instabilities of thin viscous liquid film flowing down a uniformly heated inclined plane

instabilities of a thin viscous film flowing down a uniformly heated plane are investigated in this study. the heating generates a surface tension gradient that induces thermocapillary stresses on the free surface. thus, the film is not only influenced by gravity and mean surface tension but also the thermocapillary force is acting on the free surface. moreover, the heat transfer at the free surface plays a crucial role in the evolution of the film. the main objective of this study is to scrutinize the impact of biot number bi which describes heat transfer at the free surface on instability mechanism. using the long wave expansion method, a generalized non-linear evolution equation of benney type, including the above mentioned effects, is derived for the development of the free surface. a normal mode approach and the method of multiple scales are used to obtain the linear and weakly nonlinear stability solution for the film flow. the linear stability analysis of the evolution equation shows that the biot number plays a double role; for bi < 1 it gives destabilizing effect but for bi > 1 it produces stabilization. at bi = 1, the instability is maximum. the weakly nonlinear study reveals that the impact of marangoni number mr is very strong on the bifurcation scenario even for its slight variation. this behaviour of the biot number is the consequence of the fact that the interfacialtemperature is held close to the plane temperature for bi > 1, thus weakening the marangoni effect. the weakly nonlinearstudy reveals that the impact of marangoni number mr is very strong on the bifurcation scenario even for its slight variation.