Thermal criticality for a reactive gravity driven thin film flow of a third-grade fluid with adiabatic free surface down an inclined plane
The author considers the flow of a hydrodynamically and thermally developed reactive liquid film on an inclined heated plate. The fluid is supposed to be of third-grade type. The author assumes that a lubrication theory may be used for this model. Writing the balances of momentum and of energy, and using a change of variables, the author obtains a system of two nonlinear and coupled ordinary differential equations of second-order with respect to the variable \(y\) orthogonal to the plate. The unknowns are the velocity field and the temperature. In the energy balance, a Arrhenius kinetic term is introduced which corresponds to an exothermic reaction in the fluid. These equations involve the Frank-Kamenetskii parameter \(\lambda \) and a non-newtonian \(\gamma \) among other parameters. Boundary conditions are imposed at \(y=0\) (the plate) and at \(y=1\) (the adiabatic free surface). The author uses power-series expansions in terms of \(\gamma \) (for the velocity field) or of \(\lambda \) (for the temperature) for the resolution of this problem. Using the MAPLE software, he computes the first terms of this solution. The main part of the paper describes the thermal criticality of this problem. A theoretical analysis shows that a special kind of Hermite-Padé approximation may be obtained. This theoretical analysis is then completed using the MAPLE software. The last part of the paper describes some qualitative properties of this criticality situation (study of the influence of the parameters).
scientific article; zbMATH DE number 5550604
scientific article