Z. Angew. Math. Phys. 61 (2010), 685–695 c  2010 Springer Basel AG 0044-2275/10/040685-11 published online May 11, 2010 DOI 10.1007/s00033-010-0074-3 Zeitschrift f¨ ur angewandte Mathematik und Physik ZAMP Thin film flow over a non-linear stretching sheet in presence of uniform transverse magnetic field B. S. Dandapat, Bidyut Santra and S. K. Singh Abstract. A thin viscous liquid film flow is developed over a stretching sheet under different non-linear stretching velocities in presence of uniform transverse magnetic field. Evolution equation for the film thickness is derived using long-wave approxi- mation of thin liquid film and is solved numerically by using the Newton–Kantorovich method. It is observed that all types of stretching produces film thinning, but non-monotonic stretching produces faster thinning at small distance from the origin. Effect of the transverse magnetic field is to slow down the film thinning process. Observed flow behavior is explained physically. Mathematics Subject Classification (2000). 76 · 76A20 · 76D08. Keywords. Thin liquid film · Free surface flow · Viscous flow · Non-linear stretching · Stretching sheet · MHD. 1. Introduction Boundary layer flow over a moving continuous solid surface has been intensively studied over the past two decades because of its wide applications in industry. For example, heat-treated materials that travel between feed and wind-up rollers, motion on conveyer belts, wire and fiber coating, reactor fluidization, food stuff processing, transpiration cooling, or in polymer processing industry and so on. A class of flow problems with obvious relevance to polymer extrusion is the flow induced by the stretching motion of a flat elastic sheet. For example, in a melt spinning process, the extrudate from the die is generally drawn and simultaneously stretched into a filament or sheet, which is thereafter solidified through rapid quench- ing or gradual cooling by direct contact with water or chilled metal rolls. In fact, stretching imports a unidirectional orientation to the extrudate, thereby improving its mechanical properties and the quality of the final product greatly depends on the rate of cooling. Further, the extrudate itself is of finite thickness and it flows as a thin film during stretching. It is to be noted here that most of the studies (Gupta and Gupta [1], Siddappa and Abel [2], Andersson and Dandapat [3], Andersson et al. [4], Takhar et al. [5] etc) on stretching sheet flow problem are considered as steady boundary-layer flow and the boundary conditions are prescribed at the sheet and on the fluid outside the boundary layer (at infinity). Further, it should be noted that as the stretching continues, the boundary-layer that grows near the sheet may develop and cover the entire thickness of the film; as a result, one cannot consider the boundary-layer type equation to study this problem. To the best of our knowledge, we like to state here that the study of unsteady flow due to stretching of a sheet has not yet received adequate attention when the film and the boundary layer thickness coincide, despite the extensive research made on this flow problem since last three decades. Recently, Dandapat et al. [6] and Dandapat and Maity [7] have studied the development of thin liquid film over a stretching sheet with non-planar and Dandapat and Maity [8] for planar film free surface at the onset of stretching. In these studies, they have assumed that the boundary layer covers the entire depth of the fluid and the Navier-Stokes equations are solved analytically using the matched asymptotic method and the method of characteristic to predict the variation of film thickness with space