Probability of noise-induced separatrix crossing for inertial particles in flows

The motion of weakly inertial Brownian particles, transported by steady two-dimensional fluid flows, is investigated by means of asymptotic methods. This study focuses on the phenomenon of noise-induced separatrix crossing, which can force particles to enter or exit recirculation cells in an unpredictable manner. An analytical expression for the probability of separatrix crossing is obtained. It can be applied to a wide variety of flows, provided some elementary kinematical quantities of the fluid flow are known. It does not require solving particle trajectories.

Figure 1

Sketch of a dividing streamline of the base flow ${\mathbf{u}}_0$ (thick line $AB$). (a) Convex recirculation cell; (b) concave cell. Thick lines are dividing streamlines of the fluid flow, thin lines are particle trajectories. Vector $\mathbf{n}$ is the unit vector perpendicular to the streamline, such that $({\mathbf{u}}_0,\mathbf{n},\widehat{\mathbf{z}})$ is right-handed.

Figure 2

Behavior of particles in the absence of noise, obtained by plotting the sign of $\mathrm{\Delta }{\psi }_0^0$ in the plane ($\overline{u_0^2},x_{AB}/\mathrm{Fr}$). Solid arcs represent separatrix $AB$, and typical trajectories of heavy inertial particles are sketched in short-dashed lines.

Figure 3

Deterministic particle trajectories in a test flow corresponding to three zones of the diagram of Fig. 2, obtained by solving Eq. (1) numerically. Left: centrifugal effects dominate [zone (5) of Fig. 2]. Middle: centrifugation and gravity balance each other (oblique line of Fig. 2). Right: gravity dominates centrifugation [zone (6)]. Two test particles heavier than the fluid have been used, with two different densities.

Figure 4

Upper graph: theoretical probabilities (red lines) given by Eq. (14). (a) Random $\delta {\tau }_k$; (b) constant $\delta {\tau }_k$. Symbols are the probabilities obtained from numerical simulations. Circles: Gaussian force intensity ${\xi }_{ik}$; crosses: uniform ${\xi }_{ik}$. Lower graph: variance of the stream-function jump, same symbols. Red solid lines are the theoretical law (11).