Existence of Global Weak Solutions for Some Polymeric Flow Models by Barrett, J, Schwab, C, Suli, E

“…The model consists of the unsteady incompressible Navier-Stokes equations in a bounded domain for the velocity and the pressure of the fluid, with an extra-stress tensor as right-hand side in the momentum equation. …”

Existence of global weak solutions for some polymeric flow models by Barrett, J, Schwab, C, Suli, E

“…The model consists of the unsteady incompressible Navier-Stokes equations in a bounded domain Ω ⊂ ℝd, d = 2, 3, for the velocity and the pressure of the fluid, with an extra-stress tensor as right-hand side in the momentum equation. …”

Existence and uniqueness of global weak solutions to strain-limiting viscoelasticity with Dirichlet boundary data by Bulicek, M, Patel, V, Suli, E, Sengul, Y

“…Then we focus on the class of so-called limiting strain models, i.e., models for which the magnitude of the strain tensor is known to remain small a priori, regardless of the magnitude of the Cauchy stress tensor. …”

Existence of global weak solutions to kinetic models for dilute polymers by Barrett, J, Suli, E

“…The model consists of the unsteady incompressible Navier-Stokes equations in a bounded domain in two or three space dimensions, for the velocity and the pressure of the fluid, with an elastic extra-stress tensor as right-hand side in the momentum equation. …”

Existence of large-data global weak solutions to a model of a strain-limiting viscoelastic body by Bulíček, M, Patel, V, Şengül, Y, Suli, E

“…We prove the existence of a unique large-data global-in-time weak solution to a class of models of the form utt=div T+f for viscoelastic bodies exhibiting strain-limiting behaviour, where the constitutive equation, relating the linearised strain tensor ε(u) to the Cauchy stress tensor T, is assumed to be of the form ε(ut)+αε(u)=F(T), where we define F(T)=(1+|T|a)−1aT, for constant parameters α∈(0,∞) and a∈(0,∞), in any number d of space dimensions, with periodic boundary conditions. …”

Numerical approximation of corotational dumbbell models for dilute polymers by Barrett, J, Suli, E

“…The model consists of the unsteady incompressible Navier-Stokes equations in a bounded domain Ω in R d, d=2 or 3, for the velocity and the pressure of the fluid, with an elastic extra-stress tensor as right-hand side in the momentum equation. …”

On incompressible heat-conducting viscoelastic rate-type fluids with stress-diffusion and purely spherical elastic response by Bulíček, M, Málek, J, Průša, V, Suli, E

“…In particular, the definition of weak solution is motivated by the thermodynamic basis of the model. The extra stress tensor describing the elastic response of the fluid is in our case purely spherical, which is a simplification from the physical point of view. …”

Sparse finite element approximation of high-dimensional transport-dominated diffusion problems by Schwab, C, Suli, E, Todor, R

“…We develop the numerical analysis of stabilized sparse tensor-product finite element methods for such high-dimensional, non-self-adjoint and possibly degenerate second-order partial differential equations, using piecewise polynomials of degree p > 0. …”

Existence of global weak solutions to compressible isentropic finitely extensible nonlinear bead-spring chain models for dilute polymers: the two-dimensional case by Suli, E, Barrett, J

“…The right-hand side of the Navier–Stokes momentum equation includes an elastic extra-stress tensor, which is the classical Kramers expression. …”

Existence of global weak solutions to dumbbell models for dilute polymers with microscopic cut-off by Barrett, J, Suli, E

“…The model consists of the unsteady incompressible NavierStokes equations in a bounded domain Ω ⊂ ℝd, d = 2 or 3, for the velocity and the pressure of the fluid, with an elastic extra-stress tensor as the right-hand side in the momentum equation. …”

Finite element approximation of finitely extensible nonlinear elastic dumbbell models for dilute polymers by Barrett, J, Suli, E

“…The class of models involves the unsteady incompressible Navier-Stokes equations in a bounded domain $\Omega \subset R^d$, 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. …”

Discontinuous Galerkin Finite Element Approximation of Nonlinear Second-Order Elliptic and Hyperbolic Systems by Ortner, C, Suli, E

“…An optimal order bound is derived on the discretization error in each case without requiring the global Lipschitz continuity of the tensor $S$. We then further relax our hypotheses: using a broken G{\aa}rding inequality we extend our optimal error bounds to the case of quasilinear hyperbolic systems where, instead of assuming that $S$ is uniformly monotone, we only require that the fourth-order tensor $A=\nabla S$ is satisfies a Legendre--Hadamard condition. …”

Well-posedness of the fractional Zener wave equation for heterogenous viscoelastic materials by Oparnica, L, Suli, E

“…Zener’s model for viscoelastic solids replaces Hooke’s law σ = 2με(u) + λ tr(ε(u)) I, relating the stress tensor σ to the strain tensor ε(u), where u is the displacement vector, μ > 0 is the shear modulus, and λ ≥ 0 is the first Lamé coefficient, with the constitutive law (1 + τ Dt) σ = (1 + ρ Dt)[2με(u) + λ tr(ε(u)) I], where τ > 0 is the characteristic relaxation time and ρ ≥ τ is the characteristic retardation time. …”

Existence of large-data global-in-time finite-energy weak solutions to a compressible FENE-P model by Barrett, J, Suli, E

“…We develop a priori bounds for the model, including logarithmic bounds, which guarantee the non-negativity of the conformation tensor and a bound on its trace, and we prove the existence of large-data global-in-time finite-energy weak solutions in two and three space dimensions. …”

A simple construction of a thermodynamically consistent mathematical model for non-isothermal flows of dilute compressible polymeric fluids by Dostalík, M, Málek, J, Průša, V, Suli, E

“…Our approach is based on the identification of energy storage mechanisms and entropy production mechanisms in the fluid of interest, which, in turn, leads to explicit formulae for the Cauchy stress tensor and for all of the fluxes involved. Having identified these mechanisms and derived the governing equations, we document the potential use of the thermodynamic basis of the model in a rudimentary stability analysis. …”

Finite element approximation of steady flows of colloidal solutions by Bonito, A, Girault, V, Guignard, D, Rajagopal, KR, Suli, E

“…The symmetric velocity gradient is assumed to be a monotone nonlinear function of the deviatoric part of the Cauchy stress tensor. We prove the existence of a unique weak solution to the problem, and under the additional assumption that the nonlinearity involved in the constitutive relation is Lipschitz continuous we also prove uniqueness of the weak solution. …”

McKean-Vlasov diffusion and the well-posedness of the Hookean bead-spring-chain model for dilute polymeric fluids by Yahiaoui, G

“…The limiting problem is then used to perform a rigorous derivation of the Hookean bead-spring-chain model for dilute polymeric fluids, which has the interesting feature that, if the flow domain is bounded, then so is the associated configuration space domain, and the associated Kramers stress tensor is defined by integration over this bounded configuration domain. …”

Numerical methods for simulating dilute polymeric fluids by Ye, S

“…The solvent is assumed to be a viscous incompressible Newtonian fluid, whose evolution in time is modelled by the Navier--Stokes equations; the elastic effects exhibited by the dilute polymeric fluid are modelled by the elastic extra stress tensor, whose spatial divergence appears on the right-hand side of the Navier--Stokes momentum equation. …”

Analysis of Navier--Stokes--Fokker--Planck systems for incompressible dilute polymeric fluids by He, C

“…The governing system consists of the transport equation and the Navier--Stokes equation coupled to the Fokker--Planck equation through the elastic extra-stress tensor which is defined by the Kramers expression. …”

Adaptive Galerkin approximation algorithms for partial differential equations in infinite dimensions by Schwab, C, Suli, E

“…Specifically, for the infinite-dimensional FP equation, adaptive space-time Galerkin discretizations, based on a tensorized Riesz basis, built from biorthogonal piecewise polynomial wavelet bases in time and the Hermite polynomial chaos in the Wiener-Itô decomposition of $L^{2}(H,\mu)$, are introduced and are shown to converge quasioptimally with respect to the nonlinear, best $N$-term approximation benchmark. …”