14 research outputs found
The steady state configurational distribution diffusion equation of the standard FENE dumbbell polymer model: existence and uniqueness of solutions for arbitrary velocity gradients
The configurational distribution function, solution of an evolution
(diffusion) equation of the Fokker-Planck-Smoluchowski type, is (at least part
of) the corner stone of polymer dynamics: it is the key to calculating the
stress tensor components. This can be reckoned from \cite{bird2}, where a
wealth of calculation details is presented regarding various polymer chain
models and their ability to accurately predict viscoelastic flows. One of the
simplest polymer chain idealization is the Bird and Warner's model of finitely
extensible nonlinear elastic (FENE) chains. In this work we offer a proof that
the steady state configurational distribution equation has unique solutions
irrespective of the (outer) flow velocity gradients (i.e. for both slow and
fast flows)
A numerical closure approach for kinetic models of polymeric fluids: exploring closure relations for FENE dumbbells
We propose a numerical procedure to study closure approximations for FENE
dumbbells in terms of chosen macroscopic state variables, enabling to test
straightforwardly which macroscopic state variables should be included to build
good closures. The method involves the reconstruction of a polymer distribution
related to the conditional equilibrium of a microscopic Monte Carlo simulation,
conditioned upon the desired macroscopic state. We describe the procedure in
detail, give numerical results for several strategies to define the set of
macroscopic state variables, and show that the resulting closures are related
to those obtained by a so-called quasi-equilibrium approximation
\cite{Ilg:2002p10825}
Kinetic models for polymers with inertial effects
Novel kinetic models for both Dumbbell-like and rigid-rod like polymers are
derived, based on the probability distribution function
for a polymer molecule positioned at to be oriented along direction
while embedded in a environment created by inertial effects. It is
shown that the probability distribution function of the extended model, when
converging, will lead to well accepted kinetic models when inertial effects are
ignored such as the Doi models for rod like polymers, and the Finitely
Extensible Non-linear Elastic (FENE) models for Dumbbell like polymers.Comment: 23 pages, 2 figure
Thermocapillary motion of a Newtonian drop in a dilute viscoelastic fluid
In this work we investigate the role played by viscoelasticity on the thermocapillary motion of a deformable Newtonian droplet embedded in an immiscible, otherwise quiescent non-Newtonian fluid. We consider a regime in which inertia and convective transport of energy are both negligible (represented by the limit condition of vanishingly small Reynolds and Marangoni numbers) and free from gravitational effects. A constant temperature gradient is maintained by keeping two opposite sides of the computational domain at different temperatures. Consequently the droplet experiences a motion driven by the mismatch of interfacial stresses induced by the non-uniform temperature distribution on its boundary. The departures from the Newtonian behaviour are quantified via the “thermal” Deborah number, De T and are accounted for by adopting either the Oldroyd-B model, for relatively small De T, or the FENE-CR constitutive law for a larger range of De T. In addition, the effects of model parameters, such as the concentration parameter c=1−β (where β is the viscoelastic viscosity ratio), or the extensibility parameter, L 2, have been studied numerically using a hybrid volume of fluid-level set method. The numerical results show that the steady-state droplet velocity behaves as a monotonically decreasing function of De T, whilst its shape deforms prolately. For increasing values of De T, the viscoelastic stresses show the tendency to be concentrated near the rear stagnation point, contributing to an increase in its local interface curvature
Existence of global weak solutions to some regularized kinetic models for dilute polymers
Published versio
Existence of global weak solutions to dumbbell models for dilute polymers with microscopic cut-off
Accepted versio
On diffusive 2D Fokker-Planck-Navier-Stokes systems
We study models kinetic models of polymeric fluids. We introduce a notion of
solutions which is based on moments of polymeric distributions. We prove global
existence and uniqueness of a large class of initial data for diffusive systems
of kinetic equations coupled to fluid equations. As a corollary, we obtain a
rigorous derivation of Oldroyd-B closure. We also prove decay of free energy
for all the systems considered