Influence of inertia, topography and gravity on transient axisymmetric thin‐film flow

Abstract

SUMMARYThis study examines theoretically the development of early transients for axisymmetric ow of a thin ÿlm over a stationary cylindrical substrate of arbitrary shape. The uid is assumed to emerge from an annular tube as it is driven by a pressure gradient maintained inside the annulus, and/or by gravity in the axial direction. The interplay between inertia, annulus aspect ratio, substrate topography and gravity is particularly emphasized. Initial conditions are found to have a drastic e ect on the ensuing ow. The ow is governed by the thin-ÿlm equations of the 'boundary-layer' type, which are solved by expanding the ow ÿeld in terms of orthonormal modes in the radial direction. The formulation is validated upon comparison with the similarity solution of Watson (J. Fluid Mech 1964; 20:481) leading to an excellent agreement when only 2-3 modes are included. The wave and ow structure are examined for high and low inertia. It is found that low-inertia uids tend to accumulate near the annulus exit, exhibiting a standing wave that grows with time. This behaviour clearly illustrates the di culty faced with coating high-viscosity uids. The annulus aspect is found to be in uential only when inertia is signiÿcant; there is less ow resistance for a ÿlm over a cylinder of smaller diameter. For high inertia, the free surface evolves similarly to two-dimensional ow. The substrate topography is found to have a signiÿcant e ect on transient behaviour, but this e ect depends strongly on inertia. It is observed that the ow of a high-inertia uid over a step-down exhibits the formation of a secondary wave that moves upstream of the primary wave. Gravity is found to help the ÿlm (coating) ow by halting or prohibiting the wave growth. The initial ÿlm proÿle and velocity distribution dictate whether the uid will ow downstream or accumulate near the annulus exit.

Inertial thin film flow on planar surfaces featuring topography

“…Recently, Trifonov 2007 [44], examined the stability of a viscous film flowing over a vertically aligned wavy surface, showing that there is a region of corrugation geometry (amplitude and period) where disturbances decay resulting in a stabalising effect, outside this region the flow is unstable. The reader is also directed to the recent work by Aksel et al in relation linear and nonlinear resonance of thin films (2008,2009 [45, 46]) and to the work of Kayat, Kim and Delosquer 2004 [47] who provide a detailed expose on the influence of inertia, topography and gravity on transient axisymmetric thin film flow.…”

An accurate and comprehensive model of thin fluid flows with inertia on curved substrates

“…boundary-layer equations to produce the widely used ibl model for two-dimensional flow on a flat plate (see Chang 1994, p. 110); Nguyen & Balakotaiah (2000) and Ruschak & Weinstein (2003) developed a model for the steady two-dimensional flow over a curved substrate based on assuming a cubic cross-film velocity structure; Ruyer-Quil & Manneville (2000) and Chang et al (2002) use a small number of Galerkin modes for the cross-film structure to model the flow down an inclined flat plate; and Khayat, Kim & Delosquer (2004) implement a spectral numerical method to simulate axisymmetric flows on an axisymmetric substrate. Instead, we base the derivation of the model (5)-(6) on the approach established in § 4 that is supported by centre manifold theory.…”

The transient spread of a circular liquid jet and hydraulic jump formation

“…In the present study, we seek conditions for similar long-transient behaviour to occur when the jet ceases to accelerate. Long-term transients have also been examined for thin films under the influence of flow inertia, gravity and substrate topography for Newtonian (Khayat & Welke 2001; Khayat, Kim & Delosquer 2004) and viscoelastic (Khayat & Kim 2006) fluids. Long-term transients will be explored in this study under the effect of the jet acceleration at different gravity levels.…”