Implicit-explicit and explicit projection schemes for the unsteady incompressible Navier–Stokes equations using a high-order dG method

We investigate the molecular origin of shear-thinning in melts of flexible, semiflexible and rigid oligomers with coarse-grained simulations of a sheared melt. Entanglements, alignment, stretching and tumbling modes or suppression of the latter all contribute to understanding how macroscopic flow properties emerge from the molecular level. In particular, we identify the rise and decline of entanglements with increasing chain stiffness as the major cause for the non-monotonic behaviour of the viscosity in equilibrium and at low shear rates, even for rather small oligomeric systems. At higher shear rates, chains align and disentangle, contributing to shear-thinning. By performing simulations of single chains in shear flow, we identify which of these phenomena are of collective nature and arise through interchain interactions and which are already present in dilute systems. Building upon these microscopic simulations, we identify by means of the Irving–Kirkwood formula the corresponding macroscopic stress tensor for a non-Newtonian polymer fluid. Shear-thinning effects in oligomer melts are also demonstrated by macroscopic simulations of channel flows. The latter have been obtained by the discontinuous Galerkin method approximating macroscopic polymer flows. Our study confirms the influence of microscopic details in the molecular structure of short polymers such as chain flexibility on macroscopic polymer flows.

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“…Discretisation in space is realised by the discontinuous Galerkin (dG) method [ 68 , 69 , 70 ]. Domain

is discretised by a quadrilateral mesh

with a meshsize h .…”

Section: Hybrid Multiscale Methodsmentioning

confidence: 99%

“…In this section, we illustrate the consequences of shear-thinning, whose microscopic origins have been discussed in previous sections, on two examples of macroscopic channel flows. Numerical simulations have been realised within BoSSS code, for more details see also [ 20 , 68 , 69 , 74 , 75 ].…”

Section: Two Channel Flows Of a Non-newtonian Oligomer Fluidmentioning

confidence: 99%

Shear-Thinning in Oligomer Melts—Molecular Origins and Applications

et al. 2021

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“…We proceed by describing a numerical method applied to (10a)-(10d). For time integration we apply the Chorin projection method combined with the second order Adam-Bashforth method for ordinary differential equations, see, e.g., [52], [53].…”

Section: Hybrid Multiscale Methodsmentioning

confidence: 99%

Shear-thinning in Polymer Melts -- Molecular Origins and Hybrid Multiscale Simulations

Datta,

Yelash,

Schmid

et al. 2021

Preprint

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We investigate the molecular origin of shear-thinning in melts of flexible, semiflexible and rigid polymers with coarse-grained simulations of a sheared polymer melt. Polymer alignment, stretching and tumbling modes or suppression of the latter all contribute to understanding how macroscopic flow properties emerge from the molecular level. By performing simulations of single chains in a shear flow, we identify which of these phenomena are of collective nature and arise through interchain interactions and which are already present in dilute systems. Building upon these microscopic simulations we identify by means of the Irving-Kirkwood formula the corresponding macroscopic stress tensor for a non-Newtonian polymer fluid. Shearthinning effects in polymer melts are also demonstrated by macroscopic simulations of a channel flow. The latter have been obtained by the discontinuous Galerkin method approximating macroscopic polymer flows. Our study confirms the influence of microscopic details in the molecular structure of polymers such as chain flexibility on macroscopic polymer flows.

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“…The velocity correction method combined with discontinuous Galerkin (DG) has been investigated by several works, including Hesthaven and Warburton, 11 Ferrer et al, 4 and Emamy et al 7 Here, the fixed semi-explicit dual-splitting schemes are used to solve equations (1) and (2). Within this scheme, the INS equations are split into three distinct substeps which can be solved successively.…”

Section: Velocity Correction Methods and Temporal Discretizationmentioning

confidence: 99%

“…Similar work has also been done by Joshi et al 6 to wipe the spurious divergence at element interfaces. Emamy et al 7 propose a modified version of projection scheme to reduce the influence of time step size. Steinmoeller et al 8 also review the instabilities of DG in simulating under-resolved and small viscosity cases.…”

Section: Introductionmentioning

confidence: 99%

Stability evaluation of high-order splitting method for incompressible flow based on discontinuous velocity and continuous pressure

Ren

et al. 2019

Advances in Mechanical Engineering

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In this work, we deal with high-order solver for incompressible flow based on velocity correction scheme with discontinuous Galerkin discretized velocity and standard continuous approximated pressure. Recently, small time step instabilities have been reported for pure discontinuous Galerkin method, in which both velocity and pressure are discretized by discontinuous Galerkin. It is interesting to examine these instabilities in the context of mixed discontinuous Galerkin–continuous Galerkin method. By means of numerical investigation, we find that the discontinuous Galerkin–continuous Galerkin method shows great stability at the same configuration. The consistent velocity divergence discretization scheme helps to achieve more accurate results at small time step size. Since the equal order discontinuous Galerkin–continuous Galerkin method does not satisfy inf-sup stability requirement, the instability for high Reynolds number flow is investigated. We numerically demonstrate that fine mesh resolution and high polynomial order are required to obtain a robust system. With these conclusions, discontinuous Galerkin–continuous Galerkin method is able to achieve high-order spatial convergence rate and accurately simulate high Reynolds flow. The solver is tested through a series of classical benchmark problems, and efficiency improvement is proved against pure discontinuous Galerkin scheme.

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Implicit-explicit and explicit projection schemes for the unsteady incompressible Navier–Stokes equations using a high-order dG method

Cited by 5 publications

References 20 publications

Shear-Thinning in Oligomer Melts—Molecular Origins and Applications

Shear-Thinning in Oligomer Melts—Molecular Origins and Applications

Shear-thinning in Polymer Melts -- Molecular Origins and Hybrid Multiscale Simulations

Stability evaluation of high-order splitting method for incompressible flow based on discontinuous velocity and continuous pressure

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