We investigate the break-up of Newtonian/viscoelastic droplets in a viscoelastic/Newtonian matrix under the hydrodynamic conditions of a confined shear flow. Our numerical approach is based on a combination of Lattice-Boltzmann models (LBM) and Finit
e Difference (FD) schemes. LBM are used to model two immiscible fluids with variable viscosity ratio (i.e. the ratio of the droplet to matrix viscosity); FD schemes are used to model viscoelasticity, and the kinetics of the polymers is introduced using constitutive equations for viscoelastic fluids with finitely extensible non-linear elastic dumbbells with Peterlins closure (FENE-P). We study both strongly and weakly confined cases to highlight the role of matrix and droplet viscoelasticity in changing the droplet dynamics after the startup of a shear flow. Simulations provide easy access to quantities such as droplet deformation and orientation and will be used to quantitatively predict the critical Capillary number at which the droplet breaks, the latter being strongly correlated to the formation of multiple neckings at break-up. This study complements our previous investigation on the role of droplet viscoelasticity (A. Gupta & M. Sbragaglia, {it Phys. Rev. E} {bf 90}, 023305 (2014)), and is here further extended to the case of matrix viscoelasticity.
This paper is motivated by the fact that many systems need to be maintained continually while the underlying costs change over time. The challenge is to continually maintain near-optimal solutions to the underlying optimization problems, without crea
ting too much churn in the solution itself. We model this as a multistage combinatorial optimization problem where the input is a sequence of cost functions (one for each time step); while we can change the solution from step to step, we incur an additional cost for every such change. We study the multistage matroid maintenance problem, where we need to maintain a base of a matroid in each time step under the changing cost functions and acquisition costs for adding new elements. The online version of this problem generalizes online paging. E.g., given a graph, we need to maintain a spanning tree $T_t$ at each step: we pay $c_t(T_t)$ for the cost of the tree at time $t$, and also $| T_tsetminus T_{t-1} |$ for the number of edges changed at this step. Our main result is an $O(log m log r)$-approximation, where $m$ is the number of elements/edges and $r$ is the rank of the matroid. We also give an $O(log m)$ approximation for the offline version of the problem. These bounds hold when the acquisition costs are non-uniform, in which caseboth these results are the best possible unless P=NP. We also study the perfect matching version of the problem, where we must maintain a perfect matching at each step under changing cost functions and costs for adding new elements. Surprisingly, the hardness drastically increases: for any constant $epsilon>0$, there is no $O(n^{1-epsilon})$-approximation to the multistage matching maintenance problem, even in the offline case.
We obtain, by extensive direct numerical simulations, trajectories of heavy inertial particles in two-dimensional, statistically steady, homogeneous, and isotropic turbulent flows, with friction. We show that the probability distribution function $ma
thcal{P}(kappa)$, of the trajectory curvature $kappa$, is such that, as $kappa to infty$, $mathcal{P}(kappa) sim kappa^{-h_{rm r}}$, with $h_{rm r} = 2.07 pm 0.09$. The exponent $h_{rm r}$ is universal, insofar as it is independent of the Stokes number ($rm{St}$) and the energy-injection wave number. We show that this exponent lies within error bars of their counterparts for trajectories of Lagrangian tracers. We demonstrate that the complexity of heavy-particle trajectories can be characterized by the number $N_{rm I}(t,{rm St})$ of inflection points (up until time $t$) in the trajectory and $n_{rm I} ({rm St}) equiv lim_{ttoinfty} frac{N_{rm I} (t,{rm St})}{t} sim {rm St}^{-Delta}$, where the exponent $Delta = 0.33 pm0.02$ is also universal.
