We consider the linear and nonlinear stability of two-phase density-matched but viscosity-contrasted fluids subject to laminar Poiseuille flow in a channel, paying particular attention to the formation of three-dimensional waves. A combination of Orr–Sommerfeld–Squire analysis (both modal and non-modal) with direct numerical simulation of the three-dimensional two-phase Navier–Stokes equations is used. For the parameter regimes under consideration, under linear theory, the most unstable waves are two-dimensional. Nevertheless, we demonstrate several mechanisms whereby three-dimensional waves enter the system, and dominate at late time. There exists a direct route, whereby three-dimensional waves are amplified by the standard linear mechanism; for certain parameter classes, such waves grow at a rate less than but comparable to that of the most dangerous two-dimensional mode. Additionally, there is a weakly nonlinear route, whereby a purely spanwise wave grows according to transient linear theory and subsequently couples to a streamwise mode in weakly nonlinear fashion. Consideration is also given to the ultimate state of these waves: persistent three-dimensional nonlinear waves are stretched and distorted by the base flow, thereby producing regimes of ligaments, ‘sheets’ or ‘interfacial turbulence’. Depending on the parameter regime, these regimes are observed either in isolation, or acting together.

We study droplet impact problems in a 3D cylindrical or equivalent 2D Cartesian geometry. Such structures do have an approximate experimental realization, and they are often simulated as a testbed for computational methods. We focus on droplet impact on a smooth homogeneous surface as well as head-on collision of two droplets. We perform an energy-budget analysis and introduce a correlation, which predicts the maximum spreading radius as a function of Reynolds number and Weber number. We show how the dissipation term in this analysis can be decomposed into boundary-layer dissipation in the droplet lamella (where applicable) and head loss. We use existing results in the literature (simulations and experiments) as well as our own simulation results to validate the correlation. Dissipation by head loss is a key term in the analysis: only by modeling it accurately, one can obtain good agreement between the simulations and the theory.

The Cahn-Hilliard equation with an externally prescribed chaotic shear flow is studied in two and three dimensions. The main goal is to compare and contrast the phase separation in two and three dimensions, using high-resolution numerical simulation as the basis for the study. The model flow is parametrized by its amplitudes (thereby admitting the possibility of anisotropy), length scales, and multiple time scales, and the outcome of the phase separation is investigated as a function of these parameters as well as the dimensionality. In this way, a parameter regime is identified wherein the phase separation and the associated coarsening phenomenon are not only arrested but in fact the concentration variance decays, thereby opening up the possibility of describing the dynamics of the concentration field using the theories of advection diffusion. This parameter regime corresponds to long flow correlation times, large flow amplitudes and small diffusivities. The onset of this hyperdiffusive regime is interpreted by introducing Batchelor length scales. A key result is that in the hyperdiffusive regime, the distribution of concentration (in particular, the frequency of extreme values of concentration) depends strongly on the dimensionality. Anisotropic scenarios are also investigated: for scenarios wherein the variance saturates (corresponding to coarsening arrest), the direction in which the domains align depends on the flow correlation time. Thus, for correlation times comparable to the inverse of the mean shear rate, the domains align in the direction of maximum flow amplitude, while for short correlation times, the domains initially align in the opposite direction. However, at very late times (after the passage of thousands of correlation times), the fate of the domains is the same regardless of correlation time, namely alignment in the direction of maximum flow amplitude. A theoretical model to explain these features is proposed. These features and the theoretical model carry over to the three-dimensional case, albeit that an extra degree of freedom pertains, such that the dynamics of the domain alignment in three dimensions warrant a more detailed consideration, also presented herein.

We introduce a mathematical model with a mesh-free numerical method to describe contact-line motion in lubrication theory. We show how the model resolves the singularity at the contact line, and generates smooth profiles for an evolving, spreading droplet. The model describes well the physics of droplet spreading–including Tanner’s Law for the evolution of the contact line. The model can be configured to describe complete wetting or partial wetting, and we explore both cases numerically. In the case of partial wetting, the model also admits analytical solutions for the droplet profile, which we present here. Article highlights: We formulate a mathematical model to regularize the contact-line singularity for droplet spreading. The model can be solved using a fast, accurate mesh-free numerical method. Numerical simulations confirm that the model describes the quantitative aspects of droplet spreading well.

We introduce a modified and simplified version of the pre-existing fully parallelized three-dimensional Navier–Stokes flow solver known as TPLS. We demonstrate how the simplified version can be used as a pedagogical tool for the study of computational fluid dynamics (CFDs) and parallel computing. TPLS is at its heart a two-phase flow solver, and uses calls to a range of external libraries to accelerate its performance. However, in the present context we narrow the focus of the study to basic hydrodynamics and parallel computing techniques, and the code is therefore simplified and modified to simulate pressure-driven single-phase flow in a channel, using only relatively simple Fortran 90 code with MPI parallelization, but no calls to any other external libraries. The modified code is analysed in order to both validate its accuracy and investigate its scalability up to 1000 CPU cores. Simulations are performed for several benchmark cases in pressure-driven channel flow, including a turbulent simulation, wherein the turbulence is incorporated via the large-eddy simulation technique. The work may be of use to advanced undergraduate and graduate students as an introductory study in CFDs, while also providing insight for those interested in more general aspects of high-performance computing.