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FREE-SURFACE DYNAMICS OF THIN SECOND-GRADE FLUID OVER AN UNSTEADY STRETCHING SHEET

Part of:Foundations, constitutive equations, rheology

Published online by Cambridge University Press:  05 November 2018

S. PANDAOpen the ORCID record for S. PANDA[Opens in a new window],
K. K. PATRAOpen the ORCID record for K. K. PATRA[Opens in a new window] and
M. SELLIEROpen the ORCID record for M. SELLIER[Opens in a new window]
S. PANDA*
Affiliation:
Department of Mathematics, National Institute of Technology Calicut, NIT (P.O.)-673601, Kerala, India email satyanand@nitc.ac.in, kirankumar_p140033ma@nitc.ac.in
K. K. PATRA
Affiliation:
Department of Mathematics, National Institute of Technology Calicut, NIT (P.O.)-673601, Kerala, India email satyanand@nitc.ac.in, kirankumar_p140033ma@nitc.ac.in
M. SELLIER
Affiliation:
Department of Mechanical Engineering, The University of Canterbury, Private Bag 4800, Christchurch 8140, New Zealand email mathieu.sellier@canterbury.ac.nz
*
✉satyanand@nitc.ac.in
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Abstract

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We derive an evolution equation for the free-surface dynamics of a thin film of a second-grade fluid over an unsteady stretching sheet using long-wave theory. For the numerical investigation of the viscoelastic effect on the thin-film dynamics, a finite-volume approach on a uniform grid with implicit flux discretization is applied. The present results are in excellent agreement with results available in the literature for a Newtonian fluid. We observe that the fluid thins faster with the rapid stretching rate of the sheet, but the second-grade parameter delays the thinning behaviour of the liquid film.

Keywords

thin liquid filmstretching sheetsecond-grade fluidfree-surface flowlong-wave theoryfinite-volume method

MSC classification

Primary:76A20: Thin fluid films
Secondary:76A05: Non-Newtonian fluids

Information

Type
Research Article
Information
The ANZIAM Journal , Volume 60 , Issue 2 , October 2018, pp. 249 - 268
Copyright
© 2018 Australian Mathematical Society 

References

Abbas, Z., Hayat, T., Sajid, M. and Asghar, S., “Unsteady flow of a second grade fluid film over an unsteady stretching sheet”, Math. Comput. Modelling 48 (2008) 518–526; doi:j.mcm.2007.09.015.Google Scholar
Ahmad, I., “On unsteady boundary layer flow of a second grade fluid over a stretching sheet”, Adv. Theor. Appl. Mech. 6 (2013) 95–105; doi:atam.2013.231.Google Scholar
Andersson, H. I., Aarseth, J. B. and Dandapat, B. S., “Heat transfer in a liquid film on an unsteady stretching surface”, Int. J. Heat Mass Transfer 43 (2000) 69–74; doi:S0017-9310(99)00123-4.Google Scholar
Andersson, H. I., Bech, K. H. and Dandapat, B. S., “Magnetohydrodynamic flow of a power-law fluid over a stretching sheet”, Int. J. Non-Linear Mech. 27 (1992) 929–936; doi:0020-7462(92)90045-9.Google Scholar
Crane, L. J., “Flow past a stretching plate”, Z. Angew. Math. Phys. ZAMP 21 (1970) 645–647; doi:BF01587695.Google Scholar
Dandapat, B. S., Kitamura, A. and Santra, B., “Transient film profile of thin liquid film flow on a stretching surface”, Z. Angew. Math. Phys. ZAMP 57(2006) 623–635; doi:s00033-005-0040-7.Google Scholar
Dandapat, B. S. and Maity, S., “Flow of a thin liquid film on an unsteady stretching sheet”, Phys. Fluids 18 (2006) 102101; doi:1.2360256.Google Scholar
Dandapat, B. S., Maity, S. and Kitamura, A., “Liquid film flow due to an unsteady stretching sheet”, Int. J. Non-Linear Mech. 43 (2008) 800–886; doi:j.ijnonlinmec.2008.06.003.Google Scholar
Dandapat, B. S. and Singh, S. K., “Thin film flow over a heated nonlinear stretching sheet in presence of uniform transverse magnetic field”, Int. Commun. Heat Mass Transfer 38 (2011) 324–328; doi:j.icheatmasstransfer.2010.11.009.Google Scholar
Dunn, J. E. and Fosdick, R. L., “Thermodynamics, stability and boundedness of fluids of complexity 2 and fluids of second grade”, Arch. Ration. Mech. Anal. 56 (1974) 191–252; doi:BF00280970.Google Scholar
Dunn, J. E. and Rajagopal, K. R., “Fluids of differential type: critical review and thermodynamics analysis”, Internat. J. Engrg. Sci. 33 (1995) 689–747; doi:0020-7225(94)00078-X.Google Scholar
Fosdick, R. L. and Rajagopal, K. R., “Anomalous features in the model of second order fluids”, Arch. Ration. Mech. Anal. 70 (1979) 145–152; doi:BF00250351.Google Scholar
Ha, Y., Kim, Y. J. and Myers, T. G., “On the numerical solution of a driven thin film equation”, J. Comput. Phys. 227 (2008) 7246–7263; doi:j.jcp.2008.04.007.Google Scholar
Hayat, T., Khan, M., Siddiqui, A. M. and Asghar, S., “Transient flows of a second grade fluid”, Int. J. Non-Linear Mech. 39 (2004) 1621–1633; doi:j.ijnonlinmec.2002.12.001.Google Scholar
Kalliadasis, S., Ruyer-Quil, C., Scheid, B. and Velarde, M. G., Falling liquid films, Volume 176 of Appl. Math. Sci. (Springer, London, 2012); doi:978-1-84882-367-9.Google Scholar
Levy, R., Hill, D. B., Forest, M. G. and Grotberg, J. B., “Pulmonary fluid flow challenges for experimental and mathematical modeling”, Integr. Comp. Biol. 54 (2014) 985–1000; doi:icb/icu107.Google Scholar
Liu, I. C. and Andersson, H. I., “Heat transfer in a liquid film on an unsteady stretching sheet”, Int. J. Thermal Sci. 47 (2008) 766–772; doi:j.ijthermalsci.2007.06.001.Google Scholar
Nadeem, S. and Faraz, N., “Thin film flow of a second grade fluid over a stretching/shrinking sheet with variable temperature-dependent viscosity”, Chin. Phys. Lett. 27 (2010) 034704; doi:0256-307X/27/3/034704.Google Scholar
Noor, N. F. M., Abdulaziz, O. and Hashim, I., “MHD flow and heat transfer in a thin liquid film on an unsteady stretching sheet by the homotopy analysis method”, Internat. J. Numer. Methods Fluids 63 (2010) 357–373; doi:fld.2078.Google Scholar
O’Brien, S. B. G. and Schwartz, L. W., “Theory and modeling of thin film flows”, in: Encyclopedia of surface and colloid science, (Marcel Dekker, NY, USA, 2002) 5283–5297; https://pdfs.semanticscholar.org/1cc5/1908e0dd5fbf43a8b8ed481844a1fc28f8e3.pdf.Google Scholar
Panda, S., Sellier, M., Fernando, M. C. S. and Abeyratne, M. K., “Process parameter identification in thin film flows driven by a stretching surface”, Int. J. Engng Math. 2014 (2014) 485431; doi:2014/485431.Google Scholar
Rajagopal, K. R. and Srinivasa, A. R., “On the development of fluid models of the differential type within a thermodynamic framework”, Mech. Res. Commun. 35 (2008) 483–489; doi:j.mechrescom.2008.02.004.Google Scholar
Rivlin, R. S. and Ericksen, J. L., “Stress deformation relations for isotropic materials”, J. Ration. Mech. Anal. 4 (1995) 323–425; doi:10.1007/978-1-4612-2416-7_61.Google Scholar
Santra, B. and Dandapat, B. S., “Thin-film flow over a nonlinear stretching sheet”, Z. Angew. Math. Phys. ZAMP 60 (2009) 688–700; doi:s00033-008-8015-0.Google Scholar
Sellier, M. and Panda, S., “Surface temperature reconstruction based on the thermocapillary effect”, ANZIAM J. 52 (2010) 146–159; doi:S1446181111000654.Google Scholar
Wang, C. Y., “Liquid film on an unsteady stretching sheet”, Quart. Appl. Math. 48 (1990) 601–610; doi:qam/1079908.Google Scholar