Advanced Mathematical Methods for Scientists and Engineers

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

The viscous flow induced by a shrinking sheet is studied. Existence and (non)uniqueness are proved. Exact solutions, both numerical and in closed form, are found.

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Viscous flow due to a shrinking sheet

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Principles Of Enhanced Heat Transfer

Notes on numerical fluid mechanics

Steady solutions of an inverse problem relevant to gravity-driven film flows over an undulated slippery bottom are considered. Given a target free surface shape, the goal is to obtain the corresponding bottom topography of a slippery substrate which causes the specified free surface shape for a film flowing over it. The approaches followed by Sellier (Phys Fluids 20:062106, 2008) for creeping films and by Heining and Aksel (Phys Fluids 21:083605, 2009) for inertial films for reconstructing a rigid bottom topography for a target free surface profile are extended to the reconstruction of a slippery bottom topography. The model equations for film thickness above the bottom topography are derived for creeping flows under lubrication approximation and for inertial films using the weighted-residual integral boundary layer method and are solved numerically. The influence of inertia, slip parameter and surface tension on the shape of the reconstructed bottom topography is analyzed for different prescribed free surface shapes (sinusoidal, trench and bell-shaped). It is observed that the nonlinearities that appear in the reconstructed rigid bottom substrate with no slip at the substrate are suppressed by seeking the bottom substrate to be reconstructed as a slippery substrate. A spatial linear stability analysis of the corresponding direct problem is examined using Floquet theory, and the results reveal that the slip parameter and surface tension have a high influence on the critical Reynolds number. The results provide a strategy for controlling surface defects in a gravity-driven film over a substrate; namely, in order to achieve a target free surface profile, one can design the bottom substrate to be an undulated rough/textured/grooved or a superhydrophobic surface which can be modelled as an undulated smooth substrate with velocity slip at the substrate.

Steady solution and spatial stability of gravity-driven thin-film flow: reconstruction of an uneven slippery bottom substrate

A theoretical analysis of the thermal effects on the dynamics of a thin nonuniform film of a nonvolatile incompressible viscous fluid on a heated rotating disk has been considered and the effects of temperature-dependent viscosity and surface tension have been analyzed. A nonlinear evolution equation describing the shape of the film interface has been derived as a function of space and time and its stability characteristics have been examined using linear theory. It has been observed that the infinitesimal disturbances decay for small wave numbers and are transiently stable for large wave numbers, for both zero and nonzero values of Biot number.

Dynamics and stability of a thin liquid film on a heated rotating disk film with variable viscosity

Sphere theorem for stokes' flow

A thin liquid layer of a non-Newtonian film falling down an inclined plane that is subjected to nonuniform heating has been considered. The temperature of the inclined plane is assumed to be linearly distributed and the case when the temperature gradient is positive or negative is investigated. The film flow is influenced by gravity, mean surface tension, and thermocapillary forces acting along the free surface. The coupling of thermocapillary instability and surface-wave instabilities is studied for two-dimensional disturbances. A nonlinear evolution equation is derived by applying the long-wave theory, and the equation governs the evolution of a power-law film flowing down a nonuniformly heated inclined plane. The linear stability analysis shows that the film flow system is stable when the plate temperature decreases in the downstream direction while it is less stable for increasing temperature along the plate. Weakly nonlinear stability analysis using the method of multiple scales has been investigated and this leads to a secular equation of the Ginzburg-Landau type. The analysis shows that both supercritical stability and subcritical instability are possible for the film flow system. The results indicate the existence of finite-amplitude waves, and the threshold amplitude and nonlinear speed of these waves are influenced by thermocapillarity. The nonlinear evolution equation for the film thickness is solved numerically in a periodic domain in the supercritical stable region, and the results show that the shape of the wave is influenced by the choice of wave number, non-Newtonian rheology, and nonuniform heating.

Long-Wave Instabilities in a Non-Newtonian Film on a Nonuniformly Heated Inclined Plane

Non-Newtonian fluids can present a complex rheological behaviour involving shear-thinning, viscoelastic or thixotropic effects. We focus here on the characterization of generalized Newtonian fluids as shear-thinning fluids. Rotational rheometers are torque-sensitive, which makes the measurement of viscosity at low shear-rates very difficult, in particular for low viscosity fluids. An accurate knowledge of this value is, however, particularly important for the characterization of several types of flows, in particular those featuring a free surface, a symmetry axis or a symmetry plane. The experimental determination of viscosity and surface tension is enabled from the propagation of attenuated capillary waves. An optical technique has been implemented in order to determine the shear-thinning viscosity at values of the shear-rate as small as 10 −3  s −1 from measurements of the spatial attenuation and wavelength. We have performed measurements on two different polymer solutions; we show that this technique is complementary to the classical rotational rheometer techniques since it gives access to the zero shear viscosity, which could not be measured by using rotational rheometers. The present technique is thus used to characterize the Newtonian plateau, while a rotational rheometer is used to characterize the fluid behaviour at moderate to high shear rates.

R. Usha

Papers

Steady solution and spatial stability of gravity-driven thin-film flow: reconstruction of an uneven slippery bottom substrate

Dynamics and stability of a thin liquid film on a heated rotating disk film with variable viscosity

Sphere theorem for stokes' flow

Long-Wave Instabilities in a Non-Newtonian Film on a Nonuniformly Heated Inclined Plane

Experimental determination of the viscosity at very low shear rate for shear thinning fluids by electrocapillarity