556 research outputs found
On unsteady internal flows of incompressible fluids characterized by implicit constitutive equations in the bulk and on the boundary
Long-time and large-data existence of weak solutions for initial- and
boundary-value problems concerning three-dimensional flows of
\emph{incompressible} fluids is nowadays available not only for Navier--Stokes
fluids but also for various fluid models where the relation between the Cauchy
stress tensor and the symmetric part of the velocity gradient is
\emph{nonlinear}. The majority of such studies however concerns models where
such a dependence is \emph{explicit} (the stress is a function of the velocity
gradient), which makes the class of studied models unduly restrictive. The same
concerns boundary conditions, or more precisely the slipping mechanisms on the
boundary, where the no-slip is still the most preferred condition considered in
the literature. Our main objective is to develop a robust mathematical theory
for unsteady internal flows of \emph{implicitly constituted} incompressible
fluids with implicit relations between the tangential projections of the
velocity and the normal traction on the boundary. The theory covers numerous
rheological models used in chemistry, biorheology, polymer and food industry as
well as in geomechanics. It also includes, as special cases, nonlinear slip as
well as stick-slip boundary conditions. Unlike earlier studies, the conditions
characterizing admissible classes of constitutive equations are expressed by
means of tools of elementary calculus. In addition, a fully constructive proof
(approximation scheme) is incorporated. Finally, we focus on the question of
uniqueness of such weak solutions
A semismooth Newton method for implicitly constituted non-Newtonian fluids and its application to the numerical approximation of Bingham flow
We propose a semismooth Newton method for non-Newtonian models of
incompressible flow where the constitutive relation between the shear stress
and the symmetric velocity gradient is given implicitly; this class of
constitutive relations captures for instance the models of Bingham and
Herschel-Bulkley. The proposed method avoids the use of variational
inequalities and is based on a particularly simple regularisation for which the
(weak) convergence of the approximate stresses is known to hold. The system is
analysed at the function space level and results in mesh-independent behaviour
of the nonlinear iterations.Comment: 25 page
Development of boundary layers in Euler fluids that on "activation'' respond like Navier-Stokes fluids
We consider the flow of a fluid whose response characteristics change due the
value of the norm of the symmetric part of the velocity gradient, behaving as
an Euler fluid below a critical value and as a Navier-Stokes fluid at and above
the critical value, the norm being determined by the external stimuli. We show
that such a fluid, while flowing past a bluff body, develops boundary layers
which are practically identical to those that one encounters within the context
of the classical boundary layer theory propounded by Prandtl. Unlike the
classical boundary layer theory that arises as an approximation within the
context of the Navier-Stokes theory, here the development of boundary layers is
due to a change in the response characteristics of the constitutive relation.
We study the flow of such a fluid past an airfoil and compare the same against
the solution of the Navier-Stokes equations. We find that the results are in
excellent agreement with regard to the velocity and vorticity fields for the
two cases
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