We study the flow of a thin layer of fluid over a flat surface. Commonly, the 1-D Shallow-water or Saint-Venant set of equations are used to compute the solution of such flows. These simplified equations may be obtained through the integration of the Navier–Stokes equations over the depth of the fluid, but their solution requires the introduction of constitutive relations based on strict hypothesis on the flow régime. Here, we present an approach based on a kind of boundary layer system with hydrostatic pressure. This relaxes the need for closure relations which are instead obtained as solutions of the computation. It is then demonstrated that the corresponding closures are very dependent on the type of flow considered, for example laminar viscous slumps or hydraulic jumps. This has important practical consequences as far as the applicability of standard closures is concerned.

Multiphase shear flows often show banded structures that affect the global behavior of complex fluids e.g. in microdevices. Here we investigate numerically the banding of emulsions, i.e. the formation of regions of high and low volume fractions, alternated in the vorticity direction and aligned with the flow (shear bands). These bands are associated with a decrease of the effective viscosity of the system. To understand the mechanism of experimentally observed banding, we have performed interface-resolved simulations of the two-fluid system. The experiments were performed starting with a random distribution of droplets, which under the applied shear, evolve in time resulting in a phase separation. To numerically reproduce this process, the banded structures are initialized in a narrow channel confined by two walls moving in opposite directions. We find that the initial banded distribution is stable when droplets are free to merge and unstable when coalescence is prevented. In this case, additionally, the effective viscosity of the system increases, resembling the rheological behavior of suspensions of deformable particles. Droplet coalescence, on the other hand, allows emulsions to reduce the total surface of the system and, hence, the energy dissipation associated with the deformation, which in turn reduces the effective viscosity.

A numerical and theoretical study of yield-stress fluid flows in two types of model porous media is presented. We focus on viscoplastic and elastoviscoplastic flows to reveal some differences and similarities between these two classes of flows. Small elastic effects increase the pressure drop and also the size of unyielded regions in the flow which is the consequence of different stress solutions compare to viscoplastic flows. Yet, the velocity fields in the viscoplastic and elastoviscoplastic flows are comparable for small elastic effects. By increasing the yield stress, the difference in the pressure drops between the two classes of flows becomes smaller and smaller for both considered geometries. When the elastic effects increase, the elastoviscoplastic flow becomes time-dependent and some oscillations in the flow can be observed. Focusing on the regime of very large yield stress effects in the viscoplastic flow, we address in detail the interesting limit of 'flow/no flow': yield-stress fluids can resist small imposed pressure gradients and remain quiescent. The critical pressure gradient which should be exceeded to guarantee a continuous flow in the porous media will be reported. Finally, we propose a theoretical framework for studying the 'yield limit' in the porous media.

The effect of wind on the wave breaking induced by the modulational instability is investigated numerically using the open source software Gerris. The two-phase flow is modelled by the two-dimensional Navier-Stokes equations for a single incompressible fluid and a Volume of Fluid technique is employed to capture the air-water interface. The flow is initialized as a fundamental wave component with two side-band perturbations so that the breaking is induced via the Benjamin-Feir instability mechanism. In order to investigate its effect on the wave evolution and on the breaking process, a uniform wind, twice the phase speed, is assigned in the air phase. The simulation covers the initial development of the wind profile, the growth of the modulational instability, the breaking and post breaking phases. Results show the occurrence of air flow separation from the wave crests shortly after the initial start. Pressure and tangential stress acting on the free surface are computed. It is shown that due to the flow separation there is a favourable pressure gradient about the wave crests whereas the tangential stresses are generally in favour of the wave propagation on the back of the wave but are opposed to the propagation along the forward face and in the wave trough. An initial growth of the energy content in water is observed, followed by a dissipation stage which is not related to the breaking process. In agreement with the experiments, the growth rate of the side-bands is reduced when compared to the corresponding no-wind solution. Because of the slower growth, the limiting condition for the onset of the breaking is reached with some delay. At the end of the breaking process, when the downshift is completed, the amplitude of the left side-band in the wind case is somewhat lower than that for the no-wind case. No substantial differences have been found in terms of the total energy dissipated by the whole breaking process although the dissipation rate for the wind case is noticeably higher. The higher dissipation rate observed in the wind case is found to be related to the larger amount of air entrained by the breaking process.

We present a modification of a recently developed volume of fluid method for multiphase problems (Ii et al. in J Comput Phys 231(5):2328-2358, 2012), so that it can be used in conjunction with a fractional-step method and fast Poisson solver, and validate it with standard benchmark problems. We then consider emulsions of two-fluid systems and study their rheology in a plane Couette flow in the limit of vanishing inertia. We examine the dependency of the effective viscosity on the volume fraction phi (from 10 to 30%) and the Capillary number Ca (from 0.1 to 0.4) for the case of density and viscosity ratio 1. We show that the effective viscosity decreases with the deformation and the applied shear (shear-thinning) while exhibiting a non-monotonic behavior with respect to the volume fraction. We report the appearance of a maximum in the effective viscosity curve and compare the results with those of suspensions of rigid and deformable particles and capsules. We show that the flow in the solvent is mostly a shear flow, while it is mostly rotational in the suspended phase; moreover, this behavior tends to reverse as the volume fraction increases. Finally, we evaluate the contributions to the total shear stress of the viscous stresses in the two fluids and of the interfacial force between them.

We present a numerical study of the rheology of a two-fluid emulsion in dilute and semidilute conditions. The analysis is performed for different capillary numbers, volume fractions and viscosity ratios under the assumption of negligible inertia and zero buoyancy force. The effective viscosity of the system increases for low values of the volume fraction and decreases for higher values, with a maximum for approximately 20% concentration of the disperse phase. When the dispersed fluid has lower viscosity, the normalised effective viscosity becomes smaller than 1 for high enough volume fractions. To single out the effect of droplet coalescence on the rheology of the emulsion we introduce an Eulerian force which prevents merging, effectively modelling the presence of surfactants in the system. When the coalescence is inhibited the effective viscosity is always greater than 1 and the curvature of the function representing the emulsion effective viscosity versus the volume fraction becomes positive, resembling the behaviour of suspensions of deformable particles. The reduction of the effective viscosity in the presence of coalescence is associated with the reduction of the total surface of the disperse phase when the droplets merge, which leads to a reduction of the interface tension contribution to the total shear stress. The probability density function of the flow topology parameter shows that the flow is mostly a shear flow in the matrix phase, with regions of extensional flow when the coalescence is prohibited. The flow in the disperse phase, instead, always shows rotational components. The first normal stress difference is positive, except for the smallest viscosity ratio considered, whereas the second normal difference is negative, with their ratio being constant with the volume fraction. Our results clearly show that the coalescence efficiency strongly affects the system rheology and that neglecting droplet merging can lead to erroneous predictions.

We investigate the elastoviscoplastic flow through porous media by numerical simulations. We solve the Navier–Stokes equations combined with the elastoviscoplastic model proposed by Saramito for the stress tensor evolution [1]. In this model, the material behaves as a viscoelastic solid when unyielded, and as a viscoelastic Oldroyd-B fluid for stresses higher than the yield stress. The porous media is made of a symmetric array of cylinders, and we solve the flow in one periodic cell. We find that the solution is time-dependent even at low Reynolds numbers as we observe oscillations in time of the unyielded region especially at high Bingham numbers. The volume of the unyielded region slightly decreases with the Reynolds number and strongly increases with the Bingham number; up to 70% of the total volume is unyielded for the highest Bingham numbers considered here. The flow is mainly shear dominated in the yielded region, while shear and elongational flow are equally distributed in the unyielded region. We compute the relation between the pressure drop and the flow rate in the porous medium and present an empirical closure as function of the Bingham and Reynolds numbers. The apparent permeability, normalized with the case of Newtonian fluids, is shown to be greater than 1 at low Bingham numbers, corresponding to lower pressure drops due to the flow elasticity, and smaller than 1 for high Bingham numbers, indicating larger dissipation in the flow owing to the presence of the yielded regions. Finally we investigate the effect of the Weissenberg number on the distribution of the unyielded regions and on the pressure gradient.