(583t) Stability of Mixing-Limited Patterns in Homogenous Autocatalytic Reactions

This paper presents an insight into stability of transport limited patterns that occur in isothermal homogenous autocatalytic reaction systems in tubular reactor geometries. Transport limited patterns occur when there exist large disparities between the timescale of reaction and those of diffusion and convection. Autocatalytic reactions being extremely fast naturally permit such pattern formation. The dynamics of transport limited pattern formation [1] and the effect of different transport and reaction parameters on their structure and temporal evolution [2], [3] are well understood. In this paper, we investigate the stability of such patterned states.  A patterned state is continuously under the effect of diffusion which ensures its eventual washout. Thus, conventional methods applicable for static steady systems cannot be directly used for studying the stability of a pattern, which is an evolving dynamic state. We define stability as persistence of a patterned state for two to three reactor residence times, and suitably extend bifurcation analysis of base or un-patterned steady states to dynamic, continuously evolving states. We employ a spatially averaged and regularized, low dimensional version of the complete 3-D convection-diffusion-reaction (CDR) equation that describes the spatio-temporal evolution of a pattern. It has been shown [4] that the spatial averaging (based on the Liapunov-Schmidt scheme of classical Bifurcation theory) reduces the model dimension and yet retains all relevant physics of the dimension averaged out, by preserving suitable parameters. Further, regularization increases the region of validity and accuracy of the model [1].  

In this paper, we study a co-dimension 1 bifurcation of the dynamic patterned state with Damkohler number as parameter and generate limit points. The state of a pattern toggles between stable and unstable every-time the S-shaped bifurcation diagram takes a turn at a limit point. Thus a patterned state with an odd number of limit points is stable whereas one with an even number of limit points is unstable. We cast this statement mathematically and establish the number of limit points as a function of space and time. This enables us to create false color images depicting the zones of stability and instability in contrast, and we call them stability maps. Our analysis reveals that instability spreads continuously across the 2-D circular geometry of the pattern. This helps us to understand the exact nature of decay of the patterned state. We find that the dynamics of the spread of instability in a pattern is not similar to the temporal evolution of the pattern itself. From here we conjecture that the decay nature of a pattern may be uniquely correlated with the mode number of the eigenfunction that is used as a perturbation to excite the base state into pattern formation. The maps make it possible to determine the most stable state during the time evolution of a pattern given its initial parameters and we demonstrate this by considering test cases of different mode numbers. This leads us to study the effect of the mode number on stability.

[1] Gupta A., Chakraborty S., Linear stability analysis of high and low dimensional models for describing mixing limited pattern formation in homogenous autocatalytic reactions. Chemical Engineering Journal 145(3) (2009) 399-411

[2] Chaudhury A., Chakraborty S, Dynamic simulation of mixing limited symmetric and asymmetric patterns in homogenous autocatalytic reactors. Industrial and Engineering Chemistry Research, 49(21) (2010) 10517-23

[3] Chaudhury A., Chakraborty S, Dynamics of mixing limited pattern formation in nonisothermal  homogenous autocatalytic reactors: A low dimensional computational analysis. Industrial and Engineering Chemistry Research, 50(8) (2011) 4335-43

[4] Chakraborty S., Balakotaiah V., Low dimensional models for describing mixing effects in laminar flow tubular reactors. Chemical Engineering Science, 57 (2002) 2545-2564.