  Linear and weakly nonlinear stability analyses of an externally
shear-imposed, gravity-driven falling film over a uniformly heated wavy
substrate are studied. The longwave asymptotic expansion technique is utilized
to formulate a single nonlinear free surface deflection equation. The linear
stability criteria for the onset of instability are derived using the normal
mode form in the linearized portion of the surface deformation equation. Linear
stability theory reveals that the flow-directed sturdy external shear grows the
surface wave instability by increasing the net driving force. On the contrary,
the upstream-directed imposed shear may reduce the surface mode instability by
restricting the gravity-driving force, which has the consequence of weakening
the bulk velocity of the liquid film. However, the surface mode can be
stabilized/destabilized by increasing the temperature-dependent
density/surface-tension variation. Further, the bottom steepness shows dual
behaviour on the surface instability depending upon the wavy wall's portion
(uphill/downhill). At the downhill portion, the surface wave becomes more
unstable than at the bottom substrate's uphill portion. Moreover, the
multi-scale method is incorporated to obtain the complex Ginzburg-Landau
equation in order to study the weakly nonlinear stability, confirming the
existence of various flow regions of the liquid film. At any bottom portion
(uphill/downhill), the flow-directed external shear expands the super-critical
stable zones, which causes an amplification in the nonlinear wave amplitude,
and the backflow-directed shear plays a counterproductive role. On the other
hand, the super-critical stable region decreases or increases as long as the
linear variation of density or surface tension increases with respect to the
temperature, whereas the sub-critical unstable region exhibits an inverse
trend.
