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A review of numerical algorithms for the analysis of viscous flows with moving interfaces is presented. The review is supplemented with a discussion of methods that have been introduced in the context of other classes of free boundary problems, but which can be generalized to viscous flows with moving interfaces. The available algorithms can be classified as Eulerian, Lagrangian, and mixed, ie, Eulerian-Lagrangian. Eulerian algorithms consist of fixed grid methods, adaptive grid methods, mapping methods, and special methods. Lagrangian algorithms consist of strictly Lagrangian methods, Lagrangian methods with rezoning, free Lagrangian methods and particle methods. Mixed methods rely on both Lagrangian and Eulerian concepts. The review consists of a description of the present state-of-the-art of each group of algorithms and their applications to a variety of problems. The existing methods are effective in dealing with small to medium interface deformations. For problems with medium to large deformations the methods produce results that are reasonable from a physical viewpoint; however, their accuracy is difficult to ascertain.

Falling films on inclined planes display surface and shear instability modes, the latter being important at small angles β of inclination of the plane. These modes are analyzed with particular attention paid to the effects of surface tension. The results show that the critical Reynolds number of the shear mode is nonmonotonic in either the angle β or the surface-tension parameter ζ but displays a local minimum at nonzero values of β and ζ. For large Reynolds number, an analysis shows that the shear mode is inviscidly stable, but that the surface mode is unstable.

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Linear stability of Couette flow over a wavy wall is considered. It is shown that centrifugal effects may create an instability that leads to the formation of streamwise vortices. The conditions leading to the onset of the instability depend on the amplitude and the wavelength of the waviness and can be expressed in terms of the critical Reynolds number. The global critical conditions describing the minimum critical Reynolds number and the associated wave number of wall waviness required to create the instability for the specified amplitude of the waviness are also given.

Linear stability of wall-bounded shear layers modified by distributed suction has been considered. Wall suction was introduced in order to simulate distributed surface roughness. In all cases studied, i.e. Poiseuille and Couette flows and Blasius boundary layer, wall suction was able to induce a new type of instability characterized by the appearance of streamwise vortices. Results of calculations show that a linear model of suction-induced flow modifications provides a sufficiently accurate representation of the basic state. The effects of an arbitrary suction distribution can, therefore, be assessed by decomposing this distribution into Fourier series and carrying out stability analysis on a mode-by-mode basis, i.e. once and for ever.

Low Reynolds number flow over rectangular cavities is analyzed. The problem is posed to simulate towing tank experiments of Taneda [J. Phys. Soc. Jpn. 46, 1935 (1979)]. A very good agreement exists between numerical and experimental results. The dividing streamline separates from the cavity side wall below the upper corner. The separation point moves toward this corner when the aspect ratio decreases. Flow structure inside the cavity changes considerably with the aspect ratio. Only corner vortices exist in a cavity of aspect ratio W/h=4.0. A decrease of the aspect ratio leads to the enlargement and eventual merger of these vortices. The merger begins with the formation of a stagnation point separating two vortex centers inside the cavity. These vortex centers become progressively weaker and merge to form a single central vortex in a cavity of aspect ratio W/h=2.0. Further decrease of the aspect ratio results in the enlargement of the new corner vortices and their eventual merger. This process begins in a cavity of aspect ratio between 0.6 and 0.575 and is finished in a cavity of aspect ratio 0.5, where two central vortices have been identified. The further decrease of the aspect ratio leads to the sequential repetition of this process and creation of additional central vortices.