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}.
Deep Dive into 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}.
The simulation of dilute solutions of polymers in a Newtonian solvent is a challenging modelling and numerical problem, since deformation of the polymer molecules causes stresses that result in macroscopic non-Newtonian rheological behavior. One approach is to couple the macroscopic fluid flow equations to a microscopic model for the polymers, a so-called micro-macro model [15,27,28]. The simplest microscopic models, that we will use in this paper, describe the individual polymers as non-interacting dumbbells, consisting of two beads connected by a spring that models intramolecular interaction. The state of the polymer chain is described by the end-to-end vector X t that connects both beads whose evolution is modelled using a stochastic differential equation (SDE):
where u is the velocity field of the solvent, ζ is a friction coefficient, T is the temperature, k B is the Boltzmann constant, and W t is a standard multidimensional Brownian motion. This model takes into account Stokes drag (due to the solvent velocity field), a spring force F and Brownian motion (due to collisions with solvent molecules). The left-hand side of Equation (1.1) is the convective derivative. Note that the stochastic process X t implicitly depends on the space variable x.
To specify the microscopic model (1.1) completely, we need to define the spring force. This force can be more or less complicated, depending on the effects taken into account. The simplest model is the Hookean dumbbell model for which the spring is linear elastic:
with H a spring constant. Another model, which is the focus of this paper and which is known to yield better agreement with experiments, is the finitely extensible nonlinear elastic (FENE) force [4]:
where b is a nondimensional parameter related to the maximal polymer length.
In the macroscopic part of the model, the evolution of the solvent velocity and pressure fields u and p is modeled by mass and momentum conservation equations:
with ρ the density and η s the viscosity. Equation (1.3) contains an additional stress tensor τ p due to polymer deformation, which is given via the classical Kramers’ expression
Here, n is the polymer concentration and • denotes the expectation over configuration space, which is approximated in practice by an empirical mean over a very large ensemble of realizations of X t , solutions to (1.1).
One thus obtains a coupled system (1.1)-(1.3)-(1.4) that we rewrite in a non-dimensional form as (see for example [20]):
div(u) = 0, (1.6)
)
where the nondimensional parameters are:
(1.9)
Here, U is a characteristic velocity, L = k B T /H denotes a characteristic length, λ = ζ/4H is a characteristic relaxation time for the polymers and η p = nk B T λ is a viscosity associated to the polymers. The total viscosity is η = η p + η s . The parameters Re and We are the Reynolds and Weissenberg number, respectively. The nondimensional Hookean and FENE forces write respectively:
The microscopic part of the model, i.e. (1.7)-(1.8), can equivalently be described by a diffusion equation that governs the evolution of the probability distribution ϕ(X, x, t) of the random variable X t (considered at point x in physical space):
The expectation in (1.7) then becomes an average with respect to the probability measure ϕ(X, x, t) dX: τ p (x, t) = We X ⊗ F (X) ϕ(X, x, t)dX -Id .
(1.12)
We refer for example to [4,8,34] for more details on the physical background and more complicated models.
A numerical simulation of the coupled system (1.5)-(1.8) is very expensive, since one needs to obtain the non-Newtonian stress tensor τ p at each space-time discretization node. Several approaches have been proposed in the literature [23,28]. A first approach is a deterministic micro-macro simulation. Here, one couples the Fokker-Planck equation (1.11)-(1.12) with the Navier-Stokes equations (1.5)- (1.6). The main drawback of these methods is their high computational cost, due to the high-dimensionality of the function ϕ (which depends on seven scalar variables (X, x, t) in dimension 3). This difficulty becomes all the more severe when more refined models involving higher dimensional microscopic variables X t are used to describe the polymers. Specialized techniques are currently being developed; see e.g. [1,2,7]. The micro-macro simulation can also be performed stochastically. One then discretizes the macroscopic fields (velocity, pressure, stress) on a mesh, and supplements the (macroscopic) discretization of the Navier-Stokes equations with a stochastic simulation of an ensemble of polymers using a discretization of the SDE (1.8), see [15,27]. Methods have been proposed to obtain sufficiently low-variance results [6,15,20].
Due to the very high computational cost of micro-macro simulations, another route which has been followed (see e.g. [14,16,22,32,33,35]) is to look for an approximate closure at the macroscopic level, namely a model of the form:
which is close to the microscopic model (1.7)- (
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