1,730 research outputs found
Bridging length and time scales in sheared demixing systems: from the Cahn-Hilliard to the Doi-Ohta model
We develop a systematic coarse-graining procedure which establishes the
connection between models of mixtures of immiscible fluids at different length
and time scales. We start from the Cahn-Hilliard model of spinodal
decomposition in a binary fluid mixture under flow from which we derive the
coarse-grained description. The crucial step in this procedure is to identify
the relevant coarse-grained variables and find the appropriate mapping which
expresses them in terms of the more microscopic variables. In order to capture
the physics of the Doi-Ohta level, we introduce the interfacial width as an
additional variable at that level. In this way, we account for the stretching
of the interface under flow and derive analytically the convective behavior of
the relevant coarse-grained variables, which in the long wavelength limit
recovers the familiar phenomenological Doi-Ohta model. In addition, we obtain
the expression for the interfacial tension in terms of the Cahn-Hilliard
parameters as a direct result of the developed coarse-graining procedure.
Finally, by analyzing the numerical results obtained from the simulations on
the Cahn-Hilliard level, we discuss that dissipative processes at the Doi-Ohta
level are of the same origin as in the Cahn-Hilliard model. The way to estimate
the interface relaxation times of the Doi-Ohta model from the underlying
morphology dynamics simulated at the Cahn-Hilliard level is established.Comment: 29 pages, 2 figures, accepted for publication in Phys. Rev.
Nonlinear dynamics of the viscoelastic Kolmogorov flow
The weakly nonlinear regime of a viscoelastic Navier--Stokes fluid is
investigated. For the purely hydrodynamic case, it is known that large-scale
perturbations tend to the minima of a Ginzburg-Landau free-energy functional
with a double-well (fourth-order) potential. The dynamics of the relaxation
process is ruled by a one-dimensional Cahn--Hilliard equation that dictates the
hyperbolic tangent profiles of kink-antikink structures and their mutual
interactions. For the viscoelastic case, we found that the dynamics still
admits a formulation in terms of a Ginzburg--Landau free-energy functional. For
sufficiently small elasticities, the phenomenology is very similar to the
purely hydrodynamic case: the free-energy functional is still a fourth-order
potential and slightly perturbed kink-antikink structures hold. For
sufficiently large elasticities, a critical point sets in: the fourth-order
term changes sign and the next-order nonlinearity must be taken into account.
Despite the double-well structure of the potential, the one-dimensional nature
of the problem makes the dynamics sensitive to the details of the potential. We
analysed the interactions among these generalized kink-antikink structures,
demonstrating their role in a new, elastic instability. Finally, consequences
for the problem of polymer drag reduction are presented.Comment: 26 pages, 17 figures, submitted to The Journal of Fluid Mechanic
Non-isothermal viscous Cahn--Hilliard equation with inertial term and dynamic boundary conditions
We consider a non-isothermal modified Cahn--Hilliard equation which was
previously analyzed by M. Grasselli et al. Such an equation is characterized by
an inertial term and a viscous term and it is coupled with a hyperbolic heat
equation. The resulting system was studied in the case of no-flux boundary
conditions. Here we analyze the case in which the order parameter is subject to
a dynamic boundary condition. This assumption requires a more refined strategy
to extend the previous results to the present case. More precisely, we first
prove the well-posedness for solutions with bounded energy as well as for weak
solutions. Then we establish the existence of a global attractor. Finally, we
prove the convergence of any given weak solution to a single equilibrium by
using a suitable Lojasiewicz--Simon inequality
On the Short-Time Compositional Stability of Periodic Multilayers
The short-time stability of concentration profiles in coherent periodic
multilayers consisting of two components with large miscibility gap is
investigated by analysing stationary solutions of the Cahn-Hilliard diffusion
equation. The limits of the existence and stability of periodic concentration
profiles are discussed as a function of the average composition for given
multilayer period length. The minimal average composition and the corresponding
layer thickness below which artificially prepared layers dissolve at elevated
temperatures are calculated as a function of the multilayer period length for a
special model of the composition dependence of the Gibbs free energy. For
period lengths exceeding a critical value, layered structures can exist as
metastable states in a certain region of the average composition. The phase
composition in very thin individual layers, comparable with the interphase
boundary width, deviates from that of the corresponding bulk phase.Comment: 29 pages including 7 figures, to be published in Thin Solid Film
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