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Modeling, mathematical and numerical analysis of electrorheological fluids

Michael Růžička (2004)

Applications of Mathematics

Many electrorheological fluids are suspensions consisting of solid particles and a carrier oil. If such a suspension is exposed to a strong electric field the effective viscosity increases dramatically. In this paper we first derive a model which captures this behaviour. For the resulting system of equations we then prove local in time existence of strong solutions for large data. For these solutions we finally derive error estimates for a fully implicit time-discretization.

New regularity results for a generic model equation in exterior 3D domains

Stanislav Kračmar, Patrick Penel (2005)

Banach Center Publications

We consider a generic scalar model for the Oseen equations in an exterior three-dimensional domain. We assume the case of a non-constant coefficient function. Using a variational approach we prove new regularity properties of a weak solution whose existence and uniqueness in anisotropically weighted Sobolev spaces were proved in [10]. Because we use some facts and technical tools proved in the above mentioned paper, we give also a brief review of its results and methods.

New results on the Burgers and the linear heat equations in unbounded domains.

J.I. Díaz, S. González (2005)

RACSAM

We consider the Burgers equation and prove a property which seems to have been unobserved until now: there is no limitation on the growth of the nonnegative initial datum u0(x) at infinity when the problem is formulated on unbounded intervals, as, e.g. (0 +∞), and the solution is unique without prescribing its behaviour at infinity. We also consider the associate stationary problem. Finally, some applications to the linear heat equation with boundary conditions of Robin type are also given.

Non-Newtonian fluids and function spaces

Růžička, Michael, Diening, Lars (2007)

Nonlinear Analysis, Function Spaces and Applications

In this note we give an overview of recent results in the theory of electrorheological fluids and the theory of function spaces with variable exponents. Moreover, we present a detailed and self-contained exposition of shifted N -functions that are used in the studies of generalized Newtonian fluids and problems with p -structure.

Numerical modelling of algebraic closure models of oceanic turbulent mixing layers

Anne-Claire Bennis, Tomas Chacón Rebollo, Macarena Gómez Mármol, Roger Lewandowski (2010)

ESAIM: Mathematical Modelling and Numerical Analysis

We introduce in this paper some elements for the mathematical and numerical analysis of algebraic turbulence models for oceanic surface mixing layers. In these models the turbulent diffusions are parameterized by means of the gradient Richardson number, that measures the balance between stabilizing buoyancy forces and destabilizing shearing forces. We analyze the existence and linear exponential asymptotic stability of continuous and discrete equilibria states. We also analyze the well-posedness...

On a steady flow in a three-dimensional infinite pipe

Paweł Konieczny (2006)

Colloquium Mathematicae

The paper examines the steady Navier-Stokes equations in a three-dimensional infinite pipe with mixed boundary conditions (Dirichlet and slip boundary conditions). The velocity of the fluid is assumed to be constant at infinity. The main results show the existence of weak solutions with no restriction on smallness of the flux vector and boundary conditions.

On an evolutionary nonlinear fluid model in the limiting case

Stephan Luckhaus, Josef Málek (2001)

Mathematica Bohemica

We consider the two-dimesional spatially periodic problem for an evolutionary system describing unsteady motions of the fluid with shear-dependent viscosity under general assumptions on the form of nonlinear stress tensors that includes those with p -structure. The global-in-time existence of a weak solution is established. Some models where the nonlinear operator corresponds to the case p = 1 are covered by this analysis.

On an existence theorem for the Navier-Stokes equations with free slip boundary condition in exterior domain

Rieko Shimada, Norikazu Yamaguchi (2008)

Banach Center Publications

This paper deals with a nonstationary problem for the Navier-Stokes equations with a free slip boundary condition in an exterior domain. We obtain a global in time unique solvability theorem and temporal asymptotic behavior of the global strong solution when the initial velocity is sufficiently small in the sense of Lⁿ (n is dimension). The proof is based on the contraction mapping principle with the aid of L p - L q estimates for the Stokes semigroup associated with a linearized problem, which is also...

On existence and regularity of solutions to a class of generalized stationary Stokes problem

Nguyen Duc Huy, Jana Stará (2006)

Commentationes Mathematicae Universitatis Carolinae

We investigate the existence of weak solutions and their smoothness properties for a generalized Stokes problem. The generalization is twofold: the Laplace operator is replaced by a general second order linear elliptic operator in divergence form and the “pressure” gradient ∇ p is replaced by a linear operator of first order.

On existence of solutions for the nonstationary Stokes system with boundary slip conditions

Wisam Alame (2005)

Applicationes Mathematicae

Existence of solutions for equations of the nonstationary Stokes system in a bounded domain Ω ⊂ ℝ³ is proved in a class such that velocity belongs to W p 2 , 1 ( Ω × ( 0 , T ) ) , and pressure belongs to W p 1 , 0 ( Ω × ( 0 , T ) ) for p > 3. The proof is divided into three steps. First, the existence of solutions with vanishing initial data is proved in a half-space by applying the Marcinkiewicz multiplier theorem. Next, we prove the existence of weak solutions in a bounded domain and then we regularize them. Finally, the problem with nonvanishing...

On global regular solutions to the Navier-Stokes equations with heat convection

Piotr Kacprzyk (2013)

Annales Polonici Mathematici

Global existence of regular solutions to the Navier-Stokes equations for velocity and pressure coupled with the heat convection equation for temperature in a cylindrical pipe is shown. We assume the slip boundary conditions for velocity and the Neumann condition for temperature. First we prove long time existence of regular solutions in [kT,(k+1)T]. Having T sufficiently large and imposing some decay estimates on | | f ( t ) | | L ₂ ( Ω ) , | | f , x ₃ ( t ) | | L ₂ ( Ω ) we continue the local solutions step by step up to a global one.

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