Mixing by polymers: experimental test of decay regime of mixing

By using high molecular weight fluorescent passive tracers with different diffusion coefficients and by changing the fluid velocity we study dependence of a characteristic mixing length on the Peclet number, $Pe$, which controls the mixing efficiency. The mixing length is found to be related to $Pe$ by a power law, $L_{mix}\propto Pe^{0.26\pm 0.01}$, and increases faster than expected for an unbounded chaotic flow. Role of the boundaries in the mixing length abnormal growth is clarified. The experimental findings are in a good quantitative agreement with the recent theoretical predictions.Comment: 4 pages,5 figures. accepted for publication in PR

Elastic turbulence in curvilinear flows of polymer solutions

Following our first report (A. Groisman and V. Steinberg, \sl Nature $\bf 405$, 53 (2000)) we present an extended account of experimental observations of elasticity induced turbulence in three different systems: a swirling flow between two plates, a Couette-Taylor (CT) flow between two cylinders, and a flow in a curvilinear channel (Dean flow). All three set-ups had high ratio of width of the region available for flow to radius of curvature of the streamlines. The experiments were carried out with dilute solutions of high molecular weight polyacrylamide in concentrated sugar syrups. High polymer relaxation time and solution viscosity ensured prevalence of non-linear elastic effects over inertial non-linearity, and development of purely elastic instabilities at low Reynolds number (Re) in all three flows. Above the elastic instability threshold, flows in all three systems exhibit features of developed turbulence. Those include: (i)randomly fluctuating fluid motion excited in a broad range of spatial and temporal scales; (ii) significant increase in the rates of momentum and mass transfer (compared to those expected for a steady flow with a smooth velocity profile). Phenomenology, driving mechanisms, and parameter dependence of the elastic turbulence are compared with those of the conventional high Re hydrodynamic turbulence in Newtonian fluids.Comment: 23 pages, 26 figure

Functional determinants for general Sturm-Liouville problems

Simple and analytically tractable expressions for functional determinants are known to exist for many cases of interest. We extend the range of situations for which these hold to cover systems of self-adjoint operators of the Sturm-Liouville type with arbitrary linear boundary conditions. The results hold whether or not the operators have negative eigenvalues. The physically important case of functional determinants of operators with a zero mode, but where that mode has been extracted, is studied in detail for the same range of situations as when no zero mode exists. The method of proof uses the properties of generalised zeta-functions. The general form of the final results are the same for the entire range of problems considered.Comment: 28 pages, LaTe

Elastic turbulence in von Karman swirling flow between two disks

We discuss the role of elastic stress in the statistical properties of elastic turbulence, realized by the flow of a polymer solution between two disks. The dynamics of the elastic stress are analogous to those of a small scale fast dynamo in magnetohydrodynamics, and to those of the turbulent advection of a passive scalar in the Batchelor regime. Both systems are theoretically studied in literature, and this analogy is exploited to explain the statistical properties, the flow structure, and the scaling observed experimentally. Several features of elastic turbulence are confirmed experimentally and presented in this paper: (i) saturation of the rms of the vorticity and of velocity gradients in the bulk, leading to the saturation of the elastic stress; (ii) large rms of the velocity gradients in the boundary layer, linearly growth with Wi; (iii) skewed PDFs of the injected power, with exponential tails, which indicate intermittency; PDF of the acceleration exhibit well-pronounced exponential tails too; (iv) a new length scale, i.e the thickness of the boundary layer, as measured from the profile of the rms of the velocity gradient, is found to be relevant and much smaller than the vessel size; (v) the scaling of the structure functions of the vorticity, velocity gradients, and injected power is found to be the same as that of a passive scalar advected by an elastic turbulent velocity field.Comment: submitted to Physics of Fluids; 31 pages, 29 figures (resolution reduced to screen quality

Breakdown of superfluidity of an atom laser past an obstacle

The 1D flow of a continuous beam of Bose-Einstein condensed atoms in the presence of an obstacle is studied as a function of the beam velocity and of the type of perturbing potential (representing the interaction of the obstacle with the atoms of the beam). We identify the relevant regimes: stationary/time-dependent and superfluid/dissipative; the absence of drag is used as a criterion for superfluidity. There exists a critical velocity below which the flow is superfluid. For attractive obstacles, we show that this critical velocity can reach the value predicted by Landau's approach. For penetrable obstacles, it is shown that superfluidity is recovered at large beam velocity. Finally, enormous differences in drag occur when switching from repulsive to attractive potential.Comment: 15 pages, 6 figure

Analytic and Reidemeister torsion for representations in finite type Hilbert modules

For a closed Riemannian manifold we extend the definition of analytic and Reidemeister torsion associated to an orthogonal representation of fundamental group on a Hilbert module of finite type over a finite von Neumann algebra. If the representation is of determinant class we prove, generalizing the Cheeger-M\"uller theorem, that the analytic and Reidemeister torsion are equal. In particular, this proves the conjecture that for closed Riemannian manifolds with positive Novikov-Shubin invariants, the L2 analytic and Reidemeister torsions are equal.Comment: 78 pages, AMSTe