Role of inertia in the dynamics of a thin film falling over a heated vertical cylindrical fiber.

This work investigates the linear stability of a thin liquid film flowing down a uniformly heated vertical cylindrical fiber. A fourth-order nonlinear evolution equation governing the spatiotemporal dynamics of the film thickness is derived using lubrication approximation and asymptotic expansion. The model captures the influence of gravity, inertia, surface tension, thermocapillarity, and convective heat transfer through key dimensionless parameters: Bond, Reynolds, Marangoni, and Biot numbers. Temporal stability analysis reveals that, in the absence of inertia and thermocapillarity, perturbations grow due to the classical Rayleigh-Plateau instability. Moderate inertia enhances instability, although Rayleigh-Plateau instability remain dominant over inertial instability for high surface tension fluids. The relative influence of gravity and surface tension, represented by the Bond number, tends to stabilize long-wave disturbances ( k < 1 ) while promoting the growth of short-wave modes ( k > 1 ). Thermocapillary stress stabilizes film flow on a cooled cylinder and destabilizes it on a heated one. The Biot number plays a dual role-initially amplifying instability, then reducing it as interfacial temperature gradients diminish. Spatiotemporal analysis uncovers a transition from convective to absolute instability with increasing Marangoni or Reynolds numbers. Lower Bond numbers favor absolute instability, which transitions to convective behavior as Bond number increases. Numerical simulations align well with theoretical predictions, capturing both temporal and spatiotemporal film dynamics under varying physical conditions.