An introduction to heat transfer

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

Numerical methods for Engineers and Scientists

Constructal theory is the view that the generation of flow configuration is a physics phenomenon that can be based on a physics principle (the constructal law): “For a finite-size flow system to persist in time (to survive) its configuration must evolve in such a way that it provides an easier access to the currents that flow through it” [A. Bejan, Advanced Engineering Thermodynamics, 2nd ed. (Wiley, New York, 1997); Int. J. Heat Mass Transfer, 40, 799 (1997)]. This principle predicts natural configuration across the board: river basins, turbulence, animal design (allometry, vascularization, locomotion), cracks in solids, dendritic solidification, Earth climate, droplet impact configuration, etc. The same principle yields new designs for electronics, fuel cells, and tree networks for transport of people, goods, and information. This review describes a paradigm that is universally applicable in natural sciences, engineering and social sciences.

Constructal theory of generation of configuration in nature and engineering

Hydrodynamic and hydromagnetic stability

The effect of an insoluble surfactant on the stability of the gravity-driven flow of a liquid film down an inclined plane is investigated by a normal-mode analysis. Numerical solutions of the Orr–Sommerfeld equation reveal the occurrence of a stable Marangoni mode and a possibly unstable Yih mode, and demonstrate that the primary role of the surfactant is effectively to raise the critical Reynolds number at which instability is first encountered.

Effect of surfactant on the stability of film flow down an inclined plane

The effect of an insoluble surfactant on the stability of two-layer viscous flow in an inclined channel confined by two parallel walls is considered. A lubrication-flow model applicable to long waves and low-Reynolds-number-flow is developed, and pertinent nonlinear evolution equations for the interface position and surfactant concentration are derived. Linear stability analysis based on the lubrication-flow model and the inclusive equations of Stokes flow confirm the recent findings of Frenkel & Halpern (2002) and Halpern & Frenkel (2003) that the presence of an insoluble surfactant induces a Marangoni instability in certain regions of parameter space defined by the layer thickness and viscosity ratios. Numerical simulations based on both approaches show that the interfacial waves may grow and saturate into steep profiles. The lubrication-flow model is adequate in capturing the essential features of the instability for small and moderate wavenumbers.

Effect of surfactants on the stability of two-layer channel flow

A sequence of increasingly refined interpolation grids over the triangle is proposed, with the goal of achieving uniform convergence and ensuring high interpolation accuracy. The number of interpolation nodes, N , corresponds to a complete mth-order polynomial expansion with respect to the triangle barycentric coordinates, which arises by the horizontal truncation of the Pascal triangle. The proposed grid is generated by deploying Lobatto interpolation nodes along the three edges of the triangle, and then computing interior nodes by averaged intersections to achieve three-fold rotational symmetry. Numerical computations show that the Lebesgue constant and interpolation accuracy of the proposed grid compares favorably with those of the best-known grids consisting of the Fekete points. Integration weights corresponding to the set of Lobatto triangle base points are tabulated.

A Lobatto interpolation grid over the triangle

The gravity-driven flow of a liquid film down an inclined wall with periodic indentations in the presence of a normal electric field is investigated. The film is assumed to be a perfect conductor, and the bounding region of air above the film is taken to be a perfect dielectric. In particular, the interaction between the electric field and the topography is examined by predicting the shape of the film surface under steady conditions. A nonlinear, non-local evolution equation for the thickness of the liquid film is derived using a long-wave asymptotic analysis. Steady solutions are computed for flow into a rectangular trench and over a rectangular mound, whose shapes are approximated with smooth functions. The limiting behaviour of the film profile as the steepness of the wall geometry is increased is discussed. Using substantial numerical evidence, it is established that as the topography steepness increases towards rectangular steps, trenches, or mounds, the interfacial slope remains bounded, and the film does not touch the wall. In the absence of an electric field, the film develops a capillary ridge above a downward step and a slight depression in front of an upward step. It is demonstrated how an electric field may be used to completely eliminate the capillary ridge at a downward step. In contrast, imposing an electric field leads to the creation of a free-surface ridge at an upward step. The effect of the electric field on film flow into relatively narrow trenches, over relatively narrow mounds, and down slightly inclined substrates is also considered.

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Mark Blyth

Papers

Effect of surfactant on the stability of film flow down an inclined plane

Effect of surfactants on the stability of two-layer channel flow

A Lobatto interpolation grid over the triangle

Electrified viscous thin film flow over topography

Microfluidics for pharmaceutical nanoparticle fabrication: the truth and the myth