Morten Brøns

We study numerically the streamline patterns in the incompressible, two-dimensional flow in a rectangular lid-driven cavity with an embedded cylinder. The problem is characterized by two parameters, the dimensionless cylinder radius and the Reynolds number. With the aid of bifurcation theory we identify 17 topologically distinct streamline patterns as well as bifurcation curves in the parameter plane, separating the realms of the patterns. The introduction of a small cylinder affects the streamline pattern significantly and gives rise to a complex sequence of bifurcations. For larger cylinders, the bifurcations become simpler, and the size of a corner-crossing recirculation zone decreases, such that the cylinder improves the overall circulation in the cavity.

The critical points of vorticity in a two-dimensional viscous flow are essential for identifying coherent structures in the vorticity field. Their bifurcations as time progresses can be associated with the creation, destruction or merging of vortices, and we analyse these processes using the equation of motion for these points. The equation decomposes the velocity of a critical point into advection with the fluid and a drift proportional to viscosity. Conditions for the drift to be small or vanish are derived, and the analysis is extended to cover bifurcations. We analyse the dynamics of vorticity extrema in numerical simulations of merging of two identical vortices at Reynolds numbers ranging from 5 to 1500 in the light of the theory. We show that different phases of the merging process can be identified on the basis of the balance between advection and drift of the critical points, and identify two types of merging, one for low and one for high values of the Reynolds number. In addition to local maxima of positive vorticity and minima of negative vorticity, which can be considered centres of vortices, minima of positive vorticity and maxima of negative vorticity can also exist. We find that such anti-vortices occur in the merging process at high Reynolds numbers, and discuss their dynamics.

We develop a bifurcation theory for three-dimensional vortices defined by the Q-criterion, with time as the parameter. Two types of local bifurcations are identified, denoted punching or pinching, occurring in two settings, either in the flow or close to a no-slip wall. The bifurcations represent the creation, destruction, merging, or splitting of vortices. Furthermore, in a global setting, they can also describe the transition between a spherical vortex and a vortex ring as well as the creation of a hairpin vortex from a wall. Scaling laws for the size of vortices and the distance between vortices close to bifurcation are obtained. The analysis is of a topological and general nature and will also hold for vortices defined by other local vortex criteria.

We study electrohydrodynamic (EHD) linear (in)stability of microfluidic channel flows, i.e., the stability of interface between two shearing viscous (perfect) dielectrics exposed to an electric field in large aspect ratio microchannels. We then apply our results to particular microfluidic systems known as two-liquid electroosmotic (EO) pumps. Our novel results are detailed analytical expressions for the growth rate of two-dimensional EHD modes in Couette–Poiseuille flows in the limit of small Reynolds number (R); the expansions to both zeroth and first order in R are considered. The growth rates are complicated functions of viscosity-, height-, density-, and dielectric-constant ratio, as well as of wavenumbers and voltages. To make the results useful to experimentalists, e.g., for voltage-control EO pump operations, we also derive equations for the impending voltages of the neutral stability curves that divide stable from unstable regions in voltage–wavenumber stability diagrams. The voltage equations and the stability diagrams are given for all wavenumbers. We finally outline the flow regimes in which our first-order-R voltage corrections could potentially be experimentally measured. Our work gives insight into the coupling mechanism between electric field and shear flow in parallel-planes channel flows, correcting an analogous EHD expansion to small R from the literature. We also revisit the case of pure shear instability, when the first-order-R voltage correction equals zero, and replace the renowned instability mechanism due to viscosity stratification at small R with the mechanism due to discontinuity in the slope of the unperturbed velocity profile.

We study electrohydrodynamic (EHD) linear (in)stability of microfluidic channel flows, i.e. the stability of interface between two-shearing viscous (perfect) dielectrics exposed to an electric field in large aspect ratio microchannels. We then apply our results to particular microfluidic systems known as electroosmotic (EO) pumps. Our results are detailed analytical expressions for the growth rate of two-dimensional EHD modes in Couette-Poiseuille flows in the limit of small Reynolds number ($R$); the expansions to both zeroth- and first-order-$R$ are considered. The growth rates are complicated functions of viscosity-, height-, density- and dielectric-constant ratio, as well as of wavenumbers and voltages, and to our knowledge have not been presented before in literature. To make the results more useful, e.g., for voltage-control EO pump operations, we also derive equations for the impending voltages of the neutral stability curves that divide stable from unstable regions in voltage-wavenumber stability diagrams. The voltage equations and the stability diagrams are given for all wavenumbers. We finally outline the flow regimes in which our first-order-$R$ voltage corrections could potentially be experimentally measured. Our work gives the insight into the coupling mechanism between electric field and shear flow in parallel-planes channel flows, correcting an erroneous attempt from literature. Our general analysis enables us also to refine the renowned instability mechanism due to viscosity stratification, i.e., the case of pure shear instability, when the first-order-$R$ voltage correction equals zero.

Mixed-mode oscillations consisting of alternating small- and large-amplitude oscillations are increasingly well understood and are often caused by folded singularities, canard orbits, or singular Hopf bifurcations. We show that coupling between identical nonlinear oscillators can cause mixed-mode oscillations because of symmetry breaking. This behavior is illustrated for diffusively coupled FitzHugh–Nagumo oscillators with negative coupling constant, and we show that it is caused by a singular Hopf bifurcation related to a folded saddle-node (FSN) singularity. Inspired by earlier work on models of pancreatic beta-cells [Sherman, Bull. Math. Biol. 56, 811 (1994)], we then identify a new type of bursting dynamics due to diffusive coupling of cells firing action potentials when isolated. In the presence of coupling, small-amplitude oscillations in the action potential height precede transitions to square-wave bursting. Confirming the hypothesis from the earlier work that this behavior is related to a pitchfork-of-limit-cycles bifurcation in the fast subsystem, we find that it is caused by symmetry breaking. Moreover, we show that it is organized by a FSN in the averaged system, which causes a singular Hopf bifurcation. Such behavior is related to the recently studied dynamics caused by the so-called torus canards.

We investigate vortex pair interactions at low Reynolds numbers. We base our analysis on the Q-criterion, where a vortex is defined as a region where the local rotation dominates the strain, and we make use of a topological approach to describe the qualitative changes of the vortex structure. In order to give a complete description of vortex pair interactions we further develop a general bifurcation theory for . Furthermore, we observe an excellent agreement between the core growth model and the numerical simulations for Reynolds numbers below 10. We therefore conclude that, instead of solving the Navier–Stokes equation numerically we can, for sufficiently small Reynolds numbers, apply the core growth model as a simple, analytically tractable model with a low dimension.

The flow around a rotating circular cylinder has a parameter regime with a complex pattern of periodic solutions and multiple steady states. Sierra et al. (J. Fluid Mech., vol. 905, 2020, A2) provide a complete bifurcation analysis of this regime. The numerical computations are guided by a qualitative analysis of the bifurcations stemming from a highly degenerate singular dynamical system. Surprisingly, the dynamics of the singular system itself cannot be realized as a specific flow, but acts mathematically as an organizer of the physical bifurcation diagram.

We identify an incorrect term in the expression for the rate of change of circulation of a material volume including an interface between two fluids which appears in Brøns et al. (J. Fluid Mech., vol. 758, 2014, pp. 63–93).