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We present an experimental study of gravity driven films flowing down sinusoidal bottom profiles of high waviness. We find vortices in the valleys of the undulated bottom profile. They are observed at low Reynolds numbers down to the order of 10−5. The vortices are visualized employing a particle image velocimeter with fluorescent tracers. It turns out that the vortices are generated beyond a critical film thickness. Their size tends to a finite value for thick films. The critical film thickness depends on the waviness of the bottom undulation, the inclination angle, and on the surface tension but not on the Reynolds number. Increasing the waviness, a second vortex can be generated.

The full-text may be used and/or reproduced, and given to third parties in any format or medium, without prior permission or charge, for personal research or study, educational, or not-for-prot purposes provided that:• a full bibliographic reference is made to the original source • a link is made to the metadata record in DRO • the full-text is not changed in any way The full-text must not be sold in any format or medium without the formal permission of the copyright holders.Please consult the full DRO policy for further details. An exact first integral of the full, unsteady, incompressible Navier-Stokes equations is achieved in its most general form via the introduction of a tensor potential and parallels drawn with Maxwell's theory. Subsequent to this gauge freedoms are explored, showing that when used astutely they lead to a favourable reduction in the complexity of the associated equation set and number of unknowns, following which the inviscid limit case is discussed. Finally, it is shown how a change in gauge criteria enables a variational principle for steady viscous flow to be constructed having a self-adjoint form. Use of the new formulation is demonstrated, for different gauge variants of the first integral as the starting point, through the solution of a hierarchy of classical three-dimensional flow problems; two of which are tractable analytically, the third being solved numerically. In all cases the results obtained are found to be in excellent accord with corresponding solutions available in the open literature. Concurrently, the prescription of appropriate commonly occurring physical and necessary auxiliary boundary conditions, incorporating for completeness the derivation of a first integral of the dynamic boundary condition at a free surface, is established, together with how the general approach can be advantageously reformulated for application in solving unsteady flow problems with periodic boundaries.

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.

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For physical systems the dynamics of which is formulated within the framework of Lagrange formalism, the dynamics is completely defined by only one function, namely the Lagrangian. For instance, the whole conservative Newtonian mechanics has been successfully embedded into this methodical concept. In continuum theories, however, the situation is different: no generally valid construction rule for the Lagrangian has been established in the past.In this paper general properties of Lagrangians in non-relativistic field theories are derived by considering universal symmetries, namely space-and time-translations, rigid rotations and Galilei boosts. These investigations discover the dual structure, i.e. the coexistence of two complementary representations of the Lagrangian. From the dual structure, relevant restrictions for the analytical form of the Lagrangian are derived which eventually result in a general scheme for Lagrangians. For two examples, namely Schrödinger's theory and the flow of an ideal fluid, the compatibility of the Lagrangian with the general scheme is demonstrated.The dual structure also has consequences for the balances which result from the respective symmetries by Noether's theorem: universally valid constitutive relations between the densities and the flux densities of energy, momentum, mass and centre of mass are derived. By an inverse treatment of these constitutive relations a Lagrangian for a given physical system can be constructed. This procedure is demonstrated for an elastically deforming body.

Drawing an analogy with quantum mechanics, a new Lagrangian is proposed for a variational formulation of the Navier–Stokes equations which to-date has remained elusive. A key feature is that the resulting Lagrangian is discontinuous in nature, posing additional challenges apropos the mathematical treatment of the related variational problem, all of which are resolvable. In addition to extending Lagrange's formalism to problems involving discontinuous behaviour, it is demonstrated that the associated equations of motion can self-consistently be interpreted within the framework of thermodynamics beyond local equilibrium, with the limiting case recovering the classical Navier–Stokes equations. Perspectives for applying the new formalism to discontinuous physical phenomena such as phase and grain boundaries, shock waves and flame fronts are provided.

Analytical expressions for the velocity fields corresponding to the motions of an Oldroyd-B fluid due to oscillations of an infinite flat plate as well as those induced by an oscillating pressure gradient and pressure jumps are determined by means of the Fourier sine transform. The corresponding solutions for a Maxwell and a Newtonian fluid appear as limiting cases of the solutions established here. Relevant physical properties of the flows and their dependence on material and geometry parameters are discussed.