Process Parameter Identification in Thin Film Flows Driven by a Stretching Surface

The flow of a thin liquid film over a heated stretching surface is considered in this study. Due to a potential nonuniform temperature distribution on the stretching sheet, a temperature gradient occurs in the fluid which produces surface tension gradient at the free surface of the thin film. As a result, the free surface deforms and these deformations are advected by the flow in the stretching direction. This work focuses on the inverse problem of reconstructing the sheet temperature distribution and the sheet stretch rate from observed free surface variations. This work builds on the analysis of Santra and Dandapat (2009) who, based on the long-wave expansion of the Navier-Stokes equations, formulate a partial differential equation which describes the evolution of the thickness of a film over a nonisothermal stretched surface. In this work, we show that after algebraic manipulation of a discrete form of the governing equations, it is possible to reconstruct either the unknown temperature field on the sheet and hence the resulting heat transfer or the stretching rate of the underlying surface. We illustrate the proposed methodology and test its applicability on a range of test problems. 1. Introduction The analysis of thin film flow and heat transfer over a stretching surface has been a subject of fundamental importance as it is relevant to several industrial applications such as metal and polymer extrusion, continuous casting, drawing of plastic sheets, or cable coatings to name a few. This industrial context has drawn fluid dynamists and applied mathematicians alike to study this problem from a more canonical angle. The first important contribution to the understanding of this problem is the work of Wang [1] who formulated a mathematical model and developed a solution strategy based on the homotopy analysis method (HAM). The problem has been revisited several times since this seminal work with the inclusion of additional physics or more complex rheology. Andersson et al., for example, extended Wang’s contribution by analyzing the associated heat transfer problem [2, 3], while Noor and Hashim built on the work of Dandapat et al. [4, 5] to consider the thermocapillary and magnetic field effects [6] and Aziz and Hashim that of viscous dissipation [7]. Khan et al. focused on the effect of the temperature-dependency on the viscosity and thermal conductivity on the flow in the film [8] and Andersson et al. extended the standard formulation to power-law fluids [9]. A common feature of the literature cited above is that it is implicitly assumed that the film