A spectral theory for fingering on a prewetted plane

A spectral theory is formulated to quantify the dynamic sensitivity and stability of a liquid front spreading over a prewetted inclined plane. Sensitivity of the front to surface heterogeneity is captured by the stable continuous essential spectrum of a non-normal linear operator over a practically infinite domain. Small-amplitude disturbances on the prewetted film are convected by this mechanism to the front and focused into a sharp but transient spike with a greatly magnified amplitude. A specific transverse wavelength is selected during this transient excitation. However, this amplification can only lead to fingering and select the transverse wavelength if a point eigenvalue of the discrete spectrum with the same wavelength is unstable. This discrete mode is resonantly excited by the disturbance convected and amplified by the stable essential spectrum. The preferentially excited point eigenvalue is nearly neutral due to the translational invariance of the unfingered front and has a much smaller growth rate and much shorter transverse wavelength than the most unstable discrete mode. For a range of inclination angles, it is stable for thick fronts but destabilizes far downstream when the thinning film reaches a critical thickness and a bump appears on the front. Experiments and numerical simulation of fingering on a prewetted plane verify these distinct roles of the essential and discrete spectra in this supercritical fingering instability and quantitatively confirm the predicted fingering position.