Global Solutions for Incompressible Viscoelastic Fluids

Lei, Zhen; Li, Chun; Zhou, Yi

doi:10.1007/s00205-007-0089-x

Cited by 237 publications

(244 citation statements)

References 33 publications

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“…21 We note that such a simple change to the model leads to a vast simplification mathematically, but, of course, it is not justified on physical grounds. Finally, global-in-time existence results for (P) in the case f ≡ 0 and for initial data close to equilibrium can be found in Lei et al 19 .…”

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confidence: 89%

Existence and Approximation of a (Regularized) Oldroyd-B Model

Barrett

Boyaval

2011

Math. Models Methods Appl. Sci.

126

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We consider the finite element approximation of the Oldroyd-B system of equations, which models a dilute polymeric fluid, in a bounded domain D ⊂ R d , d = 2 or 3, subject to no flow boundary conditions. Our schemes are based on approximating the pressure and the symmetric conformation tensor by either (a) piecewise constants or (b) continuous piecewise linears. In case (a) the velocity field is approximated by continuous piecewise quadratics or a reduced version, where the tangential component on each simplicial edge (d = 2) or face (d = 3) is linear. In case (b) the velocity field is approximated by continuous piecewise quadratics or the mini-element. We show that both of these types of schemes satisfy a free energy bound, which involves the logarithm of the conformation tensor, without any constraint on the time step for the backward Euler type time discretization. This extends the results of Boyaval et al. 7 on this free energy bound. There a piecewise constant approximation of the conformation tensor was necessary to treat the advection term in the stress equation, and a restriction on the time step, based on the initial data, was required to ensure that the approximation to the conformation tensor remained positive definite. Furthermore, for our approximation (b) in the presence of an additional dissipative term in the stress equation and a cut-off on the conformation tensor on certain terms in the system, similar to those introduced in Barrett and Süli 4 for the microscopic-macroscopic FENE model of a dilute polymeric fluid, we show (subsequence) convergence, as the spatial and temporal discretization parameters tend to zero, towards global-in-time weak solutions of this regularized Oldroyd-B system. Hence, we prove existence of global-in-time weak solutions to this regularized model. Moreover, in the case d = 2 we carry out this convergence in the absence of cut-offs, but with a time step restriction dependent on the spatial discretization parameter, and hence show existence of a global-in-time weak solution to the Oldroyd-B system with an additional dissipative term in the stress equation.

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confidence: 89%

Existence and Approximation of a (Regularized) Oldroyd-B Model

Barrett

Boyaval

2011

Math. Models Methods Appl. Sci.

126

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“…Besides, we would like to mention that Constantin and Kliegl [9] proved the global regularity of solutions in two dimensional case for the Oldroyd-B fluids with diffusive stress. An approach based on Lagrangian particle dynamics can be found in [19,20,21,23,24,25,30,32].…”

Section: Introductionmentioning

confidence: 99%

Global Solutions to the Oldroyd-B Model with a Class of Large Initial Data

Fang

Zi²

2016

SIAM J. Math. Anal.

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“…Many analyses of viscoelastic fluids employing fractional derivatives have been communicated in recent years. Representative studies in this regard include Lei et al [18] who derived both local and global smooth solutions to the Cauchy problem in the whole space and the periodic problem in the n-dimensional torus for incompressible viscoelastic fractional Oldroyd-B fluids, also demonstrating the validity of this approach to elastic complex fluids, magnetic rheological fluids, liquid crystal and mixture suspensions. Lin et al [19] determined both local and global existence of classical solutions for a fractional Oldroyd fluid in the absence of an artificially postulated damping mechanism.…”

Section: Introductionmentioning

confidence: 99%