We consider the dip-coating flow problem when the interface has a purely elastic response and interfacial tension is negligible. We assume the elasticity number El – the ratio of surface elasticity to viscous forces - is small and develop the solution for the free boundary as a matched asymptotic expansion in the small parameter El^{1/7}, thus determining the film thickness as a function of El. A remarkable aspect of the problem is the occurrence of multiple solutions, and five of these are found numerically. In any event, the film thickness varies as El^{4/7}, in agreement with previous experiments.

In hydraulic fracturing, specially engineered suspensions are pumped at high pressure and rate into the reservoir, causing a propagating fracture to open. When the pressure is released the fracture is supported by the grains of solid proppant that are left behind. Recent trends in the oil industry have included the use of cyclic pumping of a proppant slurry interspersed with clear frac fluid, which is found to increase the subsequent productivity. It is therefore of interest to understand how slugs of proppant pumped in a cyclic fashion can disperse in the pipe on the way and finally in the fracture.

We present a model to describe dispersion of solid particles (proppant) along the fracture in a laminar flow of shear thinning yield stress fluids. We consider two-phase governing equations, assuming that solid and fluid phases can be described as two phases of incompressible continua. We adopt the standard Hele-Shaw type scales and multi-timescale approaches to derive a 1D advection/diffusion model along the streamlines for transport and dispersion of the mean solid particle concentration. We show the effects of lift, drag and centrifugal forces on the dispersion dynamics of the particles along the fracture.

Bridgman process is one of the commonly used methods for the manufacturing and growth of semiconductor crystals. In this process, thermal convection plays an important role by affecting heat and mass transfer. Asymmetric flow patterns can lead to an inhomogeneous structure of the growing crystal and also there exist special flow patterns in a fluid domain which determine the governing transport structure and mechanics of the flow.

In this work, we analyze the bulk mixing properties of the melt flow in vertical Bridgman process through the novel approach of Lagrangian coherent structures. A mixed formulation, based on the general form of the enthalpy, is employed for the governing equations coupling Navier-Stokes, continuity and energy equations. A fully implicit second order accurate temporal and spatial finite element discretizations is also used. The obtained velocity field is utilized to identify the transport barriers in terms of finite time Lyapunov exponent ridges whose movements govern the mixing in the melt flow and Lagrangian Coherent Structures are visually revealed to show the transport barriers and flow structure.

We consider natural convection of viscoplastic fluids in a vertical channel with differentially heated side walls. Further to the horizontal gradient, a linear vertical temperature gradient is imposed on both walls to facilitate the study of the buoyancy layers forming on the walls. We show that for sufficiently small yield stress, a boundary layer forms on each wall. Asymptotic techniques are then used to describe the behaviour of the boundary layer as yield stress and stratification parameter vary.

The penalty method is a classical and widespread method for the numerical treatment of boundary conditions such as Dirichlet boundary conditions or unilateral contact boundary conditions. This approach leads to an unconstrained problems and avoids the introduction of additional unknowns in the form of Lagrange multipliers.

In the case of slip boundary conditions for fluid flows or elastic deformations, one of the main obstacle to the efficiency of the finite element method is that a Babuška’s type paradox occurs. Observed first by Sapondzyan [2] and Babuška [1] on the plate equation in a disk with simple support boundary conditions, Babuška’s paradox can be stated as follows: on a sequence of polygonal domains converging to the domain with a smooth boundary, the solutions of the corresponding problems do not converge to the solution of the problem on the limit domain.

Our presentation will focus on the finite element approximation of Stokes equations with slip boundary conditions imposed with the penalty method in two and three space dimensions. For a smooth curved boundary, we prove convergence estimates in terms of both the penalty and discretization parameters. A numerical example is presented confirming this analysis.

[1] Babuška I. Stability of domains with respect to basic problems in the theory of partial differential equations, mainly the theory of elasticity I. (Russian) Czechoslovak Math. J. 11:76–105, 1961.

[2] Sapondzyan O.M. Bending of a simply supported polygonal plate (Russian). Akad. Nauk Armyan. SSR. Izv. Fiz.-Mat. Estest. Tehn. Nauki, 5:29–46, 1952.