Time-dependent mean velocities and dispersion coefficients are evaluated for a general two-dimensional laminar flow. A Lagrangian method is adopted by which a Brownian particle is traced in an artificially restructured velocity field. Asymptotic expressions for short, medium and long periods of time are obtained for Couette flow, plane Poiseuille flow and open-channel flow over an inclined flat surface. A new formula is suggested by which the Taylor dispersion coefficient can be evaluated from purely kinematical considerations. Within an error of less than one percent, over the entire time domain and for various flow fields, a very simple analytical expression is derived for the time-dependent dispersion coefficient
\[
\tilde{D}(\tau) = D + D^T\left(1-\frac{1-{\rm e}^{-\alpha\tau}}{a\tau}\right),
\]
where D is the molecular diffusion coefficient, DT denotes the Taylor dispersion coefficient, τ stands for the non-dimensional time π2Dt/Y/, Y is the distance between walls and a = (N + 1)2 is an integer which is determined by the number of symmetry planes N that the flow field possesses. For Couette and open-channel flow there are no planes of symmetry and a = 1; for Poiseuille flow there is one plane of symmetry and a = 4.
Motility-Induced Microphase and Macrophase Separation in a Two-Dimensional Active Brownian Particle System
