Sains Malaysiana 42(7)(2013): 975–980

Travelling-Wave Similarity Solutions for an Unsteady Shear-Stress-Driven

Dry Patch in a Flowing Film

(Penyelesaian Keserupaan Gelombang Menjalar bagi Aliran Tak Mantap di Sekitar

Tompokan Kering yang Didorong oleh Tegasan Ricih)

Yazariah M.Yatim*

School of Mathematical Sciences, Universiti Sains Malaysia, 11800 Penang, Malaysia

Brian R. Duffy & Stephen K. Wilson

Department of Mathematics and Statistics, University of Strathclyde

G1 1XHGlasgow, United Kingdom

Diserahkan: 1 Februari 2012/Diterima: 16 Januari 2013

ABSTRACT

We investigate unsteady flow of a thin film of Newtonian fluid around a symmetric slender dry patch moving with constant velocity on an inclined planar substrate, the flow being driven by a prescribed constant shear stress at the free surface of the film (which would be of uniform thickness in the absence of the dry patch). We obtain a novel unsteady travelling-wave similarity solution which predicts that the dry patch has a parabolic shape and that the film thickness increases monotonically away from the dry patch.

Keywords: Dry patch; similarity solution; thin-film flow; travelling-wave

ABSTRAK

Kami mengkaji aliran tak mantap filem nipis bendalir Newtonian di sekitar tompokan kering simetri yang bergerak dengan halaju yang malar di atas substrat yang condong. Aliran tersebut didorong oleh tegasan ricih yang malar pada permukaan bebas filem itu (yang mempunyai ketebalan yang seragam sekiranya tompokan kering tidak wujud). Kami memperoleh penyelesaian keserupaan gelombang menjalar bagi tompokan kering yang menunjukkan tompokan kering tersebut berbentuk parabola dan ketebalan filem menjauhi tompokan kering didapati bertambah secara monotonik.

Kata kunci: Aliran filem nipis; gelombang menjalar; penyelesaian keserupaan; tompokan kering

RUJUKAN

Hartley, D.E. & Murgatroyd, W. 1964. Criteria for the break-up of thin liquid layers flowing isothermally over solid surfaces.Int. J. Heat Mass Transfer 7: 1003-1015.

Murgatroyd, W. 1965. The role of shear and form forces in the stability of a dry patch in two-phase film flow. Int. J. Heat Mass Transfer 8: 297-301.

Podgorski,T., Flesselles, J-M. & Limat, L. 1999. Dry arches within flowing films. Phys. Fluids 11: 845-852.

Podgorski, T., Flesselles, J-M. & Limat, L. 2001. Curvature of a dry patch boundary in a flowing film. C.R. Acad. Sci., Ser IV 2: 1361-1367.

Ponter, A.B., Davies, G.A., Ross, T.K. & Thornley, P.G. 1967.The influence of mass transfer on liquid film breakdown. Int. J. Heat Mass Transfer 10: 349-359.

Rio, E., Daerr, A. & Limat, L. 2004. Probing with a laser sheet the contact angle distribution along a contact line. J. Colloid Interface Sci. 269: 164-170.

Rio, E. & Limat, L. 2006. Wetting hysteresis of a dry patch left inside a flowing film. Phys. Fluids 18: 032102-1–032102-8.

Sćbilleau, J., Lebon, L. & Limat, L. 2009. Stability of a dry patch in a viscous flowing film. Eur. Phys. J. Special Topics 166: 139-142.

Wilson, S.K., Duffy, B.R. & Davis, S.H. 2001. On a slender dry patch in a liquid film draining under gravity down an inclined plane. Eur. J. Appl. Math. 12: 233-252.

Yatim, Y.M., Duffy, B.R. & Wilson, S.K. 2012. Travelling-wave similarity solutions for an unsteady gravity-driven dry patch. In Progress in Industrial Mathematics at ECMI 2010.Heidelberg: Springer-Verlag. pp. 441-447.

*Pengarang untuk surat-menyurat; email: yazariahmy@usm.my