A representation theorem for time-scale functionals
On uniqueness for bounded channel flows of viscoelastic fluids
It was conjectured in [1] that there is at most one bounded channel flow for a viscoelastic fluid whose stress relaxation function is positive, integrable, and strictly convex. In this paper we prove the uniqueness of bounded channel flows, assuming to be non-negative, integrable, and convex, but different from a very specific piecewise linear function. Furthermore, whenever these hypotheses apply, the unbounded channel flows, if any, must grow in time faster than any polynomial.
On bounded channel flows of viscoelastic fluids
We show that the smooth bounded channel flows of a viscoelastic fluid exhibit the following qualitative feature: Whenever the channel is sufficiently wide, any bounded velocity field satisfying the homogeneous equation of motion is such that if the flow stops at some time, then the flow is never unidirectional throughout the channel. We first demonstrate the qualitative property of the bounded channel flows. Then we show explicitly how a piecewise linear approximation of a relaxation function can...
On bounded channel flows of viscoelastic fluids
We show that the smooth bounded channel flows of a viscoelastic fluid exhibit the following qualitative feature: Whenever the channel is sufficiently wide, any bounded velocity field satisfying the homogeneous equation of motion is such that if the flow stops at some time, then the flow is never unidirectional throughout the channel. We first demonstrate the qualitative property of the bounded channel flows. Then we show explicitly how a piecewise linear approximation of a relaxation function can...
On uniqueness for bounded channel flows of viscoelastic fluids
It was conjectured in [1] that there is at most one bounded channel flow for a viscoelastic fluid whose stress relaxation function is positive, integrable, and strictly convex. In this paper we prove the uniqueness of bounded channel flows, assuming to be non-negative, integrable, and convex, but different from a very specific piecewise linear function. Furthermore, whenever these hypotheses apply, the unbounded channel flows, if any, must grow in time faster than any polynomial.
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