Numerical Investigation of Two-Dimensional Oldroyd-B Fluid Flows Over A Straight Rectangular Domain

In this paper, non-Newtonian viscoelastic Oldroyd-B fluid flows in two-dimensional rectangular domain is numerically investigated, where the flow between two rigid walls is driven by a pressure difference along -direction (horizontal). The numerical results of the nonlinear system of partial differential equations are obtained by decoupling the system into Navier-Stokes system and tensorial transport equation. Computational Fluid Dynamics (CFD) simulations are done by using the finite element method. The numerical simulations are presented in terms of the contours of velocity, pressure and extra stress tensor. The Hood-Taylor finite element method is used for the approximation of the velocity and the pressure while the discontinuous Galerkin method is used to approximate the stress tensor. All the meshes and simulations are carried out by the general finite element solver FreeFem++, which has been found as a potential tool to provide a reasonably good numerical simulations of complicated flow behavior.

The Cauchy problem for an Oldroyd-B model in three dimensions

We consider an Oldroyd-B model which is derived in Ref. 4 [J. W. Barrett, Y. Lu and E. Süli, Existence of large-data finite-energy global weak solutions to a compressible Oldroyd-B model, Commun. Math. Sci. 15 (2017) 1265–1323] via micro–macro-analysis of the compressible Navier–Stokes–Fokker–Planck system. The global well posedness of strong solutions as well as the associated time-decay estimates in Sobolev spaces for the Cauchy problem are established near an equilibrium state. The terms related to [Formula: see text], in the equation for the extra stress tensor and in the momentum equation, lead to new technical difficulties, such as deducing [Formula: see text]-norm dissipative estimates for the polymer number density and its spatial derivatives. One of the main objectives of this paper is to develop a way to capture these dissipative estimates via a low–medium–high-frequency decomposition.

Implementation of partial slip boundary conditions in an open-source finite-volume-based computational library

This paper reports the implementation of slip boundary conditions in the open-source computational library OpenFOAM. The linear and nonlinear Navier slip laws, which are newly implemented in this paper, can be used both for Newtonian and viscoelastic constitutive models. For the former case, the Couette flow assumption near the wall is employed, and for the latter, the cell-centered extra-stress tensor components are linearly extrapolated to the wall. The validation is performed by comparing the numerical results obtained for Newtonian and simplified Phan-Thien-Tanner constitutive model fluids in Couette and Poiseuille flows, with existing analytical solutions. The results obtained using different slip factors were shown to be in agreement with the analytical solutions, even for the most extreme cases where the slip factor is high enough to induce a plug flow pattern for the velocity field. The newly implemented boundary conditions are also used to study the influence of slip in polymer processing, namely in the production of an extruded profile. The results obtained show that the developed slip boundary conditions are able to deal with complex geometrical problems, and are an important tool to support the search of a balanced flow distribution in the design of profile extrusion dies.

On convergent schemes for two-phase flow of dilute polymeric solutions

We construct a Galerkin finite element method for the numerical approximation of weak solutions to a recent micro-macro bead-spring model for two-phase flow of dilute polymeric solutions derived by methods from nonequilibrium thermodynamics ([Grün, Metzger, M3AS 26 (2016) 823–866]). The model consists of Cahn-Hilliard type equations describing the evolution of the fluids and the unsteady incompressible Navier-Stokes equations in a bounded domain in two or three spatial dimensions for the velocity and the pressure of the fluids with an elastic extra-stress tensor on the right-hand side in the momentum equation which originates from the presence of dissolved polymer chains. The polymers are modeled by dumbbells subjected to a finitely extensible, nonlinear elastic (FENE) spring-force potential. Their density and orientation are described by a Fokker-Planck type parabolic equation with a center-of-mass diffusion term. We perform a rigorous passage to the limit as the spatial and temporal discretization parameters simultaneously tend to zero, and show that a subsequence of these finite element approximations converges towards a weak solution of the coupled Cahn-Hilliard-Navier-Stokes-Fokker-Planck system. To underline the practicality of the presented scheme, we provide simulations of oscillating dilute polymeric droplets and compare their oscillatory behaviour to the one of Newtonian droplets.

Analysis of an augmented fully-mixed formulation for the coupling of the Stokes and heat equations

We introduce and analyse an augmented mixed variational formulation for the coupling of the Stokes and heat equations. More precisely, the underlying model consists of the Stokes equation suggested by the Oldroyd model for viscoelastic flow, coupled with the heat equation through a temperature-dependent viscosity of the fluid and a convective term. The original unknowns are the polymeric part of the extra-stress tensor, the velocity, the pressure, and the temperature of the fluid. In turn, for convenience of the analysis, the strain tensor, the vorticity, and an auxiliary symmetric tensor are introduced as further unknowns. This allows to join the polymeric and solvent viscosities in an adimensional viscosity, and to eliminate the polymeric part of the extra-stress tensor and the pressure from the system, which, together with the solvent part of the extra-stress tensor, are easily recovered later on through suitable postprocessing formulae. In this way, a fully mixed approach is applied, in which the heat flux vector is incorporated as an additional unknown as well. Furthermore, since the convective term in the heat equation forces both the velocity and the temperature to live in a smaller space than usual, we augment the variational formulation by using the constitutive and equilibrium equations, the relation defining the strain and vorticity tensors, and the Dirichlet boundary condition on the temperature. The resulting augmented scheme is then written equivalently as a fixed-point equation, so that the well-known Schauder and Banach theorems, combined with the Lax-Milgram theorem and certain regularity assumptions, are applied to prove the unique solvability of the continuous system. As for the associated Galerkin scheme, whose solvability is established similarly to the continuous case by using the Brouwer fixed-point and Lax–Milgram theorems, we employ Raviart–Thomas approximations of order k for the stress tensor and the heat flux vector, continuous piecewise polynomials of order ≤ k + 1 for velocity and temperature, and piecewise polynomials of order ≤ k for the strain tensor and the vorticity. Finally, we derive optimal a priori error estimates and provide several numerical results illustrating the good performance of the scheme and confirming the theoretical rates of convergence.

NUMERICAL SIMULATION OF A ELASTO-VISCOPLASTIC FLUID FLOW INSIDE A CAVITY

This article addresses finite element approximations for elasto-viscoplastic flows. Numerical simulations aiming at investigating the role of elasticity for inertialess flows of viscoplastic materials within lid-driven cavity.The mechanical model is made up of the usual governing equations for incompressible fluids coupled with a Oldroyd-B type equation (de Souza Mendes, 2011) modified to incorporate the dependency both of relaxation and retardation time as the viscoplastic viscosity on the strain rate. These parameters depend on the material microstructure, which level is described by an structure parameter . This model is approximated by a multi-field Galerkin least-squares formulation (Behr et al., 1993) in terms of extra-stress tensor, the pressure field and the velocity vector. Results, focused on the determination of yield surface topology, investigate the influence of elastic and viscous governing parameters on the flow pattern.

Existence of global weak solutions to compressible isentropic finitely extensible bead-spring chain models for dilute polymers

We prove the existence of global-in-time weak solutions to a general class of models that arise from the kinetic theory of dilute solutions of nonhomogeneous polymeric liquids, where the polymer molecules are idealized as bead-spring chains with finitely extensible nonlinear elastic (FENE) type spring potentials. The class of models under consideration involves the unsteady, compressible, isentropic, isothermal Navier–Stokes system in a bounded domain [Formula: see text] in [Formula: see text], [Formula: see text] or [Formula: see text], for the density [Formula: see text], the velocity [Formula: see text] and the pressure [Formula: see text] of the fluid, with an equation of state of the form [Formula: see text], where [Formula: see text] is a positive constant and [Formula: see text]. The right-hand side of the Navier–Stokes momentum equation includes an elastic extra-stress tensor, which is the sum of the classical Kramers expression and a quadratic interaction term. The elastic extra-stress tensor stems from the random movement of the polymer chains and is defined through the associated probability density function that satisfies a Fokker–Planck-type parabolic equation, a crucial feature of which is the presence of a center-of-mass diffusion term.

Peristaltic Transport of Visco-Elasto-Plastic Fluids in a Planar Channel

We numerically investigate peristaltic transport of incompressible visco-elasto-plastic fluids in a two-dimensional symmetric channel. The constitutive equation used for extra stress tensor is of more general form as it includes a number of well-known models like Maxwell A and B, Johnson–Segalman, Oldroyd-B, and Bingham models as its special cases. A detailed mathematical modelling of the problem is presented. The flow equations in the wave frame reduce to a single nonlinear ordinary differential equation in stream function by the implication of widely taken assumptions of long wavelength and low Reynolds number. The solution of the problem is obtained by two ways; namely, shooting method and Matlab built in routine bvp4c, and their comparison shows an excellent agreement. A parametric study based on bvp4c solution is performed to see the effects of parameters on velocity profile, pressure rise per wavelength, frictional forces, and trapping phenomenon.

EXISTENCE AND EQUILIBRATION OF GLOBAL WEAK SOLUTIONS TO KINETIC MODELS FOR DILUTE POLYMERS II: HOOKEAN-TYPE MODELS

We show the existence of global-in-time weak solutions to a general class of coupled Hookean-type bead-spring chain models that arise from the kinetic theory of dilute solutions of polymeric liquids with noninteracting polymer chains. The class of models involves the unsteady incompressible Navier–Stokes equations in a bounded domain in ℝd, d = 2 or 3, for the velocity and the pressure of the fluid, with an elastic extra-stress tensor appearing on the right-hand side in the momentum equation. The extra-stress tensor stems from the random movement of the polymer chains and is defined by the Kramers expression through the associated probability density function that satisfies a Fokker–Planck-type parabolic equation, a crucial feature of which is the presence of a center-of-mass diffusion term. We require no structural assumptions on the drag term in the Fokker–Planck equation; in particular, the drag term need not be corotational. With a square-integrable and divergence-free initial velocity datum ṵ0 for the Navier–Stokes equation and a non-negative initial probability density function ψ0 for the Fokker–Planck equation, which has finite relative entropy with respect to the Maxwellian M, we prove, via a limiting procedure on certain regularization parameters, the existence of a global-in-time weak solution t ↦ (ṵ(t), ψ(t)) to the coupled Navier–Stokes–Fokker–Planck system, satisfying the initial condition (ṵ(0), ψ(0)) = (ṵ0, ψ0), such that t ↦ ṵ(t) belongs to the classical Leray space and t ↦ ψ(t) has bounded relative entropy with respect to M and t ↦ ψ(t)/M has integrable Fisher information (with respect to the measure [Formula: see text]) over any time interval [0, T], T>0. If the density of body forces [Formula: see text] on the right-hand side of the Navier–Stokes momentum equation vanishes, then a weak solution constructed as above is such that t ↦ (ṵ(t), ψ(t)) decays exponentially in time to [Formula: see text] in the [Formula: see text]-norm, at a rate that is independent of (ṵ0, ψ0) and of the center-of-mass diffusion coefficient. Our arguments rely on new compact embedding theorems in Maxwellian-weighted Sobolev spaces and a new extension of the Kolmogorov–Riesz theorem to Banach-space-valued Sobolev spaces.

EXISTENCE AND EQUILIBRATION OF GLOBAL WEAK SOLUTIONS TO KINETIC MODELS FOR DILUTE POLYMERS I: FINITELY EXTENSIBLE NONLINEAR BEAD-SPRING CHAINS

We show the existence of global-in-time weak solutions to a general class of coupled FENE-type bead-spring chain models that arise from the kinetic theory of dilute solutions of polymeric liquids with noninteracting polymer chains. The class of models involves the unsteady incompressible Navier–Stokes equations in a bounded domain in ℝd, d = 2 or 3, for the velocity and the pressure of the fluid, with an elastic extra-stress tensor appearing on the right-hand side in the momentum equation. The extra-stress tensor stems from the random movement of the polymer chains and is defined by the Kramers expression through the associated probability density function that satisfies a Fokker–Planck-type parabolic equation, a crucial feature of which is the presence of a centre-of-mass diffusion term. We require no structural assumptions on the drag term in the Fokker–Planck equation; in particular, the drag term needs not be corotational. With a square-integrable and divergence-free initial velocity datum ṵ0 for the Navier–Stokes equation and a non-negative initial probability density function ψ0 for the Fokker–Planck equation, which has finite relative entropy with respect to the Maxwellian M, we prove, via a limiting procedure on certain regularisation parameters, the existence of a global-in-time weak solution t ↦ (ṵ(t), ψ(t)) to the coupled Navier–Stokes–Fokker–Planck system, satisfying the initial condition (ṵ(0), ψ(0)) = (ṵ0, ψ0), such that t ↦ ṵ(t) belongs to the classical Leray space and t ↦ ψ(t) has bounded relative entropy with respect to M and t ↦ ψ(t)/M has integrable Fisher information (w.r.t. the measure [Formula: see text] over any time interval [0, T], T > 0. If the density of body forces [Formula: see text] on the right-hand side of the Navier–Stokes momentum equation vanishes, then a weak solution constructed as above is such that t ↦ (ṵ(t), ψ(t)) decays exponentially in time to [Formula: see text] in the [Formula: see text] norm, at a rate that is independent of (ṵ0, ψ0) and of the centre-of-mass diffusion coefficient.