Existence and stability of solutions of a delay-differential system

“…vt Jo ae:L1(0, 00) and A is locally integrable, the solution X of (4.8) is known to exist locally (see Driver [8]). The functional 9 in (4.10) is of 'higher order', which means that 9(0, t) = 0, and 9(X, t) = o(IIXII), IIXII~o.…”

Section: C:~f (R T) + H(f(r T))e-t + Rt H(f(r T) -F(r T))e-(t-'t)mentioning

confidence: 99%

Transient behaviour and stability points of the Poiseuille flow of a KBKZ-fluid

Aarts

Ven

1995

J Eng Math

View full text Add to dashboard Cite

The Poiseuille flow of a KBKZ-fluid, being a nonlinear viscoelastic model for a polymeric fluid, is studied. The flow starts from rest and especially the transient phase of the flow is considered. It is shown that under certain conditions the steady flow equation has three different equilibrium points. The stability of these points is investigated. It is proved that two points are stable, whereas the remaining one is unstable, leading to several peculiar phenomena such as discontinuities in the velocity gradient near the wall of the pipe ('spurt') and hysteresis. Our theoretical results are confirmed by numerical calculations of the velocity gradient.

show abstract

“…To specify a solution of (2) we require a to > a and a bounded continuous function <f>: [a, to] -► Rn ; we then obtain a solution x(t, to, cf>) satisfying (2) on an interval [to, to + P) with x(t, to, <j>) = <fi(t) for a < t < to ■ For details see Driver [5] or Burton [2]. To make the presentation here parallel that for finite delay equations, for each t > a we consider the function space C(t) with 4> £ C(t) if (j>:[a,t]->Rn is bounded and continuous.…”

Section: Ii) V^(t X)mentioning

confidence: 99%