29 research outputs found
Spectrum of the Fokker-Planck operator representing diffusion in a random velocity field
We study spectral properties of the Fokker-Planck operator that represents
particles moving via a combination of diffusion and advection in a
time-independent random velocity field, presenting in detail work outlined
elsewhere [J. T. Chalker and Z. J. Wang, Phys. Rev. Lett. {\bf 79}, 1797
(1997)]. We calculate analytically the ensemble-averaged one-particle Green
function and the eigenvalue density for this Fokker-Planck operator, using a
diagrammatic expansion developed for resolvents of non-Hermitian random
operators, together with a mean-field approximation (the self-consistent Born
approximation) which is well-controlled in the weak-disorder regime for
dimension d>2. The eigenvalue density in the complex plane is non-zero within a
wedge that encloses the negative real axis. Particle motion is diffusive at
long times, but for short times we find a novel time-dependence of the
mean-square displacement, in dimension d>2, associated
with the imaginary parts of eigenvalues.Comment: 8 pages, submitted to Phys Rev
Optimization of chaotic micromixers using finite time Lyapunov exponents
In microfluidics mixing of different fluids is a highly non-trivial task due to the absence of turbulence. The dominant process allowing mixing at low Reynolds number is therefore diffusion, thus rendering mixing in plain channels very inefficient. Recently, passive chaotic micromixers such as the staggered herringbone mixer were developed, allowing efficient mixing of fluids by repeated stretching and folding of the fluid interfaces. The optimization of the geometrical parameters of such mixer devices is often performed by time consuming and expensive trial and error experiments. We demonstrate that the application of the lattice Boltzmann method to fluid flow in highly complex mixer geometries together with standard techniques from statistical physics and dynamical systems theory can lead to a highly efficient way to optimize micromixer geometries. The strategy applies massively parallel fluid flow simulations inside a mixer, where massless and noninteracting tracer particles are introduced. By following their trajectories we can calculate finite time Lyapunov exponents in order to quantify the degree of chaotic advection inside the mixer. The current report provides a review of our results published in [1] together with additional details on the simulation methodology