Three-dimensional disturbances in a conducting liquid film flowing down an inclined plane in the presence of constant electromagnetic field

Three-dimensional stability features of an electrically conducting viscous liquid film flowing down an inclined plane are investigated under the action of gravitational force, surface tension as well as magnetic and coupling influence of electric and magnetic (electromagnetic) field of forces. Our (linear and nonlinear stability) analysis, based on the three-dimensional wave equation which is nonlinear in nature and obtained by using the classical momentum-integral method, indicates the reality of four different (explosive, supercritical, unconditional and subcritical) flow zones of this flow problem depending upon the sign of the curves ωi+ (temporal growth rate) and J2 (second Landau constant) which contain electric parameter E, magnetic parameter M, Reynolds number Re, Weber number We, angle of inclination θ, and angle of propagation φ. The effects of these parameters on four different flow zones as well as on the threshold (supercritical and subcritical) amplitudes ςa and nonlinear phase speed Ncr are explored in detail. The explosive unstable flow zone of this flow problem increases or decreases accordingly as the increase in E or M and φ, irrespective of the values of Re, We, and θ, which confirm the destabilizing role of E and the stabilizing influence of M and φ on this flow dynamics as well. The analysis reveals the necessary conditions for the nonexistence of unconditional stable zone (kj≤kc) as well as the existence of four different flow zones (kj > kc) and the second explosive unstable zone (kj < kc) of this flow problem which are essentially the relations between the cut-off wavenumbers kc (for which ωi+ is zero) and kj (for which J2 is zero) depending upon the values of E, M, Re, We, θ, and φ.