Pattern formation revisited within nonequilibrium thermodynamics: Burgers’-type equation

Abstract

We revisit the description of reaction–diffusion phenomena within nonequilibrium thermodynamics and investigate the role of a nonstandard splitting of the entropy balance into the entropy production and the divergence of entropy flux. As previously reported by Pavelka et al. (Int J Eng Sci 78:192-217, 2014), a new term is identified following from the kinetic energy of diffusion. This newly appearing term acts as a thermodynamic force driving the reaction kinetics. Using the standard constitutive relations within the linear nonequilibrium thermodynamics, the governing equations for a reaction–diffusion problem in a two-species system are derived. They turn out to be linked to Burgers’ equation. It is shown that the onset of stability is not altered, but a non-periodic pattern can emerge. The latter follows from the relation of the governing equation to Burger’s equation with a source term. Hence, transients formed by glued and merging parabolic profiles are expected to appear at least in certain parameter regimes. We explore the significance of this effect and observe that for a comparable magnitude of the diffusion and of the new term stemming from the kinetic energy of diffusion, the solution is expected to be linked to the saw-tooth like solution to Burger’s equation rather than to the eigenmodes of the Laplacian. We conclude that the reaction–diffusion model proposed by Turing is robust to the addition of this effect of the kinetic energy of diffusion, at least when this new term is sufficiently small. As the governing equations can be rewritten into the classical reaction–diffusion problem but with reaction kinetics outside of the classical law of mass action, the analysis presented in this study suggests that a yet richer behaviour of the classical reaction–diffusion problems can be expected, if nonstandard reaction kinetics are considered.

Appendix A: Details of the numerical solution

We include particular details for the discussion of the seemingly stationary profile to the governing equation (16). The final profile shown in Fig. 1 satisfies well the governing equation everywhere with the exception of boundary points, see Fig. 6. As this error does not scale with mesh size, we conclude that it is not an admissible smooth solution to the problem at hand.