39,317 research outputs found
Approximation of 2D Euler Equations by the Second-Grade Fluid Equations with Dirichlet Boundary Conditions
The second-grade fluid equations are a model for viscoelastic fluids, with
two parameters: , corresponding to the elastic response, and , corresponding to viscosity. Formally setting these parameters to
reduces the equations to the incompressible Euler equations of ideal fluid
flow. In this article we study the limits of solutions of
the second-grade fluid system, in a smooth, bounded, two-dimensional domain
with no-slip boundary conditions. This class of problems interpolates between
the Euler- model (), for which the authors recently proved
convergence to the solution of the incompressible Euler equations, and the
Navier-Stokes case (), for which the vanishing viscosity limit is
an important open problem. We prove three results. First, we establish
convergence of the solutions of the second-grade model to those of the Euler
equations provided , as , extending
the main result in [19]. Second, we prove equivalence between convergence (of
the second-grade fluid equations to the Euler equations) and vanishing of the
energy dissipation in a suitably thin region near the boundary, in the
asymptotic regime ,
as . This amounts to a convergence criterion similar to the
well-known Kato criterion for the vanishing viscosity limit of the
Navier-Stokes equations to the Euler equations. Finally, we obtain an extension
of Kato's classical criterion to the second-grade fluid model, valid if , as . The proof of all these results
relies on energy estimates and boundary correctors, following the original idea
by Kato.Comment: 20pages,1figur
Convergence of the 2D Euler- to Euler equations in the Dirichlet case: indifference to boundary layers
In this article we consider the Euler- system as a regularization of
the incompressible Euler equations in a smooth, two-dimensional, bounded
domain. For the limiting Euler system we consider the usual non-penetration
boundary condition, while, for the Euler- regularization, we use
velocity vanishing at the boundary. We also assume that the initial velocities
for the Euler- system approximate, in a suitable sense, as the
regularization parameter , the initial velocity for the limiting
Euler system. For small values of , this situation leads to a boundary
layer, which is the main concern of this work. Our main result is that, under
appropriate regularity assumptions, and despite the presence of this boundary
layer, the solutions of the Euler- system converge, as ,
to the corresponding solution of the Euler equations, in in space,
uniformly in time. We also present an example involving parallel flows, in
order to illustrate the indifference to the boundary layer of the limit, which underlies our work.Comment: 22page
Cancer survivors' experiences of group cognitive behavioural therapy
Section A: Prevalence of anxiety amongst cancer survivors (CS) is high and growing evidence suggests benefits of individual, or group, CBT in reducing anxiety. However, previous reviews were either cancer specific or specified cancer severity. Hence, there is a lack of a review looking at group CBT effectiveness on anxiety across different cancers and severity that this review aims to explore. A systematic review was conducted and twelve RCT studies were reviewed. Results indicated that group CBT interventions were effective in improving anxiety in CS across cancer types. Shortterm interventions also produced positive results. Implications for future research were discussed.
Section B: Cancer diagnosis impacts significantly on patients’ anxiety and quality of life. Although studies investigating the effectiveness of group CBT in CS have increased, there remains a paucity of data exploring CS experiences. This study aims to investigate CS’ experiences of receiving group CBT for anxiety. Qualitative grounded theory methodology was applied. Thirteen CS attended a telephone or face-to-face interview. A framework was developed and findings indicated that group CBT seemed acceptable amongst CS, a range of positive and negative experiences were reported and anxiety improved. Some of the mechanisms of change were understanding anxiety, connection with others, accepting cancer, greater hope about the future and access to CBT tools
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