Wavy regime of a colloidal falling film

In this paper, we study the linear stability and the subsequent formation of nonlinear waves in a colloidal, gravity-driven falling film flow. We first investigate the system's stability by performing a linear stability analysis and observe that the presence of colloidal particles stabilizes both the long-wave surface and the short-wave shear instability modes. The stabilization is attributed to Brownian diffusion equilibrating the leading order particle volume fraction, causing a uniform increase in viscosity. With the particle evolution equation decoupled from the momentum equation in the linear limit, we analytically study the damped yet intricate evolution of perturbations to the particle concentration field. The particle mode decays via three different asymptotic routes: Brownian diffusion, anomalous diffusion, and Taylor dispersion. To study the nonlinear waves, we then derive nonlinear models in the framework of long-wave theory using Benney's gradient expansion approach and the integral-boundary-layer approach. Subsequently, we use a central-manifold approach to derive a depth-averaged equation for the particle volume fraction evolution, incorporating Brownian diffusion and Taylor dispersion. Comparisons of the linear predictions of the nonlinear models with the linear stability calculations show good agreement in the limit of small wave numbers.