4,212 research outputs found
Continuous, Semi-discrete, and Fully Discretized Navier-Stokes Equations
The Navier--Stokes equations are commonly used to model and to simulate flow
phenomena. We introduce the basic equations and discuss the standard methods
for the spatial and temporal discretization. We analyse the semi-discrete
equations -- a semi-explicit nonlinear DAE -- in terms of the strangeness index
and quantify the numerical difficulties in the fully discrete schemes, that are
induced by the strangeness of the system. By analyzing the Kronecker index of
the difference-algebraic equations, that represent commonly and successfully
used time stepping schemes for the Navier--Stokes equations, we show that those
time-integration schemes factually remove the strangeness. The theoretical
considerations are backed and illustrated by numerical examples.Comment: 28 pages, 2 figure, code available under DOI: 10.5281/zenodo.998909,
https://doi.org/10.5281/zenodo.99890
Asymptotic behaviour of a rigid body with a cavity filled by a viscous liquid
We consider the system of equations modeling the free motion of a rigid body
with a cavity filled by a viscous (Navier-Stokes) liquid. We give a rigorous
proof of Zhukovskiy's Theorem, which states that in the limit of time going to
infinity, the relative fluid velocity tends to zero and the rigid velocity of
the full structure tends to a steady rotation around one of the principle axes
of inertia.
The existence of global weak solutions for this system was established
previously. In particular, we prove that every weak solution of this type is
subject to Zhukovskiy's Theorem. Independently of the geometry and of
parameters, this shows that the presence of fluid prevents precession of the
body in the limit. In general, we cannot predict which axis will be attained,
but we show stability of the largest axis and provide criteria on the initial
data which are decisive in special cases.Comment: 18 pages, 0 figure
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