274 research outputs found
Existence for the steady problem of a mixture of two power-law fluids
The steady problem resulting from a mixture of two distinct fluids of
power-law type is analyzed in this work. Mathematically, the problem results
from the superposition of two power laws, one for a constant power-law index
with other for a variable one. For the associated boundary-value problem, we
prove the existence of very weak solutions, provided the variable power-law
index is bounded from above by the constant one. This result requires the
lowest possible assumptions on the variable power-law index and, as a
particular case, extends the existence result by Ladyzhenskaya dated from 1969
to the case of a variable exponent and for all zones of the pseudoplastic
region. In a distinct result, we extend a classical theorem on the existence of
weak solutions to the case of our problem.Comment: 22. arXiv admin note: substantial text overlap with arXiv:1203.679
Finite element approximation of an incompressible chemically reacting non-Newtonian fluid
We consider a system of nonlinear partial differential equations modelling
the steady motion of an incompressible non-Newtonian fluid, which is chemically
reacting. The governing system consists of a steady convection-diffusion
equation for the concentration and the generalized steady Navier-Stokes
equations, where the viscosity coefficient is a power-law type function of the
shear-rate, and the coupling between the equations results from the
concentration-dependence of the power-law index. This system of nonlinear
partial differential equations arises in mathematical models of the synovial
fluid found in the cavities of moving joints. We construct a finite element
approximation of the model and perform the mathematical analysis of the
numerical method in the case of two space dimensions. Key technical tools
include discrete counterparts of the Bogovski\u{\i} operator, De Giorgi's
regularity theorem in two dimensions, and the Acerbi-Fusco Lipschitz truncation
of Sobolev functions, in function spaces with variable integrability exponents.Comment: 40 page
Weak Solutions for a Non-Newtonian Diffuse Interface Model with Different Densities
We consider weak solutions for a diffuse interface model of two non-Newtonian
viscous, incompressible fluids of power-law type in the case of different
densities in a bounded, sufficiently smooth domain. This leads to a coupled
system of a nonhomogenouos generalized Navier-Stokes system and a Cahn-Hilliard
equation. For the Cahn-Hilliard part a smooth free energy density and a
constant, positive mobility is assumed. Using the -truncation method
we prove existence of weak solutions for a power-law exponent
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Finite element approximation of steady flows of generalized Newtonian fluids with concentration-dependent power-law index
We consider a system of nonlinear partial differential equations describing
the motion of an incompressible chemically reacting generalized Newtonian fluid
in three space dimensions. The governing system consists of a steady
convection-diffusion equation for the concentration and a generalized steady
power-law-type fluid flow model for the velocity and the pressure, where the
viscosity depends on both the shear-rate and the concentration through a
concentration-dependent power-law index. The aim of the paper is to perform a
mathematical analysis of a finite element approximation of this model. We
formulate a regularization of the model by introducing an additional term in
the conservation-of-momentum equation and construct a finite element
approximation of the regularized system. We show the convergence of the finite
element method to a weak solution of the regularized model and prove that weak
solutions of the regularized problem converge to a weak solution of the
original problem.Comment: arXiv admin note: text overlap with arXiv:1703.0476
Existence theory for stochastic power law fluids
We consider the equations of motion for an incompressible Non-Newtonian fluid
in a bounded Lipschitz domain during the time intervall
together with a stochastic perturbation driven by a Brownian motion
. The balance of momentum reads as where is the velocity,
the pressure and an external volume force. We assume the common power law
model and
show the existence of weak (martingale) solutions provided
. Our approach is based on the -truncation and a
harmonic pressure decomposition which are adapted to the stochastic setting
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