Theory of weakly nonlinear multidimensional detonations

We derive an asymptotic model for the dynamics of weakly nonlinear multi-dimensional detonations from the compressible reactive Navier-Stokes equations. It is assumed that activation energy is large, heat release is small, evolution is slow, and $\gamma-1$ is small. The resultant model in 2D in dimensionless form is given by \begin{eqnarray*} u_{t}+uu_{x}+v_{y} & = &-\frac{1}{2}T_{x}+\mu u_{xx}\\ v_{x} & = &u_{y}\\ \lambda_{x} & = &-k(1-\lambda) e^{\theta T}-d\lambda_{xx}\\ \kappa T_{x}+T & = &u+q\lambda+qd\lambda_{x}. \end{eqnarray*} where $u, v$ is the velocity field, $T$ is the temperature, and $\lambda \in[0,1]$ is the reaction progress variable, $q$ heat release, and $\mu$, $\kappa$, $d$ are coefficients of viscosity, heat conduction, and diffusion, respectively. This system is a generalization of the models of small disturbance unsteady transonic flow, weakly nonlinear acoustics (Zabolotskaya-Khokhlov (ZK) equation), and water waves (dispersionless Kadomtsev-Petviashvili (KP) equation). The model predicts regular and irregular multi-dimensional patterns, and in 1D exhibits transition from steady and stable traveling waves to oscillatory traveling waves through a Hopf bifurcation as $\theta$ is increased. Period-doubling bifurcations leading to chaos are also observed.