The formation and presence of eddies within thick gravity-driven free-surface film flow over a corrugated substrate are considered, with the governing equations solved semianalytically using a complex variable method for Stokes flow and numerically via a full finite element formulation for the more general problem when inertia is significant. The effect of varying geometry ͑involving changes in the film thickness or the amplitude and wavelength of the substrate͒ and inertia is explored separately. For Stokes-like flow and varying geometry, excellent agreement is found between prediction and existing flow visualizations and measured eddy center locations associated with the switch from attached to locally detached flow. It is argued that an appropriate measure of the influence of inertia at the substrate is in terms of a local Reynolds number based on the characteristic corrugation length scale. Since, for small local Reynolds numbers, the local flow structure there becomes effectively decoupled from the inertia-dominated overlying film and immune from instabilities at the free-surface; the influence of inertia manifests itself as a skewing of the dividing streamline ͑separatrix͒. It is shown that the formation and presence of eddies can be manipulated in one of two ways. While an decrease/increase in the corrugation steepness leads to the disappearance/appearance of kinematically induced eddies, an increase/decrease in the inertia present in the system leads to the appearance/disappearance of inertially induced eddies. A critical corrugation steepness for a given film thickness is defined, demarking the transition from a kinematically to an inertially induced local eddy flow structure and vice versa.

Although it is well known that Bernoulli's equation is obtained as the first integral of Euler's equations in the absence of vorticity and that in the case of non-vanishing vorticity a first integral of them can be found using the Clebsch transformation for inviscid flow, generalization of the procedure for viscous flow has remained elusive. Accordingly, in this paper, a first integral of the Navier-Stokes equations for steady flow is constructed. In the case of a two-dimensional flow, this is made possible by formulating the governing equations in terms of complex variables and introducing a new scalar potential. Associated boundary conditions are considered, and an extension of the theory to three dimensions is proposed. The capabilities of the new approach are demonstrated by calculating a Reynolds number correction to the laminar shear flow generated in the narrow gap between a flat moving and a stationary wavy wall, as is often encountered in lubrication problems. It highlights the first integral as a suitable tool for the development of new analytical and numerical methods in fluid dynamics.

Eddy formation and presence in a plane laminar shear flow configuration consisting of two infinitely long plates orientated parallel to each other is investigated theoretically. The upper plate, which is planar, drives the flow; the lower one has a sinusoidal profile and is fixed. The governing equations are solved via a full finite element formulation for the general case and semi-analytically at the Stokes flow limit. The effects of varying geometry (involving changes in the mean plate separation or the amplitude and wavelength of the lower plate) and inertia are explored separately. For Stokes flow and varying geometry, excellent agreement between the two methods of solution is found. Of particular interest with regard to the flow structure is the importance of the clearance that exists between the upper plate and the tops of the corrugations forming the lower one. When the clearance is large, an eddy is only present at sufficiently large amplitudes or small wavelengths. 1However, as the plate clearance is reduced, a critical value is found which triggers the formation of an eddy in an otherwise fully attached flow for any finite amplitude and arbitrarily large wavelength. This is a precursor to the primary eddy to be expected in the lid-driven cavity flow which is formed in the limit of zero clearance between the plates.The influence of the flow driving mechanism is assessed by comparison with corresponding solutions for the case of gravity-driven fluid films flowing over an undulating substrate.When inertia is present, the flow generally becomes asymmetrical. However, it is found that for large mean plate separations the flow local to the lower plate becomes effectively decoupled from the inertia dominated overlying flow if the wavelength of the lower plate is sufficiently small. In such cases the local flow retains its symmetry. A local Reynolds number based on the wavelength is shown to be useful in characterising these large-gap flows. As the mean plate separation is reduced, the form of the asymmetry caused by inertia changes, and becomes strongly dependent on the plate separation. For lower plate wavelengths which do not exhibit a kinematically induced secondary eddy, an inertially induced secondary eddy can be created if the mean plate separation is sufficiently small and the global Reynolds number sufficiently large.

We study the effects of side walls on the primary instability of a gravity-driven thin liquid film flowing down a flat channel. The influences of different capillary boundary layer effects at the side walls on the instability of the free surface are resolved experimentally, by varying the crosswise side wall distance of the measurement positions between 5 mm and the channel center-line. The height of the capillary elevation and, thus, the resulting pretensioning of the free surface and the magnitude of a possible velocity overshoot have been adjusted by changing the contact angle between the liquid and the side wall. The influence of the contact angle on the stability of the flow, and especially its range, is remarkable. The difference of the neutral stability curves for the two investigated contact angles is up to 25% and remains significant even up to a side wall distance of 17 times the capillary length. Irrespective of the contact angle, the type of the free surface instability undergoes a transition from long-wave in the center of the channel to short-wave, as is well known for boundary layer flows, when the side wall distance is reduced. Furthermore, we have found that the presence of a velocity overshoot tends to destabilize the free surface.

The increasing demand for thinner films in scientific and technological applications requires a better knowledge of the effect of side walls on such flows, especially as far as the formation of a capillary boundary layer is concerned. In this paper gravity-driven thin film flow in an open channel is investigated, highlighting the competing effects of the no-slip condition and velocity overshoot due to capillary elevation at the bounding side walls. Their influence on the flow rate, the velocity field, the Reynolds number, and the free surface shape is studied for two flow types: (i) the case with vanishingly small capillary elevation at the side walls compared to the film thickness at the center of the channel; (ii) the situation when capillary elevation at the side walls dominates over the film thickness at the center of the channel. For both, large deviations from the two-dimensional reference system occur. The theoretical predictions are compared to experimental observations, for the case of side walls with different wetting properties defined in terms of the static contact angle there.