to be done in this area. Face recognition is a problem that scarcely clamoured for attention before the computer age but, having surfaced, has involved a wide range of techniques and has attracted the attention of some fine minds (David Mumford was a Fields Medallist in 1974). This singular application of mathematical modelling to a messy applied problem of obvious utility and importance but with no unique solution is a pretty one to share with students: perhaps, returning to the source of our opening quotation, we may invert Duncan's earlier observation, 'There is an art to find the mind's construction in the face!'.

Linear and nonlinear waves, by G. B. Whitham. Pp.636. £50. 1999. ISBN 0 471 35942 4 (Wiley).

The boundary layer equations for plane, incompressible, and steady flow are 

$$\matrix{ {u{{\partial u} \over {\partial x}} + v{{\partial u} \over {\partial y}} = - {1 \over \varrho }{{\partial p} \over {\partial x}} + v{{{\partial ^2}u} \over {\partial {y^2}}},} \cr {0 = {{\partial p} \over {\partial y}},} \cr {{{\partial u} \over {\partial x}} + {{\partial v} \over {\partial y}} = 0.} \cr }$$

Boundary Layer Theory

There is a special reason for reviewing this book at this time: it is the 50th edition of a compendium that is known and used frequently in most chemical and physical laboratories in many parts of the world. Surely, a publication that has been published for 56 years, withstanding the vagaries of science in this century, must have had something to offer. There is another reason: while the book is a standard fixture in most chemical and physical laboratories, including those in medical centers, it is not as frequently seen in the laboratories of physician's offices (those either in solo or group practice). I believe that the Handbook can be useful in those laboratories. One of the reasons, among others, is that the various basic items of information it offers may be helpful in new tests, either physical or chemical, which are continuously being published. The basic information may relate

Handbook of Chemistry and Physics

The dynamics and stability of thin liquid films have fascinated scientists over many decades: the observations of regular wave patterns in film flows down a windowpane or along guttering, the patterning of dewetting droplets, and the fingering of viscous flows down a slope are all examples that are familiar in daily life. Thin film flows occur over a wide range of length scales and are central to numerous areas of engineering, geophysics, and biophysics; these include nanofluidics and microfluidics, coating flows, intensive processing, lava flows, dynamics of continental ice sheets, tear-film rupture, and surfactant replacement therapy. These flows have attracted considerable attention in the literature, which have resulted in many significant developments in experimental, analytical, and numerical research in this area. These include advances in understanding dewetting, thermocapillary- and surfactant-driven films, falling films and films flowing over structured, compliant, and rapidly rotating substrates, and evaporating films as well as those manipulated via use of electric fields to produce nanoscale patterns. These developments are reviewed in this paper and open problems and exciting research avenues in this thriving area of fluid mechanics are also highlighted.

Dynamics and stability of thin liquid films

We review flows of dense cohesionless granular materials, with a special focus on the question of constitutive equations. We first discuss the existence of a dense flow regime characterized by enduring contacts. We then emphasize that dimensional analysis strongly constrains the relation between stresses and shear rates, and show that results from experiments and simulations in different configurations support a description in terms of a frictional visco-plastic constitutive law. We then discuss the successes and limitations of this empirical rheology in light of recent alternative theoretical approaches. Finally, we briefly present depth-averaged methods developed for free surface granular flows.

Flows of Dense Granular Media

New models of film flows down inclined planes have been derived by combining a gradient expansion at first or second order to weighted residual techniques with polynomials as test functions. The two-dimensional formulation has been extended to account for three-dimensional flows as well. The full second-order two-dimensional model can be expressed as a set of four coupled evolution equations for four slowly varying fields, the thickness h, the flow rate q and two other quantities measuring the departure from the flat-film semi-parabolic velocity profile. A simplified model has been obtained in terms of h and q only. Including viscous dispersion effects properly, it closely sticks to the asymptotic expansion in the appropriate limit. Our new models improve over previous ones in that they remain valid deep into the strongly nonlinear regime, as shown by the comparison of our results relative to travelling-wave and solitary-wave solutions with those of both direct numerical simulations and experiments.

Improved modeling of flows down inclined planes

Falling Liquid Films gives a detailed review of state-of-the-art theoretical, analytical and numerical methodologies, for the analysis of dissipative wave dynamics and pattern formation on the surface of a film falling down a planar inclined substrate. This prototype is an open-flow hydrodynamic instability, that represents an excellent paradigm for the study of complexity in active nonlinear media with energy supply, dissipation and dispersion. It will also be of use for a more general understanding of specific events characterizing the transition to spatio-temporal chaos and weak/dissipative turbulence. Particular emphasis is given to low-dimensional approximations for such flows through a hierarchy of modeling approaches, including equations of the boundary-layer type, averaged formulations based on weighted residuals approaches and long-wave expansions. Whenever possible the link between theory and experiment is illustrated, and, as a further bridge between the two, the development of order-of-magnitude estimates and scaling arguments is used to facilitate the understanding of basic, underlying physics.

Falling Liquid Films

A new model of film flow down an inclined plane is derived by a method combining results of the classical long wavelength expansion to a weighted-residuals technique. It can be expressed as a set of three coupled evolution equations for three slowly varying fields, the thickness h, the flow-rate q, and a new variable Ƭ that measures the departure of the wall shear from the shear predicted by a parabolic velocity profile. Results of a preliminary study are in good agreement with theoretical asymptotic properties close to the instability threshold, laboratory experiments beyond threshold and numerical simulations of the full Navier-Stokes equations.

Modeling film flows down inclined planes

We study the reliability of two-dimensional models of film flows down inclined planes obtained by us [Ruyer-Quil and Manneville, Eur. Phys. J. B 15, 357 (2000)] using weighted-residual methods combined with a standard long-wavelength expansion. Such models typically involve the local thickness h of the film, the local flow rate q, and possibly other local quantities averaged over the thickness, thus eliminating the cross-stream degrees of freedom. At the linear stage, the predicted properties of the wave packets are in excellent agreement with exact results obtained by Brevdo et al. [J. Fluid Mech. 396, 37 (1999)]. The nonlinear development of waves is also satisfactorily recovered as evidenced by comparisons with laboratory experiments by Liu et al. [Phys. Fluids 7, 55 (1995)] and with numerical simulations by Ramaswamy et al. [J. Fluid Mech. 325, 163 (1996)]. Within the modeling strategy based on a polynomial expansion of the velocity field, optimal models have been shown to exist at a given order in the long-wavelength expansion. Convergence towards the optimum is studied as the order of the weighted-residual approximation is increased. Our models accurately and economically predict linear and nonlinear properties of film flows up to relatively high Reynolds numbers, thus offering valuable theoretical and applied study perspectives.

/pdf/further-accuracy-and-convergence-results-on-the-modeling-of-4x596yuvd2.pdf

Further accuracy and convergence results on the modeling of flows down inclined planes by weighted-residual approximations

In a previous work, two-dimensional film flows were modelled using a weightedresidual approach that led to a four-equation model consistent at order e2. A twoequation model resulted from a subsequent simplification but at the cost of lowering the degree of the approximation to order e only. A Pade approximant technique is applied here to derive a refined two-equation model consistent at order e2. This model, formulated in terms of coupled evolution equations for the film thickness h and the flow rate q, accounts for inertia effects due to the deviations of the velocity profile from the parabolic shape, and closely follows the asymptotic long-wave expansion in the appropriate limit. Comparisons of two-dimensional wave properties with experiments and direct numerical simulations show good agreement for the range of parameters in which a two-dimensional wavy motion is reported in experiments. The stability of two-dimensional travelling waves to three-dimensional perturbations is investigated based on the extension of the models to include spanwise dependence. The secondary instability is found to be not very selective, which explains the widespread presence of the synchronous instability observed in the experiments by Liu et al. (1995) whereas Floquet analysis predicts a subharmonic scenario in most cases. Three-dimensional wave patterns are computed next assuming periodic boundary conditions. Transition from two- to three-dimensional flows is shown to be strongly dependent on initial conditions. The herringbone patterns, the synchronously deformed fronts and the three-dimensional solitary waves observed in experiments are recovered using our regularized model, which is found to be an excellent compromise between the complete model, which has seven equations, and the simplified model, which does not include the second-order inertia corrections. Those corrections are found to play a role in the selection of the type of secondary instability as well as of the spanwise wavelength of the emerging pattern.

/pdf/wave-patterns-in-film-flows-modelling-and-three-dimensional-263n544z44.pdf

Christian Ruyer-Quil

Papers

Improved modeling of flows down inclined planes

Falling Liquid Films

Modeling film flows down inclined planes

Further accuracy and convergence results on the modeling of flows down inclined planes by weighted-residual approximations

Wave patterns in film flows: Modelling and three-dimensional waves