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

Abstract

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 θ and the angle of propagation of the interfacial disturbances ϕ. The temporal growth rate ωi+ and second Landau constant J2, based on which various (explosive, supercritical, unconditional, subcritical) stability zones of this flow problem are categorized, contain the shape factors B and β 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 θ, any stability that arises in two-dimensional disturbances (ϕ = 0) must also be present in three-dimensional disturbances. For ϕ = 0, there exists a second explosive unstable zone (instead of unconditional stable zone) after a certain value of Re (or θ) due to the involvement of B and β in the expression of J2. This explosive unstable zone vanishes after a certain value of ϕ depending upon the values of Re, We and θ, which confirms the stabilizing influence of ϕ on the thin film flow dynamics irrespective of the values of Re, We and θ.