36,353 research outputs found
Approximation by planar elastic curves
We give an algorithm for approximating a given plane curve segment by a
planar elastic curve. The method depends on an analytic representation of the
space of elastic curve segments, together with a geometric method for obtaining
a good initial guess for the approximating curve. A gradient-driven
optimization is then used to find the approximating elastic curve.Comment: 18 pages, 10 figures. Version2: new section 5 added (conclusions and
discussions
B\'ezier curves that are close to elastica
We study the problem of identifying those cubic B\'ezier curves that are
close in the L2 norm to planar elastic curves. The problem arises in design
situations where the manufacturing process produces elastic curves; these are
difficult to work with in a digital environment. We seek a sub-class of special
B\'ezier curves as a proxy. We identify an easily computable quantity, which we
call the lambda-residual, that accurately predicts a small L2 distance. We then
identify geometric criteria on the control polygon that guarantee that a
B\'ezier curve has lambda-residual below 0.4, which effectively implies that
the curve is within 1 percent of its arc-length to an elastic curve in the L2
norm. Finally we give two projection algorithms that take an input B\'ezier
curve and adjust its length and shape, whilst keeping the end-points and
end-tangent angles fixed, until it is close to an elastic curve.Comment: 13 pages, 15 figure
Weakly nonlinear subcritical instability of visco-elastic Poiseuille flow
It is well known that the Poiseuille flow of a visco-elastic polymer fluid
between plates or through a tube is linearly stable in the zero Reynolds number
limit, although the stability is weak for large Weissenberg numbers. In this
paper we argue that recent experimental and theoretical work on the instability
of visco-elastic fluids in Taylor-Couette cells and numerical work on channel
flows suggest a scenario in which Poiseuille flow of visco-elastic polymer
fluids exhibits a nonlinear "subcritical" instability due to normal stress
effects, with a threshold which decreases for increasing Weissenberg number.
This proposal is confirmed by an explicit weakly nonlinear stability analysis
for Poiseuille flow of an UCM fluid. Our analysis yields explicit predictions
for the critical amplitude of velocity perturbations beyond which the flow is
nonlinearly unstable, and for the wavelength of the mode whose critical
amplitude is smallest. The nonlinear instability sets in quite abruptly at
Weissenberg numbers around 4 in the planar case and about 5.2 in the
cylindrical case, so that for Weissenberg numbers somewhat larger than these
values perturbations of the order of a few percent in the wall shear stress
suffice to make the flow unstable. We have suggested elsewhere that this
nonlinear instability could be an important intrinsic route to melt fracture
and that preliminary experiments are both qualitatively and quantitatively in
good agreement with these predictions.Comment: 20 pages, 16 figures. Accepted for publication in J. of Non-Newtonian
Fluid Mechanic
Mesoscopic mechanism of the domain wall interaction with elastic defects in ferroelectrics
The role of elastic defects on the kinetics of 180-degree uncharged
ferroelectric domain wall motion is explored using continuum time-dependent LGD
equation with elastic dipole coupling. In one dimensional case, ripples, steps
and oscillations of the domain wall velocity appear due to the wall-defect
interactions. While the defects do not affect the limiting-wall velocity vs.
field dependence, they result in the minimal threshold field required to
activate the wall motions. The analytical expressions for the threshold field
are derived and the latter is shown to be much smaller than the thermodynamic
coercive field. The threshold field is linearly proportional to the
concentration of defects and non-monotonically depends on the average distance
between them. The obtained results provide the insight into the mesoscopic
mechanism of the domain wall pinning by elastic defects in ferroelectrics.Comment: 18 pages, 6 figures, 1 appendi
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