  We study the dynamics of polymers and elastic manifolds in non potential
static random flows. We find that barriers are generated from combined effects
of elasticity, disorder and thermal fluctuations. This leads to glassy trapping
even in pure barrier-free divergenceless flows $v {f \to 0}{\sim} f^\phi$
($\phi > 1$). The physics is described by a new RG fixed point at finite
temperature. We compute the anomalous roughness $R \sim L^\zeta$ and dynamical
$t\sim L^z$ exponents for directed and isotropic manifolds.
