User:Abdusalam aili

Evolution of tracer concentration fields in chaotic advection
Concentration field is a molecular or mass density distribution of a particle dispersed into a fluid in the its whole domain. As an example of a concentration field, we can consider a dye poured into water, or any transparent fluid. Our objective is to study how the dye concentration evolves during the mixing procedure. An example is shown on the figure "Mixing patterns obtained from the chaotic advection of a blob of dye", which is on the right. The concentration field evolves according to the advection-diffusion equation. That means there will be a competetion between advection which decrease the width of filament and the diffusion which tends to increase the width of filament while decreasing the concetration fluctuation. The advection-diffusion equation, called also Convection–diffusion equation, is deduced by combining diffusion equation and advection equation to describe the particle concentration field in a moving fluid. It expresse the relationship between the sum of the diffusion and advection, and the gradient of concentration field, which is linked by Diffusion coefficient. In the beginning of mixing process, the advection term dominates the evoluation of the concentration field of particles, and decreases the width of filament until Batchelor scale, where the diffusion begins to play a significant role. The figure (c) and (f) gives an image of Batchelor scale. From (c) to (f), there is no change of filament length scale, but only the concentration. It is the smallest lengthscale that can be observed in the concentration pattern, since diffusion smears out any finer detail. It is defined as the square root of the ratio between Diffusion coefficient and Lyapunov Exponent. This scale mesures the balance between the stretching and diffusion effect on the evolution of our concentration field.
 * Advection-diffusion equation
 * Batchelor scale

w_B = \sqrt{\frac{D}{\lambda}} $$ Where: \lambda $$ is the Lyapunov Exponent when most filament reaches the Batchelor scale, the diffusion begins to decrease significantly the contrast of concentration between filament and the other part of the fluid domain. The resolution of the advection–diffusion equation shows that after mixing time tm, the decrease of concentration is exponantial. The rapid decay of concentration field is given by the following formula:
 * D is the Diffusion coefficient
 * Rapid decay of concentration fluctuations

c(0, t) \simeq \sqrt{\frac{\lambda s_0^2}{2\pi D}} \exp{-\lambda t} $$ Where: \lambda $$ is the Lyapunov Exponent s_0 $$ is the initial length scale of concentration
 * c(0,t) is the concentration field at the centre.
 * D is the Diffusion coefficient