On the local strong solutions for a system describing the flow of a viscoelastic fluid

Ondřej Kreml; Milan Pokorný

Displaying similar documents to “On the local strong solutions for a system describing the flow of a viscoelastic fluid”

Global existence of smooth solutions for the compressible viscous fluid flow with radiation in $ℝ^{3}$

Hyejong O, Hakho Hong, Jongsung Kim (2023)

Applications of Mathematics

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This paper is concerned with the 3-D Cauchy problem for the compressible viscous fluid flow taking into account the radiation effect. For more general gases including ideal polytropic gas, we prove that there exists a unique smooth solutions in $[0, \infty)$ , provided that the initial perturbations are small. Moreover, the time decay rates of the global solutions are obtained for higher-order spatial derivatives of density, velocity, temperature, and the radiative heat flux.

A regularity criterion for the 2D MHD and viscoelastic fluid equations

Zhuan Ye (2015)

Annales Polonici Mathematici

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This paper is dedicated to a regularity criterion for the 2D MHD equations and viscoelastic equations. We prove that if the magnetic field B, respectively the local deformation gradient F, satisfies $\nabla B, \nabla F \in L^{q} (0, T; L^{p} (ℝ ²))$ for 1/p + 1/q = 1 and 2 < p ≤ ∞, then the corresponding local solution can be extended beyond time T.

On uniqueness for bounded channel flows of viscoelastic fluids

Marshall J. Leitman, Epifanio G. Virga (1988)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

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It was conjectured in [1] that there is at most one bounded channel flow for a viscoelastic fluid whose stress relaxation function $G$ is positive, integrable, and strictly convex. In this paper we prove the uniqueness of bounded channel flows, assuming $G$ to be non-negative, integrable, and convex, but different from a very specific piecewise linear function. Furthermore, whenever these hypotheses apply, the unbounded channel flows, if any, must grow in time faster than any polynomial. ...

On the global existence for a regularized model of viscoelastic non-Newtonian fluid

Ondřej Kreml, Milan Pokorný, Pavel Šalom (2015)

Colloquium Mathematicae

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We study the generalized Oldroyd model with viscosity depending on the shear stress behaving like ${μ (D) \sim | D |}^{p - 2}$ (p > 6/5), regularized by a nonlinear stress diffusion. Using the Lipschitz truncation method we prove global existence of a weak solution to the corresponding system of partial differential equations.

Global existence of solutions for incompressible magnetohydrodynamic equations

Wisam Alame, W. M. Zajączkowski (2004)

Applicationes Mathematicae

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Global-in-time existence of solutions for incompressible magnetohydrodynamic fluid equations in a bounded domain Ω ⊂ ℝ³ with the boundary slip conditions is proved. The proof is based on the potential method. The existence is proved in a class of functions such that the velocity and the magnetic field belong to $W_{p}^{2, 1} (Ω \times (0, T))$ and the pressure q satisfies $\nabla q \in L_{p} (Ω \times (0, T))$ for p ≥ 7/3.

Consistent streamline residual-based artificial viscosity stabilization for numerical simulation of incompressible turbulent flow by isogeometric analysis

Bohumír Bastl, Marek Brandner, Kristýna Slabá, Eva Turnerová (2022)

Applications of Mathematics

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In this paper, we propose a new stabilization technique for numerical simulation of incompressible turbulent flow by solving Reynolds-averaged Navier-Stokes equations closed by the SST $k$ - $ω$ turbulence model. The stabilization scheme is constructed such that it is consistent in the sense used in the finite element method, artificial diffusion is added only in the direction of convection and it is based on a purely nonlinear approach. We present numerical results obtained by our in-house...

On uniqueness for bounded channel flows of viscoelastic fluids

Marshall J. Leitman, Epifanio G. Virga (1988)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti

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Numerical comparison of unsteady compressible viscous flow in convergent channel

Pořízková, Petra, Kozel, Karel, Horáček, Jaromír

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This study deals with a numerical solution of a 2D flows of a compressible viscous fluids in a convergent channel for low inlet airflow velocity. Three governing systems – Full system, Adiabatic system, Iso-energetic system $b a s e d o n t h e N a v i e r - S t o k e s e q u a t i o n s f o r l a m i n a r f l o w a r e t e s t e d . T h e n u m e r i c a l s o l u t i o n i s r e a l i z e d b y f i n i t e v o l u m e m e t h o d a n d t h e p r e d i c t o r - c o r r e c t o r M a c C o r m a c k s c h e m e w i t h J a m e s o n a r t i f i c i a l v i s c o s i t y u s i n g a g r i d o f q u a d r i l a t e r a l c e l l s . T h e u n s t e a d y g r i d o f q u a d r i l a t e r a l c e l l s i s c o n s i d e r e d i n t h e f o r m o f c o n s e r v a t i o n l a w s u s i n g A r b i t r a r y L a g r a n g i a n - E u l e r i a n m e t h o d . T h e n u m e r i c a l r e s u l t s, a c q u i r e d f r o m a d e v e l o p e d p r o g r a m, a r e p r e s e n t e d f o r i n l e t v e l o c i t y$ u=4.12 ms-1 $a n d R e y n o l d s n u m b e r R e =$ 4 103 $.$

Estimates of lower order derivatives of viscous fluid flow past a rotating obstacle

Reinhard Farwig (2005)

Banach Center Publications

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Consider the problem of time-periodic strong solutions of the Stokes system modelling viscous incompressible fluid flow past a rotating obstacle in the whole space ℝ³. Introducing a rotating coordinate system attached to the body yields a system of partial differential equations of second order involving an angular derivative not subordinate to the Laplacian. In a recent paper [2] the author proved $L^{q}$ -estimates of second order derivatives uniformly in the angular and translational velocities,...

THE Navier-stokes flow around a rotating obstacle with time-dependent body force

Toshiaki Hishida (2009)

Banach Center Publications

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We study the motion of a viscous incompressible fluid filling the whole three-dimensional space exterior to a rigid body, that is rotating with constant angular velocity ω, under the action of external force f. By using a frame attached to the body, the equations are reduced to (1.1) in a fixed exterior domain D. Given f = divF with $F \in B U C (ℝ; L_{3 / 2, \infty} (D))$ , we consider this problem in D × ℝ and prove that there exists a unique solution $u \in B U C (ℝ; L_{3, \infty} (D))$ when F and |ω| are sufficiently small. If, in particular, the external...

On the spectral instability of parallel shear flows

Emmanuel Grenier, Yan Guo, Toan T. Nguyen (2014-2015)

Séminaire Laurent Schwartz — EDP et applications

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This short note is to announce our recent results [2,3] which provide a complete mathematical proof of the viscous destabilization phenomenon, pointed out by Heisenberg (1924), C.C. Lin and Tollmien (1940s), among other prominent physicists. Precisely, we construct growing modes of the linearized Navier-Stokes equations about general stationary shear flows in a bounded channel (channel flows) or on a half-space (boundary layers), for sufficiently large Reynolds number $R \to \infty$ . Such an instability...

Remark on regularity criterion for weak solutions to the shear thinning fluids

Jae-Myoung Kim (2024)

Mathematica Bohemica

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J. Q. Yang (2019) established a regularity criterion for the 3D shear thinning fluids in the whole space $ℝ^{3}$ via two velocity components. The goal of this short note is to extend this result in viewpoint of Lorentz space.

On new characterization of inextensible flows of space-like curves in de Sitter space

Mustafa Yeneroğlu (2016)

Open Mathematics

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Elastica and inextensible flows of curves play an important role in practical applications. In this paper, we construct a new characterization of inextensible flows by using elastica in space. The inextensible flow is completely determined for any space-like curve in de Sitter space [...] S 1 3 $𝕊_{1}^{3}$ . Finally, we give some characterizations for curvatures of a space-like curve in de Sitter space [...] S 1 3 $𝕊_{1}^{3}$ .

The nonlinear future stability of the FLRW family of solutions to the irrotational Euler-Einstein system with a positive cosmological constant

Igor Rodnianski, Jared Speck (2013)

Journal of the European Mathematical Society

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In this article, we study small perturbations of the family of Friedmann-Lemaître-Robertson-Walker cosmological background solutions to the coupled Euler-Einstein system with a positive cosmological constant in $1 + 3$ spacetime dimensions. The background solutions model an initially uniform quiet fluid of positive energy density evolving in a spacetime undergoing exponentially accelerated expansion. Our nonlinear analysis shows that under the equation of state $p = c^{2} ρ, 0 < c^{2} < 1 / 3$ , the background metric + fluid...

Cauchy problem for the non-newtonian viscous incompressible fluid

Milan Pokorný (1996)

Applications of Mathematics

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We study the Cauchy problem for the non-Newtonian incompressible fluid with the viscous part of the stress tensor $τ^{V} (𝕖) = τ (𝕖) - 2 μ_{1} Δ 𝕖$ , where the nonlinear function $τ (𝕖)$ satisfies $τ_{i j} (𝕖) e_{i j} \geq c {| 𝕖 |}^{p}$ or $τ_{i j} (𝕖) e_{i j} \geq {c (| 𝕖 |}^{2} + {| 𝕖 |}^{p})$ . First, the model for the bipolar fluid is studied and existence, uniqueness and regularity of the weak solution is proved for $p > 1$ for both models. Then, under vanishing higher viscosity $μ_{1}$ , the Cauchy problem for the monopolar fluid is considered. For the first model the existence of the weak solution is proved for $p > \frac{3 n}{n + 2}$ , its uniqueness...

Weighted L² and $L^{q}$ approaches to fluid flow past a rotating body

R. Farwig, S. Kračmar, M. Krbec, Š. Nečasová, P. Penel (2009)

Banach Center Publications

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Consider the flow of a viscous, incompressible fluid past a rotating obstacle with velocity at infinity parallel to the axis of rotation. After a coordinate transform in order to reduce the problem to a Navier-Stokes system on a fixed exterior domain and a subsequent linearization we are led to a modified Oseen system with two additional terms one of which is not subordinate to the Laplacean. In this paper we describe two different approaches to this problem in the whole space case....

Relaxation of the incompressible porous media equation

László Székelyhidi Jr (2012)

Annales scientifiques de l'École Normale Supérieure

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It was shown recently by Córdoba, Faraco and Gancedo in [1] that the 2D porous media equation admits weak solutions with compact support in time. The proof, based on the convex integration framework developed for the incompressible Euler equations in [4], uses ideas from the theory of laminates, in particular $T 4$ configurations. In this note we calculate the explicit relaxation of IPM, thus avoiding $T 4$ configurations. We then use this to construct weak solutions to the unstable interface...