Markus Scholle

Vortices in film flow over strongly undulated bottom profiles at low Reynolds numbers

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.

Competing geometric and inertial effects on local flow structure in thick gravity-driven fluid films

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 substrate) 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 i...

Eddy genesis and manipulation in plane laminar shear flow

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 semianalytically 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. However, as the plate clearance ...

A first integral of Navier–Stokes equations and its applications

Although it is well known that Bernoullis equation is obtained as the first integral of Eulers 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.

Construction of Lagrangians in continuum theories

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 Schrödingers 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 Noethers 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.

On a potential-velocity formulation of Navier-Stokes equations

Computational methods in continuum mechanics, especially those encompassing fluid dynamics, have emerged as an essential investigative tool in nearly every field of technology. Despite being underpinned by a well-developed mathematical theory and the existence of readily available commercial software codes, computing solutions to the governing equations of fluid motion remains challenging: in essence due to the non-linearity involved. Additionally, in the case of free surface film flows the dynamic boundary condition at the free surface complicates the mathematical treatment notably. Recently, by introduction of an auxiliary potential field, a first integral of the two-dimensional Navier-Stokes equations has been constructed leading to a set of equations, the differential order of which is lower than that of the original Navier-Stokes equations. In this paper a physical interpretation is provided for the new potential, making use of the close relationship between plane Stokes flow and plane linear elasticity. Moreover, it is shown that by application of this alternative approach to free surface flows the dynamic boundary condition is reduced to a standard Dirichlet-Neumann form, which allows for an elegant numerical treatment. A least squares finite element method is applied to the problem of gravity driven film flow over corrugated substrates in order to demonstrate the capabilities of the new approach. Encapsulating non-Newtonian behaviour and extension to three-dimensional problems is discussed briefly.

Line–shaped objects and their balances related to gauge symmetries in continuum theories

Within the Lagrange formalism Noethers theorem is a well–known tool for connecting symmetries of a physical system with homogeneous balance equations. In the context of this paper we call them balance equations of the volume type. They are associated with symmetry groups of the Lie type. However, in physics there are a lot of different balance equations which we call balance equations of the area type. Physically, they are associated with the dynamics of line–shaped objects. In this paper a general theory is presented which supplements Noethers theorem in so far as the area–type balances are associated with regauging symmetry groups of the non–Lie type. The theory is demonstrated for three prominent examples: for the Helmholtz laws of the vortex dynamics in an ideal fluid, for the dislocation dynamics in the dynamical eigenstress problem of elastic crystals, and for the homogeneous Maxwell equations. The theory will also be a valuable tool for solving inverse variational problems, i.e. to construct Lagrangians for physical systems.

A non-conventional discontinuous Lagrangian for viscous flow

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 Lagranges 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.

A complex-valued first integral of Navier-Stokes equations: Unsteady Couette flow in a corrugated channel system

For a two-dimensional incompressible viscous flow, a first integral of the governing equations of motion is constructed based on a reformulation of the unsteady Navier-Stokes equations in terms of complex variables and the subsequent introduction of a complex potential field; complementary solid and free surface boundary conditions are formulated. The methodology is used to solve the challenging problem of unsteady Couette flow between two sinusoidally varying corrugated rigid surfaces utilising two modelling approaches to highlight the versatility of the first integral. In the Stokes flow limit, the results obtained in the case of steady flow are found to be in excellent agreement with corresponding investigations in the open literature. Similarly, for unsteady flow, the results are in accord with related investigations, exploring material transfer between trapped eddies and the associated bulk flow, and vice versa. It is shown how the work relates to the classical complex variable method for solving the...

Exact integration of the unsteady incompressible Navier-Stokes equations, gauge criteria, and applications

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.