Resonant mode scanning to compute the spectrum of capillary surfaces with dynamic wetting effects

A capillary surface bound by a solid rectangular channel exhibits dynamic wetting effects characterized by a constitutive law relating the dynamic contact-angle to the contact-line speed through the contact-line mobility $$\Lambda $$ parameter. Limiting cases correspond to the free ( $$\Lambda =0$$ ) and pinned ( $$\Lambda =\infty $$ ) contact-line. Viscous potential flow is used to derive the governing integrodifferential equation from a boundary integral approach. The spectrum is determined from a boundary value problem where the eigenvalue parameter appears in the boundary condition. Here we introduce a new frequency scan approach to compute the spectrum, whereby we scan the complex frequency plane and compute the system response from which we identify the complex resonant frequency. Damping effects due to viscosity and Davis dissipation from finite $$\Lambda $$ do not attenuate signal response, but rather shift the response poles into the complex plane. Our new approach is verified against an analytical solution in the appropriate limit. We identify the critical mobility that maximizes Davis dissipation and the critical Ohnesorge number (viscosity) where the transition from underdamped to overdamped oscillations occurs, as it depends upon the static contact-angle $$\alpha $$ . Our approach is applied to a rectangular channel, but is suitable for a myriad of geometric supports.