Beyond-Newtonian dynamics of a planar circular restricted three-body problem with Kerr-like primaries

ABSTRACT The dynamics of the planar circular restricted three-body problem with Kerr-like primaries in the context of a beyond-Newtonian approximation is studied. The beyond-Newtonian potential is developed by using the Fodor–Hoenselaers–Perjés procedure. An expansion in the Kerr potential is performed and terms up to the first non-Newtonian contribution of both the mass and spin effects are included. With this potential, a model for a test particle of infinitesimal mass orbiting in the equatorial plane of the two primaries is examined. The introduction of a parameter, ϵ, allows examination of the system as it transitions from the Newtonian to the beyond-Newtonian regime. The evolution and stability of the fixed points of the system as a function of the parameter ϵ is also studied. The dynamics of the particle is studied using the Poincaré map of section and the Maximal Lyapunov Exponent as indicators of chaos. Intermediate values of ϵ seem to be the most chaotic for the two cases of primary mass ratios (=0.001, 0.5) examined. The amount of chaos in the system remains higher than the Newtonian system as well as for the planar circular restricted three-body problem with Schwarzschild-like primaries for all non-zero values of ϵ.

What is Cartesianism?

Although Bernard le Bovier de Fontenelle has often been treated as merely a “popularizer” of Descartes’s philosophy, it is shown here that Fontenelle’s Cartesianism is a peculiar one. Cartesianism is mainly treated by Fontenelle as an exemplary modern “way of reasoning”, and absolutely not as a coherent “system” that one should adhere to. This has two consequences: first, this way of reasoning, the “geometrical spirit”, can be applied to domains that the Cartesian system would have forbidden (poetic, politics, etc.); and second, this way of reasoning, supposedly relying on higher norms of clarity, is used by Fontenelle to oppose the Newtonian system of attraction at a time when all his contemporaries are adopting it.

Droplet Formation and Fission in Shear-Thinning/Newtonian Multiphase System Using Bilayer Bifurcating Microchannel

With a novel platform of bilayer polydimethylsiloxane microchannel formed by bifurcating junction, we aim to investigate droplet formation and fission in a multiphase system with complex three-dimensional (3D) structure and understand the variations in mechanism associated with droplet formation and fission in the microstructure between shear-thinning/Newtonian system versus Newtonian/Newtonian system. The investigation concentrates on shear-thinning fluid because it is one of the most ubiquitous rheological properties of non-Newtonian fluids. Sodium carboxymethyl cellulose (CMC) solution and silicone oil have been used as model fluids and numerical model has been established to characterize the shear-thinning effect in formation of CMC-in-oil emulsions, as well as breakup dynamics when droplets flow through 3D bifurcating junction. The droplet volume and generation rate have been compared between two systems at the same Weber number and capillary number. Variation in droplet fission has been found between two systems, demonstrating that the shear-thinning property and confining geometric boundaries significantly affect the deformation and breakup of each mother droplet into two daughter droplets at bifurcating junction. The understanding of the droplet fission in the novel microstructure will enable more versatile control over the emulsion formation and fission when non-Newtonian fluids are involved. The model systems in the study can be further developed to investigate the mechanical property of emulsion templated particles such as drug encapsulated microcapsules when they flow through complex media structures, such as blood capillaries or the porous tissue structure, which feature with bifurcating junction.

Effects of matrix viscoelasticity on the lateral migration of a deformable drop in a wall-bounded shear

The dynamics of a drop deforming, orienting and moving in a shear flow of a viscoelastic liquid near a wall is numerically investigated using a front-tracking finite-difference method and a semi-analytic theory. The viscoelasticity is modelled using the modified FENE-CR constitutive equation. In a Newtonian system, deformation in a drop breaks the reversal symmetry of the system resulting in a migration away from the wall. This study shows that the matrix elasticity reduces the migration velocity, the reduction scaling approximately linearly with viscoelasticity (product of the Deborah numberDeand the ratio of polymer viscosity to total viscosity$\beta $). Similar to a Newtonian system, for small Deborah numbers, the dynamics quickly reaches a quasi-steady state where deformation, inclination, as well as migration and slip velocities become independent of the initial drop–wall separation. They all approximately scale inversely with the square of the instantaneous separation except for deformation which scales inversely with the cube of separation. The deformation shows a non-monotonic variation with increasing viscoelasticity similar to the case of a drop in an unbounded shear and is found to influence little the change in migration. Two competing effects due to matrix viscoelasticity on drop migration are identified. The first stems from the reduced inclination angle of the drop with increasing viscoelasticity that tries to enhance migration velocity. However, it is overcome by the second effect inhibiting migration that results from the normal stress differences from the curved streamlines around the drop; they are more curved on the side away from the wall compared with those in the gap between the wall and the drop, an effect that is also present for a rigid particle. A perturbative theory of migration is developed for small ratio of the drop size to its separation from the wall that clearly shows the migration to be caused by the image stresslet field due to the drop in presence of the wall. The theory delineates the two competing viscoelastic effects, their relative magnitudes, and predicts migration that matches well with the simulation. Using the simulation results and the stresslet theory, we develop an algebraic expression for the quasi-steady migration velocity as a function ofCa,Deand$\beta $. The transient dynamics of the migrating drop is seen to be governed by the finite time needed for development of the viscoelastic stresses. For larger capillary numbers, in both Newtonian and viscoelastic matrices, a viscous drop fails to reach a quasi-steady state independent of initial drop–wall separation. Matrix viscoelasticity tends to prevent drop breakup. Drops that break up in a Newtonian matrix are stabilized in a viscoelastic matrix if it is initially far away from the wall. Initial proximity to the wall enhances deformation and aids in drop breakup.