Effect of mutual location and the shape of heaters on the stability of thin films flowing over locally heated surfaces

We consider the motion of a liquid film falling down a locally heated planar substrate. Marangoni effect due to the local temperature gradients at the free surface induces a three dimensional interfacial deformation near the vicinity of the heater. The objective of the present paper is to study the influence of streamwise arrangement of the heaters of rectangular shape or with infinite spanwise lengths on the structure of interfacial deformation and the stability of the film. The problem is studied in the framework of longwave theory. We studied the influence of mutual location of heaters on the steady state of the film. As the film is heated by heaters with infinite/finite spanwise lengths, 2D/3D steady states exist. A linear stability analysis is performed with respected to both the 2D and 3D basic state. The results show that the mutual location of heater plays an important role in the stability and transient behavior of the film for 2D steady state. However, the 3D steady state of a film heated by rectangular heaters is linear stable. This means that the 3D steady state is a coherent structure in films heated by rectangular heaters.