vanden-Broeck JM (1987) Free-surface flow over an obstruction in a channel. The Physics of fluids 30(8):2315–2317. https://doi.org/10.1063/1.866121
Wiryanto L (2019) Thin film flow over a bump on an inclined channel. In: Journal of Physics: Conference Series, p 022058, https://doi.org/10.1088/1742-6596/1321/2/022058
Wiryanto L, Mungkasi S (2014) A boussinesq-type model for waves generated by flow over a bump. Applied Mathematical Sciences 8(106):5293–5302. https://doi.org/10.12988/ams.2014.47528
Yih CS (1963) Stability of liquid flow down an inclined plane. The physics of Fluids 6(3):321–334. https://doi.org/10.1063/1.1706737
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T, M., Panda, S. Generalized second-grade fluid flow over a tilted plate with bump topography. Rheol Acta 63, 267–282 (2024). https://doi.org/10.1007/s00397-024-01438-y
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DOI: https://doi.org/10.1007/s00397-024-01438-y