Markus Scholle

A potential–based, first–integral formulation of the Navier–Stokes equations is used as a platform for modelling solitary waves in gravity‐driven film flow, giving rise to a nonlinear partial differential equation in complex form. Its subsequent modal decomposition—leading to an infinite set of ordinary differential equations (ODEs), followed by a bimodal approximation and elimination of unknowns—finally results in just two coupled ODEs to be solved. The latter is achieved both analytically and numerically using a Ritz–Galerkin methodology. Illustrative solitary wave results are provided, which exhibit the expected internal recirculating flow pattern and corresponding pronounced, “hump”–like, free–surface deviation—typically observed experimentally and consistent with other analytical/numerical results in the literature. Despite the numerous simplifying assumptions involved in arriving at such a minimalist model, the parameter study undertaken shows that it is capable—based on the number of test functions employed—of providing valid predictions apropos solitary waves; namely, regarding the existence of multiple solutions, having different size and propagation speed, for a given film thickness and effective inclination angle.

A new potential-based first-integral approach is proposed and formulated, facilitating the efficient generation of base-flow solutions for a particular class of bilayer Couette and film flows and a subsequent linear stability analysis of the same. A complicating feature of the flows investigated is the presence of surface topography. The methodology employed gives rise to (i) an advantageous dynamic boundary condition at a layer-separating interface or free-surface; (ii) base-flow solutions having a semi-analytic form; and (iii) the convenient mapping of the associated flow domains to regular rectangular counterparts. Typical base-flow results are provided for periodically repeating surface topography of differing amplitude. The novel stability analyses performed compare results obtained when surface topography is present with those for the corresponding benchmark case when surface topography is absent.

Two bilayer flow arrangements, involving either (i) shear- or (ii) gravity-driven fluids, are explored in the presence of a periodically repeating surface topography. Key simplifying assumptions are that the flow is considered two-dimensional and inertialess. The solutions are obtained utilising a novel first-integral representation of the Navier–Stokes equations, leading to a highly efficient methodology for generating the accompanying flow structure within the layers in tandem with the disturbance experienced by the internal interface separating them—for configuration (ii) this extends also to the upper layer’s free-surface shape. Representative examples of each flow type, with a change of key parameters, are provided with particular emphasis on (ii).

With the aim of describing compressible viscous flows by means of a variational principle that takes into account heat conduction, a recently proposed Lagrangian is subjected to a detailed linear wave analysis that stems directly from the Lagrangian. The accompanying thermodynamic equation of state employed leads to a natural decomposition of the conduction term into three contributions, with the importance of each accessed through a detailed analysis employing a recently developed perturbation methodology giving rise to a favorable system of governing Jacobi equations. In addition to the model Lagrangian itself, three potential model scenarios—based on different combinations of the contributions forming the Lagrangian—are rigorously evaluated and appraised, regarding the occurrence, or otherwise, of dissipation recognizable by an attenuation of harmonic waves. Results reveal that two of the four models are suitable candidates, and suggest one in particular.

The shear–driven, bilayer flow of two immiscible liquids sandwiched between a moving planar and a stationary corrugated plate, in parallel alignment and under Stokes flow conditions, is explored. The motivation, partly inspired by biomimetic considerations, is the use of surface texturing/topography to reduce friction and wear while increasing load carrying capacity in lubrication problems. The starting point of the associated mathematical formulation is a variational principle which, in addition to the classical fluid mechanical field quantities, includes potential fields enabling an elegant reformulation of the dynamic condition at the interface separating the two layers. Two methods of solution are utilised: (i) an analytic one of the field equations, boundary and interface conditions by invoking the long–wave approximation; (ii) an exact semi–analytic one, obtained after transformation to complex variables and application of a spectral method. For validation and comparison purposes a FE formulation of the classical continuity and Navier–Stokes equations is solved for. Resulting streamline patterns, interface shapes and wall shear stresses are discussed.

A rigorous analysis is undertaken based on the analysis of both Galilean and Lorentz (Poincaré) invariance in field theories in general in order to (i) identify a unique analytical scheme for canonical pairs of Lagrangians, one of them having Galilean, the other one Poincaré invariance; and (ii) to obtain transition conditions for the state function purely from Hamilton’s principle and extended Noether’s theorem applied to the aforementioned symmetries. The general analysis is applied on Schrödinger and Klein–Gordon theory, identifying them as a canonical pair in the sense of (i) and exemplified for the scattering problem for both theories for a particle beam against a potential step in order to show that the transition conditions that result according to (ii) in a ‘natural’ way reproduce the well-known ‘methodical’ continuity conditions commonly found in the literature, at least in relevant cases, closing a relevant argumentation gap in quantum mechanical scattering problems.