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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Notes on numerical fluid mechanics

The flow of a thin Newtonian fluid layer on a porous inclined plane is considered. Applying the long-wave theory, a nonlinear evolution equation for the thickness of the film is obtained. It is assumed that the flow through the porous medium is governed by Darcy’s law. The critical conditions for the onset of instability of a fluid layer flowing down an inclined porous wall, when the characteristic length scale of the pore space is much smaller than the depth of the fluid layer above, are obtained. The results of the linear stability analysis reveal that the film flow system on a porous inclined plane is more unstable than that on a rigid inclined plane and that increasing the permeability of the porous medium enhances the destabilizing effect. A weakly nonlinear stability analysis by the method of multiple scales shows that there is a range of wave numbers with a supercritical bifurcation, and a range of larger wave numbers with a subcritical bifurcation. Numerical solution of the evolution equation in a...

Thin Newtonian film flow down a porous inclined plane: Stability analysis

The axisymmetric motion of a fluid caused by an unsteady stretching surface that has relevance in extrusion process and bioengineering has been investigated. It has been shown that if the unsteady stretching velocity is prescribed by rb/(1 − αt), then the problem admits a similarity solution which gives much insight to the character of solutions. The asymptotic and numerical solutions are obtained and they could be used in the testing of computer codes or analytical models of more realistic engineering systems. The results are governed by a nondimensional unsteady parameter S and it has been observed that no similarity solutions exist for S > 4

The Axisymmetric Motion of a Liquid Film on an Unsteady Stretching Surface

The velocity and pressure in Stokes flow are written in terms of two functions A and B, where A is biharmonic and B is harmonic. Lamb's (1) general solution of Stokes's equations and Oseen's (2) solution due to a Stokeslet in the presence of a no-slip spherical boundary have the same structure as our representation. Ranger's (3) representation follows as a special case of our result. A sphere theorem for non-axisymmetric flow outside or inside a sphere is stated and proved. Collins's theorem (4) for axisymmetric flow follows as a special case of our theorem. A few illustrative examples are given and in each case the drag and torque on the sphere are calculated.

Lamb's solution of stokes's equations: a sphere theorem

The tear film on the front of the eye is critical to proper eyesight; in many mathematical models of the tear film, the tear film is assumed to be on a flat substrate. We re-examine this assumption by studying the effect of a substrate which is representative of the human cornea. We study the flow of a thin fluid film on a prolate spheroid which is a good approximation to the shape of the human cornea. Two lubrication models for the dynamics of the film are studied in prolate spheroidal coordinates which are appropriate for this situation. One is a self-consistent leading-order hyperbolic partial differential equation (PDE) valid for relatively large substrate curvature; the other retains the next higher-order terms resulting in a fourth-order parabolic PDE for the film dynamics. The former is studied for both Newtonian and Ellis (shear thinning) fluids; for typical tear film parameter values, the shear thinning is too small to be significant in this model. For larger shear thinning, we find a significant effect on finite-time singularities. The second model is studied for a Newtonian fluid and allows for a meniscus at one end of the domain. We do not find a strong effect on the thinning rate at the center of the cornea. We conclude that the corneal shape does not have a significant effect on the thinning rate of the tear film for typical conditions.

http://www.math.udel.edu/~driscoll/papers/2011-BraunUshaMcFaddenEtal-121.pdf

Thin film dynamics on a prolate spheroid with application to the cornea

http://iopscience.iop.org/article/10.1016/0169-5983(96)00002-0/pdf

R. Usha

Papers

Thin Newtonian film flow down a porous inclined plane: Stability analysis

The Axisymmetric Motion of a Liquid Film on an Unsteady Stretching Surface

Lamb's solution of stokes's equations: a sphere theorem

Thin film dynamics on a prolate spheroid with application to the cornea

Arbitrary squeezing of a viscous fluid between elliptic plates