Stability of a viscous liquid film flowing down an inclined plane with respect to three-dimensional disturbances

An analysis is presented for the stability of a viscous liquid film flowing down an inclined plane with respect to three-dimensional disturbances under the action of gravity and surface tension. Using momentum-integral method, the nonlinear free surface evolution equation is derived by introducing the self-similar semiparabolic velocity profiles along the flow (x- and y-axis) directions. A normal mode technique and the method of multiple scales are used to obtain the theoretical (linear and nonlinear stability) results of this flow problem, which conceive the physical parameters: Reynolds number Re, Weber number We, angle of inclination of the plane theta and the angle of propagation of the interfacial disturbances phi. The temporal growth rate omega(+)(i) and second Landau constant J(2), based on which various (explosive, supercritical, unconditional, subcritical) stability zones of this flow problem are categorized, contain the shape factors B and beta owing to the non-zero steady basic flow along the y-axis direction. A novel result which emerges from the linear stability analysis is that for any given value of Re, We and theta, any stability that arises in two-dimensional disturbances (phi = 0) must also be present in three-dimensional disturbances. For phi = 0, there exists a second explosive unstable zone (instead of unconditional stable zone) after a certain value of Re (or theta) due to the involvement of B and beta in the expression of J(2). This explosive unstable zone vanishes after a certain value of phi depending upon the values of Re, We and theta, which confirms the stabilizing influence of phi on the thin film flow dynamics irrespective of the values of Re, We and theta.