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On the Cauchy problems for certain regularized models to the incompressible viscoelastic flow

Hua Qiu, Zheng‐an Yao

Mathematical Methods in the Applied Sciences > 43 > 6 > 2999 - 3017

In this paper, we study the Cauchy problems for certain regularized models to the incompressible viscoelastic flow in n space dimensions with n=2,3. Firstly, we establish a regularity condition for the solution under

\nabla u \in L^{1} (0, T; L^{\infty} (R^{n}))

. Furthermore, we obtain a regularity condition to the smooth solution for the inviscid regularized models in two space dimensions. Finally, we prove a global existence...

article

Stability analysis of several first order schemes for the Oldroyd model with smooth and nonsmooth initial data

Tong Zhang, Yanxia Qian

Numerical Methods for Partial Differential Equations > 34 > 6 > 2180 - 2216

In this article, we develop several first order fully discrete Galerkin finite element schemes for the Oldroyd model and establish the corresponding stability results for these numerical schemes with smooth and nonsmooth initial data. The stable mixed finite element method is used to the spatial discretization, and the temporal treatments of the spatial discrete Oldroyd model include the first order...

article

Ideal density‐dependent incompressible viscoelastic flow in the critical Besov spaces

Qiu Hua, Shaomei Fang

Mathematical Methods in the Applied Sciences > 41 > 10 > 3913 - 3933

In this paper, we consider the well‐posedness issue for the density‐dependent incompressible viscoelastic fluids of the Oldroyd model for the ideal case in space dimension greater than 2. We obtain the local well‐posedness of this model under the assumption that the initial density is bounded away from zero in the critical Besov spaces by means of the Littlewood‐Paley theory and Bony's paradifferential...

article

Stationary Oldroyd model with diffusive stress in thin pipes

Laurent Chupin, Andrei Ichim, Sébastien Martin

ZAMM ‐ Journal of Applied Mathematics and Mechanics / Zeitschrift für... > 98 > 1 > 147 - 172

The aim of this paper is to study the stationary flow of an Oldroyd fluid with particular diffusive stress, flowing through a thin 3D pipe. An asymptotic expansion with respect to a small parameter describing the ratio between the diameter of the cross section and the length is calculated and rigorously justified. The result obtained is compared against a numerical solution in the particular case...

article

The Cauchy problem for a Bardina–Oldroyd model to the incompressible viscoelastic flow

Hua Qiu

Nonlinear Analysis: Real World Applications > 2017 > 34 > C > 172-187

In this paper, we study the Cauchy problem for a regularized viscoelastic fluid model in space dimension two, the Bardina–Oldroyd model, which is inspired by the simplified Bardina model for the turbulent flows of fluids, introduced by Cao et al. (2006). In particular, we obtain the local existence of smooth solutions to this model via the contraction mapping principle. Furthermore, we prove the global...

article

Viscoelastic flows in a rough channel: A multiscale analysis

Laurent Chupin, Sébastien Martin

Annales de l'Institut Henri Poincare (C) Non Linear Analysis > 2017 > 34 > 2 > 483-508

We investigate the influence of the rough boundaries on viscoelastic flows, described by the diffusive Oldroyd model. The fluid domain has a rough wall modeled by roughness patterns of size ε≪1. We present and rigorously justify an asymptotic expansion with respect to ε, at any order, based upon the definition of elementary problems: Oldroyd-type problems at the global scale defined on a smoothened...

article

Electrohydrodynamic peristaltic flow of a viscoelastic Oldroyd fluid with a mild stenosis: Application of an endoscope

N. T. M. El-dabe, G. M. Moatimid, M. A. Hassan, D. R. Mostapha

Journal of Applied Mechanics and Technical Physics > 2016 > 57 > 1 > 38-54

The effect of a vertical alternating current, electric field, and heat transfer on a peristaltic flow of a dielectric viscoelastic Oldroyd fluid is studied. This analysis involves uniform and nonuniform annuli having a mild stenosis. The analytical solutions of equations of motion are based on the perturbation technique. This technique depends on two parameters: amplitude ratio and small wave number...

article

Stationary Oldroyd model with diffusive stress: Mathematical analysis of the model and vanishing diffusion process

Laurent Chupin, Sébastien Martin

Journal of Non-Newtonian Fluid Mechanics > 2015 > 218 > Complete > 27-39

In this paper we address the stationary Oldroyd model with diffusive stress in two ways: first we present the mathematical analysis of the model. Second, as the link between the diffusive model and the standard one is questionable from the mathematical point of view, we discuss, by means of numerical simulations, the behaviour of the model with respect to vanishing diffusion. In particular, numerical...

article

Local well‐posedness for the ideal incompressible density‐dependent viscoelastic flow

Hua Qiu, Shuguang Shao

Mathematical Methods in the Applied Sciences > 38 > 6 > 1081 - 1088

In this paper, we study the ideal incompressible density‐dependent Oldroyd model in

R^{n} (n = 2, 3)

article

The Oldroyd- $$\alpha $$ α model for the incompressible viscoelastic flow

Hua Qiu, Shuanghu Zhang

Monatshefte für Mathematik > 2014 > 174 > 1 > 125-139

In this paper, we introduce and study a new model, which is called the Lagrangian-averaged Oldroyd- $$\alpha $$ α (LAO- $$\alpha $$ α ) model in two space dimensions. Such a model is inspired by the Lagrangian-averaged Navier–Stokes- $$\alpha $$ α model (also known as the viscous Camassa–Holm equations). We obtain global existence result for the Cauchy problem of the LAO- ...

article

Stabilized finite element method for the viscoelastic Oldroyd fluid flows

Kun Wang, Zhiyong Si, Yanfang Yang

Numerical Algorithms > 2012 > 60 > 1 > 75-100

In this paper, a new stabilized finite element method based on two local Gauss integrations is considered for the two-dimensional viscoelastic fluid motion equations, arising from the Oldroyd model for the non-Newtonian fluid flows. This new stabilized method presents attractive features such as being parameter-free, or being defined for non-edge-based data structures. It confirms that the lowest...

article

A finite element penalty method for the linearized viscoelastic Oldroyd fluid motion equations

Kun Wang, Yueqiang Shang, Hongbo Wei

Computers and Mathematics with Applications > 2011 > 62 > 4 > 1814-1827

In this paper, a fully discrete finite element penalty method is considered for the two-dimensional linearized viscoelastic fluid motion equations, arising from the Oldroyd model for the non-Newton fluid flows. With the finite element method for the spatial discretization and the backward Euler scheme for the temporal discretization, the velocity and pressure are decoupled in this method, which leads...

article

On error estimates of the penalty method for the viscoelastic flow problem I: Time discretization

Kun Wang, Yinnian He, Xinlong Feng

Applied Mathematical Modelling > 2010 > 34 > 12 > 4089-4105

In this paper, we investigate the two-dimensional viscoelastic fluid motion problem arising in the Oldroyd model. By applying the dual method, the Helmholtz decomposition and some other techniques, we deduce the long-time optimal error estimates for its penalty method. Furthermore, the global optimal error bounds in the L ² and H ¹ -norms for the time semi-discrete of the penalty system...

article

Well-posedness in critical spaces for incompressible viscoelastic fluid system

Jianzhen Qian

Nonlinear Analysis > 2010 > 72 > 6 > 3222-3234

In this paper, we study the well-posedness in critical spaces of incompressible viscoelastic fluid system of Oldroyd model. Precisely, for this system with initial data in Besov space of critical regularity, we prove the existence and uniqueness of the local solution, which is also shown to exist globally in time provided the initial data is small under certain norms.

article

Remarks on the blowup criteria for Oldroyd models

Zhen Lei, Nader Masmoudi, Yi Zhou

Journal of Differential Equations > 2010 > 248 > 2 > 328-341

We provide a new method to prove and improve the Chemin–Masmoudi criterion for viscoelastic systems of Oldroyd type in [J.Y. Chemin, N. Masmoudi, About lifespan of regular solutions of equations related to viscoelastic fluids, SIAM J. Math. Anal. 33 (1) (2001) 84–112] in two space dimensions. Our method is much easier than the one based on the well-known losing a priori estimate and is expected to...

article

Two-phase viscoelastic jetting

Jiun-Der Yu, Shinri Sakai, J.A. Sethian

Journal of Computational Physics > 2007 > 220 > 2 > 568-585

A coupled finite difference algorithm on rectangular grids is developed for viscoelastic ink ejection simulations. The ink is modeled by the Oldroyd-B viscoelastic fluid model. The coupled algorithm seamlessly incorporates several things: (1) a coupled level set-projection method for incompressible immiscible two-phase fluid flows; (2) a higher-order Godunov type algorithm for the convection terms...

article

Global Existence of Classical Solutions for Some Oldroyd-B Model via the Incompressible Limit*

Zhen Lei

Chinese Annals of Mathematics, Series B > 2006 > 27 > 5 > 565-580

In this paper, we prove local and global existence of classical solutions for a system of equations concerning an incompressible viscoelastic fluid of Oldroyd-B type via the incompressible limit when the initial data are sufficiently small.

article

Existence results for the flow of a nonhomogeneous visco-elastic fluid

R. Salvi

Nonlinear Analysis > 2005 > 63 > 5-7 > e2121-e2127

In this paper, we consider a two-phase problem consisting of viscous incompressible fluid and visco-elastic particles. We pose the full system of equations modelling the motion of the fluid/solid system. The unique solvability is established locally and globally, in time with so-called smallness restrictions. Further, existence of a weak solution is treated.

article

Boundary layers for stress diffusive perturbation in viscoelastic fluids

Laurent Chupin

Applied Mathematics Letters > 2005 > 18 > 6 > 641-647

In this paper, we study a stress diffusive perturbation of the system describing a viscoelastic flow. We analyse the boundary layer which arises near the boundary and we observe in particular that there is no boundary layer on the velocity at the first order.

article

Numerical study of viscoelastic mixtures through a Cahn-Hilliard flow model

Franck Boyer, Laurent Chupin, Pierre Fabrie

European Journal of Mechanics / B Fluids > 2004 > 23 > 5 > 759-780

In this paper we are concerned with the numerical study of viscoelastic diphasic flows.We first propose a complete model and give some formal justifications including the case where both or one phase is a non-Newtonian one. Then, we have implemented a numerical method taking into account the Lagrangian transport in the resolution of the extra-stress equation of Oldroyd type. We describe this fully-practical...

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Search results

On the Cauchy problems for certain regularized models to the incompressible viscoelastic flow

Stability analysis of several first order schemes for the Oldroyd model with smooth and nonsmooth initial data

Ideal density‐dependent incompressible viscoelastic flow in the critical Besov spaces

Stationary Oldroyd model with diffusive stress in thin pipes

The Cauchy problem for a Bardina–Oldroyd model to the incompressible viscoelastic flow

Viscoelastic flows in a rough channel: A multiscale analysis

Electrohydrodynamic peristaltic flow of a viscoelastic Oldroyd fluid with a mild stenosis: Application of an endoscope

Stationary Oldroyd model with diffusive stress: Mathematical analysis of the model and vanishing diffusion process

Local well‐posedness for the ideal incompressible density‐dependent viscoelastic flow

The Oldroyd- $$\alpha $$ α model for the incompressible viscoelastic flow

Stabilized finite element method for the viscoelastic Oldroyd fluid flows

A finite element penalty method for the linearized viscoelastic Oldroyd fluid motion equations

On error estimates of the penalty method for the viscoelastic flow problem I: Time discretization

Well-posedness in critical spaces for incompressible viscoelastic fluid system

Remarks on the blowup criteria for Oldroyd models

Two-phase viscoelastic jetting

Global Existence of Classical Solutions for Some Oldroyd-B Model via the Incompressible Limit*

Existence results for the flow of a nonhomogeneous visco-elastic fluid

Boundary layers for stress diffusive perturbation in viscoelastic fluids

Numerical study of viscoelastic mixtures through a Cahn-Hilliard flow model

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