RESUMO
The effect of shear flow on mode selection and the length scale of patterns formed in a nonlinear autocatalytic reaction-diffusion model is investigated. We predict analytically the existence of transverse and longitudinal modes. The type of the selected mode strongly depends on the difference in the flow rates of the participating species, quantified by the differential flow parameter. Spatial structures are obtained by varying the length scale of individual modes and superposing them via the differential flow parameter. Our predictions are in line with numerical results obtained from lattice Boltzmann simulations.
RESUMO
Pattern formation in reaction-diffusion systems is of great importance in surface micropatterning [Grzybowski et al., Soft Matter 1, 114 (2005)], self-organization of cellular micro-organisms [Schulz et al., Annu. Rev. Microbiol. 55, 105 (2001)], and in developmental biology [Barkai et al., FEBS Journal 276, 1196 (2009)]. In this work, we apply the lattice Boltzmann method to study pattern formation in reaction-diffusion systems. As a first methodological step, we consider the case of a single species undergoing transformation reaction and diffusion. In this case, we perform a third-order Chapman-Enskog multiscale expansion and study the dependence of the lattice Boltzmann truncation error on the diffusion coefficient and the reaction rate. These findings are in good agreement with numerical simulations. Furthermore, taking the Gray-Scott model as a prominent example, we provide evidence for the maturity of the lattice Boltzmann method in studying pattern formation in nonlinear reaction-diffusion systems. For this purpose, we perform linear stability analysis of the Gray-Scott model and determine the relevant parameter range for pattern formation. Lattice Boltzmann simulations allow us not only to test the validity of the linear stability phase diagram including Turing and Hopf instabilities, but also permit going beyond the linear stability regime, where large perturbations give rise to interesting dynamical behavior such as the so-called self-replicating spots. We also show that the length scale of the patterns may be tuned by rescaling all relevant diffusion coefficients in the system with the same factor while leaving all the reaction constants unchanged.
RESUMO
We study scaling laws characterizing the interdiffusive zone between two miscible fluids flowing side by side in a Y-shape laminar micromixer using the lattice Boltzmann method. The lattice Boltzmann method solves the coupled three-dimensional (3D) hydrodynamics and mass transfer equations and incorporates intrinsic features of 3D flows related to this problem. We observe the different power-law regimes occurring at the center of the channel and close to the top/bottom wall. The extent of the interdiffusive zone scales as the square root of the axial distance at the center of the channel. At the top/bottom wall, we find an exponent 1/3 at early stages of mixing as observed in the experiments of Ismagilov [Appl. Phys. Lett. 76, 2376 (2000)]. At a larger distance from the entrance, the scaling exponent close to the walls changes to 1/2 [J.-B. Salmon and A. Adjari, J. Appl. Phys. 101, 074902 (2007)]. Here, we focus on the effect of the finite aspect ratio on diffusive broadening. Interestingly, we find the same scaling laws regardless of the channel's aspect ratio. However, the point at which the exponent 1/3 characterizing the broadening at the top/bottom wall reverts to the normal diffusive behavior downstream strongly depends on the aspect ratio. We propose an interpretation of this observation in terms of the shear rate at the side walls. A criterion for the range of aspect ratios with non-negligible effect on diffusive broadening is also provided.
